Geometric Brownian Motion (GBM) is a widely used model in finance, particularly for the simulation of stock prices, because it incorporates the essential features of asset price dynamics - randomness and compound returns. The GBM is described by the following stochastic differential equation (SDE):
dSt = μStdt + σStdWt
where St is the asset price at time t, μ is the expected return rate (drift), σ is the volatility of the asset, and dWt is an infinitesimal increment of the standard Brownian motion, also known as the Wiener process.
To simulate a Geometric Brownian Motion, we discretize the time interval [0, T] into N equally spaced increments, with $\Delta t = \frac{T}{N}$, and use the Euler-Maruyama numerical method.
Given:
- Initial asset price S0
- Expected return rate μ
- Volatility σ
- Time horizon T
- Number of time steps N
The simulation algorithm proceeds as follows:
1. Generate N independent and identically distributed (i.i.d.) standard normal random variables Z1, …, ZN (e.g., using the Box-Muller method).
2. Compute the asset price for each time step, i.e., ti = iΔt, i = 1, …, N, according to the following recursion formula:
$$S_{t_i} = S_{t_{i-1}} \exp((\mu - \frac{1}{2} \sigma^2) \Delta t + \sigma \sqrt{\Delta t} Z_i)$$
GBM is widely used in financial modeling, most notably in the Black-Scholes option pricing model, because it exhibits some key properties:
1. Continuously compounding returns: GBM’s logarithmic increments are normally distributed and independent, indicating that the returns are log-normal and exhibit continuously compounding nature.
log (ST/S0) ∼ N(μT, σ2T)
2. Martingale properties: Under the risk-neutral probabilistic measure, the discounted asset price process is a martingale, which is a key concept in pricing financial derivatives.
3. Analytical tractability: GBM allows for closed-form solutions for various financial products, including European options, which makes it easier to understand and derive the pricing models.
4. Keeps asset prices positive: By definition, the Geometric Brownian Motion generates positive paths for asset prices, which is a reasonable assumption in finance.
It’s important to note that GBM has its limitations in capturing the complexities of real-world financial markets, such as discontinuous jumps, stochastic volatility, and fat-tailed return distributions. Due to these limitations, more advanced models have been developed, such as the Heston model, the stochastic volatility model, and the jump-diffusion model. Nonetheless, GBM remains an essential starting point for understanding stochastic processes in finance.