Simulating Stock Prices using Geometric
Brownian Motion
October 5, 2023
Sajad Ahmad Sheikh,University of Kashmir
1 Introduction
Geometric Brownian Motion (GBM) is one of the most popular methods for simulating stock prices. It
is based on the Brownian motion model, which describes the random motion of particles suspended in a
fluid.
2 Mathematical Model
The GBM model for stock prices is given by the stochastic differential equation:
dSt=µStdt +σStdWt
Where:
•Stis the stock price at time t
•µis the expected return (drift)
•σis the volatility
•dWtis a Wiener process or Brownian motion
The solution to this equation is:
St=S0exp µ−
σ2
2t+σWt
3 Algorithm for Simulating Stock Prices
1. Initialize the stock price S0, expected return µ, volatility σ, and time increment ∆t.
2. For each time step tuntil the desired time horizon:
•Generate a random number Zfrom a standard normal distribution.
•Calculate the change in stock price ∆Susing the formula:
∆S=µSt∆t+σStZ√∆t
•Update the stock price: St+∆t=St+ ∆S
3. Repeat the above steps for multiple simulations to generate different stock price paths.
4 Significance
•Risk Assessment: By simulating various stock price paths, we can assess the potential risk
associated with a stock.
•Option Pricing: GBM is fundamental in the Black-Scholes option pricing model.
•Portfolio Optimization: Simulating stock prices helps in optimizing portfolio allocations.
1
