The geometric Brownian motion (GBM) model is a widely used stochastic process in finance that describes the evolution of a variable over time, such as the price of a stock or other financial instrument. The GBM model is particularly useful for modeling stock prices, where the assumption is made that the stock price changes randomly over time and that the rate of change is proportional to the stock price itself.
Mathematically, the GBM model is expressed as:
dS(t)β=βΞΌS(t)dtβ
+β
ΟS(t)dW(t)
where S(t) is the stock price at time t, ΞΌ is the expected return of the stock, Ο is the volatility of the stock, dt is a small time increment, dW(t) is a Wiener process (i.e., a continuous-time stochastic process that models random fluctuations), and the differential equation dS(t) represents the instantaneous rate of change of the stock price.
In the context of option pricing, the GBM model is used to calculate the expected future value of an underlying asset, such as a stock, that an option is based on. This expected future value is known as the "forward price", and is used to determine the value of the option at expiration.
The Black-Scholes model is a widely used pricing model for European call and put options that uses the GBM model as its underlying assumption. The Black-Scholes model assumes that the stock prices follow a GBM process and that the risk-free rate is constant. Using this model, the value of a call option can be calculated using the formula:
Cβ=βS(t)N(d1)β
ββ
Ke(β
ββ
r(Tβ
ββ
t))N(d2)
where C is the price of the call option, S(t) is the current spot price of the underlying asset, N(.) is the standard normal cumulative distribution function, K is the strike price of the option, r is the risk-free interest rate, T is the time until expiration, t is the current time, and d1 and d2 are calculated using the Black-Scholes formula:
$$\begin{aligned}
d1 &= (ln(S(t)/K) + (r + \sigma^2/2)(T-t)) / (\sigma(T-t)^{1/2})\\
d2 &= d1 - \sigma(T-t)^{1/2}\end{aligned}$$
In summary, the GBM model is a widely used stochastic process in finance that describes the evolution of a variable over time, such as stock prices, and is used extensively in option pricing models like the Black-Scholes model.