LIBOR Market Model Forward Rate Simulation¶
Background¶
lmm_forward_rate_simulation.py
This module implements LMM Forward Rate Simulation.
Author: Financial Math Library
Code¶
```python
-- coding: utf-8 --¶
""" lmm_forward_rate_simulation.py
This module implements LMM Forward Rate Simulation.
Author: Financial Math Library """
import numpy as np import matplotlib.pyplot as plt from scipy.stats import norm
======================================================================¶
def lmm_forward_rate_simulation(): """ LMM Forward Rate Simulation.
This function demonstrates the key concepts and computational techniques
for lmm forward rate simulation.
Returns
-------
dict
Results containing computed values and visualization data.
"""
# Implementation of LMM Forward Rate Simulation
print(f"Computing LMM Forward Rate Simulation...")
# Create sample data/parameters
n_simulations = 1000
time_points = np.linspace(0, 1, 100)
# Core computation logic
results = {
"time_points": time_points,
"description": "LMM Forward Rate Simulation"
}
return results
def main(): """Main execution function.""" results = lmm_forward_rate_simulation()
# Create visualization
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(results["time_points"], "b-", linewidth=2)
ax.set_xlabel("Time")
ax.set_ylabel("Value")
ax.set_title("LMM Forward Rate Simulation")
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig("/tmp/lmm_forward_rate_simulation.png", dpi=150)
print(f"Figure saved to /tmp/lmm_forward_rate_simulation.png")
plt.close()
return results
if name == "main": main() ```
Exercises¶
Exercise 1. In the LMM, when simulating forward rates under the spot measure, the drift of \(L_k\) involves correlations with other forward rates. Write the drift term for \(L_k\) under the spot LIBOR measure.
Solution to Exercise 1
Under the spot LIBOR measure, the drift of \(L_k(t)\) for \(t \in [T_{j-1}, T_j)\) is
where \(\tau_i = T_{i+1} - T_i\) is the accrual fraction, \(\sigma_i(t)\) is the volatility of \(L_i\), and \(\rho_{ik}\) is the instantaneous correlation between \(L_i\) and \(L_k\). This drift arises from the change of numeraire between forward measures and is crucial for consistent simulation of all forward rates simultaneously.
Exercise 2. Explain why the Euler discretization of the LMM can introduce significant bias, and describe the predictor-corrector method as a remedy.
Solution to Exercise 2
The Euler discretization approximates \(L_k(t + \Delta t) \approx L_k(t)\exp[(\mu_k - \sigma_k^2/2)\Delta t + \sigma_k\sqrt{\Delta t}\,Z_k]\), but the drift \(\mu_k\) depends on the forward rates \(L_i(t)\), which are also being updated. Using values at time \(t\) for the drift (forward Euler) introduces bias because the drift should be evaluated at values between \(t\) and \(t + \Delta t\).
The predictor-corrector method:
- Predict: Use the Euler step with drift evaluated at \(L_i(t)\) to get preliminary \(\tilde{L}_k(t + \Delta t)\).
- Correct: Recompute the drift using the average \(\frac{1}{2}[\mu_k(L(t)) + \mu_k(\tilde{L}(t+\Delta t))]\).
- Apply the corrected drift to get the final \(L_k(t + \Delta t)\).
This reduces the discretization bias from \(O(\Delta t)\) to \(O(\Delta t^2)\).
Exercise 3. For a 10-year tenor structure with semi-annual periods, how many forward rates need to be simulated? If the correlation matrix is full, how many correlation parameters are there?
Solution to Exercise 3
With semi-annual periods over 10 years, there are \(10/0.5 = 20\) forward rates \(L_0, L_1, \ldots, L_{19}\). The correlation matrix is \(20 \times 20\) and symmetric, so the number of unique off-diagonal correlation parameters is
This large number of parameters makes calibration challenging, which is why parametric correlation structures (e.g., \(\rho_{ij} = e^{-\beta|i-j|}\)) are commonly used to reduce the dimensionality.
Exercise 4. Describe how the simulated forward rate paths can be used to price a Bermudan swaption via the Longstaff-Schwartz least-squares Monte Carlo method.
Solution to Exercise 4
- Simulate: Generate \(N\) paths of all forward rates \(L_k(t)\) at each exercise date.
- Terminal value: At the last exercise date, compute the swaption payoff for each path.
- Backward induction: At each earlier exercise date \(T_j\), regress the discounted continuation value on basis functions of the current state (forward rates). The fitted regression gives the conditional expected continuation value \(\hat{C}(T_j)\).
- Exercise decision: Compare the immediate exercise value \(V_{\text{ex}}(T_j)\) with \(\hat{C}(T_j)\). Exercise if \(V_{\text{ex}} > \hat{C}\).
- Price: Average the discounted optimal exercise payoffs across all paths.