Riccati Ordinary Differential Equation Solver (Vasicek, Cox-Ingersoll-Ross, Heston)¶

Background¶

riccati_ode_solver.py

This module implements Riccati ODE Solver.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" riccati_ode_solver.py

This module implements Riccati ODE Solver.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt from scipy.sparse import diags

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

def riccati_ode_solver(): """ Riccati ODE Solver.

This function demonstrates the key concepts and computational techniques
for riccati ode solver.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Riccati ODE Solver
print(f"Computing Riccati ODE Solver...")

# Create sample data/parameters
n_simulations = 1000
time_points = np.linspace(0, 1, 100)

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Riccati ODE Solver"
}

return results

def main(): """Main execution function.""" results = riccati_ode_solver()

# 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("Riccati ODE Solver")
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig("/tmp/riccati_ode_solver.png", dpi=150)
print(f"Figure saved to /tmp/riccati_ode_solver.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 \(\text{Var}(r_T | r_0)\).

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 \(\kappa = 0.5\), \(\theta = 0.05\), \(\sigma = 0.1\).

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.

Exercise 4. For the Vasicek model, derive the zero-coupon bond price \(P(0,T)\) given \(r_0 = 0.03\), \(\kappa = 0.5\), \(\theta = 0.05\), \(\sigma = 0.02\), \(T = 5\).