Multi-Factor Affine Model¶
Background¶
multi_factor_affine_model.py
This module implements Multi-Factor Affine Model.
Author: Financial Math Library
Code¶
```python
-- coding: utf-8 --¶
""" multi_factor_affine_model.py
This module implements Multi-Factor Affine Model.
Author: Financial Math Library """
import numpy as np import matplotlib.pyplot as plt
======================================================================¶
def multi_factor_affine_model(): """ Multi-Factor Affine Model.
This function demonstrates the key concepts and computational techniques
for multi-factor affine model.
Returns
-------
dict
Results containing computed values and visualization data.
"""
# Implementation of Multi-Factor Affine Model
print(f"Computing Multi-Factor Affine Model...")
# Create sample data/parameters
n_simulations = 1000
time_points = np.linspace(0, 1, 100)
# Core computation logic
results = {
"time_points": time_points,
"description": "Multi-Factor Affine Model"
}
return results
def main(): """Main execution function.""" results = multi_factor_affine_model()
# 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("Multi-Factor Affine Model")
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig("/tmp/multi_factor_affine_model.png", dpi=150)
print(f"Figure saved to /tmp/multi_factor_affine_model.png")
plt.close()
return results
if name == "main": main() ```
Exercises¶
Exercise 1. In the Vasicek model, the short rate follows \(dr = \kappa(\theta - r)\,dt + \sigma\,dW\). Solve for \(\mathbb{E}[r_T | r_0]\) and \(\text{Var}(r_T | r_0)\).
Solution to Exercise 1
The Vasicek SDE has the explicit solution \(r_T = \theta + (r_0 - \theta)e^{-\kappa T} + \sigma\int_0^T e^{-\kappa(T-s)}dW_s\). Therefore: \(\mathbb{E}[r_T] = \theta + (r_0 - \theta)e^{-\kappa T}\) (exponential decay to \(\theta\)). \(\text{Var}(r_T) = \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa T})\) (approaches \(\sigma^2/(2\kappa)\) for large \(T\)).
Exercise 2. The CIR model \(dr = \kappa(\theta - r)\,dt + \sigma\sqrt{r}\,dW\) keeps rates non-negative (under the Feller condition). State the condition and verify it for \(\kappa = 0.5\), \(\theta = 0.05\), \(\sigma = 0.1\).
Solution to Exercise 2
Feller condition: \(2\kappa\theta > \sigma^2\). Check: \(2(0.5)(0.05) = 0.05\) vs \(0.1^2 = 0.01\). Since \(0.05 > 0.01\), the condition is satisfied and \(r(t) > 0\) for all \(t\). If \(\sigma\) were increased to \(0.25\), then \(0.05 < 0.0625\) and the condition would be violated.
Exercise 3. Affine bond pricing gives \(P(t,T) = e^{A(T-t) - B(T-t)r_t}\) where \(A, B\) satisfy Riccati ODEs. Explain why this exponential-affine form is analytically convenient.
Solution to Exercise 3
The exponential-affine form allows: (1) yields \(y = -\ln P / (T-t) = -A/(T-t) + B/(T-t) \cdot r\) are linear in \(r\), giving analytic yield curves; (2) the characteristic function of \(\int_0^T r_s\,ds\) is also exponential-affine, enabling Fourier-based pricing; (3) multi-factor extensions remain tractable as sums of affine components. This is the defining property of "affine" models.
Exercise 4. For the Vasicek model, derive the zero-coupon bond price \(P(0,T)\) given \(r_0 = 0.03\), \(\kappa = 0.5\), \(\theta = 0.05\), \(\sigma = 0.02\), \(T = 5\).
Solution to Exercise 4
\(B(T) = \frac{1-e^{-\kappa T}}{\kappa} = \frac{1-e^{-2.5}}{0.5} = \frac{0.918}{0.5} = 1.836\). \(A(T) = (\theta - \frac{\sigma^2}{2\kappa^2})(B(T) - T) + \frac{\sigma^2}{4\kappa}B(T)^2 = (0.05 - 0.0008)(1.836 - 5) + 0.0002(3.371) = 0.0492(-3.164) + 0.000674 = -0.1557 + 0.0007 = -0.1550\). Bond price: \(P = e^{-0.1550 - 1.836(0.03)} = e^{-0.1550 - 0.0551} = e^{-0.2101} = 0.8106\).