Ho-Lee Zero-Coupon Bonds (Grzelak)¶

Background¶

Ho-Lee model simulation and zero-coupon bond pricing.

Simulates the Ho-Lee model and computes zero-coupon bond prices P(0,t).

Based on "Financial Engineering" course by L.A. Grzelak. The course is based on the book "Mathematical Modeling and Computation in Finance: With Exercises and Python and MATLAB Computer Codes", by C.W. Oosterlee and L.A. Grzelak, World Scientific Publishing Europe Ltd, 2019. @author: Lech A. Grzelak

Code¶

```python """ Ho-Lee model simulation and zero-coupon bond pricing.

Simulates the Ho-Lee model and computes zero-coupon bond prices P(0,t).

Based on "Financial Engineering" course by L.A. Grzelak. The course is based on the book "Mathematical Modeling and Computation in Finance: With Exercises and Python and MATLAB Computer Codes", by C.W. Oosterlee and L.A. Grzelak, World Scientific Publishing Europe Ltd, 2019. @author: Lech A. Grzelak """ import numpy as np import matplotlib.pyplot as plt

======================================================================¶

Functions / Classes¶

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def f0t(t, p0t): """ Calculate forward rate at time t.

Parameters
----------
t : float
    Time point
p0t : callable
    Zero-coupon bond pricing function P(0,T)

Returns
-------
float
    Forward rate f(0,t)
"""
dt = 0.01
expr = -(np.log(p0t(t + dt)) - np.log(p0t(t - dt))) / (2 * dt)
return expr

def generate_paths_ho_lee_euler(num_paths, num_steps, T, p0t, sigma): """ Generate Ho-Lee model paths using Euler scheme.

Parameters
----------
num_paths : int
    Number of simulation paths
num_steps : int
    Number of time steps
T : float
    Time to maturity
p0t : callable
    Zero-coupon bond pricing function
sigma : float
    Volatility parameter

Returns
-------
dict
    Dictionary with keys 'time', 'R', 'M' containing time grid, rates, and bank account
"""
dt = T / float(num_steps)

# Initial interest rate is forward rate at time t->0
r0 = f0t(0.01, p0t)

def theta(t):
    """Theta function for Ho-Lee model."""
    return ((f0t(t + dt, p0t) - f0t(t - dt, p0t)) / (2.0 * dt) +
            sigma ** 2.0 * t)

z = np.random.normal(0.0, 1.0, (num_paths, num_steps))
w = np.zeros((num_paths, num_steps + 1))
r = np.zeros((num_paths, num_steps + 1))
m = np.zeros((num_paths, num_steps + 1))
m[:, 0] = 1.0
r[:, 0] = r0
time = np.zeros(num_steps + 1)

for i in range(0, num_steps):
    # Ensure samples from normal have mean 0 and variance 1
    if num_paths > 1:
        z[:, i] = (z[:, i] - np.mean(z[:, i])) / np.std(z[:, i])
    w[:, i + 1] = w[:, i] + np.sqrt(dt) * z[:, i]
    r[:, i + 1] = r[:, i] + theta(time[i]) * dt + sigma * (w[:, i + 1] -
                                                             w[:, i])
    m[:, i + 1] = m[:, i] * np.exp((r[:, i + 1] + r[:, i]) * 0.5 * dt)
    time[i + 1] = time[i] + dt

paths = {"time": time, "R": r, "M": m}
return paths

def plot_zcb_comparison(time_grid, p0t_market, p0t_mc): """ Plot comparison of market and Monte Carlo zero-coupon bond prices.

Parameters
----------
time_grid : array
    Time points
p0t_market : array
    Market ZCB prices
p0t_mc : array
    Monte Carlo ZCB prices
"""
plt.figure(1)
plt.grid()
plt.xlabel('T')
plt.ylabel('P(0,T)')
plt.plot(time_grid, p0t_market)
plt.plot(time_grid, p0t_mc, '--r')
plt.legend(['P(0,t) market', 'P(0,t) Monte Carlo'])
plt.title('ZCBs from Ho-Lee Model')

def main(): """Run Ho-Lee model simulation and ZCB pricing.""" # ============= Parameters ============= num_paths = 25000 num_steps = 500 sigma = 0.007 T = 40

# Define a ZCB curve (obtained from market)
p0t = lambda T: np.exp(-0.1 * T)

# ============= Monte Carlo Simulation =============
paths = generate_paths_ho_lee_euler(num_paths, num_steps, T, p0t, sigma)
m = paths["M"]
ti = paths["time"]

# Compare price of ZCB from Monte Carlo and analytical expression
p_t = np.zeros(num_steps + 1)
for i in range(0, num_steps + 1):
    p_t[i] = np.mean(1.0 / m[:, i])

# ============= Plotting =============
plot_zcb_comparison(ti, p0t(ti), p_t)

======================================================================¶

Main¶

======================================================================¶

if name == "main": main() ```

Exercises¶

Exercise 1. The Ho-Lee model SDE is \(dr(t) = \theta(t)\,dt + \sigma\,dW(t)\). If the market zero-coupon bond curve is \(P(0,T) = e^{-0.1T}\), compute the forward rate \(f(0,T)\) and the drift function \(\theta(t)\) for \(\sigma = 0.007\).

Exercise 2. Explain how the money market account \(M(t)\) is computed in the code and why the ZCB price is estimated as \(P(0,t) \approx \mathbb{E}[1/M(t)]\).

Exercise 3. For \(\sigma = 0.007\), \(T = 40\), and \(N = 25{,}000\) paths, the code compares analytical and Monte Carlo ZCB prices. Estimate the standard error of the Monte Carlo estimate for \(P(0,20)\) if the variance of \(1/M(20)\) is approximately \(0.01\).

Exercise 4. The Ho-Lee model allows negative interest rates. For \(\sigma = 0.007\) and a flat forward curve at \(10\%\), estimate the probability that \(r(40) < 0\) assuming \(r(40)\) is approximately normally distributed.