Hull-White Zero Bond Pricing¶
Hull-White ZB Dynamics¶
where
Zero Bond Price¶
where
Proof (Expectation)
where
Proof (PDE)
Since the Hull-White short rate dynamics is
the ZCB PDE becomes
Substituting the affine ansatz \(P(t,T)=e^{A(\tau)+B(\tau)r(t)}\) with \(\tau=T-t\) leads to the Riccati ODEs for \(A\) and \(B\).
Hull-White Model Recovers Market ZB Curve¶
The Hull-White model, by construction, exactly recovers the market zero-coupon bond curve \(P^M(0,T)\):
```python def main(): hw = HullWhite(sigma=0.01, lambd=0.01, P=P_market) t, R, M = hw.generate_sample_paths( num_paths=20_000, num_steps=100, T=30, seed=42 )
# Monte Carlo ZCB prices
P_MC = []
for T_i in T_grid:
idx = int(T_i / dt)
P_mc = np.mean(1.0 / M[:, idx])
P_MC.append(P_mc)
# Compare with analytic formula
r0 = compute_r0(P_market)
P_analytic = [hw.compute_ZCB(0, T_i, r0) for T_i in T_grid]
# Both should match P_market(T_i)
```
Hull-White Yield Curve Simulation¶
The Hull-White model can simulate future yield curves by computing \(P(t_i, T)\) for a range of maturities \(T\) at each simulated time \(t_i\):
```python def main(): hw = HullWhite(sigma=0.01, lambd=0.01, P=P_market) t, R, M = hw.generate_sample_paths( num_paths=7, num_steps=100, T=10, seed=42 )
# For each path, compute yield curve at T = 10
for path_idx in range(7):
r_T = R[path_idx, -1]
yields = []
for tau in tau_grid:
P_val = hw.compute_ZCB(10, 10 + tau, r_T)
y = -np.log(P_val) / tau
yields.append(y)
plt.plot(tau_grid, yields)
```
Exercises¶
Exercise 1. Verify that the bond volatility \(\sigma_P(t,T) = -\frac{\sigma}{\lambda}(1 - e^{-\lambda(T-t)})\) is negative for \(T > t\). Explain the economic intuition: why does a bond price decrease when the short rate increases?
Solution to Exercise 1
The bond volatility is \(\sigma_P(t,T) = -\frac{\sigma}{\lambda}(1 - e^{-\lambda(T-t)})\). For \(T > t\), the time to maturity \(\tau = T - t > 0\), so \(e^{-\lambda\tau} \in (0,1)\), which gives \(1 - e^{-\lambda\tau} > 0\). Since \(\sigma > 0\) and \(\lambda > 0\), the entire expression is negative:
Economic intuition: When the short rate \(r(t)\) increases, future discount factors \(e^{-\int_t^T r(s)ds}\) decrease on average, reducing the present value of the bond's unit payoff at maturity. The negative bond volatility reflects this inverse relationship: a positive shock to the Brownian motion \(dW^{\mathbb{Q}}\) raises the short rate (through \(dr = \cdots + \sigma\,dW^{\mathbb{Q}}\)), which decreases bond prices. This is the standard interest rate risk mechanism -- bond prices fall when rates rise.
Exercise 2. Show that in the limit \(\lambda \to 0\), the function \(B(t,T)\) reduces to \(T - t\) and the bond price formula recovers the Ho-Lee model form. What happens to \(A(t,T)\) in this limit?
Solution to Exercise 2
Limit of \(B(t,T)\): Taylor expand the exponential for small \(\lambda\):
Therefore:
In the limit \(\lambda \to 0\), we obtain \(B(t,T) \to T - t\), which is the Ho-Lee duration function.
Limit of \(A(t,T)\): From the formula:
As \(\lambda \to 0\), we have \(B(t,T) \to \tau\) and \(\frac{1 - e^{-2\lambda t}}{4\lambda} \to \frac{2t}{4} = \frac{t}{2}\). Therefore:
This is the Ho-Lee deterministic factor. The convexity correction becomes \(-\frac{\sigma^2}{2}\tau^2 t\), which grows with \(t\) and \(\tau^2\) without the dampening provided by mean reversion.
Exercise 3. Consider a Hull-White model with \(\sigma = 0.01\), \(\lambda = 0.03\), and a flat forward curve \(f^M(0,t) = 0.05\). Compute \(P(0, 10)\) using the bond price formula and verify that it equals the market discount factor \(e^{-0.05 \times 10}\).
Solution to Exercise 3
Given: \(\sigma = 0.01\), \(\lambda = 0.03\), \(f^M(0,t) = 0.05\) (flat), and \(r(0) = f^M(0,0) = 0.05\).
Step 1: Compute \(B(0,10)\).
Step 2: Compute \(A(0,10)\).
With a flat forward curve, \(P^M(0,T) = e^{-0.05T}\):
At \(t = 0\): \(P^M(0,0) = 1\), and \(1 - e^{-2\lambda \cdot 0} = 0\), so the convexity term vanishes:
Wait -- let us be more careful. Since \(t = 0\):
Step 3: Compute \(P(0,10)\).
The market discount factor is \(P^M(0,10) = e^{-0.05 \times 10} = e^{-0.5} \approx 0.6065\). They match exactly, confirming that the Hull-White model recovers the market discount curve at \(t = 0\).
Exercise 4. Starting from the affine ansatz \(P(t,T) = e^{A(\tau) + B(\tau)r(t)}\) with \(\tau = T - t\), substitute into the PDE
and derive the Riccati ODE for \(B(\tau)\).
Solution to Exercise 4
Starting from the affine ansatz \(P(t,T) = e^{A(\tau) + B(\tau)r(t)}\) with \(\tau = T - t\), compute the partial derivatives:
Substituting into the PDE \(\frac{\partial P}{\partial t} + \lambda(\theta(t) - r)\frac{\partial P}{\partial r} + \frac{1}{2}\sigma^2\frac{\partial^2 P}{\partial r^2} = rP\) and dividing by \(P > 0\):
Collecting terms proportional to \(r\):
Since this must hold for all \(r\), the coefficient of \(r\) must vanish:
This is the Riccati ODE for \(B(\tau)\). It is a first-order linear ODE with constant coefficients. Using the integrating factor \(e^{\lambda\tau}\):
Integrating from \(0\) to \(\tau\) with \(B(0) = 0\):
In the positive-\(B\) convention used in the bond price formula \(P = A\,e^{-B\,r}\), we write \(B(t,T) = \frac{1}{\lambda}(1 - e^{-\lambda\tau})\).
Exercise 5. Explain why the Hull-White model exactly recovers the market zero-coupon bond curve \(P^M(0,T)\) for every maturity \(T\). Which parameter in the model is responsible for this calibration, and how is it determined?
Solution to Exercise 5
The Hull-White model recovers \(P^M(0,T)\) because the time-dependent drift function \(\theta(t)\) is explicitly calibrated to the market term structure.
Which parameter: The function \(\theta(t)\) (the time-dependent mean reversion level) is responsible for exact calibration. Unlike the Vasicek model where the mean reversion target \(b\) is a constant, Hull-White promotes \(b\) to a deterministic function \(\theta(t)\).
How it is determined: Setting \(P(0,T) = P^M(0,T)\) for all \(T\) and using the bond price formula at \(t = 0\):
Since \(A(0,T)\) depends on \(\theta(\cdot)\) through the integral \(\int_0^T \theta(u)B(u,T)du\), this equation implicitly defines \(\theta(t)\). Differentiating twice with respect to \(T\) and solving yields:
This formula expresses \(\theta(t)\) entirely in terms of market observables (\(f^M(0,t)\) and its derivative) and model parameters (\(\lambda\), \(\sigma\)). The function \(\theta(t)\) acts as a free function that absorbs the entire initial term structure, ensuring perfect calibration at \(t = 0\).
Exercise 6. In the Monte Carlo verification code, the expected value \(\mathbb{E}^{\mathbb{Q}}[1/M(T)]\) is estimated by \(\frac{1}{N}\sum_{i=1}^{N} 1/M_i(T)\). Show that this estimator is unbiased for \(P(0,T)\) and discuss the sources of Monte Carlo error (discretization bias vs. sampling error).
Solution to Exercise 6
The estimator is \(\hat{P}(0,T) = \frac{1}{N}\sum_{i=1}^N \frac{1}{M_i(T)}\) where \(M_i(T) = e^{\int_0^T r^{(i)}(s)ds}\) is the money market account along path \(i\).
Unbiasedness: By the risk-neutral pricing formula:
Since the paths \(M_1(T), \ldots, M_N(T)\) are i.i.d. draws under \(\mathbb{Q}\):
So the estimator is unbiased.
Sources of error:
-
Sampling error (statistical): The finite sample mean \(\frac{1}{N}\sum 1/M_i\) deviates from the true expectation with standard error \(\text{std}(1/M)/\sqrt{N}\). This decreases as \(\sqrt{N}\) and is the dominant error source for large \(N\).
-
Discretization bias (systematic): The short rate paths are simulated using discrete time steps (e.g., Euler scheme), so \(\int_0^T r(s)ds\) is approximated by \(\sum_k r(t_k)\Delta t\). This introduces a systematic bias that does not vanish with more paths \(N\) but decreases with smaller time steps \(\Delta t\). For the Hull-White model, the exact distribution of \(r\) is known (Gaussian), so exact simulation is possible, eliminating discretization bias entirely.
Exercise 7. The yield curve simulation code computes future yield curves \(R(T, T+\tau)\) for different short rate realizations \(r(T)\). Prove that all simulated yield curves must pass through the same long-maturity asymptote as \(\tau \to \infty\), regardless of \(r(T)\).
Solution to Exercise 7
The yield at maturity \(\tau\) from the simulated curve at time \(T\) is:
where \(b(\tau) = \frac{1 - e^{-\lambda\tau}}{\lambda\tau}\).
As \(\tau \to \infty\):
Therefore the \(r(T)\)-dependent term vanishes:
This limit is deterministic -- it depends only on the market curve \(P^M(0,\cdot)\) and the model parameters \(\lambda, \sigma\), but not on the realization \(r(T)\). Explicitly:
As \(\tau \to \infty\), \(B(T,T+\tau) \to 1/\lambda\) (bounded), so the \(B^2\) correction term divided by \(\tau\) vanishes, and the dominant term is \(-\frac{1}{\tau}\log P^M(0,T+\tau)\), which converges to the asymptotic forward rate \(\bar{f} = \lim_{T\to\infty} f^M(0,T)\).
Hence all simulated yield curves share the same long-maturity asymptote \(\bar{f}\), regardless of the realized short rate \(r(T)\). This is a direct consequence of the single-factor structure and the boundedness of \(B(t,T)\) under mean reversion.