Geometric Brownian Motion

Definition

Geometric Brownian motion (GBM) is a stochastic process that models the evolution of financial prices over time, characterized by continuous paths and the properties of Brownian motion. This model is widely used in finance, particularly for stock price modeling, as it incorporates both the deterministic trend and the random fluctuations in asset prices, making it essential for understanding various financial applications.

5 Must Know Facts For Your Next Test

1. GBM assumes that the logarithm of asset prices follows a normal distribution, which allows for continuous compounding of returns.
2. The mathematical representation of GBM is given by the stochastic differential equation: $$dS_t = \mu S_t dt + \sigma S_t dB_t$$, where $\mu$ is the drift, $\sigma$ is the volatility, and $B_t$ is a standard Brownian motion.
3. In GBM, asset prices cannot become negative, making it a more realistic model for financial applications compared to other processes that allow negative values.
4. The parameters of drift and volatility are typically estimated from historical price data, making GBM adaptable to various assets and market conditions.
5. GBM is foundational for the Black-Scholes model used in option pricing, which assumes that stock prices follow this specific stochastic process.

Review Questions

• How does geometric Brownian motion differ from standard Brownian motion in terms of its application to financial modeling?
  • Geometric Brownian motion extends standard Brownian motion by incorporating both a deterministic trend (drift) and randomness (volatility) into the modeling of asset prices. While standard Brownian motion has paths that can be anywhere on the real line, GBM ensures that asset prices remain positive and follows a log-normal distribution. This makes GBM particularly suitable for financial contexts where prices cannot go negative and helps better capture market behavior.
• Discuss how Ito's Lemma is applied in deriving properties of geometric Brownian motion.
  • Ito's Lemma plays a critical role in deriving the behavior of functions involving stochastic processes like geometric Brownian motion. It allows us to compute the changes in functions of GBM by taking into account the drift and volatility components. By applying Ito's Lemma to functions of the price process, we can derive important results such as pricing formulas and risk management strategies relevant in finance.
• Evaluate how geometric Brownian motion contributes to modern financial theories and practices, particularly in option pricing models.
  • Geometric Brownian motion serves as a cornerstone for many modern financial theories and practices, most notably in option pricing models like Black-Scholes. By assuming that stock prices follow GBM, these models can derive closed-form solutions for pricing options based on expected future movements of asset prices. This incorporation of both drift and volatility allows practitioners to make informed decisions regarding hedging strategies and risk assessments. The success of these models highlights the importance of GBM in linking theoretical finance with practical application.

Related terms

Ito's Lemma:

A fundamental theorem in stochastic calculus that provides a way to find the differential of a function of a stochastic process, crucial for working with processes like GBM.

Drift and Volatility: Drift refers to the expected return of an asset over time, while volatility measures the degree of variation in asset prices; both are key parameters in modeling GBM.

Risk-neutral Valuation: A pricing approach in which all investors are assumed to be indifferent to risk, allowing for simpler calculations of expected asset values under GBM.