Discuss the approach to ‘Geometric Brownian Motion’ with its applications to the Mean Reversion Models.

Geometric Brownian Motion (GBM):

Geometric Brownian Motion is a stochastic process widely used in finance and mathematics to model the behavior of continuous-time, continuous-state variables. It's a mathematical model that describes the random motion of a variable over time. In finance, GBM is commonly used to model the price movement of assets, particularly in the context of option pricing and risk management.

The GBM model is defined by the following stochastic differential equation:

��=����+����dS=μSdt+σSdW

Where:

• ��dS is the change in the asset price.
• �μ is the drift rate, representing the expected rate of return of the asset.
• �σ is the volatility, representing the standard deviation of the asset's returns.
• ��dW is a Wiener process (Brownian motion), representing random shocks.

Applications of GBM to Mean Reversion Models:

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