09. Simulate stock price trajectories

Programming the GBM Simulate Function

Creating a simulate function for the GBM model provides essential tools for testing trading algorithms against different market scenarios.

Function Elements Include:

• N: Number of steps in each trajectory.
• K: Total number of trajectories.
• Dt: Time increment (Δt) per step.
• S₀: Initial stock price.

Simulation Process:

1. Trajectory Allocation

   • Use a 2D array with N+1 rows and K columns.
   • Initialize with "np.nan" values.

2. Formula Application

   • Utilize the GBM formula adapting its components using the logarithm.

3. Array Generation Using Linspace

   • Obtain a series: Δt, 2Δt, 3Δt, …
   • Multiply by (μ - \frac{σ^2}{2}) for adjustments.

4. Brownian Motion Simulation

   • Generate n standard normal random variables.
   • Scale by Δt, and compute cumulative sums.

5. Assembling

   • Combine terms into the exponential function argument.
   • Include initial price S₀ to initiate trajectories.

With these steps, price trajectories are successfully simulated, enabling scenario analysis for trading strategies.

QUESTION:

Generate a trajectory of a geometric Brownian motion with parameters mu = 0.4 and sigma = 0.5. The trajectory should include 200 prices with a time increment of 1/252 year. Start with the price equal to 1.