Options mathematics is the study of asymmetric contingent claims: contracts that create a right without forcing the holder to exercise it. Calls, puts, intrinsic value, time value, put–call parity, binomial trees, risk-neutral probabilities, Black–Scholes–Merton, delta, gamma, theta, vega, rho, implied volatility, volatility skew, smiles and dynamic hedging all grow from one idea: the option payoff is nonlinear, so value depends on the distribution and path of the underlying in ways a forward does not.
For readers searching for options mathematics, call option formula, put option formula, option payoff, intrinsic value, time value, put call parity, binomial option pricing, Black Scholes formula, Black Scholes Merton, option Greeks, delta gamma theta vega rho, implied volatility, volatility smile, volatility skew, option hedging or risk neutral option pricing, the strongest learning route is payoff → arbitrage bounds → parity → one-period replication → multi-step binomial tree → continuous-time BSM → Greeks and implied volatility.
CFA Institute’s 2026 contingent-claims material explicitly follows this sequence: no-arbitrage and replication, binomial valuation, Black–Scholes–Merton, Black’s model, Greeks and implied volatility. The mathematics is used globally across equity, currency, interest-rate and futures options. This page is educational and not a recommendation to buy, sell or write options.
50-Second Router
- Call: right to buy underlying at strike K; expiry payoff max(S_T−K,0).
- Put: right to sell underlying at K; expiry payoff max(K−S_T,0).
- Intrinsic value: immediate exercise value.
- Time value: option price minus intrinsic value; reflects remaining uncertainty and carry.
- European: exercise only at expiry; American: exercise allowed earlier under contract.
- Put–call parity: European call/put/stock/bond payoff identity.
- Binomial model: value options by state-by-state replication and backward induction.
- Risk-neutral probability: pricing weight derived from no-arbitrage, not a forecast probability.
- BSM: continuous-time model for European options under strong assumptions.
- Greeks: derivatives of option value with respect to market inputs.
- Implied volatility: volatility input that makes model price match market price.
- Verification: check payoff bounds, parity, monotonicity, convexity and hedging consistency.
The Central Proposition: Nonlinearity Is the Source of Option Value
A forward has a linear payoff: if the underlying rises by one unit, the long forward payoff rises by one unit. A call behaves differently. Below strike, expiry payoff is zero. Above strike, payoff rises one-for-one with the underlying. That kink is nonlinearity. It creates asymmetric exposure and makes volatility valuable to the holder.
Because downside is truncated at zero payoff while upside remains open for a long call, a wider distribution of future prices can increase expected option payoff under the pricing measure even when the forward price is unchanged. That is why option value depends on volatility while a plain forward price does not require a volatility assumption.
Adrian’s first exercise is to draw the payoff before reading any formula. The graph already explains intrinsic value, convexity, delta changes and why a long option holder can benefit from larger moves.
1. Call option
Call option is contract giving holder right to buy underlying at strike K by/at expiry under exercise terms. It creates asymmetric upside exposure. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced.
Mathematics. Payoff=max(S_T−K,0). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point.
Failure mode. Option payoff is not the same as profit because premium paid matters. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output.
Connection. This feeds directly into calls. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge.
2. Put option
Put option is right to sell underlying at strike K. It creates asymmetric downside protection/speculation. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced.
Mathematics. Payoff=max(K−S_T,0). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point.
Failure mode. Long put payoff is bounded above by K in simple European stock setup if stock cannot go below zero. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output.
Connection. This feeds directly into puts. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge.
3. Strike price
Strike price is contracted exercise price. It determines where payoff kink occurs. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced.
Mathematics. K fixed by contract. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point.
Failure mode. Strike is not market forecast or current spot. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output.
Connection. This feeds directly into option contract. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge.
4. Expiry
Expiry is final date on which option rights are determined/exercised under terms. Time to expiry is a major value input. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced.
Mathematics. T decreases to zero. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point.
Failure mode. Last trading/exercise/settlement conventions can differ. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output.
Connection. This feeds directly into time value. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge.
5. European option
European option is option exercisable only at expiry. It enables clean parity and closed-form models. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced.
Mathematics. Value based on terminal exercise only. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point.
Failure mode. ‘European’ describes exercise style, not trading location. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output.
Connection. This feeds directly into parity. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge.
6. American option
American option is option exercisable at any time up to expiry. It adds optimal early-exercise decision. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced.
Mathematics. V=max(intrinsic, continuation) at each node. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point.
Failure mode. BSM European formula is not generally sufficient for American puts/certain dividend calls. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output.
Connection. This feeds directly into early exercise. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge.
7. Bermudan option
Bermudan option is option exercisable on specified discrete dates. It sits between European and American styles. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced.
Mathematics. Backward induction checks exercise dates only. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point.
Failure mode. Exercise schedule matters materially. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output.
Connection. This feeds directly into structured options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge.
8. Intrinsic value
Intrinsic value is immediate exercise payoff. It is lower bound for American option and part of market premium decomposition. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced.
Mathematics. Call intrinsic=max(S−K,0); put=max(K−S,0). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point.
Failure mode. European option before expiry may trade below immediate intrinsic-like amount adjusted for carry because immediate exercise unavailable. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output.
Connection. This feeds directly into option value. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge.
9. Time value
Time value is option price minus relevant intrinsic value convention. It reflects remaining optionality and carry. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced.
Mathematics. TimeValue=Premium−Intrinsic. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point.
Failure mode. Time value can be affected by rates/dividends, not volatility alone. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output.
Connection. This feeds directly into option premium. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge.
10. In the money
In the money is option with positive intrinsic value under standard spot comparison. It describes moneyness, not profitability. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced.
Mathematics. Call ITM if S>K; put if S Failure mode. An ITM option can still be losing versus premium paid. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into moneyness. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. At the money is strike near underlying price/forward depending convention. It is region of high time value and gamma for many options. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. K≈S or forward-based ATM convention. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. FX options often use specialised delta/ATM conventions. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into moneyness. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Out of the money is zero current intrinsic value. It can still have significant time value. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Call OTM if S Failure mode. OTM does not mean worthless before expiry. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into moneyness. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Moneyness is relative relationship between spot/forward and strike. It organises option behaviour. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. S/K, log(S/K) or forward moneyness are common measures. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Different markets use different conventions. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into volatility surface. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Option premium is price paid by buyer to seller. It is maximum initial cash cost for long vanilla option before fees. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Profit=payoff−premium accumulated appropriately. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Premium is not margin collateral. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into option trading. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Option writer is seller granting option right. It receives premium and assumes contingent obligation. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Short option payoff=-long payoff. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Premium received does not cap potential call-writing loss on uncovered call. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into risk. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Covered call is short call combined with underlying ownership. It caps upside above strike in exchange for premium. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff=S_T−max(S_T−K,0)+premium effects. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Covered does not mean risk-free; stock can fall. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into strategies. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Protective put is long stock plus long put. It floors terminal stock value near strike before premium/carry. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff=max(S_T,K). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Protection has premium cost and expiry. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into insurance. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Call spread is long lower-strike call and short higher-strike call. It creates bounded upside exposure. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff grows between strikes then caps. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Net premium and assignment/exercise terms matter. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into strategies. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Put spread is long higher-strike put and short lower-strike put for bearish protection/spread. It creates bounded downside payoff. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff between strikes. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Protection is limited below lower strike. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into strategies. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Straddle is long call and put at same strike/expiry. It gains from large moves in either direction after premium hurdle. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff=|S_T−K| before premium. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Long volatility strategy can lose if realised move is small. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into volatility. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Strangle is long OTM call and OTM put. It is cheaper than comparable ATM straddle but requires larger move. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Piecewise payoff outside two strikes. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Skew makes call/put wing premiums asymmetric. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into volatility. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Butterfly is option combination producing tent-shaped terminal payoff. It isolates range/curvature exposure. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Classic call butterfly long K1,long K3,short 2 K2 for equally spaced strikes. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Static arbitrage imposes nonnegative butterfly values under standard setup. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into surface. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Risk reversal is long one wing option and short opposite wing. It expresses skew/directional-volatility exposure. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. FX/equity conventions vary. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Not a pure volatility trade because delta exposure can remain. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into skew. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Collar is underlying plus protective put funded partly by short call. It bounds downside/upside. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff constrained between strikes. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Zero-cost collar still sacrifices upside. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into hedging. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Option lower bound is no-arbitrage minimum value. It rules out obvious underpricing. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. European call≥max(0,S−KD) simple nondividend; put≥max(0,KD−S). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Dividends/exercise style modify bounds. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into arbitrage. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Option upper bound is no-arbitrage maximum under simple setup. It follows payoff dominance. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Call≤S; European put≤KD. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. American put upper bound can differ. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into arbitrage. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Put-call parity is European option identity from identical terminal payoffs. It is model-independent under matching terms. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. C+KD=P+S for nondividend stock. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Mismatch in expiry/strike/exercise/dividend breaks equality. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into parity. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Synthetic forward is long call and short put with same K,T. It produces S_T−K terminal payoff. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. C−P=S−KD. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Funding leg determines forward delivery PV. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into replication. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Synthetic stock is long call, short put and bond. It reproduces stock terminal payoff. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. S=C−P+KD. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Dividend-paying stock requires adjustment. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into replication. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. One-period binomial model is two-state underlying model used to derive option price by replication. It makes no-arbitrage mechanics transparent. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. S_u=uS, S_d=dS. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Physical up probability is not needed for arbitrage price. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into binomial. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Up factor is multiplicative stock move in binomial tree. It defines upper node. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. S_u=uS. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. u must be consistent with no-arbitrage relative to risk-free growth and d. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into tree. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Down factor is multiplicative downward move. It defines lower node. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. S_d=dS. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. If risk-free gross return lies outside [d,u], simple tree contains arbitrage. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into tree. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Risk-free gross return is accumulation over tree step. It anchors risk-neutral probability. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. R=1+rΔt or e^{rΔt}. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Rate/step units must match. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into tree. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Risk-neutral probability is derived pricing probability making discounted stock a martingale. It prices state claims. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. q=(R-d)/(u-d). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. q is not subjective forecast probability. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into binomial. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Binomial delta is underlying units in replicating portfolio. It matches option payoff difference across states. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Δ=(V_u−V_d)/(S_u−S_d). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Delta changes node by node. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into hedging. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Bond position is risk-free borrowing/lending amount in replication. Together with delta it matches both states. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. B chosen from V_u=ΔS_u+BR etc. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Sign indicates borrowing versus lending. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into replication. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Backward induction is valuation from final payoffs backward through tree. It extends one-step replication to many periods. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. V=D[qV_u+(1−q)V_d] with exercise test for American. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Node probabilities/discount rates must match step conventions. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into trees. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Recombining tree is tree where up-then-down equals down-then-up. It reduces computational node growth. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. ud=du. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Some processes require nonrecombining trees. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into numerical methods. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. CRR tree is Cox-Ross-Rubinstein binomial parameterisation. It approximates lognormal diffusion as steps shrink. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. u=e^{σ√Δt}, d=1/u. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Calibration/step count affect convergence. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into option pricing. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. American exercise value is intrinsic value available at node. It competes with continuation value. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. V=max(Intrinsic,Continuation). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Early exercise policy can be model-sensitive for dividends/rates. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into American options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Continuation value is value of keeping option alive rather than exercising. It is discounted risk-neutral expected future node value. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Continuation=D[qV_u+(1−q)V_d]. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Ignoring exercise comparison misprices American claims. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into trees. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Early exercise premium is extra value of American option over otherwise comparable European option. It arises from exercise flexibility. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. American≥European under same terms. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. For nondividend call with positive rates, early exercise is generally not optimal in standard model. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into American options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Black-Scholes-Merton model is continuous-time arbitrage-free model for European options under geometric Brownian motion and frictionless assumptions. It is foundational closed-form option pricing. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. C=S e^{-qT}N(d1)−K e^{-rT}N(d2). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Constant volatility/lognormal assumptions are imperfect empirically. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into option pricing. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Geometric Brownian motion is continuous stochastic process with lognormal prices. It is BSM underlying dynamic. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. dS/S=(μ−q)dt+σdW under physical measure. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Real prices jump and volatility changes. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into BSM. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Lognormal price is positive price distribution implied by GBM. It prevents negative stock price. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. ln(S_T) normal under model. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Returns exhibit fat tails/skew in data. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into BSM. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Risk-free rate is discount/financing input in BSM consistent with model/collateral context. It affects strike PV and carry. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. K e^{-rT}. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Modern markets require curve-consistent discounting rather than one universal rate. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into BSM. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Dividend yield is continuous yield q representing income paid by underlying. It reduces forward price and call value relative to no-dividend case. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Spot term S e^{-qT}. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Discrete dividends require more careful modelling. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into equity options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. d1 is standardised BSM quantity combining moneyness, rates, dividend yield, volatility and time. It appears in delta and pricing. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. d1=[ln(S/K)+(r−q+0.5σ²)T]/(σ√T). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. d1 is not literally probability stock finishes ITM under physical measure. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into BSM. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. d2 is d1 minus σ√T. It relates to risk-neutral exercise probability in European call interpretation. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. d2=d1−σ√T. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. N(d2) requires model context; it is not real-world probability. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into BSM. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Black-Scholes call is closed-form European call value. It represents dynamically replicated claim under assumptions. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. C=S e^{-qT}N(d1)−K e^{-rT}N(d2). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Market option prices imply varying volatilities, violating constant σ across strikes. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into BSM. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Black-Scholes put is European put via formula or parity. It is symmetric under parity. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. P=K e^{-rT}N(−d2)−S e^{-qT}N(−d1). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Sign/carry mistakes are common. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into BSM. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Black model is variant pricing European options on forwards/futures/rates under lognormal forward assumption. It is widely used for futures options and swaptions. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Option value uses discounted [F N(d1)−K N(d2)]. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Negative/near-zero rates motivated normal alternatives in some markets. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into rates options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Bachelier model is normal model for underlying/forward allowing negative values. It is common for normal-vol quoted rate options Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Price based on normal distribution rather than lognormal. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Normal and lognormal vol numbers are not directly comparable. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into rates. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Delta is first derivative of option price with respect to underlying. It is local directional sensitivity and hedge ratio. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Δ=∂V/∂S. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Delta is not constant; gamma changes it. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into Greeks. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Call delta is positive sensitivity, generally between 0 and 1 for standard long call. It approaches 1 deep ITM and 0 deep OTM under standard setup. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. BSM call Δ=e^{-qT}N(d1). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. FX/forward option delta conventions differ. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into delta. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Put delta is negative sensitivity for long standard put. It approaches -1 deep ITM and 0 deep OTM under standard setup. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Put Δ=e^{-qT}(N(d1)-1). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Premium-adjusted delta conventions can differ in FX. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into delta. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Gamma is second derivative with respect to underlying. It measures curvature and how delta changes. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Γ=∂²V/∂S². Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Long vanilla options generally have positive gamma; short options negative. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into Greeks. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Theta is sensitivity to passage of time, holding other inputs fixed. It captures time decay/carry effects. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Θ=∂V/∂t or conventionally per day. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Theta sign can vary by option/moneyness/rates/dividends; not universally negative in every structure. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into Greeks. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Vega is sensitivity to implied/model volatility. It measures volatility exposure. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Vega=∂V/∂σ. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Vega is not a Greek letter but market convention uses the name. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into Greeks. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Rho is sensitivity to interest rate. It measures discount/carry rate exposure. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. ρ=∂V/∂r. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. In multi-curve markets rate sensitivities are richer than one rho. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into Greeks. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Charm is sensitivity of delta to passage of time. It matters for hedge drift. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Charm=∂Δ/∂t. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Higher-order Greeks depend strongly on convention/model. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into second-order Greeks. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Vanna is sensitivity of delta to volatility or vega to spot. It links spot and volatility changes. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Vanna=∂²V/(∂S∂σ). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Important for skewed FX/equity books but model-dependent. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into higher Greeks. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Volga/Vomma is sensitivity of vega to volatility. It captures convexity in volatility. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. ∂²V/∂σ². Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Large vol shocks make linear vega approximation weak. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into higher Greeks. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Speed is derivative of gamma with respect to spot. It measures gamma change as underlying moves. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. ∂Γ/∂S. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Useful for large option books, not introductory standalone risk. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into higher Greeks. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Delta-neutral portfolio is portfolio with near-zero first-order spot sensitivity. It isolates higher-order exposures locally. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. ΣΔ_i×positions_i≈0. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Delta-neutral is not risk-neutral; gamma, vega, theta remain. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into hedging. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Gamma-neutral portfolio is portfolio constructed to offset second-order spot curvature. It stabilises delta across small moves. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. ΣΓ_i×positions_i≈0. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Requires options/other convex instruments, not underlying alone. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into hedging. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Vega-neutral portfolio is portfolio with offsetting volatility sensitivity. It reduces first-order implied-vol moves. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. ΣVega_i×positions_i≈0. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Skew/term-structure vol can still move nonuniformly. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into hedging. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Dynamic delta hedge is rebalancing underlying position as delta changes. It approximates continuous replication. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Hold −Δ against option exposure, update over time. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Discrete hedging leaves gamma/jump/slippage risk. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into BSM replication. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Gamma scalping is rebalancing delta around a long-gamma position to monetise realised movement relative to option decay/cost. It links realised volatility and hedging P&L. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Hedge gains depend on path and transaction costs. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Not guaranteed profit; implied versus realised vol and costs matter. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into options trading. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Implied volatility is σ solving model price equal to observed market option price. It translates option premium into volatility coordinates. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Find σ: ModelPrice(σ)=MarketPrice. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. IV is model-specific and differs across strikes/maturities. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into volatility. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Historical volatility is volatility estimated from past underlying returns. It is a statistical forecast input, not option price. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Sample std of returns annualised. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Historical vol and implied vol answer different questions. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into volatility. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Realised volatility is volatility actually experienced over a future/past period. It is path-dependent and measurable ex post. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Often sqrt of sum squared high-frequency/daily returns under convention. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Option P&L depends on path, not just one realised-vol number. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into volatility. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Volatility smile is implied vols high in wings relative to ATM. It reveals non-lognormal pricing/tail demand. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. IV(K) U-shaped in simple depiction. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Equity markets more often show skew/smirk rather than symmetric smile. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into surface. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Volatility skew is systematic implied-vol difference by strike, often higher OTM put vol in equities. It reflects asymmetric tails/demand and model adjustments. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. IV decreases/increases with strike depending market. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. One flat BSM volatility cannot fit skewed option prices. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into surface. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Term structure of volatility is implied volatility varying by expiry. It reflects horizon-specific uncertainty and events. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. IV(T) curve. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Annualised vols across maturities are not additive. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into surface. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Volatility surface is implied volatility across strike/moneyness and maturity. It is the market’s option-price representation. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. σ_imp(K,T). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Surface must satisfy static-arbitrage constraints. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into calibration. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Sticky-strike is heuristic assumption that implied vol by strike stays fixed as spot moves. It is one smile-dynamics model. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. σ(K) constant locally. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Real smile dynamics often differ. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into risk. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Sticky-delta is heuristic that implied vol by delta/moneyness stays fixed. Common in FX market intuition. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. σ(delta) stable locally. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. It changes Greeks relative to sticky strike. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into FX options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Local volatility is state/time-dependent volatility surface calibrated to option prices. It fits one-time marginal distributions under model. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. σ_loc(S,t). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Perfect surface fit does not guarantee realistic forward smile dynamics. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into advanced options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Stochastic volatility is model where volatility itself is random. It captures volatility clustering/smile dynamics. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Heston-style variance process. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Calibration and parameter stability are challenging. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into advanced options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Jump diffusion is model adding discontinuous price jumps. It captures gap risk/fat tails. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. dS includes Poisson jump term. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Delta hedging cannot eliminate jump risk continuously. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into advanced options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Barrier option is option activated/terminated when underlying crosses barrier. It is path-dependent. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff depends on whether barrier was hit. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Discrete monitoring creates missed-crossing numerical issues. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into exotics. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Asian option is option payoff based on average underlying price. It reduces sensitivity to single terminal price. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff=max(Average−K,0). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Average distribution complicates closed-form valuation. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into exotics. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Lookback option is payoff depends on maximum/minimum observed price. It is strongly path-dependent. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff references path extrema. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Requires rich path modelling. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into exotics. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Digital option is pays fixed amount if condition satisfied. It has discontinuous payoff at strike. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff=1_{S_T>K}×cash. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. High gamma near strike creates hedging difficulty. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into exotics. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Chooser option is holder chooses call/put form at future date. It embeds future optionality choice. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Value depends on choice date/state. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Cannot price by simple vanilla premium sum. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into exotics. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Compound option is option on another option. It nests contingent claims. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Exercise first option to receive second. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Multiple exercise dates increase complexity. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into exotics. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Option-adjusted spread is spread measure for bonds with embedded options after modelling option value. It separates credit/liquidity spread from option effect. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. OAS added to curve in option-aware pricing. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Model-dependent cash flows make OAS not directly comparable across models. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into fixed income. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Callable bond is bond containing issuer call option. Investor is short option, limiting upside as rates fall. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Bond=option-free bond−issuer call. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Negative convexity can emerge. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into fixed income. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Putable bond is bond containing holder put. Investor owns protective option. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Bond=option-free bond+put. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Put affects duration/convexity and spread. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into fixed income. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Convertible bond is bond with equity conversion option. It mixes credit and equity optionality. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Value≈straight bond+conversion option−other features. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Credit/equity dependence complicates. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into hybrids. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Warrant is option-like security issued by company, often creating dilution on exercise. It resembles long-dated call with capital-structure effects. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Dilution distinguishes from exchange-traded call. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Using ordinary option formula without dilution can misprice. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into corporate options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Employee stock option is compensation option with vesting/nontransferability/exercise behaviour. It differs from traded vanilla call. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Valuation incorporates forfeiture/exercise assumptions. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Black-Scholes with contractual maturity alone can overstate effective life. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into accounting. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Option parity arbitrage is trade exploiting violation of put-call parity. It is static and model-independent under conditions. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Compare C+KD vs P+S. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Bid/ask/dividends/borrow can remove executable profit. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into arbitrage. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Box spread is combination of bull call spread and bear put spread creating fixed payoff. It synthetically lends/borrows at implied rate. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff=K2−K1 at expiry. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Early exercise/fees can complicate American-option boxes. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into arbitrage. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Conversion is long stock + long put + short call. It produces bond-like payoff under parity. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff=K at expiry. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Dividend and borrow matter. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into arbitrage. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Reversal is short stock + short put + long call. Opposite conversion Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Produces borrowing-like structure. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Short-sale constraints matter. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into arbitrage. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Pin risk is uncertainty near strike at expiry about whether option finishes ITM/exercises. It creates hedge uncertainty. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Small spot move flips assignment/exercise. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Operational risk can exceed model value late in day. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into expiry. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Assignment risk is risk short option is exercised/assigned under contract rules. It matters especially for American options. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Short writer may receive exercise notice. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Early exercise behaviour creates operational positions. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into options operations. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Exercise settlement is physical or cash settlement after exercise. It determines resulting cash/asset positions. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Contract terms govern. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Model payoff must match settlement convention. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into operations. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Volatility risk premium is difference between option-implied compensation for volatility/tail risk and expected/realised volatility under physical measure. It can make implied vol systematically differ from future realised vol. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. IV−forecast RV is one crude lens. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Not a guaranteed trading profit after jumps/costs. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into options markets. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Smile arbitrage is static inconsistency in option surface across strikes/maturities. It can imply negative risk-neutral densities or calendar violations. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Enforce monotonicity/convexity/term constraints. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Raw noisy quotes need cleaning before calibration. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into surface. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Surface calibration is fit model parameters to observed option prices or implied vols. It creates internally consistent pricing inputs. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Minimise weighted pricing error subject to constraints. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Best fit today can have poor dynamics tomorrow. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into model risk. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Model-free implied variance is variance-swap-style extraction using strip of option prices. It aggregates option surface without one BSM volatility. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Integral/sum of OTM option prices under assumptions. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Finite strikes/liquidity require extrapolation. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into volatility derivatives. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Variance swap is contract paying realised variance minus strike variance. It gives direct volatility exposure. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff=N_var(RealisedVar−K_var). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Variance is not volatility; units are squared. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into volatility derivatives. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Volatility swap is contract paying realised volatility minus strike volatility. It differs from variance due to square-root nonlinearity. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff=N_vol(RealisedVol−K_vol). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Cannot price by simply square-rooting fair variance due to convexity. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into volatility derivatives. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. VIX-style index is option-implied expected variance measure over defined horizon/methodology. It summarises option surface, not historical volatility. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Built from option prices across strikes. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. VIX level is annualised volatility points, not probability. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into volatility. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Option P&L attribution is decomposition of option value change into delta, gamma, vega, theta, rates and residual. It connects Greeks to realised P&L. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. ΔV≈ΔS·delta+0.5ΓΔS²+vegaΔσ+thetaΔt+… Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Large moves/skew shifts require full repricing. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into risk management. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Hedge slippage is difference between theoretical hedge execution and realised trade prices/timing. It reduces replication performance. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. ActualP&L−ModelHedgeP&L. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. High-frequency rebalancing can increase costs. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into hedging. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Jump risk is discontinuous underlying movement. It defeats continuous delta replication assumption. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Price gap occurs before hedge can rebalance. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Short gamma portfolios can suffer large jump losses. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into options risk. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Gap risk is overnight/event jump exposure. It is operational form of jump risk. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Opening price differs from previous close. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Stop-loss orders do not guarantee execution price. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into risk. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Volatility-of-volatility is randomness of volatility itself. It affects smile and vega convexity. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Vol processes have own σ_v parameter. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. BSM constant vol cannot capture. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into advanced options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Correlation skew is strike-dependent correlation implied from multi-asset/index options. It affects basket/index option pricing. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Implied dependence changes in tails. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Constant correlation can misprice crash states. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into multi-asset options. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Quanto option is option payoff on foreign asset converted at fixed FX rate. It embeds covariance adjustment between asset and FX. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Pricing drift adjusted by asset-FX covariance. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Ignoring correlation creates wrong value. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into cross-asset. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Basket option is option on weighted portfolio of assets. Its value depends strongly on correlations. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Underlying basket B=Σw_iS_i. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Moment matching can miss tail dependence. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into multi-asset. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Spread option is option on difference between two prices. It depends on both volatilities and correlation. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff=max(S1−S2−K,0). Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. No simple universal BSM closed form. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into commodities. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Rainbow option is option on multiple underlyings with max/min/best-of payoffs. It is correlation-sensitive. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Payoff depends on joint order statistics. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. Marginal vol alone insufficient. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into multi-asset. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Model risk is risk chosen option model misrepresents dynamics, tails, exercise or surface. It affects value and hedge. Option mathematics is easiest when payoff geometry is understood before valuation inputs are introduced. Mathematics. Compare models and hedge residuals. Keep strike, spot, forward, maturity, rates, dividends and volatility on consistent units and conventions. For Greeks, always state whether sensitivities are per unit, per 1% volatility point, per day or per basis point. Failure mode. A perfect calibration today can hedge poorly tomorrow. Jo’s diagnostic is to check payoff bounds, parity and monotonic direction before accepting any model output. Connection. This feeds directly into governance. Ryan would compare model value with binomial/replication or market implied-volatility representation. The purpose is to tie option price to both payoff and hedge. Buy European call with K=100 for premium 8. At expiry if S_T=120, payoff=20 and simple profit before financing/fees=12. If S_T=90, payoff=0 and loss=8. Break-even at expiry is K+premium=108 in this simplified cash-premium view. The option can be ITM between 100 and 108 yet still produce negative net profit relative to premium. This distinguishes moneyness from investment profitability. Buy stock at 100 and one-year put K=90 for premium 3. Terminal combined payoff is S_T+max(90−S_T,0), so value before premium is never below 90. The put creates a floor but costs 3 upfront. The protected position still loses from 100 toward 90 plus premium effects, but catastrophic downside below 90 is transferred to put seller. This is insurance-like asymmetry, not free protection. S=100, K=105, one-year discount factor 0.96, call price 7. European non-dividend parity gives put P=C+KD−S=7+100.8−100=7.8. If comparable put were 10 with frictionless executable markets, the two equivalent terminal portfolios would be mispriced. Parity is a payoff identity and does not depend on choosing volatility. Stock 100 can become 120 or 90 in one year. Risk-free gross return is 1.05. Call K=100 pays 20 up, 0 down. Delta=(20−0)/(120−90)=2/3. Risk-neutral q=(1.05−0.90)/(1.20−0.90)=0.5. Call value=(0.5×20+0.5×0)/1.05≈9.5238. Replication and risk-neutral expectation give the same result. Physical up probability is irrelevant to the arbitrage price. At a binomial node, put intrinsic value is 15 while discounted continuation value is 12. For an American put, node value is max(15,12)=15 and early exercise is optimal in the model. For a European put, value at same node would remain 12 because exercise is unavailable before expiry. This single comparison is the core of American-option backward induction. For a non-dividend stock, BSM call uses S, K, r, σ and T through d1 and d2. The formula can be read economically as a leveraged underlying position minus a discounted strike-payment position, with N(d1) and N(d2) weighting those components under the pricing model. The stock’s physical expected return does not appear because dynamic replication removes it from the arbitrage-free price under model assumptions. This is one of the deepest lessons in derivative pricing: price and forecast are separate. You are short 100 calls, each with delta 0.60 and multiplier 1. Net option delta=-60 shares. Buy 60 shares to become approximately delta-neutral. If spot moves, call delta changes because gamma is nonzero. The hedge must be rebalanced to remain neutral. A delta hedge reduces first-order spot risk, not gamma, vega, theta or jump risk. A delta-neutral option position has gamma +0.02 per currency unit squared. Spot jumps by 5. Second-order P&L contribution≈0.5×0.02×25=+0.25 per unit before theta/vega and hedge slippage. Long gamma benefits from sufficiently large realised movement, while often paying negative theta to own that convexity. This is why option books are frequently described through Greek trade-offs rather than one directional view. Market call trades at 12, but BSM using 20% volatility gives price 9. Increase σ until model price reaches 12; the solving σ is implied volatility. The market is not literally announcing future realised volatility. It is providing an option price which, under the chosen model, maps to a volatility coordinate. Different strikes usually map to different implied vols, producing skew/smile. Suppose one-month 90-strike put implies 30% vol, ATM option 22%, and 110-strike call 20%. A flat 22% BSM model underprices the downside put relative to market. The skew can reflect asymmetric crash risk, demand for protection and model limitations. Traders therefore manage a volatility surface rather than one σ. Greeks calculated with surface dynamics can differ from flat-vol Greeks. An option has delta 0.4, gamma 0.03, vega 0.12 per 1 vol point and theta -0.05 per day. Over one day, spot rises 2, implied vol rises 1 point. Approx P&L=0.4×2+0.5×0.03×4+0.12×1−0.05=0.93 per unit, before higher-order terms. Full repricing may differ because Greeks themselves change. The residual is useful model/approximation information. This is the practical role of Greeks: local P&L explanation and hedge sizing. Own stock at 100 and sell call K=110 for premium 4. If stock ends 130, stock gain 30 but call payoff -20, leaving 10 plus premium 4=14. If stock ends 80, call expires worthless and total loss is -20+4=-16. The strategy converts some upside into premium but does not protect deeply against stock decline. Calling it “income generation” without showing capped upside and downside is incomplete. A forward payoff is linear, so spreading future outcomes around the same forward mean does not change discounted expected payoff under risk-neutral pricing. A call payoff is convex: upside states contribute increasingly while downside payoff floors at zero. Jensen-style convexity intuition means more dispersion can raise option value. That is why vega is positive for long standard calls and puts. Volatility is itself a priced state variable. Mira’s intuition test is to imagine the underlying either barely moving or making huge two-sided moves while keeping the forward level similar. Which world gives the option holder more opportunity? BSM assumes continuous trading, frictionless markets, geometric Brownian motion, constant volatility and specified rate/dividend structure. Real markets jump, trade discretely, face transaction costs and display volatility smiles. Yet BSM remains foundational because it provides a coherent benchmark, interpretable Greeks and the language of implied volatility. Modern option practice often uses BSM as a quoting coordinate while calibrating richer local/stochastic-volatility/jump models for risk and exotic pricing. The model’s enduring value is not that every assumption is true; it is that the framework shows exactly how replication transforms uncertainty into price under a defined world. Premium and financing/fees matter. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. An ITM option can still lose relative to premium. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Early exercise changes value and parity/bounds. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Risk-neutral q is derived from no-arbitrage. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Standard replication removes μ from formula. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Market surfaces vary by strike and expiry. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Gamma, vega, theta, jumps and basis remain. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Often quoted per 1 volatility percentage point, not per 100% change. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Some structures/conditions produce nonstandard theta behavior. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Forward moneyness can matter more than spot moneyness. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Contracts must align exactly. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Dividend/carry conditions determine exercise economics. The repair is to return to payoff geometry, contract terms and no-arbitrage bounds before recalculating model value. Situation. Investor fears a crash but wants upside. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Buy protective put; compare premium, floor and delta/vega exposure. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. Protection is temporary and priced. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Investor sells covered call. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Plot combined payoff and compare with unhedged stock. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. Premium does not eliminate downside. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Importer buys currency call rather than forward. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Compare guaranteed maximum exchange cost plus premium with forward lock. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. Option preserves favourable currency move at premium cost. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Deep ITM put with positive rates approaches exercise decision. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Use binomial continuation-versus-intrinsic test at each node. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. European formula cannot decide early exercise. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Short-dated implied vol rises before company announcement. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Separate event volatility from ordinary time decay. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. High IV may collapse after event even if stock barely moves. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Trader owns options and dynamically hedges spot. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Track realised movement, theta and transaction costs. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. Long gamma profits are not guaranteed; implied/realised relation matters. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Dealer sells options and collects premium. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Measure negative gamma, vega and jump stress. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. Stable small gains can hide rare large losses. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Portfolio holds downside puts across strikes. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Use surface Greeks and stress skew steepening/flattening. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. One vega number cannot capture strike-dependent vol moves. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Investor owns bond but is short issuer call. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Value option-free bond minus call and inspect negative convexity. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. Yield-to-maturity alone misses exercise economics. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Currency pair has pronounced risk reversal/skew. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Use correct FX delta/vol quote convention and two interest-rate curves. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. Equity-option conventions cannot be copied blindly into FX. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Index put IV reflects correlation/crash risk across constituents. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Compare index variance with weighted single-stock variance and implied correlation. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. Correlation itself becomes an option-market risk factor. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. Situation. Protection disappears or activates at barrier. The mathematical task is to choose a payoff that matches the asymmetry desired. Method. Simulate/solve path monitoring and stress gaps through barrier. Adrian draws payoff, Jo checks parity/bounds, Aisha chooses binomial or continuous valuation, and Ryan calculates Greeks and stress P&L. Boundary. Vanilla delta/vega alone may miss discontinuity. Mira then identifies what model feature—volatility dynamics, jumps, early exercise, correlation or path dependence—can make simple vanilla intuition fail. An option is a nonlinear right. Its price is the cost of replicating or risk-neutrally valuing that asymmetry under a model, and its Greeks describe how that cost changes locally as the market moves. Payoff geometry comes first. Put–call parity gives model-independent consistency. Binomial trees make replication visible. Black–Scholes–Merton compresses continuous dynamic replication into a celebrated closed form. Implied volatility translates market prices back into a common model coordinate. Greeks turn that value into a risk map. World-class option mathematics does not stop at the formula. It asks whether the option is American or European, whether dividends and rates are modelled correctly, whether the volatility surface is arbitrage-free, whether hedges survive jumps and discrete trading, and whether higher-order risks explain P&L that delta missed. With this page, the `BTT-BFM-WORLD-150` through `-180` market-pricing foundation is complete: FX → derivatives → no-arbitrage → options. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Assign delta, gamma, vega, theta and rho to an option. Apply spot, vol, time and rate changes. Compare Taylor approximation with full repricing and analyse residual. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose several strikes and expiries with option prices. Solve implied vols, plot smile/skew and term structure, then check monotonicity and butterfly/calendar arbitrage conditions conceptually. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. For call, put, covered call, protective put, straddle and spread, draw terminal payoff before premium and then profit after premium. Mark strike, break-even, max gain and max loss where finite. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Choose S,u,d,r,K. Calculate terminal payoffs, derive q, work backward, and at each American node compare continuation with exercise. Verify European value is no greater than American counterpart. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes. Use a realistic option chain with bid/ask placeholders. Construct synthetic stock/forward and determine whether apparent mid parity gap survives executable spreads and stock borrow/dividend assumptions. Complete the lab first under a flat-volatility BSM or binomial assumption, then change one structural feature—dividend, early exercise, skew, jump or stochastic volatility. Ben should reconcile the payoff, Clara should record market conventions, and Ethan should identify the hedge residual created by the richer feature. Finish with a model-independent check wherever possible: parity, bounds, monotonicity or convexity in strike. These checks remain valuable even when the chosen stochastic model changes.11. At the money
12. Out of the money
13. Moneyness
14. Option premium
15. Option writer
16. Covered call
17. Protective put
18. Call spread
19. Put spread
20. Straddle
21. Strangle
22. Butterfly
23. Risk reversal
24. Collar
25. Option lower bound
26. Option upper bound
27. Put-call parity
28. Synthetic forward
29. Synthetic stock
30. One-period binomial model
31. Up factor
32. Down factor
33. Risk-free gross return
34. Risk-neutral probability
35. Binomial delta
36. Bond position
37. Backward induction
38. Recombining tree
39. CRR tree
40. American exercise value
41. Continuation value
42. Early exercise premium
43. Black-Scholes-Merton model
44. Geometric Brownian motion
45. Lognormal price
46. Risk-free rate
47. Dividend yield
48. d1
49. d2
50. Black-Scholes call
51. Black-Scholes put
52. Black model
53. Bachelier model
54. Delta
55. Call delta
56. Put delta
57. Gamma
58. Theta
59. Vega
60. Rho
61. Charm
62. Vanna
63. Volga/Vomma
64. Speed
65. Delta-neutral portfolio
66. Gamma-neutral portfolio
67. Vega-neutral portfolio
68. Dynamic delta hedge
69. Gamma scalping
70. Implied volatility
71. Historical volatility
72. Realised volatility
73. Volatility smile
74. Volatility skew
75. Term structure of volatility
76. Volatility surface
77. Sticky-strike
78. Sticky-delta
79. Local volatility
80. Stochastic volatility
81. Jump diffusion
82. Barrier option
83. Asian option
84. Lookback option
85. Digital option
86. Chooser option
87. Compound option
88. Option-adjusted spread
89. Callable bond
90. Putable bond
91. Convertible bond
92. Warrant
93. Employee stock option
94. Option parity arbitrage
95. Box spread
96. Conversion
97. Reversal
98. Pin risk
99. Assignment risk
100. Exercise settlement
101. Volatility risk premium
102. Smile arbitrage
103. Surface calibration
104. Model-free implied variance
105. Variance swap
106. Volatility swap
107. VIX-style index
108. Option P&L attribution
109. Hedge slippage
110. Jump risk
111. Gap risk
112. Volatility-of-volatility
113. Correlation skew
114. Quanto option
115. Basket option
116. Spread option
117. Rainbow option
118. Model risk
Worked Example 1: Call Payoff and Profit
Worked Example 2: Protective Put
Worked Example 3: Put–Call Parity
Worked Example 4: One-Step Binomial Call
Worked Example 5: American Put Early Exercise
Worked Example 6: BSM Call
Worked Example 7: Delta Hedge
Worked Example 8: Gamma P&L
Worked Example 9: Implied Volatility
Worked Example 10: Volatility Skew
Worked Example 11: Option P&L Attribution
Worked Example 12: Covered Call
Why Volatility Matters More for Options Than Forwards
Why BSM Is Foundational but Not Literal Reality
A Professional Option-Mathematics Workflow
Common Failure Modes
1. Payoff confused with profit
2. ITM confused with profitable
3. European and American treated identically
4. Physical probability used in binomial price
5. BSM physical expected return inserted
6. Flat volatility assumed from one option
7. Delta-neutral called risk-free
8. Vega units misread
9. Theta sign universalised
10. Option quote compared without carry/dividends
11. Parity used with mismatched strikes/expiries
12. American call early exercise rule overgeneralised
Formula Map
Concept Simplified formula Meaning Call payoff max(S_T−K,0) Right to buy at strike. Put payoff max(K−S_T,0) Right to sell at strike. Put-call parity C+KD=P+S European non-dividend payoff identity. Binomial delta (V_u−V_d)/(S_u−S_d) Replicating underlying holding. Risk-neutral probability (R−d)/(u−d) One-step pricing weight. Binomial value D[qV_u+(1−q)V_d] Discounted risk-neutral continuation value. BSM call S e^{-qT}N(d1)−K e^{-rT}N(d2) European call benchmark model. Delta ∂V/∂S First-order spot sensitivity. Gamma ∂²V/∂S² Curvature / delta sensitivity. Vega ∂V/∂σ Volatility sensitivity. Authoritative Reference Map
Connected Banking And Finance Mathematics Route
Applied Case Study 1: Protecting a stock position
Applied Case Study 2: Generating capped equity income
Applied Case Study 3: Hedging an importer
Applied Case Study 4: American put
Applied Case Study 5: Earnings event option
Applied Case Study 6: Delta-hedged long gamma
Applied Case Study 7: Short volatility book
Applied Case Study 8: Volatility skew hedge
Applied Case Study 9: Callable bond
Applied Case Study 10: FX option
Applied Case Study 11: Index versus single-stock options
Applied Case Study 12: Barrier option
Final Principle
Deep Practice Lab 1: Draw payoff and profit separately
Deep Practice Lab 2: Build a two-step binomial tree
Deep Practice Lab 3: Check put-call parity across quotes
Deep Practice Lab 4: Greek P&L explain
Deep Practice Lab 5: Build an implied-vol surface
Deep Practice Lab 6: Draw payoff and profit separately
Deep Practice Lab 7: Build a two-step binomial tree
Deep Practice Lab 8: Check put-call parity across quotes
Deep Practice Lab 9: Greek P&L explain
Deep Practice Lab 10: Build an implied-vol surface
Deep Practice Lab 11: Draw payoff and profit separately
Deep Practice Lab 12: Build a two-step binomial tree
Deep Practice Lab 13: Check put-call parity across quotes
Deep Practice Lab 14: Greek P&L explain
Deep Practice Lab 15: Build an implied-vol surface
Deep Practice Lab 16: Draw payoff and profit separately
Deep Practice Lab 17: Build a two-step binomial tree
Deep Practice Lab 18: Check put-call parity across quotes
Deep Practice Lab 19: Greek P&L explain
Deep Practice Lab 20: Build an implied-vol surface
Deep Practice Lab 21: Draw payoff and profit separately
Deep Practice Lab 22: Build a two-step binomial tree
Deep Practice Lab 23: Check put-call parity across quotes
Deep Practice Lab 24: Greek P&L explain
Deep Practice Lab 25: Build an implied-vol surface
Deep Practice Lab 26: Draw payoff and profit separately
Deep Practice Lab 27: Build a two-step binomial tree
Deep Practice Lab 28: Check put-call parity across quotes
Deep Practice Lab 29: Greek P&L explain
Deep Practice Lab 30: Build an implied-vol surface
Deep Practice Lab 31: Draw payoff and profit separately
Deep Practice Lab 32: Build a two-step binomial tree
Deep Practice Lab 33: Check put-call parity across quotes
Deep Practice Lab 34: Greek P&L explain
Deep Practice Lab 35: Build an implied-vol surface
Deep Practice Lab 36: Draw payoff and profit separately
Deep Practice Lab 37: Build a two-step binomial tree
Deep Practice Lab 38: Check put-call parity across quotes
Deep Practice Lab 39: Greek P&L explain
Deep Practice Lab 40: Build an implied-vol surface
Deep Practice Lab 41: Draw payoff and profit separately
Deep Practice Lab 42: Build a two-step binomial tree
Deep Practice Lab 43: Check put-call parity across quotes
Deep Practice Lab 44: Greek P&L explain
Deep Practice Lab 45: Build an implied-vol surface
Deep Practice Lab 46: Draw payoff and profit separately
Deep Practice Lab 47: Build a two-step binomial tree
Deep Practice Lab 48: Check put-call parity across quotes
Deep Practice Lab 49: Greek P&L explain
Deep Practice Lab 50: Build an implied-vol surface
Deep Practice Lab 51: Draw payoff and profit separately
Deep Practice Lab 52: Build a two-step binomial tree
Deep Practice Lab 53: Check put-call parity across quotes
Deep Practice Lab 54: Greek P&L explain
Deep Practice Lab 55: Build an implied-vol surface
Deep Practice Lab 56: Draw payoff and profit separately
Deep Practice Lab 57: Build a two-step binomial tree
Deep Practice Lab 58: Check put-call parity across quotes
Deep Practice Lab 59: Greek P&L explain
Deep Practice Lab 60: Build an implied-vol surface
Deep Practice Lab 61: Draw payoff and profit separately
Deep Practice Lab 62: Build a two-step binomial tree
Deep Practice Lab 63: Check put-call parity across quotes
Deep Practice Lab 64: Greek P&L explain
Deep Practice Lab 65: Build an implied-vol surface
Deep Practice Lab 66: Draw payoff and profit separately
Deep Practice Lab 67: Build a two-step binomial tree
Deep Practice Lab 68: Check put-call parity across quotes
Deep Practice Lab 69: Greek P&L explain
Deep Practice Lab 70: Build an implied-vol surface
Deep Practice Lab 71: Draw payoff and profit separately
Deep Practice Lab 72: Build a two-step binomial tree
Deep Practice Lab 73: Check put-call parity across quotes
Deep Practice Lab 74: Greek P&L explain
Deep Practice Lab 75: Build an implied-vol surface
Deep Practice Lab 76: Draw payoff and profit separately
Deep Practice Lab 77: Build a two-step binomial tree
Deep Practice Lab 78: Check put-call parity across quotes
Deep Practice Lab 79: Greek P&L explain
Deep Practice Lab 80: Build an implied-vol surface
Deep Practice Lab 81: Draw payoff and profit separately
Deep Practice Lab 82: Build a two-step binomial tree
Deep Practice Lab 83: Check put-call parity across quotes
Deep Practice Lab 84: Greek P&L explain
Deep Practice Lab 85: Build an implied-vol surface
Deep Practice Lab 86: Draw payoff and profit separately
Deep Practice Lab 87: Build a two-step binomial tree
Deep Practice Lab 88: Check put-call parity across quotes