Institutional Fixed Income & Lognormal Interest Rate Lattices

Black-Karasinski (1991) Model Lab

Construct calibrated lognormal short-rate trinomial trees via Arrow-Debreu forward induction, eliminate negative interest rates, price callable & puttable bonds, and analyze negative convexity under the landmark Black-Karasinski (1991) framework.

Interactive BK Simulator Mathematical Foundations Mastery Quiz
Root Short Rate r(0)

3.00%

Strictly positive $r > 0$
Option-Free Bond Price

$102.45

Clean straight bond
Callable Bond Price

$100.15

Capped at call price
Embedded Call Option

$2.30

Issuer surrender value
Model Parameters & Bond Calibration
BK Model Parameters: $d \ln r = [\theta(t) - a \ln r]dt + \sigma dW$
Initial Yield Curve $P(0, T)$ (Nelson-Siegel)
5-Year Bond & Embedded Option Specs
Price-Yield relationship comparing Straight Bond vs Callable Bond across yield curve shocks ($\pm 200\text{ bps}$). Notice the classic concave bend (negative convexity) as yields decline and prices cap at the call price.
Recombining Black-Karasinski trinomial tree envelope over time steps $t_i$. Central node $r_{i, 0} = \exp(\alpha_i)$ shifts dynamically via Arrow-Debreu forward induction to fit market bond prices exactly.
Arrow-Debreu state price distribution $Q_{i, j}$ across tree nodes at mid-horizon. Strictly positive rates with right-skewed lognormal density.
Fixed Income Valuation & Embedded Option Decomposition
Straight Option-Free Bond
Clean Price: $102.450
Par Value: $100.000
Convexity: Positive
Callable Corporate Bond
Callable Price: $100.150
Embedded Call: $2.300
Convexity: Negative
Puttable Bond & Swaption
Puttable Price: $103.200
Embedded Put: $0.750
1Y Swaption: 85.2 bps
Sensitivity Grid: Mean-Reversion $a$ vs Volatility $\sigma$ on Embedded Call Option ($)
Shows value of the issuer's surrender option ($V_{\text{straight}} - V_{\text{callable}}$). Higher volatility dramatically elevates call option value.
Sensitivity Grid: Call Price vs Coupon Rate on Callable Bond Value ($)
Evaluates callable bond value across coupon rate incentives and call penalty levels.
Mathematical Foundations of the Black-Karasinski (1991) Model
1. Lognormal Diffusion and Positivity Guarantee

Published by Fischer Black and Piotr Karasinski (1991), the model resolved a critical dilemma in term structure theory: Gaussian models (such as Vasicek and Hull-White) generate negative interest rates with non-zero probability, whereas square-root diffusions (such as CIR) restrict volatility to scale with $\sqrt{r}$. Black and Karasinski postulated that the short rate is lognormally distributed by assuming that its natural logarithm $x(t) = \ln r(t)$ follows an Ornstein-Uhlenbeck process:

$$d \ln r(t) = \left[ \theta(t) - a(t) \ln r(t) \right] dt + \sigma(t) \, dW(t)$$

Applying the transformation $r(t) = \exp(x(t))$, the short rate is strictly positive for all $t$ with probability 1:

$$r(t) > 0 \quad \forall t \ge 0$$
2. Recombining Trinomial Lattice Construction

Because lognormal short-rate models lack closed-form analytical zero-coupon bond formulas, pricing relies on a discrete recombining trinomial tree (Hull & White 1994, 2000). First, a standard tree is built for the state variable $x_{i, j} = j \Delta x$ with spatial grid step $\Delta x = \sigma \sqrt{3 \Delta t}$.

From node $(i, j)$, mean reversion creates a drift increment $M_{i, j} = -a j \Delta x \Delta t$. Defining $k = \text{round}(M_{i, j} / \Delta x)$ and $\eta = M_{i, j} - k \Delta x$, branching probabilities are:

$$p_u = \frac{1}{6} + \frac{\eta^2}{2 \Delta x^2} + \frac{\eta}{2 \Delta x}, \quad p_m = \frac{2}{3} - \frac{\eta^2}{\Delta x^2}, \quad p_d = \frac{1}{6} + \frac{\eta^2}{2 \Delta x^2} - \frac{\eta}{2 \Delta x}$$
3. Arrow-Debreu Forward Induction Calibration

To ensure the tree matches the market zero-coupon bond curve $P(0, t_{i+1})$ exactly, the short rate at node $(i, j)$ is shifted by a time-dependent constant $\alpha_i$:

$$r_{i, j} = \exp\left( \alpha_i + j \Delta x \right)$$

Let $Q_{i, j}$ be the Arrow-Debreu state price (the present value at $t=0$ of receiving $\$1$ if node $(i, j)$ is reached). At each time step $i$, $\alpha_i$ is solved via a 1D Newton-Raphson root finder satisfying:

$$\sum_j Q_{i, j} \exp\left(-e^{\alpha_i + j \Delta x} \Delta t\right) = P(0, t_{i+1})$$
4. Negative Convexity in Callable Bonds

Standard option-free bonds exhibit positive convexity ($\frac{d^2 P}{d y^2} > 0$), meaning price rises more when yields fall than it drops when yields rise. However, for a callable bond, the issuer holds an embedded call option. By backward dynamic programming:

$$V_{\text{callable}}(i, j) = \min\left( e^{-r_{i, j} \Delta t} \mathbb{E}[V(i+1)] + C, \, \text{Call Price} \right)$$

When rates fall, the continuation value exceeds the call price, capping price upside. This creates a concave region ($\frac{d^2 P}{d y^2} < 0$) known as negative convexity, causing callable bonds to underperform straight bonds during interest rate rallies.

Black-Karasinski Term Structure Mastery Quiz
1. What is the fundamental assumption of the Black-Karasinski (1991) short-rate model?
2. What major advantage does Black-Karasinski have over the Black-Derman-Toy (BDT 1990) model?
3. Why do callable corporate bonds exhibit negative convexity when market interest rates fall?
4. How does Hull-White forward induction calibrate the BK trinomial tree to the market yield curve?
Frequently Asked Questions

The Black-Karasinski (BK 1991) model, authored by Fischer Black and Piotr Karasinski in the Financial Analysts Journal, models the natural logarithm of the short rate as an Ornstein-Uhlenbeck mean-reverting diffusion process: d ln(r) = [θ(t) - a ln(r)]dt + σ dW. By modeling ln(r) rather than r directly, the short rate r(t) = exp(x(t)) is guaranteed to remain strictly positive at all times.

Because the underlying state variable x(t) = ln(r(t)) is mapped to the short rate via the exponential function r(t) = exp(x(t)), the short rate can never reach zero or become negative, regardless of the magnitude of downward Brownian motion shocks. This eliminates the unrealistic negative rate probabilities present in Gaussian models like Vasicek and Hull-White.

Unlike the Black-Derman-Toy (1990) model—where short-rate volatility and mean-reversion are coupled and cannot be specified independently—Black-Karasinski explicitly separates the mean-reversion speed 'a' and volatility 'σ'. Unlike Hull-White (1990)—which is Gaussian and permits negative interest rates with closed-form analytical formulas—Black-Karasinski is lognormal and strictly positive, requiring numerical lattices (trinomial trees) for calibration and pricing.

Negative convexity is the phenomenon where a bond's price-yield curve becomes concave (bends downward) as market interest rates decline. For a callable bond, as yields drop, the probability of the issuer exercising the call option increases, capping the bond's price at the call price. On the Black-Karasinski trinomial tree, backward induction dynamically caps node values at the call price, vividly illustrating this downward price compression.

Calibration employs Hull-White forward induction. At each time step i, the state price Q_{i, j} (the discounted value of receiving $1 if node (i, j) is reached) is known from previous steps. A 1D Newton-Raphson root solver finds the unique horizontal drift shift α_i such that the sum of state prices discounted at r_{i, j} = exp(α_i + j Δx) exactly matches the market zero-coupon bond price P(0, t_{i+1}).
Explore Quantitative Interest Rate & Term Structure Simulations
Black-Derman-Toy (BDT) Lab

Construct lognormal binomial lattices fitting both term structure and yield volatilities without negative rates.

Two-Factor Hull-White (G2++) Lab

Explore correlated two-factor Gaussian short rates, curve twist/slope dynamics, and 1D swaption quadrature.

HJM Forward Rate Lab

Simulate instantaneous forward rate SDEs, no-arbitrage drift, and multi-factor term structure Monte Carlo paths.

LIBOR Market Model (BGM) Lab

Model discrete forward rates, Spot LIBOR numeraire drift stepping, and correlation decay across swaptions.