ASSISTANT
a. To find the solution of the SDE, we first perform the transformation 𝑦 = 𝑟𝑒^𝛾𝑡. Let's apply Ito's lemma to find the dynamics of y.
dy = (∂y/∂r) dr + (1/2) (∂²y/∂r²) (dr)² + (∂y/∂t) dt
Since y = re^(𝛾t), we have:
∂y/∂r = e^(𝛾t)
∂²y/∂r² = 0
∂y/∂t = 𝛾re^(𝛾t)
Now, we can substitute these derivatives and the given SDE for the spot rate into Ito's lemma equation:
dy = e^(𝛾t) ((𝛼 - 𝛾r)dt + √𝛽dW) + (1/2) * 0 * (√𝛽dW)² + 𝛾re^(𝛾t) dt
After simplifying, we have:
dy = [𝛼e^(𝛾t) + 𝛾r(e^(𝛾t) - 1)]dt + e^(𝛾t)√𝛽dW
Now, let's integrate this equation on [0, T] to get the solution for y:
y(T) = y(0) + ∫[𝛼e^(𝛾t) + 𝛾r(e^(𝛾t) - 1)] dt + ∫e^(𝛾t)√𝛽 dW
b. To find the conditional expectation and variance, we first need to derive the SDE for y:
dy = [𝛼e^(𝛾t) + 𝛾r(e^(𝛾t) - 1)]dt + e^(𝛾t)√𝛽dW
Taking expectation on both sides:
E(dy) = [𝛼e^(𝛾t) + 𝛾r(e^(𝛾t) - 1)] E(dt)
Divide by dt:
E(dy/dt) = 𝛼e^(𝛾t) + 𝛾re^(𝛾t) - 𝛾r
Now, we need to find the variance.
Var(dy) = e^(2𝛾t)𝛽 Var(dW)
Dividing by dt:
Var(dy/dt) = e^(2𝛾t)𝛽
c. In order to simulate y(T) using Monte Carlo, we will generate paths using the Euler-Maruyama method. First, import the necessary libraries and set the parameters:
```python
import numpy as np
import matplotlib.pyplot as plt
alpha = 0.1
gamma = 0.1
beta = 0.02
T = 1
dt = 0.001
n_paths = 1000
n_steps = int(T/dt)
```
Next, create paths for y using the Euler-Maruyama method:
```python
np.random.seed(42)
y0 = 0.1 * np.exp(gamma * 0)
time_grid = np.linspace(0, T, n_steps)
y_paths = np.zeros((n_paths, n_steps))
y_paths[:, 0] = y0
for path in range(n_paths):
for step in range(1, n_steps):
dW = np.random.normal(0, np.sqrt(dt))
y_previous = y_paths[path, step-1]
r_previous = y_previous / np.exp(gamma * time_grid[step-1])
factor = np.exp(gamma * time_grid[step])
dy = (alpha * factor + gamma * r_previous * (factor - 1)) * dt + factor * np.sqrt(beta) * dW
y_paths[path, step] = y_previous + dy
```
Now, calculate the mean and variance of the y_paths:
```python
y_mean = np.mean(y_paths, axis=0)
y_var = np.var(y_paths, axis=0)
```
Finally, for comparison with (b), we can calculate the expected value and variance of y according to the derived formulas:
```python
y_expected = alpha / gamma * (np.exp(gamma * time_grid) - 1) + y0 * np.exp(gamma * time_grid)
y_variance = beta * (np.exp(2 * gamma * time_grid) - 1) / (2 * gamma)
```
To visualize and compare the results, we can plot the Monte Carlo paths, mean, and variance alongside the theoretical mean and variance:
```python
plt.figure(figsize=(12, 6))
# Plot multiple paths
for path in y_paths[:50]:
plt.plot(time_grid, path, color='lightgray', lw=0.5)
# Plot Monte Carlo mean and variance
plt.plot(time_grid, y_mean, label='Monte Carlo Mean', linestyle='dotted', color='blue')
plt.fill_between(time_grid, y_mean - np.sqrt(y_var), y_mean + np.sqrt(y_var), color='blue', alpha=0.1, label='Monte Carlo Std Dev')
# Plot theoretical mean and variance
plt.plot(time_grid, y_expected, label='Theoretical Mean', linestyle='dashed', color='red')
plt.fill_between(time_grid, y_expected - np.sqrt(y_variance), y_expected + np.sqrt(y_variance), color='red', alpha=0.1, label='Theoretical Std Dev')
plt.xlabel('Time')
plt.ylabel('y(t)')
plt.legend()
plt.show()
```
The simulated mean and variance are close to the theoretical values from (b), which validates our calculations using the Monte Carlo simulation.