Hull-White (1-Factor) Interest Rate Model Lab

Simulate the Hull-White (1990) extended Vasicek short-rate model. Calibrate the time-dependent drift $\theta(t)$ to match any initial market term structure with zero error, price European bond options and interest rate caplets analytically, and analyze trinomial lattice dynamics.

Hull-White (1990) Extended Vasicek Exact Curve Calibration Bond Option Analytical Pricing Jamshidian Decomposition
Calibrated Market Regimes:
1. Model & Curve Parameters
Hull-White SDE: $dr_t = [\theta(t) - a r_t]dt + \sigma dW_t$
Half-life: 11.55 yrs.
%
Annual volatility of short rate.
Initial Market Term Structure $f^M(0, t)$
Calibrates deterministic drift $ heta(t)$ to match initial discount factors.
%
Initial rate $f^M(0,0)$.
%
Long-end curve spread.
Derivative Target Specification
Option maturity date.
Underlying zero bond tenor.
Strike price per $1 par bond.
%
Caplet annual strike rate.
2. Exact Calibration & Analytical Derivative Pricing
EXACT MARKET MATCH (0 bps)
Bond Option Vol ($sigma_P$)
3.12%
T1=2Y, T2=5Y
Zero Bond Call ($)
$0.0245
Per $1.00 Par
Zero Bond Put ($)
$0.0382
Per $1.00 Par
IR Caplet Value ($)
$43,150
$1M Notional
Hull-White Pricing Architecture: Exact Analytical Jamshidian Formula
Component Analytical Formula Value Description
Bond Duration Factor $B(T_1, T_2)$ (1 - exp(-a*(T2 - T1))) / a 2.755 Sensitivity of $P(T_1, T_2)$ to the short rate $r(T_1)$
Integrated Bond Vol $sigma_P$ (sigma/a)*(1 - e^{-a(T2-T1)}) * √[(1 - e^{-2aT1})/(2a)] 0.0312 Total forward volatility of zero-coupon bond price
Forward Zero Ratio $F_P$ P^M(0, T2) / P^M(0, T1) 0.8654 Forward price of the $T_2$ bond delivered at $T_1$
IR Floorlet Value ($1M Notional) N * (1 + au*R_K) * Call(T1, T2, K') $28,420 Equivalent floorlet payoff protecting floating borrowers
Arbitrage-Free Property: Unlike Vasicek or CIR where initial market discount bonds $P^M(0,T)$ have non-zero pricing errors, Hull-White automatically satisfies $P(0, T) equiv P^M(0, T)$ for every maturity $T$ because $ heta(t)$ absorbs the slope and curvature of the forward curve.
3. Term Structure & Short-Rate Diffusion
Yield Curve (0 to 30 Years): Blue curve depicts the market spot yield $Y(0,T)$, red dashed curve shows the instantaneous forward rate $f(0,T)$, and purple line tracks the bond option volatility structure $sigma_P(0, T)$.
4. Calibrated Market Schedule & Model Verification
Zero Pricing Error
Maturity Market Zero ($P^M) Spot Yield ($Y$) Forward ($f$) Model Price ($P) Error ($epsilon$)
Market discount factor: $P^M(0,T) = exp(-int_0^T f^M(0,u)du)$. Note error $epsilon = 0.0000$ across all tenors.
5. Sensitivity: Mean Reversion ($a$) vs. Volatility ($sigma$) on Call Option ($)
Shows how increasing mean reversion $a$ dampens long-term bond volatility $sigma_P$, lowering option prices.
6. Sensitivity: Caplet Strike ($R_K$) vs. Option Expiry ($T_1$) on Caplet Value ($)
Values a $1,000,000 notional caplet across varying strike interest rates and forward expiration tenors.
7. Hull-White Trinomial Tree Architecture & Lattice Pricing
Stage 1: Building the Zero-Mean Standardized Tree

To price American, Bermudan, and path-dependent interest rate derivatives (such as Bermudan swaptions or callable bonds), Hull and White introduced a recombining trinomial lattice. First, consider an auxiliary zero-drift process $R_t = r_t - alpha(t)$ satisfying:

$$dR_t = -a R_t dt + sigma dW_t$$

At time step $Delta t$, the state space space step is set to $Delta R = sigma sqrt{3 Delta t}$. From state $(i, j)$ where $R = j Delta R$, the branch probabilities for standard nodes are:

$$p_u = rac{1}{6} + rac{a^2 j^2 Delta t^2 - a j Delta t}{2}, quad p_m = rac{2}{3} - a^2 j^2 Delta t^2, quad p_d = rac{1}{6} + rac{a^2 j^2 Delta t^2 + a j Delta t}{2}$$
Stage 2: Forward Induction via Arrow-Debreu State Prices

In Stage 2, the tree is shifted at each time slice $i$ by a deterministic displacement $alpha_i$ so that the short rate is $r_{i, j} = alpha_i + j Delta R$. The shift $alpha_i$ is solved inductively using Arrow-Debreu state prices $Q_{i, j}$:

$$P^M(0, t_{i+1}) = sum_{j} Q_{i, j} expleft( -(alpha_i + j Delta R) Delta t ight)$$

Solving this one-dimensional non-linear root gives the exact $alpha_i$ that guarantees perfect alignment with the market term structure $P^M(0, t_{i+1})$ without any numerical discrepancy.

8. Hull-White Term Structure Mathematics & Jamshidian Decomposition
Analytical Time-Varying Drift Calibration $ heta(t)$

In the Vasicek model, the mean reversion drift is $a(b - r)$, forcing the model yield curve into fixed asymptotic shapes that almost never match current market prices. Hull & White solved this by replacing $a cdot b$ with a calibrated time-dependent function $ heta(t)$:

$$ heta(t) = rac{partial f^M(0, t)}{partial t} + a f^M(0, t) + rac{sigma^2}{2a}left( 1 - e^{-2at} ight)$$

Here, $f^M(0, t) = - rac{partial ln P^M(0, t)}{partial t}$ is the instantaneous market forward rate. The term $ rac{sigma^2}{2a}(1 - e^{-2at})$ represents the convexity adjustment required to prevent Jensen's inequality arbitrage.

Jamshidian's Decomposition for Swaptions & Caps

Because discount bond prices $P(T_1, T_2)$ are strictly decreasing monotonic functions of $r(T_1)$, Farshid Jamshidian (1989) proved that European options on coupon bonds or swaps decompose into independent options on each coupon zero-coupon bond:

$$ ext{Call}(T_1, T_2, K) = P^M(0, T_2) N(d_1) - K P^M(0, T_1) N(d_2)$$ $$d_1 = rac{ln(P^M(0, T_2) / [K P^M(0, T_1)]) + rac{1}{2}sigma_P^2}{sigma_P}, quad d_2 = d_1 - sigma_P$$

For an interest rate caplet on $[T_1, T_2]$, the payoff is mathematically isomorphic to $(1 + au R_K)$ European put options on the underlying zero bond with strike $K' = rac{1}{1 + au R_K}$, enabling instantaneous closed-form hedging.

9. Hull-White Model Mastery Self-Assessment Quiz
10. Related Quantitative Finance & Fixed Income Labs
BDT Interest Rate Lattice Lab · CIR Interest Rate Model Lab

Explore square-root diffusion, Feller condition verification, and strictly non-negative rate dynamics.

Vasicek Short-Rate Model Lab

Simulate Ornstein-Uhlenbeck Gaussian diffusion, analytical bond pricing, and term structure curves.

Bond Portfolio Immunization Lab

Model classical Redington duration matching, convexity surplus, and Fong-Vasicek structural risk.

11. Frequently Asked Questions (FAQ)

Introduced in 1990 by John Hull and Alan White, the Hull-White one-factor model extends the classical Vasicek (1977) model by replacing the constant long-term mean reversion level with a time-dependent drift function theta(t). This extension enables the model to achieve an exact arbitrage-free fit to the currently observed market term structure of interest rates while retaining the mathematical tractability of Gaussian affine term structure models.

The drift theta(t) is chosen analytically to satisfy theta(t) = d f^M(0,t)/dt + a * f^M(0,t) + (sigma^2 / 2a)*(1 - exp(-2at)), where f^M(0,t) is the instantaneous market forward rate. Because theta(t) absorbs the entire shape and slope of today's market forward curve, theoretical zero-coupon bond prices generated by the model at time t=0 match baseline market discount factors P^M(0,T) with zero pricing error.

Farshid Jamshidian (1989) demonstrated that because discount bond prices are strictly monotonic functions of the short rate in one-factor affine models, a European option on any portfolio of cash flows (such as a coupon bond or interest rate swap) can be decomposed into a portfolio of European options on the underlying zero-coupon bonds. Under Hull-White, each zero-coupon bond option has an exact analytical Black-Scholes-like closed-form solution.

An interest rate cap is a portfolio of caplets. Under Hull-White, each caplet on the forward rate spanning [T1, T2] with strike R_K is mathematically equivalent to (1 + tau*R_K) European put options on a zero-coupon bond maturing at T2 with strike 1 / (1 + tau*R_K) expiring at T1. European swaptions are priced by finding the unique critical short rate r* that equates the swap NPV to zero at option expiry and evaluating the corresponding bond options.

Yes. This simulator provides a fully sanitized RFC 4180 compliant CSV export containing calibrated Hull-White drift parameters, market discount factors, analytical bond option prices, caplet and floorlet valuations, and dual 5x5 sensitivity matrices with formula injection defense.