Geometric Brownian Motion for Asset Prices

Understanding Geometric Brownian Motion

Geometric Brownian Motion is defined by the following stochastic differential equation:

$dS_t = \mu S_t dt + \sigma S_t dW_t$

Where:

• $S_t$ is the asset price at time t
• $\mu$ is the drift (expected return)
• $\sigma$ is the volatility
• $dW_t$ is a Wiener process (standard Brownian motion)

The solution to this equation gives the asset price at any future time:

$S_t = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)$

Properties of GBM in financial markets

1. Continuous paths: Asset prices follow continuous trajectories without jumps
2. Proportional returns: Price changes are proportional to the current price
3. Log-normal distribution: Future prices follow a log-normal distribution
4. Independent increments: Price changes are independent of past movements

These properties make GBM particularly suitable for modeling liquid financial markets where price changes are relatively smooth.

Applications in derivatives pricing

GBM serves as the foundation for many financial models, most notably the Black-Scholes Model for Option Pricing. The assumption of geometric Brownian motion enables:

1. Option pricing formulas: Closed-form solutions for vanilla options
2. Delta hedging: Dynamic hedging strategies based on continuous price paths
3. Risk metrics: Calculation of the Greeks Delta Gamma Theta Vega Rho

Limitations and extensions

While GBM provides a tractable framework for financial modeling, it has several limitations:

1. Constant volatility assumption: Does not capture volatility clustering
2. No jumps: Cannot model sudden price changes
3. Tail behavior: Underestimates extreme events

These limitations have led to more sophisticated models such as:

• Jump-Diffusion Models Merton's Model
• Heston Model for Stochastic Volatility

Risk management implications

Understanding GBM is crucial for:

1. Portfolio optimization: Modeling expected returns and risks
2. Value at Risk: Computing probabilistic risk measures
3. Scenario analysis: Simulating potential price paths

The mathematical framework of GBM enables the calculation of key risk metrics like Value at Risk VaR Models and supports Monte Carlo Simulations for Risk Estimation.

Implementation considerations

When implementing GBM in trading systems:

1. Parameter estimation: Robust methods for estimating drift and volatility
2. Discretization: Appropriate time steps for numerical simulation
3. Random number generation: High-quality random number generators for Monte Carlo

Market microstructure considerations

At shorter time scales, price movements may deviate from GBM due to:

1. Bid-ask bounce: Price oscillation between bid and ask
2. Market impact: Large trades affecting price dynamics
3. Tick size: Discrete price levels

These effects are particularly important for High-Frequency Trading Risk management and Market Microstructure analysis.