Geometric Brownian Motion - Use of GBM in Finance

Use of GBM in Finance

Geometric Brownian Motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior.

Some of the arguments for using GBM to model stock prices are:

• The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality.
• A GBM process only assumes positive values, just like real stock prices.
• A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices.
• Calculations with GBM processes are relatively easy.

However, GBM is not a completely realistic model, in particular it falls short of reality in the following points:

• In real stock prices, volatility changes over time (possibly stochastically), but in GBM, volatility is assumed constant.
• In real stock prices, returns are usually not normally distributed (real stock returns have higher kurtosis ('fatter tails'), which means there is a higher chance of large price changes).