Reader question: How can an interest-rate tree preserve mean reversion while keeping the short rate positive and still fit today’s yield curve exactly?
The Black–Karasinski model makes the logarithm of the short rate mean reverting. A recombining lattice is built for the log-rate state, then a time-dependent shift is calibrated node layer by node layer so that model discount factors match the observed term structure.
This article owns one precise computational job: building and calibrating a Black–Karasinski lognormal short-rate tree for fixed-income derivative pricing. It does not own Black–Derman–Toy, Hull–White Gaussian trees, G2++, HJM, LMM, or generic binomial-tree mathematics.
This is public mathematical and computational education. It is not a forecast of rates, and a tree that fits today’s curve exactly can still have weak future dynamics.
1. The state variable is log short rate
A standard continuous-time Black–Karasinski specification can be written schematically as:
d ln rt = [θ(t) − a ln rt]dt + σ dWt.
The ingredients are:
- a — mean-reversion speed;
- σ — volatility of log short rate;
- θ(t) — time-dependent drift/shift chosen to fit the initial term structure.
Because rt = exp(xt) for xt=ln rt, the modeled short rate remains positive.
2. Why positivity mattered historically
When short rates were assumed to stay positive, a lognormal model was attractive because negative-rate states were ruled out structurally rather than clipped after simulation.
That strength can become a weakness in regimes where negative short rates are economically possible. Model architecture must match the regime, not nostalgia.
3. Mean reversion in log space
If xt=ln rt, then x follows an Ornstein–Uhlenbeck-like process with time-dependent center.
Shocks to log rates decay roughly at speed a. The half-life intuition is:
half-life ≈ ln 2 / a.
A larger a pulls the log short rate back more quickly after shocks.
4. Why a tree is natural
Many fixed-income options contain early exercise or path dependence. A recombining tree provides:
- state-by-state discounting;
- backward induction;
- exercise comparison;
- coupon cash-flow handling;
- callable/putable bond logic.
Black–Karasinski is therefore often implemented as a calibrated short-rate lattice rather than only as a continuous-time equation.
5. Build the lattice in x = ln r
For timestep Δt, choose a state spacing such as:
Δx = σ√(3Δt)
in a trinomial implementation.
The unshifted state at node (i,j) can be represented as:
xi,j = jΔx
before adding the calibrated time-layer shift.
6. Transition probabilities encode mean reversion
At each node, up/middle/down probabilities are chosen so that the discrete tree matches the first moments of the mean-reverting x process over Δt.
The expected next x state should approximate:
E[xt+Δt|xt] ≈ xt + (m(t)−a xt)Δt.
The conditional variance should approximate σ²Δt to first order.
Exact probability formulas depend on the chosen lattice construction, but probabilities must remain between zero and one. Boundary handling is part of the numerical method.
7. The calibrated shift
Let the lattice state be decomposed into:
xi,j = αi + ~xi,j,
where ~x is the mean-reverting tree coordinate and αi is a deterministic layer shift.
The short rate at the node is:
ri,j = exp(αi + ~xi,j).
The calibration chooses αi so that the model reproduces the market discount factor to the next maturity.
8. State prices make calibration efficient
Let Qi,j denote the Arrow–Debreu/state price reaching node (i,j).
The model price today of a zero-coupon bond maturing one step beyond layer i is:
Pmodel(0,ti+1) = Σj Qi,j exp[−ri,jΔt].
Because ri,j depends on αi, solve:
Pmodel(0,ti+1;αi) = Pmarket(0,ti+1).
This is a one-dimensional root-finding problem for each time layer.
9. Why the root is usually well behaved
Increasing α raises every short rate on that layer, which lowers the discounted bond value.
The model bond price is therefore typically monotone in α, allowing robust bracketing methods such as bisection/Brent.
Falsifier: if the calibration function is not monotone in α under a standard implementation, inspect state prices, discounting and rate transformation.
10. Forward calibration proceeds one layer at a time
- Build the unshifted log-rate lattice and transition probabilities.
- Set initial state price Q0,0=1.
- At layer i, solve for αi to match the next market discount factor.
- Compute node short rates.
- Propagate discounted state prices to layer i+1.
- Repeat until the required horizon.
At the end, the tree is calibrated to today’s zero curve by construction.
11. Price a bond by backward induction
At maturity, set the bond payoff. At each earlier node:
Vi,j = e−ri,jΔt E[Vi+1|i,j] + coupon.
The expectation uses the node’s transition probabilities.
12. Callable and putable bonds
At an exercise date, compare continuation value with the contract’s call or put price.
For a callable bond:
V = min(continuation value, call price)
subject to contract conventions.
This is why a calibrated tree remains useful even when closed-form European formulas exist elsewhere.
13. Calibrating a and σ
The layer shifts αi fit today’s curve. The dynamics parameters a and σ must be calibrated from option information or chosen under a documented regime.
Typical targets can include:
- caps/floors;
- swaptions;
- bond options;
- historical dynamics as a secondary diagnostic.
A good curve fit says nothing about whether a and σ are well identified.
14. Positive rates are both feature and assumption
Since r=exp(x), rates cannot be negative.
That can be attractive in strictly positive-rate regimes and inappropriate when negative short rates are observed or materially possible.
Robustness test: a beautiful structural guarantee is not a virtue if the market has already falsified the guarantee.
15. Lognormal tails
Because log r is approximately Gaussian, r has a right-skewed lognormal-style distribution.
This creates asymmetric rate outcomes and can produce high-rate tails different from Gaussian short-rate models.
The distributional assumption affects option prices, especially for long maturities and higher volatility.
16. Inputs and outputs
Inputs can include:
- discount curve;
- time grid;
- a and σ;
- tree branching rule;
- transition-probability construction;
- option calibration targets;
- coupon/exercise schedules;
- root-finding tolerances.
Outputs can include:
- layer shifts αi;
- node short rates;
- state prices;
- reproduced discount curve;
- bond/option values;
- exercise boundaries;
- tree-convergence diagnostics;
- option repricing residuals;
- rate-distribution summaries.
17. Evidence polarity
Evidence for confidence includes:
- discount curve reproduced to tolerance;
- option calibration residuals stable;
- prices converge as Δt shrinks;
- state prices remain nonnegative;
- transition probabilities stay within [0,1];
- tree values agree with simpler benchmarks;
- exercise boundaries stabilize under refinement.
Evidence against confidence includes:
- curve mismatch;
- negative/invalid transition probabilities;
- large timestep sensitivity;
- unstable a/σ calibration;
- persistent option-smile residuals;
- market regimes allowing negative rates;
- far-tail results dominated by lattice boundaries.
18. Counterexample: negative-rate market
If the observed short end of the curve is negative, r=exp(x) cannot reproduce that state naturally.
Falsifier: do not force-fit by distorting unrelated parameters. Use a model whose state support matches the regime.
19. Counterexample: coarse tree
A coarse time grid can reproduce the initial curve exactly yet misprice options because exercise and diffusion dynamics are under-resolved.
Falsifier: halve Δt repeatedly and require price/exercise-boundary convergence.
20. Counterexample: curve fit hides volatility failure
The α shifts can always be recalibrated to the curve for a wide range of a and σ.
Therefore exact curve fit cannot validate dynamics.
Falsifier: hold out option instruments and compare dynamics out of sample.
21. Counterexample: boundary compression
Finite recombining trees need boundary rules. If high-rate states are truncated or reflected too aggressively, long-dated convex payoffs can be biased.
Falsifier: widen the lattice and compare tail-sensitive prices.
22. Black–Karasinski versus Black–Derman–Toy
Black–Derman–Toy also uses lognormal short rates but has a different structure and calibration tradition.
Black–Karasinski adds explicit mean reversion in log-rate dynamics, making the long-run behavior more controlled.
23. Black–Karasinski versus Hull–White
Hull–White uses Gaussian short-rate dynamics and therefore permits negative rates.
Black–Karasinski applies mean reversion to log rates, enforcing positivity but losing Gaussian affine tractability.
24. Black–Karasinski versus G2++
G2++ adds a second Gaussian factor to enrich curve covariance.
Black–Karasinski remains one-factor but changes the distributional support and tree geometry.
25. Weak links
- negative-rate incompatibility;
- coarse timestep;
- tree boundary handling;
- invalid transition probabilities;
- weak a/σ identification;
- option-smile limitation;
- curve interpolation inconsistencies;
- root-finding tolerances;
- mistaking exact curve fit for dynamic validation.
26. What would falsify confidence?
Confidence should be withdrawn if the market requires negative rates; if tree prices do not converge; if option residuals remain systematic; if a and σ are unstable; if lattice boundaries drive long-dated values; or if a simpler model performs equally well for the intended job.
27. Verification and update triggers
Preserve the zero curve, interpolation, timestep, lattice spacing, transition rules, α calibration roots, option targets, a/σ parameters, and convergence tables.
Revalidate when the rate regime changes, negative rates become relevant, curve construction changes, option volatility structure changes, timestep/lattice implementation changes, or new callable instruments are added.
Research anchors
- Fischer Black and Piotr Karasinski, Bond and Option Pricing When Short Rates Are Lognormal, Financial Analysts Journal, 1991.
- Standard fixed-income numerical-method references and implementation literature on calibrated trinomial short-rate trees.
Educational boundary: Black–Karasinski earns its place when positive lognormal short rates and mean reversion are useful assumptions. The calibrated tree is an edition of those assumptions—not a guarantee about the future rate world.