Hull-White Swap Pricing¶

Hull-White Recovers Swap Pricing¶

The Hull-White model can price interest rate swaps by computing ZCB prices at each payment date:

```python def main(): # Swap setting CP = OptionTypeSwap.PAYER # payer swap pays fixed rate notional = 10000.0 Ks = np.linspace(0.0, 0.08, 30)

hw = HullWhite(sigma=0.01, lambd=0.01, P=P_market)

# Method 1: Analytic using ZCB curve
swap_analytic = []
for K in Ks:
    swap = compute_SwapPrice(P_market, t=0, notional=notional,
                             K=K, Ti=1.0, Tm=10.0, n=10, CP=CP)
    swap_analytic.append(swap)

# Method 2: Monte Carlo using Hull-White
num_paths = 50_000
t, R, M = hw.generate_sample_paths(num_paths, num_steps=100, T=10)

swap_mc = []
for K in Ks:
    # Compute swap payoff on each path
    payoffs = []
    for path in range(num_paths):
        swap_val = hw.compute_SwapPrice(
            t=0, r_t=R[path, 0], notional=notional,
            K=K, Ti=1.0, Tm=10.0, n=10, CP=CP
        )
        payoffs.append(swap_val)
    swap_mc.append(np.mean(payoffs))

# Compare
plt.figure(figsize=(10, 5))
plt.plot(Ks, swap_analytic, 'b-', label='Analytic (ZCB Curve)')
plt.plot(Ks, swap_mc, 'r--', label='Hull-White MC')
plt.xlabel('Fixed Rate K')
plt.ylabel('Swap Value')
plt.title('Hull-White Recovers Swap Pricing')
plt.legend()
plt.grid(True)
plt.show()

```

The payer swap value is:

\[\begin{array}{lllllllll} \displaystyle {\bf\text{IRS}}^{\text{Payer}}(t,{\cal T},N,K) &=& N\left(P(t,T_m)-P(t,T_n)\right)-NK\sum_{k=m+1}^n \tau_k P(t,T_k) \end{array}\]

The Hull-White model computes each $P(t,T_k)=e^{A(t,T_k)+B(t,T_k)r(t)}$ using the calibrated functions $A$ and $B$, which by construction match the market ZCB curve. Therefore, the Hull-White swap price agrees with the market swap price.

Exercises¶

Exercise 1. A payer swap has notional $N = \$10{,}000$, fixed rate $K = 0.05$, and annual payments from year 1 to year 10. Write out the formula for the swap value at $t = 0$ in terms of ZCB prices $P(0, T_k)$.

Solution to Exercise 1

The payer swap value at $t = 0$ with notional $N = \$10{,}000$, fixed rate $K = 0.05$, and annual payments from year 1 to year 10 ($T_m = 0$, $T_n = 10$, $\tau_k = 1$ for all $k$) is:

\[ \text{IRS}^{\text{Payer}}(0) = N\!\left(P(0, T_m) - P(0, T_n)\right) - NK\sum_{k=m+1}^{n}\tau_k P(0, T_k) \]

With $T_m = 0$ (so $P(0,0) = 1$):

\[ \text{IRS}^{\text{Payer}}(0) = 10{,}000\!\left(1 - P(0, 10)\right) - 10{,}000 \times 0.05 \sum_{k=1}^{10} P(0, k) \]

\[ = 10{,}000\!\left(1 - P(0,10) - 0.05\sum_{k=1}^{10} P(0,k)\right) \]

The floating leg value is $N(P(0,0) - P(0,10)) = N(1 - P(0,10))$, and the fixed leg value is $NK\sum_{k=1}^{10} P(0,k)$. The payer swap pays fixed and receives floating, so its value is the floating leg minus the fixed leg.

Exercise 2. Show that the par swap rate $K^*$ (the rate that makes the swap value zero at inception) is given by $K^* = \frac{P(0, T_m) - P(0, T_n)}{\sum_{k=m+1}^n \tau_k P(0, T_k)}$. Compute $K^*$ for a 5-year annual swap when $P(0, k) = e^{-0.04k}$ for $k = 1, \ldots, 5$.

Solution to Exercise 2

The par swap rate $K^*$ is the fixed rate that makes the swap value zero:

\[ \text{IRS}^{\text{Payer}}(0) = N(P(0,T_m) - P(0,T_n)) - NK^*\sum_{k=m+1}^n \tau_k P(0,T_k) = 0 \]

Solving for $K^*$:

\[ K^* = \frac{P(0,T_m) - P(0,T_n)}{\sum_{k=m+1}^n \tau_k P(0,T_k)} \]

For a 5-year annual swap with $P(0,k) = e^{-0.04k}$, $T_m = 0$, $T_n = 5$, $\tau_k = 1$:

\[ P(0,0) = 1, \quad P(0,5) = e^{-0.20} \approx 0.81873 \]

\[ \sum_{k=1}^5 P(0,k) = e^{-0.04} + e^{-0.08} + e^{-0.12} + e^{-0.16} + e^{-0.20} \]

\[ \approx 0.96079 + 0.92312 + 0.88692 + 0.85214 + 0.81873 = 4.44170 \]

Therefore:

\[ K^* = \frac{1 - 0.81873}{4.44170} = \frac{0.18127}{4.44170} \approx 0.04081 \]

The par swap rate is approximately 4.08%, which is close to (but slightly above) the continuous rate of 4% because of the difference between continuous and simple compounding.

Exercise 3. Explain why the Hull-White model recovers the same swap price as direct computation from the market ZCB curve. What property of the Hull-White calibration guarantees this?

Solution to Exercise 3

The Hull-White model recovers the same swap price as the market ZCB curve because of the exact calibration property. Specifically, the Hull-White model calibrates the time-dependent function $\theta(t)$ (or equivalently the drift function) so that the model-implied zero-coupon bond prices match the market curve exactly:

\[ P^{\text{HW}}(0, T) = P^{\text{market}}(0, T) \quad \text{for all } T \]

Since the swap value at $t = 0$ depends only on ZCB prices $P(0, T_k)$ at the payment dates:

\[ \text{IRS}^{\text{Payer}}(0) = N(P(0,T_m) - P(0,T_n)) - NK\sum_{k=m+1}^n \tau_k P(0,T_k) \]

and the Hull-White model reproduces these ZCB prices exactly, the model-implied swap price matches the market swap price. This is guaranteed by the calibration of $\theta(t)$ to fit the initial yield curve, which is the defining feature of the Hull-White (extended Vasicek) framework.

Exercise 4. In the Monte Carlo approach, why are multiple paths simulated even though the swap at $t = 0$ is model-independent? Discuss how Monte Carlo becomes essential for pricing path-dependent swap variants.

Solution to Exercise 4

At $t = 0$, the plain vanilla swap price is determined entirely by the current ZCB curve through the formula $\text{IRS}(0) = N(P(0,T_m) - P(0,T_n)) - NK\sum \tau_k P(0,T_k)$. This is model-independent -- no simulation is needed.

The Monte Carlo approach in the code simulates multiple Hull-White paths primarily for validation: it confirms that the Hull-White model reproduces the analytic swap price, verifying the implementation.

Monte Carlo becomes essential for pricing path-dependent swap variants where the payoff depends on the evolution of rates, not just their terminal values:

Callable swaps / Bermudan swaptions: The exercise decision at each date depends on the rate path, requiring dynamic programming or regression-based methods.
Range accrual swaps: The floating coupon accrues only on days when the rate falls within a specified range, making the payoff path-dependent.
CMS swaps: The coupon depends on swap rates at future dates, which are nonlinear functions of the yield curve.
Amortizing swaps: The notional changes based on prepayment behavior, which may depend on the rate path.

In all these cases, the payoff cannot be expressed solely in terms of $P(0, T_k)$, and simulation of the short rate dynamics is required.

Exercise 5. Derive the receiver swap value from the payer swap value using the identity $\text{IRS}^{\text{Receiver}} = -\text{IRS}^{\text{Payer}}$. For what value of $K$ are the payer and receiver swap values equal?

Solution to Exercise 5

The receiver swap pays fixed and receives floating, which is the opposite of the payer swap. Therefore:

\[ \text{IRS}^{\text{Receiver}}(t_0) = NK\sum_{k=m+1}^n \tau_k P(t_0,T_k) - N(P(t_0,T_m) - P(t_0,T_n)) = -\text{IRS}^{\text{Payer}}(t_0) \]

The payer and receiver swap values are equal when both are zero:

\[ \text{IRS}^{\text{Payer}}(t_0) = \text{IRS}^{\text{Receiver}}(t_0) \iff \text{IRS}^{\text{Payer}}(t_0) = 0 \]

This occurs when the fixed rate $K$ equals the par swap rate:

\[ K = K^* = \frac{P(t_0,T_m) - P(t_0,T_n)}{\sum_{k=m+1}^n \tau_k P(t_0,T_k)} \]

At this rate, both the payer and receiver swaps have zero value, and the swap is said to be "at par."

Exercise 6. A forward-starting swap begins at $T_m = 5$ with payments at years 6 through 10. Express its value at $t = 0$ in terms of ZCB prices and explain how the Hull-White model prices this instrument.

Solution to Exercise 6

A forward-starting swap beginning at $T_m = 5$ with annual payments at years 6, 7, 8, 9, 10 has the payer value at $t = 0$:

\[ \text{IRS}^{\text{Payer}}(0) = N\!\left(P(0, 5) - P(0, 10)\right) - NK\sum_{k=6}^{10} \tau_k P(0, T_k) \]

With annual payments ($\tau_k = 1$):

\[ \text{IRS}^{\text{Payer}}(0) = N\!\left(P(0,5) - P(0,10) - K\sum_{k=6}^{10} P(0,k)\right) \]

Note that the floating leg starts at $T_m = 5$, so $P(0,T_m) = P(0,5)$, not $P(0,0) = 1$. The first floating payment is at $T_6 = 6$ based on the rate observed at $T_5 = 5$.

Hull-White pricing: The Hull-White model prices this using the same formula, since the model-implied ZCB prices $P^{\text{HW}}(0,T_k)$ match the market curve by calibration. The value depends on $P(0,5), P(0,6), \ldots, P(0,10)$, all of which are reproduced exactly. The forward-starting feature introduces no additional complexity for the Hull-White model at $t = 0$ -- the analytic formula applies directly. However, if one needs the swap value at a future date $t > 0$ (e.g., for swaption pricing), the Hull-White dynamics of $r(t)$ determine the future ZCB prices $P(t, T_k)$ and hence the swap value conditional on $r(t)$.