Black-Derman-Toy (BDT) Interest Rate Lattice Lab

Simulate the Black-Derman-Toy (1990) lognormal short-rate binomial lattice. Calibrate the tree to match initial spot yields and yield volatility, price straight and callable bonds via backward induction, and quantify embedded option value and negative convexity.

BDT (1990) Lattice Lognormal Rates ($r > 0$) Arrow-Debreu Calibration Callable Bond Valuation Negative Convexity
Calibrated Market Regimes:
1. Yield Curve & Bond Specifications
Market Spot Rates & Volatility
%
Initial short rate $r_{0, 0}$.
%
Annual lognormal volatility.
bps
Spot rate progression per year.
Total binomial tree depth.
Callable Bond Target Terms
%
Annual coupon on par $100.
$
Issuer redemption price.
Years before call eligibility.
$
Investor put option floor.
2. BDT Lattice Valuation & Embedded Options
NEGATIVE CONVEXITY ACTIVE
Straight Bond
$101.42
No embedded call
Callable Bond
$100.18
Capped by $K_{ ext{call}}
Call Option Value
$1.24
Straight - Callable
Yield to Maturity
4.46%
Straight Bond YTM
BDT Lognormal Lattice Properties & Step Parameters
Step Baseline Rate $r_{i, 0}$ Up-Step Ratio $u_i$ Arrow-Debreu Total $sum Q$ Market Zero $P^M(0, t_{i+1})$
Non-Negative Guarantee: Because $r_{i, j} = r_{i, 0} cdot u_i^j$ with $u_i = e^{2sigmasqrt{Delta t}} > 1$, every rate on the BDT tree is strictly positive ($r_{i, j} > 0$), satisfying institutional risk bounds.
3. Recombining Short-Rate Binomial Lattice
Lattice Nodes ($r_{i, j}$): Displays the calibrated recombining short rates at each node. Red borders indicate nodes where the issuer's call option is in-the-money and early redemption is exercised.
4. Step-by-Step Calibration Audit
Inductive Solution
Time $t$ Spot Yield Lowest Rate Highest Rate Spread
State prices $Q_{i, j}$ satisfy $sum_{j} Q_{i, j} cdot D_{i, j} = P^M(0, t_{i+1})$ with zero residual error.
5. Sensitivity: Volatility ($sigma$) vs. Yield Shift on Embedded Call Value ($)
Shows how lower yields and higher volatility dramatically increase the value of the issuer's embedded call option.
6. Sensitivity: Call Strike ($K_{ ext{call}}$) vs. Coupon ($c$) on Callable Price ($)
Evaluates callable bond pricing across differing issuer strike provisions and contractual coupon cash flows.
7. Black-Derman-Toy Lattice Mathematics & Negative Convexity
The BDT Lognormal Short-Rate Lattice

Published in the Financial Analysts Journal in 1990 by Fischer Black, Emanuel Derman, and William Toy, the BDT model discretizes the interest rate process on a recombining binomial tree. At time slice $i$, the short rate at state $j in {0, 1, dots, i}$ is given by:

$$r_{i, j} = r_{i, 0} cdot e^{2 j sigma_i sqrt{Delta t}} = r_{i, 0} cdot u_i^j$$

Because the exponential function is strictly positive, $r_{i, j} > 0$ for all nodes. This eliminated the negative rate anomaly that plagued Gaussian models like Vasicek.

Arrow-Debreu State Prices Forward Induction

The tree is calibrated iteratively using Arrow-Debreu state prices $Q_{i, j}$, which represent the value at $t=0$ of receiving $$1$ at node $(i, j)$:

$$Q_{i+1, k} = sum_{j} Q_{i, j} cdot rac{1}{1 + r_{i, j} Delta t} cdot pi(j o k)$$

The baseline rate $r_{i, 0}$ is solved at each step $i$ to satisfy: $sum_{j=0}^{i} Q_{i, j} / (1 + r_{i, 0} u_i^j Delta t) = P^M(0, t_{i+1})$.

Callable Bond Pricing & Backward Induction

Callable bonds give the issuer the right to buy back the debt at a specified call price $K_{ ext{call}}$. At maturity $N$, the bond is worth $F + C$. Moving backward through the lattice:

$$V_{ ext{hold}}(i, j) = rac{0.5 V(i+1, j+1) + 0.5 V(i+1, j)}{1 + r_{i, j} Delta t} + C$$ $$V_{ ext{callable}}(i, j) = minleft( V_{ ext{hold}}(i, j), K_{ ext{call}} + C ight)$$
Negative Convexity Dynamics

For a straight bond, price is a convex function of yield ($ rac{d^2 P}{dy^2} > 0$), meaning price rises more when yields fall than it drops when yields rise. However, for a callable bond, as yields fall below the coupon rate, the call option moves deep into the money, pulling the price toward $K_{ ext{call}}$.

$$ ext{Callable Price} = ext{Straight Price} - ext{Call Option Value}$$

This capping causes the price-yield curve to flatten and become concave downward ($ rac{d^2 P}{dy^2} < 0$), exposing investors to prepayment and reinvestment risk.

8. BDT Model Mastery Self-Assessment Quiz
9. Related Quantitative Finance & Fixed Income Labs
Hull-White Model Lab

Explore exact term structure drift calibration $ heta(t)$ and analytical bond options.

CIR Interest Rate Model Lab

Simulate square-root diffusion, Feller condition verification, and non-negative short rates.

CRR Binomial Option Lab

Price American equity options and early exercise boundaries on recombining lattices.

10. Frequently Asked Questions (FAQ)

Introduced in 1990 by Fischer Black, Emanuel Derman, and William Toy, the BDT model is a discrete-time, one-factor binomial interest rate model. The short rate at step i and state j is modeled lognormally as r_{i, j} = r_{i, 0} * exp(2*j*sigma*sqrt(dt)). The tree is calibrated inductively using Arrow-Debreu state prices to match the market term structure of spot yields and interest rate volatilities.

Because interest rates in the BDT tree are generated as exponential transformations of an underlying Gaussian random walk (r_{i, j} = r_{i, 0} * u^j where u = exp(2*sigma*sqrt(dt)) > 1 and r_{i, 0} > 0), the rates are strictly positive across all nodes and all future time horizons, avoiding the negative interest rate problem of Vasicek and Hull-White.

At terminal maturity, the bond pays face value plus final coupon. Moving backward through time, the holding value at each node is the discounted risk-neutral expectation of future values. For a callable bond, the value is capped at min(V_hold, Call_Price). For a puttable bond, it is bounded from below at max(V_hold, Put_Price). The difference between straight and callable bond values yields the embedded option price.

Negative convexity occurs when a bond's price appreciation slows down and levels off as yields fall, due to the increasing likelihood that the issuer will call the bond away to refinance at lower rates. On a price-yield graph, the curve bends concave downward rather than convex upward, capping upside potential for bondholders.

Yes. This simulator provides a fully sanitized RFC 4180 compliant CSV export containing calibrated short rates across the entire binomial lattice, Arrow-Debreu state prices, straight and callable bond node values, embedded option metrics, and dual 5x5 sensitivity matrices with formula injection defense.