No-arbitrage mathematics is the consistency engine of modern finance. It says that if two portfolios deliver exactly the same future cash flows in every relevant state, they should have the same current value once market conventions and frictions are controlled. From that principle come forward prices, bond discount factors, state prices, put–call parity, risk-neutral probabilities, binomial option values, swap rates and much of derivative valuation.
For readers searching for no arbitrage finance, arbitrage pricing, law of one price, replication finance, state prices, Arrow Debreu prices, risk neutral probability, risk neutral valuation, discount factor mathematics, put call parity, forward replication, self financing portfolio, fundamental theorem of asset pricing or arbitrage-free valuation, the key idea is not “markets are always efficient”. It is narrower and more mathematical: when a payoff can be replicated exactly by traded instruments, inconsistent prices create a mechanical trading strategy with no net future risk under the model.
CFA Institute’s 2026 derivatives and fixed-income readings place arbitrage, replication and the law of one price at the foundation of forward commitments, contingent claims and arbitrage-free fixed-income valuation. This page develops that foundation step by step, while the existing specialist algorithm library retains deeper implementation articles. It is educational mathematics, not a claim that real markets are frictionless or free of temporary dislocations.
50-Second Router
- Law of one price: identical future cash flows should have identical current prices in an arbitrage-free market.
- Arbitrage: a self-financing strategy with no net downside under the model and positive payoff in at least one state, or equivalent strict definitions.
- Replication: construct one payoff from other traded assets.
- Self-financing: after initial setup, strategy changes are funded from within the portfolio rather than new external cash.
- Discount factor: current price of one unit of certain future cash.
- State price: current price of one unit paid only in a specified future state.
- Risk-neutral probability: transformed probability used for pricing so discounted traded prices are martingales under assumptions; it is not a real-world forecast probability.
- Complete market: every relevant contingent payoff can be replicated; arbitrage-free price is unique.
- Incomplete market: some payoffs cannot be perfectly replicated; no-arbitrage may give a price range rather than one unique value.
- Put–call parity: equality created by two portfolios with identical terminal option/asset cash flows.
- Verification: compare state-by-state payoff tables, not expected payoffs alone.
The Central Proposition: Equal Payoffs Require Equal Prices
Suppose Portfolio A and Portfolio B produce exactly the same cash flow at every future date and in every possible state included in the model. If A costs S$100 and B costs S$95 today, buy B and sell A. The initial net inflow is S$5, while all future cash flows cancel state by state. Under frictionless assumptions, that is arbitrage.
The logic does not require a forecast about which state will occur. That is its power. Pricing by replication is fundamentally different from saying “the expected future payoff is X, so I think the price should be Y”. Replication asks whether another traded portfolio already produces the same payoff.
Adrian’s test is therefore visual: write a table with one row per state and one column per portfolio. If every terminal cell matches, current prices must match in the idealised model.
1. Law of one price
Law of one price is principle that identical future cash flows should have the same current value in an arbitrage-free market. It is the simplest relative-pricing theorem. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. If CF_A(s)=CF_B(s) for all states s, then P_A=P_B. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Equal expected payoff is not enough; payoffs must match state by state for exact replication. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into relative valuation. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
2. Arbitrage
Arbitrage is trading strategy generating nonnegative payoff in all states and positive payoff in at least one state with zero/nonpositive initial cost under common definitions. It identifies inconsistent relative prices. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. Initial cost≤0, terminal payoff≥0 all states and >0 some state. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Real-world funding, short-sale and transaction constraints can prevent textbook arbitrage. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into pricing bounds. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
3. Pure arbitrage
Pure arbitrage is strict zero-risk profit under the ideal market model. It differs from high-Sharpe or statistical trades. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. No modelled state has loss. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Calling a risky convergence trade arbitrage abuses the term. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into derivative pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
4. Statistical arbitrage
Statistical arbitrage is strategy seeking positive expected return from statistical patterns rather than state-by-state guaranteed profit. It is not no-arbitrage in the theorem sense. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. Expected P&L>0 under model but losses remain possible. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Do not use statistical arbitrage evidence to derive exact price identities. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into trading. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
5. Self-financing strategy
Self-financing strategy is portfolio whose rebalancing is funded by selling/buying assets within the strategy after initial capital. It prevents hidden external cash from creating fake profit. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. Changes in holdings satisfy value conservation before gains/losses. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Injecting cash midway invalidates a claimed arbitrage proof unless included. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into dynamic replication. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
6. Replication
Replication is constructing a portfolio whose payoff equals a target payoff in all model states. It converts valuation into a portfolio-cost problem. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. Find weights w such that Xw=payoff vector. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Approximate hedging is not exact replication. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into derivatives. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
7. Replicating portfolio
Replicating portfolio is specific traded-asset holdings reproducing target cash flows. Its current cost gives target arbitrage-free value in a complete market. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. V_target=w’P. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. If multiple exact replicas have different costs, arbitrage exists. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into valuation. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
8. Dominance
Dominance is one payoff is never worse and sometimes better than another. It imposes price inequality rather than exact equality. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. If A payoff≥B payoff all states, then P_A≥P_B under no-arbitrage. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Ignoring timing can break dominance comparisons. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into bounds. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
9. Monotonicity
Monotonicity is higher state-by-state payoff cannot have lower price under positive pricing function. It is a basic arbitrage-free property. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. X≥Y ⇒ P(X)≥P(Y). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Negative state prices would violate monotonicity. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into state prices. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
10. Linearity of price
Linearity of price is price of combined replicable cash flows is sum of component prices under frictionless linear markets. It enables portfolio decomposition. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. P(aX+bY)=aP(X)+bP(Y). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Nonlinear funding/margin/market-impact costs can break practical linearity. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into portfolio valuation. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
11. Additivity
Additivity is price of two cash-flow streams together equals sum when no interaction or constraints alter economics. It underlies bond decomposition. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. P(X+Y)=P(X)+P(Y). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Embedded options/covenants can make legal package not simply separable. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into fixed income. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
12. Homogeneity
Homogeneity is scaling payoff by factor scales price by same factor under linear pricing. It makes per-unit prices meaningful. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. P(aX)=aP(X). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Large positions can face market impact in reality. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into notional scaling. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
13. Discount factor
Discount factor is current price of one unit of certain currency paid at future date T. It is the fundamental time-value price. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. D(0,T)=P(today of 1 at T). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Discount factor is not the same as discount rate. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into bonds. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
14. Zero-coupon bond
Zero-coupon bond is tradable claim paying one unit at maturity. It operationalises the discount factor. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. Price=D(0,T) per unit redemption. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Credit-risky zero is not risk-free discount factor. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into term structure. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
15. Present value
Present value is sum of future cash flows multiplied by appropriate discount factors. It is arbitrage-free when factors come from consistent tradable prices. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. PV=ΣCF_tD_t. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. One YTM for all dates is a compression, not general discounting. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into cash-flow valuation. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
16. State
State is mutually exclusive future outcome in a finite-state model. It allows payoff vectors to be compared exactly. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. Payoff X=(X_1,…,X_n). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. State definitions must be exhaustive for the model. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into state pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
17. State-contingent claim
State-contingent claim is claim paying only under specified state conditions. It is the building block of complete-market pricing. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. Payoff vector with state-specific entries. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Real markets rarely trade pure state claims directly. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into Arrow securities. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
18. Arrow-Debreu security
Arrow-Debreu security is theoretical claim paying one unit in one state and zero in all others. Its price is a state price. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. q_s=P(pay 1 in state s). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. State prices are model constructs unless directly replicated. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into asset pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
19. State price
State price is current price per unit payoff in a particular state. It combines time value and risk pricing. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. P(X)=Σq_sX_s. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. State price is not a physical probability because it includes discounting/risk adjustment. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into risk-neutral measure. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
20. Positive state price
Positive state price is requirement under standard arbitrage-free finite models when state payoff can be isolated. It preserves monotonicity. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. q_s>0. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Zero/negative state prices can signal redundant/impossible states or arbitrage. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into fundamental theorem. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
21. Pricing kernel
Pricing kernel is random variable transforming future payoff into current price under a physical probability measure. It generalises state prices. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. P(X)=E_P[mX]. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Kernel is not directly observable without model/asset data. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into asset pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
22. Stochastic discount factor
Stochastic discount factor is another name/framework for pricing kernel. It prices risk by covariance with marginal value of wealth. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. P=E[mX]. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Using one constant discount rate ignores state dependence. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into asset pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
23. Risk-neutral probability
Risk-neutral probability is probability measure under which discounted traded asset prices are martingales, when such measure exists. It simplifies arbitrage-free pricing. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. P_0=D×E_Q[Payoff]. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Q probabilities are pricing weights, not real-world forecasts. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into derivatives. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
24. Physical probability
Physical probability is real-world/statistical probability measure. It models actual outcome frequencies. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. E_P governs forecasting/risk. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Physical expected return is not needed for many replication prices. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into forecasting. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
25. Equivalent martingale measure
Equivalent martingale measure is risk-neutral measure assigning zero probability to same impossible events as physical measure and making discounted prices martingales. Its existence is linked to no arbitrage. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. Discounted S_t is Q-martingale. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Uniqueness depends on market completeness. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into fundamental theorem. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
26. Fundamental theorem of asset pricing
Fundamental theorem of asset pricing is link between no-arbitrage and existence of an equivalent martingale measure under technical conditions. It formalises the pricing system. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. No arbitrage ↔ existence; completeness ↔ uniqueness in standard finite models. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. The theorem’s exact mathematical conditions matter in continuous models. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into mathematical finance. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
27. Complete market
Complete market is market where every contingent claim in model state space can be replicated. It yields unique arbitrage-free prices. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. Payoff matrix spans state space. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space.
Failure mode. Real markets can be incomplete due to missing instruments/frictions. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete.
Connection. This feeds directly into unique pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check.
28. Incomplete market
Incomplete market is market where not every contingent payoff can be replicated. No-arbitrage alone may yield interval/set of prices. The discipline is to compare entire payoff vectors rather than one expected scenario.
Mathematics. Payoff matrix rank Failure mode. Choosing one price requires preferences/model assumptions beyond pure replication. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into pricing bounds. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Spanning is ability of traded asset payoff vectors to generate target payoff vector. It is linear algebra behind replication. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Target lies in column space of payoff matrix. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. More securities do not guarantee spanning if payoffs are redundant. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into linear algebra. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Redundant security is security whose payoff is replicable from others. Its price is pinned by law of one price. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Payoff vector is linear combination of others. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. If its market price differs from replica cost, arbitrage exists. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into relative pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Payoff matrix is matrix of asset cash flows across states. It turns replication into linear equations. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Xw=y. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Poorly defined states/payoffs create algebra that looks precise but models wrong world. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into finite-state pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Rank is dimension of independent payoff directions spanned by assets. It determines completeness in finite-state settings. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. rank(X)=number states for full spanning. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Numerical near-collinearity can make replication unstable. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into linear algebra. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Linear system is equations solved for replicating holdings. It is core computational step. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Xw=y. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. A mathematical solution may require prohibited short positions in constrained markets. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into replication. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Short selling is selling borrowed asset to create negative position. It is often required in arbitrage replication. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Weight w<0. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Short-sale constraints widen arbitrage-free bounds. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into market frictions. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Borrowing/lending is risk-free financing used to shift cash across time. It creates discount-factor leg of replication. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Borrow now, repay at risk-free accumulation. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Borrowing rate may exceed lending rate, creating pricing bands. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into funding frictions. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Transaction cost is bid-offer, fees and market impact paid to trade. It turns exact equalities into no-arbitrage intervals. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Arbitrage profit must exceed total costs. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Ignoring costs creates phantom opportunities. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into real markets. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Funding spread is difference between participant borrowing cost and benchmark discount rate. It changes replication economics. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Carry uses actual available funding under institutional context. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Textbook risk-free borrowing may be unavailable. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into derivatives. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Collateral is assets/cash posted against derivative exposure. It changes funding and discounting in modern markets. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Collateral remuneration curve can influence discount factors. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Ignoring CSA can make clean replication inconsistent with trade terms. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into OTC pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Market impact is price movement caused by executing large trades. It breaks linear scaling. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Execution price depends on quantity. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Homogeneity assumptions are local approximations. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into large trades. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Liquidity constraint is inability to trade required quantity instantly at quoted price. It can prevent arbitrage despite theoretical mispricing. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Feasible set restricts weights/trades. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. No-arbitrage model assumes tradability of replicating instruments. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into limits. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Model arbitrage is apparent arbitrage created by inconsistent models/inputs rather than executable market prices. It signals system inconsistency. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Cross-model prices violate parity. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Before trading, reconcile conventions, collateral, timestamps and liquidity. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into validation. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Cash-and-carry is replication of forward by buying spot and financing carry. It derives forward upper/lower relations. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Long spot + borrow ↔ long forward payoff structure. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Income/storage/shortability must be included. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into forwards. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Forward price is delivery price making forward zero value at inception. It is determined by replication under assumptions. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. F=S/D adjusted for income/carry. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Expected spot is unnecessary for pure no-arbitrage pricing. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into forwards. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Forward value is difference between contract terms and current replicating terms. It is current asset/liability value. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. V=D(F_current−K). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Do not confuse zero inception value with zero risk. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into valuation. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Covered interest parity is FX no-arbitrage relation across two money markets and forward exchange. It is international cash-and-carry. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. F=S A_dom/A_for. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Cross-currency basis/funding frictions create real deviations. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into FX. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Put-call parity is identity linking European call, put, underlying and bond with same strike/maturity. It comes from exact terminal payoff replication. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. C+K D=P+S for non-dividend stock, simple form. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Mismatched strike, expiry or exercise style invalidates equality. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Synthetic forward is long call plus short put at same strike/expiry. It replicates long forward payoff at maturity. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. C−P=S−KD in PV terms. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. American exercise/dividends can alter simple relation. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Protective put is long stock plus long put. Its payoff has floor at strike. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. S+P equals call+bond under parity. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Premium makes protection costly. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Fiduciary call is long call plus risk-free bond paying strike. It replicates protective put terminal payoff. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. C+KD=S+P. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Bond amount must be PV of strike, not strike today. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into parity. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Binomial state model is discrete up/down future price model. It makes state-by-state replication transparent. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. S_u,S_d with two states. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Up/down probabilities for pricing are derived, not forecasted physical probabilities. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Hedge ratio in one-step binomial is underlying units needed to match option payoff difference across states. It solves replication. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Δ=(C_u−C_d)/(S_u−S_d). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Delta is state/time dependent and changes in multi-step tree. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into option replication. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Risk-neutral up probability is pricing weight making expected underlying growth equal risk-free rate in binomial model. It is derived from no-arbitrage. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. q=(R-d)/(u-d) under simple gross-return notation. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. q outside [0,1] signals arbitrage/inconsistent tree assumptions. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into binomial pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Risk-neutral option value is discounted expected payoff under derived risk-neutral probabilities. It equals replicating portfolio cost. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. C_0=D[qC_u+(1-q)C_d]. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. This is not a forecast of actual option payoff frequency. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Dynamic replication is rebalancing replicating portfolio through time as state variables change. It underlies continuous-time option pricing intuition. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Hold Δ_t underlying plus cash position. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Discrete hedging leaves residual gamma/jump risk. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into Black-Scholes. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Delta hedging is local replication using underlying sensitivity. It neutralises first-order price moves over small interval. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Hedge underlying amount=−Δ option exposure. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Delta changes; hedge is not set-and-forget. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Gamma is rate at which delta changes with underlying. It measures curvature and hedge instability. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Γ=∂²V/∂S². In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Delta-neutral portfolio can still lose on large moves due to gamma. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Martingale is process whose conditional expected future value equals current value under specified measure. Discounted traded prices are martingales under risk-neutral measure. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. E_Q[discounted S_T|F_t]=discounted S_t. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Martingale under Q does not mean real-world expected return is risk-free. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into asset pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Numeraire is benchmark asset in units of which other prices are expressed. Changing numeraire changes associated risk-neutral measure. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Price/Numeraire is martingale under corresponding measure. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Numeraire technique is mathematical, not currency conversion alone. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into advanced pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Change of numeraire is switching pricing benchmark to simplify expectations. It is powerful for rates/FX derivatives. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Associated Radon-Nikodym derivative transforms measure. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Wrong measure/numeraire pairing creates drift errors. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into quant finance. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Risk-neutral drift is drift under pricing measure consistent with numeraire/discounting. It replaces physical expected return in replication pricing. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Equity drift becomes r−q in simple BSM. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. It is not prediction of actual return. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into Black-Scholes. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. No-arbitrage bounds is upper/lower price constraints even when exact replication unavailable. They exclude obvious dominance trades. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Option bounds from intrinsic/discounted strike and stock. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Bounds are not unique fair values. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into incomplete markets. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Call lower bound is minimum European call value implied by arbitrage. It prevents call price below discounted intrinsic forward relation. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. C≥max(0,S−KD) for non-dividend simple case. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Dividends modify bound. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Put lower bound is minimum European put value under parity/dominance. It prevents underpricing relative to strike bond/stock. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. P≥max(0,KD−S). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Exercise style/dividends matter. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Call upper bound is call cannot exceed underlying price under simple non-dividend setup. Dominance pins maximum. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. C≤S. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Certain dividends/other underlyings require care. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Put upper bound is European put cannot exceed PV of strike in simple setup. Maximum payoff is strike. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. P≤KD. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. American put can have different upper-bound treatment because early exercise. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Arbitrage band is range of prices consistent with transaction costs/borrowing-lending spreads. It replaces exact equality in frictional markets. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Lower≤Price≤Upper. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Small parity deviations may be non-tradable. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into real markets. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Static arbitrage is arbitrage using positions set now and held without dynamic rebalancing. Parity violations often create static trades. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Initial portfolio then wait to maturity. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Options surfaces have static-arbitrage constraints across strikes/maturities. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into volatility surfaces. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Dynamic arbitrage is arbitrage requiring trading over time. It appears in continuous-time consistency conditions. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Self-financing strategy adjusts holdings. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Execution/frictions make dynamic arbitrage harder in reality. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into continuous finance. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Butterfly arbitrage is option-price inconsistency across strikes implying negative state density or riskless payoff trade. It constrains option smile curvature. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Call price convex in strike under standard conditions. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Noisy quotes can violate after transaction costs. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into option surfaces. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Calendar arbitrage is option-price inconsistency across maturities under comparable conditions. Longer maturity option should respect time-value monotonicity in specified setup. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Total variance/option prices subject to maturity constraints. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Dividends/rates/early exercise complicate simple rules. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into option surfaces. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Monotonicity in strike is call prices decrease as strike rises; put prices increase, under standard setup. It follows payoff dominance. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. K1 Failure mode. Violations can signal bad data or arbitrage. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Convexity in strike is European call price is convex in strike. It follows nonnegative butterfly payoff/state density. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Second strike derivative relates to risk-neutral density under conditions. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Discrete bid/ask data require tolerance. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into Breeden-Litzenberger. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Risk-neutral density is distribution of future underlying under pricing measure inferred from option prices under smoothness assumptions. It links state prices and option surface. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Second derivative of call price w.r.t strike gives discounted density. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Density is risk-neutral, not physical forecast. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Breeden-Litzenberger relation is link between option strike derivatives and risk-neutral state-price density. It operationalises state pricing from market options. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. ∂²C/∂K²≈D×f_Q(K). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Smoothing/noise can create negative estimated densities. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into advanced options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Replication error is difference between target payoff/value and replicating strategy outcome. It measures model/discretisation/friction limitations. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Error=Target−Replica. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Small backtest error under one regime does not guarantee future robustness. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into model risk. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Hedging error is P&L residual after applying model hedge. It reveals jumps, discrete hedging and model misspecification. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Residual P&L=option+hedge cash flows. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Transaction costs and slippage contribute. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Super-replication is portfolio payoff at least as large as target in every state. It gives an upper price bound in incomplete/constrained markets. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Xw≥target statewise. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Cheapest superhedge can be conservative. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into robust pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Sub-replication is portfolio payoff no greater than target in every state. It gives lower bound. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Xw≤target. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Bounds can be wide in incomplete markets. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into robust pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Arbitrage-free interval is range between best sub- and super-replication prices. It characterises incomplete-market no-arbitrage prices. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Lower≤P≤Upper. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Choosing one price inside interval requires additional modelling/preferences. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into incomplete markets. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Market completeness is ability to span all state-contingent payoffs. It determines uniqueness of pricing measure. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Unique EMM ↔ completeness in standard finite setup. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Continuous markets can be incomplete with stochastic volatility/jumps. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into theory. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Market incompleteness is presence of unhedgeable risk factors. It creates multiple martingale measures/pricing choices. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Additional criteria select a measure/price. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Model choice embeds assumptions about unspanned risk. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into quant finance. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Equivalent claim is portfolio with exactly same payoff as target. It is the object of law-of-one-price comparison. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Equivalent payoff, equal value. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Same mean/variance is not equivalence. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into relative pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Synthetic asset is derivative/cash combination reproducing another asset exposure. It demonstrates replication relationships. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Synthetic stock=call−put+bond in parity setup. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Synthetic position inherits derivative counterparty/liquidity features. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Synthetic bond is asset/derivative combination producing fixed payoff. It exposes discount-factor consistency. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Stock+put−call=PV strike bond under parity. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Dividends/exercise style alter simple relation. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Synthetic option is underlying/bond/other option combination reproducing option payoff. It pins option price. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Call=stock+put−bond. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Contract details must match. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into parity. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Static replication is replication using fixed positions until maturity. It is robust when exact payoff identity exists. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Put-call parity is static. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Many exotic options require dynamic/approximate replication. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Model-independent identity is pricing relation based on payoff algebra rather than stochastic assumptions. It is especially reliable conceptually. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Put-call parity needs no volatility forecast. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Market frictions still affect executable equality. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into derivatives. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Model-dependent price is value requiring assumptions about underlying dynamics because payoff not statically spanned. Options under incomplete/state-rich markets need a model. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. BSM/binomial provide dynamics. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Calibration fit does not prove future dynamics. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into option pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Discounted expectation is present value expressed as discounted expected payoff under pricing measure. It is equivalent to replication when conditions hold. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. V=D E_Q[X_T]. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Using physical expected payoff with risk-free discounting generally misprices risk. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into risk-neutral valuation. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Expected payoff fallacy is pricing by ordinary expected payoff alone without risk adjustment/replication. It ignores covariance with priced risk The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Price is not generally E_P[X]/(1+r). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Lottery-like payoffs show expectation can differ greatly from price. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into asset pricing. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Arbitrage-free fixed-income valuation is pricing bond cash flows using state/discount factors consistent with current curve. It avoids inconsistent single-yield assumptions. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Price=ΣCF_tD_t for deterministic CFs. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Credit/option cash flows need state-dependent valuation. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into fixed income. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Binomial interest-rate tree is state model calibrated to term structure for rate-dependent claims. It extends arbitrage-free valuation to path/state-dependent bonds/options. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Backward induction with risk-neutral transition probabilities. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Tree calibration/model dynamics affect results. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into rates. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Backward induction is valuation from terminal payoffs backward one step at a time. It combines continuation values and discounting. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. V_t=D E_Q[V_{t+1}] with exercise comparison for American claims. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. Wrong discount/transition pairing creates inconsistencies. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into trees. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Early exercise comparison is for American options compare immediate exercise with continuation value at each node. It is statewise optimal-control logic. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. V=max(Intrinsic,Continuation). In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. European options omit early exercise decision. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into American options. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. Replication cost is current market cost of exact hedging portfolio. It is the arbitrage-free value of replicated claim. The discipline is to compare entire payoff vectors rather than one expected scenario. Mathematics. Price=Σw_iP_i. In finite-state problems, write traded-asset payoffs as columns of a matrix and the target as a payoff vector. Replication becomes linear algebra; discounting and state pricing become linear functionals on that payoff space. Failure mode. If trade execution costs differ, practical value becomes a band. Jo’s diagnostic is to test every state, not just the most likely state. If one state leaves a residual payoff, the proposed arbitrage or replication is incomplete. Connection. This feeds directly into valuation. Ryan would price the same claim both by replication and by risk-neutral expectation. Agreement is a powerful consistency check. A risk-free bond costs 0.95 today and pays 1 in both up/down states. A stock costs 100 and pays 120 in up state, 90 in down state. Let state prices be q_u and q_d. Then q_u+q_d=0.95 and 120q_u+90q_d=100. Solving gives q_u=(100−90×0.95)/30≈0.48333 and q_d≈0.46667. Both are positive, consistent with no arbitrage. Any claim paying X_u,X_d has price q_uX_u+q_dX_d in this complete two-state market. Using the state prices above, total discount factor is 0.95. Risk-neutral up probability is Q_u=q_u/0.95≈0.50877; down probability≈0.49123. Price of a call paying 20 in up state and 0 in down is 0.95×(0.50877×20)=S$9.6667. Same answer comes from state price 20×q_u. Q_u is a pricing probability. It does not assert a 50.877% real-world chance that stock rises. Call payoff is 20 up, 0 down. Find Δ shares and B risk-free terminal cash such that 120Δ+B=20 and 90Δ+B=0. Subtract equations: 30Δ=20, so Δ=2/3. Then B=−60 terminal cash. Today, stock cost is 66.6667 and PV of −60 terminal cash is −57 at discount factor 0.95, giving call cost 9.6667. State-price, risk-neutral and replication methods agree because they are three representations of the same no-arbitrage system. For a non-dividend stock, European options with same K,T satisfy C+K D=P+S. Suppose S=100, K=100, D=0.95 and C=12. Then P=C+95−100=7. If market put were 10 with all else equal and frictionless trading, buy cheap portfolio and sell expensive equivalent portfolio to lock the difference. Parity does not require a volatility assumption because terminal payoff identity is exact. Spot asset costs 100; one-year discount factor is 0.96, no income. Fair forward price is 100/0.96≈104.1667. Buy asset for 100 financed by borrowing 100 today; debt repayment is 104.1667. Deliver asset into short forward at that price. Terminal asset and cash obligations cancel. If forward market quoted 110 and all assumptions were tradable, cash-and-carry profit would exist. A European call on a non-dividend stock cannot cost more than the stock itself because call payoff max(S_T−K,0)≤S_T in every state. If call cost 105 while stock cost 100, buy stock and short call; initial cash inflow 5 and terminal stock value always covers call obligation with nonnegative residual. This is a bound derived from payoff dominance, not a stochastic model. Similar inequalities create quick diagnostics for option-data quality. Imagine three possible future states but only a risk-free bond and one risky asset with independent payoff directions. Two securities cannot generally span every three-state payoff. A new exotic claim may have no exact replicating portfolio. No-arbitrage can still impose upper/lower bounds through sub- and super-replication, but one unique price requires extra assumptions or another traded security. This is the conceptual reason stochastic-volatility/jump risks can make real option markets incomplete. Textbook forward relation assumes one risk-free rate for borrowing and lending. Suppose investor can lend at 3% but borrow at 5%. Cash-and-carry and reverse cash-and-carry use different financing rates, creating a no-arbitrage interval rather than one exact forward price. Observed forward can sit inside that band without executable arbitrage. Real market frictions do not invalidate no-arbitrage reasoning; they change equality into inequality. Rearrange put-call parity: S=C−P+KD. A long call, short put and risk-free bond paying strike replicate the stock at maturity under the simple European non-dividend setup. This identity helps understand option combinations and validates option chains. If synthetic and cash stock differ beyond carrying/friction effects, investigate quote timestamps and contract details. Synthetic positions carry derivative counterparty/margin/liquidity characteristics even when terminal payoff matches stock. Suppose a stock has high real-world expected return because investors require compensation for risk. A derivative price based on exact replication does not need that physical expected return. Under the pricing measure, discounted traded assets earn the risk-free rate in expectation. This is why Black–Scholes pricing does not contain the stock’s physical expected return μ in the standard formula. Replication eliminates it from the pricing problem. Forecasting and pricing are different mathematical tasks. Real markets have bid–offer spreads, inventory constraints, financing limits, short-sale restrictions, capital costs, taxes, settlement risk and latency. A price difference smaller than trading frictions is not executable arbitrage. A large difference may persist if balance-sheet capacity is scarce. The theorem layer is still useful because it defines the frictionless centre and reveals which frictions must explain deviations. Cross-currency basis after the global financial crisis is a classic example: textbook CIP equality can deviate because bank balance sheets, funding and collateral constraints matter. Mira’s question is not “does no-arbitrage fail?” but “which assumption prevents the replicating trade from being freely executable?” Exact replication requires state-by-state equality. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. They are pricing weights under a transformed measure. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. State price combines discounting and probability/risk pricing. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. Replication makes it unnecessary in basic model. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. Bid/ask, funding and shorting can remove profit. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. External cash injections invalidate zero-cost arbitrage claims. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. Unique risk-neutral measure may not exist. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. Strike, expiry, exercise style, dividends and settlement must match. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. Constraints create bounds rather than exact relations. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. Funding asymmetry widens no-arbitrage bands. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. Some claims require dynamic trading or richer instruments. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. A calibrated model can still have wrong dynamics. The repair is to return to payoff tables, replication feasibility and executable trading assumptions. Situation. Observed forward differs materially from spot carry relation. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Build cash-and-carry and reverse cash-and-carry cash-flow tables using executable funding/shorting terms. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. A theoretical gap becomes arbitrage only if all legs are tradable after costs. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. European call and put share strike/expiry. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Construct protective-put and fiduciary-call payoffs state by state. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. Different exercise styles or dividend assumptions invalidate simple parity. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. Two traded assets span two future states. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Solve two linear equations for state prices and use them to price third claim. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. Negative state price signals inconsistent data or model setup. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. Only bond and stock trade across three states. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Show one exotic payoff outside their span and derive sub/super-replication bounds. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. No-arbitrage alone need not produce unique price. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. Two money-market investments plus FX forward should produce same hedged terminal cash. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Solve forward by replication and compare market quote. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. Funding/balance-sheet basis can explain persistent real-market deviations. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. Coupon bond price differs from sum of discounted cash flows using zero curve. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Check cash-flow dates, discount factors and clean/dirty price. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. A curve that cannot reprice its calibration instruments is internally inconsistent. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. Call-put-bond package replicates stock. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Compare market cost of synthetic and cash stock including dividends/borrow. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. Synthetic terminal equality can coexist with different margin/liquidity profiles. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. Call trades below max(0,S−KD). The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Construct arbitrage with call, stock and bond in textbook setup. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. Executable spread/borrow constraints determine whether violation is tradeable. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. Three option strikes imply negative butterfly cost/payoff inconsistency. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Construct long wings/short middle payoff. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. Static-arbitrage cleaning is essential before fitting volatility surfaces. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. Delta hedge of option leaves P&L after large jump. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Attribute residual to gamma/jump/discrete rebalancing. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. No-arbitrage model does not promise perfect hedge under violated dynamics. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. Trader borrows at 6% and lends at 3%. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Derive upper/lower forward bounds rather than one exact rate. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. Textbook one-rate equality becomes a band. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. Situation. CSA remunerates collateral at overnight rate. The task is to identify whether the claim is exactly spanned or only bounded by available instruments. Method. Use collateral-consistent discount curve in clean valuation. Adrian writes the payoff matrix, Jo checks self-financing and trade feasibility, Aisha solves replica/state prices, and Ryan compares with risk-neutral expectation. Boundary. Ignoring collateral terms creates apparent arbitrage between desks/models. Mira then identifies which friction or unspanned risk prevents theorem-level equality from becoming an executable market trade. No-arbitrage is not a forecast. It is a consistency condition on prices of tradable payoffs. Law of one price gives equality for identical cash flows. Replication converts that equality into a pricing method. State prices and risk-neutral probabilities are alternative coordinates for the same linear pricing system. Completeness determines whether the price is unique. Frictions turn sharp equalities into executable bands. That theorem layer explains why forward prices can be derived without forecasting, why put–call parity survives changes in volatility expectations, and why binomial option values emerge from hedging rather than subjective probabilities. The next owner applies this machinery to contingent claims in detail: calls, puts, binomial trees, Black–Scholes–Merton, Greeks and implied volatility. Create two states and two independent traded assets. Solve state prices, check positivity, then price three new claims. Reprice each claim via direct replication and risk-neutral expectation. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Invent two portfolios with identical terminal payoffs but different initial prices. Write initial and state-by-state cash flows for buy-cheap/sell-expensive strategy and verify nonnegative payoff everywhere. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Use three states and only two independent securities. Choose a target payoff outside their span. Find cheap sub-replica and super-replica to create a no-arbitrage interval. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Start from put-call parity, then add bid/ask, stock borrow fee and different lending/borrowing rates. Calculate an executable parity band rather than exact equality. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Choose a binomial stock with a physical up probability different from derived Q probability. Calculate real-world expected return and option price separately. Explain why both probabilities can be internally correct for different tasks. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Create two states and two independent traded assets. Solve state prices, check positivity, then price three new claims. Reprice each claim via direct replication and risk-neutral expectation. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Invent two portfolios with identical terminal payoffs but different initial prices. Write initial and state-by-state cash flows for buy-cheap/sell-expensive strategy and verify nonnegative payoff everywhere. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Use three states and only two independent securities. Choose a target payoff outside their span. Find cheap sub-replica and super-replica to create a no-arbitrage interval. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Start from put-call parity, then add bid/ask, stock borrow fee and different lending/borrowing rates. Calculate an executable parity band rather than exact equality. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Choose a binomial stock with a physical up probability different from derived Q probability. Calculate real-world expected return and option price separately. Explain why both probabilities can be internally correct for different tasks. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Create two states and two independent traded assets. Solve state prices, check positivity, then price three new claims. Reprice each claim via direct replication and risk-neutral expectation. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Invent two portfolios with identical terminal payoffs but different initial prices. Write initial and state-by-state cash flows for buy-cheap/sell-expensive strategy and verify nonnegative payoff everywhere. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Use three states and only two independent securities. Choose a target payoff outside their span. Find cheap sub-replica and super-replica to create a no-arbitrage interval. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Start from put-call parity, then add bid/ask, stock borrow fee and different lending/borrowing rates. Calculate an executable parity band rather than exact equality. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Choose a binomial stock with a physical up probability different from derived Q probability. Calculate real-world expected return and option price separately. Explain why both probabilities can be internally correct for different tasks. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Create two states and two independent traded assets. Solve state prices, check positivity, then price three new claims. Reprice each claim via direct replication and risk-neutral expectation. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Invent two portfolios with identical terminal payoffs but different initial prices. Write initial and state-by-state cash flows for buy-cheap/sell-expensive strategy and verify nonnegative payoff everywhere. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Use three states and only two independent securities. Choose a target payoff outside their span. Find cheap sub-replica and super-replica to create a no-arbitrage interval. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Start from put-call parity, then add bid/ask, stock borrow fee and different lending/borrowing rates. Calculate an executable parity band rather than exact equality. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Choose a binomial stock with a physical up probability different from derived Q probability. Calculate real-world expected return and option price separately. Explain why both probabilities can be internally correct for different tasks. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Create two states and two independent traded assets. Solve state prices, check positivity, then price three new claims. Reprice each claim via direct replication and risk-neutral expectation. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Invent two portfolios with identical terminal payoffs but different initial prices. Write initial and state-by-state cash flows for buy-cheap/sell-expensive strategy and verify nonnegative payoff everywhere. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Use three states and only two independent securities. Choose a target payoff outside their span. Find cheap sub-replica and super-replica to create a no-arbitrage interval. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Start from put-call parity, then add bid/ask, stock borrow fee and different lending/borrowing rates. Calculate an executable parity band rather than exact equality. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Choose a binomial stock with a physical up probability different from derived Q probability. Calculate real-world expected return and option price separately. Explain why both probabilities can be internally correct for different tasks. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Create two states and two independent traded assets. Solve state prices, check positivity, then price three new claims. Reprice each claim via direct replication and risk-neutral expectation. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Invent two portfolios with identical terminal payoffs but different initial prices. Write initial and state-by-state cash flows for buy-cheap/sell-expensive strategy and verify nonnegative payoff everywhere. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Use three states and only two independent securities. Choose a target payoff outside their span. Find cheap sub-replica and super-replica to create a no-arbitrage interval. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Start from put-call parity, then add bid/ask, stock borrow fee and different lending/borrowing rates. Calculate an executable parity band rather than exact equality. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Choose a binomial stock with a physical up probability different from derived Q probability. Calculate real-world expected return and option price separately. Explain why both probabilities can be internally correct for different tasks. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Create two states and two independent traded assets. Solve state prices, check positivity, then price three new claims. Reprice each claim via direct replication and risk-neutral expectation. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Invent two portfolios with identical terminal payoffs but different initial prices. Write initial and state-by-state cash flows for buy-cheap/sell-expensive strategy and verify nonnegative payoff everywhere. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Use three states and only two independent securities. Choose a target payoff outside their span. Find cheap sub-replica and super-replica to create a no-arbitrage interval. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Start from put-call parity, then add bid/ask, stock borrow fee and different lending/borrowing rates. Calculate an executable parity band rather than exact equality. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Choose a binomial stock with a physical up probability different from derived Q probability. Calculate real-world expected return and option price separately. Explain why both probabilities can be internally correct for different tasks. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Create two states and two independent traded assets. Solve state prices, check positivity, then price three new claims. Reprice each claim via direct replication and risk-neutral expectation. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Invent two portfolios with identical terminal payoffs but different initial prices. Write initial and state-by-state cash flows for buy-cheap/sell-expensive strategy and verify nonnegative payoff everywhere. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Use three states and only two independent securities. Choose a target payoff outside their span. Find cheap sub-replica and super-replica to create a no-arbitrage interval. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Start from put-call parity, then add bid/ask, stock borrow fee and different lending/borrowing rates. Calculate an executable parity band rather than exact equality. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Choose a binomial stock with a physical up probability different from derived Q probability. Calculate real-world expected return and option price separately. Explain why both probabilities can be internally correct for different tasks. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Create two states and two independent traded assets. Solve state prices, check positivity, then price three new claims. Reprice each claim via direct replication and risk-neutral expectation. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Invent two portfolios with identical terminal payoffs but different initial prices. Write initial and state-by-state cash flows for buy-cheap/sell-expensive strategy and verify nonnegative payoff everywhere. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Use three states and only two independent securities. Choose a target payoff outside their span. Find cheap sub-replica and super-replica to create a no-arbitrage interval. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Start from put-call parity, then add bid/ask, stock borrow fee and different lending/borrowing rates. Calculate an executable parity band rather than exact equality. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Choose a binomial stock with a physical up probability different from derived Q probability. Calculate real-world expected return and option price separately. Explain why both probabilities can be internally correct for different tasks. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Create two states and two independent traded assets. Solve state prices, check positivity, then price three new claims. Reprice each claim via direct replication and risk-neutral expectation. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete. Invent two portfolios with identical terminal payoffs but different initial prices. Write initial and state-by-state cash flows for buy-cheap/sell-expensive strategy and verify nonnegative payoff everywhere. Complete the lab with state-by-state payoff rows before solving equations. Ben should check linear algebra, Clara should record trading/funding assumptions, and Ethan should state which conclusion survives when transaction costs or constraints are introduced. Then change the physical probability while leaving traded prices unchanged. Any exact replication price should remain unchanged. This exercise makes the separation between forecasting probabilities and arbitrage-free pricing probabilities concrete.29. Spanning
30. Redundant security
31. Payoff matrix
32. Rank
33. Linear system
34. Short selling
35. Borrowing/lending
36. Transaction cost
37. Funding spread
38. Collateral
39. Market impact
40. Liquidity constraint
41. Model arbitrage
42. Cash-and-carry
43. Forward price
44. Forward value
45. Covered interest parity
46. Put-call parity
47. Synthetic forward
48. Protective put
49. Fiduciary call
50. Binomial state model
51. Hedge ratio in one-step binomial
52. Risk-neutral up probability
53. Risk-neutral option value
54. Dynamic replication
55. Delta hedging
56. Gamma
57. Martingale
58. Numeraire
59. Change of numeraire
60. Risk-neutral drift
61. No-arbitrage bounds
62. Call lower bound
63. Put lower bound
64. Call upper bound
65. Put upper bound
66. Arbitrage band
67. Static arbitrage
68. Dynamic arbitrage
69. Butterfly arbitrage
70. Calendar arbitrage
71. Monotonicity in strike
72. Convexity in strike
73. Risk-neutral density
74. Breeden-Litzenberger relation
75. Replication error
76. Hedging error
77. Super-replication
78. Sub-replication
79. Arbitrage-free interval
80. Market completeness
81. Market incompleteness
82. Equivalent claim
83. Synthetic asset
84. Synthetic bond
85. Synthetic option
86. Static replication
87. Model-independent identity
88. Model-dependent price
89. Discounted expectation
90. Expected payoff fallacy
91. Arbitrage-free fixed-income valuation
92. Binomial interest-rate tree
93. Backward induction
94. Early exercise comparison
95. Replication cost
Worked Example 1: One-Period State Prices
Worked Example 2: Risk-Neutral Probability
Worked Example 3: Replicate the Call
Worked Example 4: Put–Call Parity
Worked Example 5: Forward Replication
Worked Example 6: Dominance Bound
Worked Example 7: Incomplete Market
Worked Example 8: Borrowing/Lending Spread Creates a Band
Worked Example 9: Synthetic Stock
Worked Example 10: Risk-Neutral Expectation Versus Physical Expectation
No-Arbitrage Does Not Mean Markets Never Misprice
A Professional Arbitrage-Free Valuation Workflow
Common Failure Modes
1. Expected payoff equality mistaken for payoff equality
2. Risk-neutral probabilities interpreted as forecasts
3. Discount factor treated as probability
4. Physical expected return used in forward pricing
5. Arbitrage declared before costs
6. Self-financing ignored
7. Incomplete market priced as if complete
8. Put-call parity used on mismatched contracts
9. Short-sale feasibility assumed
10. Borrowing rate equals lending rate assumed in reality
11. Static replication used for path-dependent claim
12. Model calibration mistaken for theorem
Formula Map
Concept Simplified formula Meaning State-price valuation P(X)=Σq_sX_s Price as state-contingent cash-flow value. Risk-neutral valuation P=D·E_Q[X] Discounted expected payoff under pricing measure. Replication Xw=y; Price=w’P_assets Linear algebra of payoff spanning. Forward price F=S/D No-income asset carried to maturity. Put-call parity C+KD=P+S European non-dividend parity. Binomial delta (C_u−C_d)/(S_u−S_d) One-step replicating stock holding. Risk-neutral up probability (R−d)/(u−d) Pricing weight in simple binomial model. Authoritative Reference Map
Connected Banking And Finance Mathematics Route
Applied Case Study 1: Mispriced forward
Applied Case Study 2: Put-call parity check
Applied Case Study 3: State-price recovery
Applied Case Study 4: Incomplete three-state market
Applied Case Study 5: Cross-currency parity
Applied Case Study 6: Bond curve consistency
Applied Case Study 7: Synthetic stock
Applied Case Study 8: Option lower-bound violation
Applied Case Study 9: Butterfly quote error
Applied Case Study 10: Dynamic hedge residual
Applied Case Study 11: Funding asymmetry
Applied Case Study 12: Collateralised derivative
Final Principle
Deep Practice Lab 1: Solve a payoff matrix
Deep Practice Lab 2: Build an arbitrage table
Deep Practice Lab 3: Create an incomplete market
Deep Practice Lab 4: Stress parity with frictions
Deep Practice Lab 5: Compare physical and risk-neutral probabilities
Deep Practice Lab 6: Solve a payoff matrix
Deep Practice Lab 7: Build an arbitrage table
Deep Practice Lab 8: Create an incomplete market
Deep Practice Lab 9: Stress parity with frictions
Deep Practice Lab 10: Compare physical and risk-neutral probabilities
Deep Practice Lab 11: Solve a payoff matrix
Deep Practice Lab 12: Build an arbitrage table
Deep Practice Lab 13: Create an incomplete market
Deep Practice Lab 14: Stress parity with frictions
Deep Practice Lab 15: Compare physical and risk-neutral probabilities
Deep Practice Lab 16: Solve a payoff matrix
Deep Practice Lab 17: Build an arbitrage table
Deep Practice Lab 18: Create an incomplete market
Deep Practice Lab 19: Stress parity with frictions
Deep Practice Lab 20: Compare physical and risk-neutral probabilities
Deep Practice Lab 21: Solve a payoff matrix
Deep Practice Lab 22: Build an arbitrage table
Deep Practice Lab 23: Create an incomplete market
Deep Practice Lab 24: Stress parity with frictions
Deep Practice Lab 25: Compare physical and risk-neutral probabilities
Deep Practice Lab 26: Solve a payoff matrix
Deep Practice Lab 27: Build an arbitrage table
Deep Practice Lab 28: Create an incomplete market
Deep Practice Lab 29: Stress parity with frictions
Deep Practice Lab 30: Compare physical and risk-neutral probabilities
Deep Practice Lab 31: Solve a payoff matrix
Deep Practice Lab 32: Build an arbitrage table
Deep Practice Lab 33: Create an incomplete market
Deep Practice Lab 34: Stress parity with frictions
Deep Practice Lab 35: Compare physical and risk-neutral probabilities
Deep Practice Lab 36: Solve a payoff matrix
Deep Practice Lab 37: Build an arbitrage table
Deep Practice Lab 38: Create an incomplete market
Deep Practice Lab 39: Stress parity with frictions
Deep Practice Lab 40: Compare physical and risk-neutral probabilities
Deep Practice Lab 41: Solve a payoff matrix
Deep Practice Lab 42: Build an arbitrage table
Deep Practice Lab 43: Create an incomplete market
Deep Practice Lab 44: Stress parity with frictions
Deep Practice Lab 45: Compare physical and risk-neutral probabilities
Deep Practice Lab 46: Solve a payoff matrix
Deep Practice Lab 47: Build an arbitrage table