Geometric Brownian Ito Drift

Worked Example

Consider a variable S following the stochastic differential equation (SDE):
dS = μ * S * dt + σ * S * dW

Where S0 = 1, μ = 0, σ = 0.5, and T = 1. Let us calculate the statistics at terminal time T.

We calculate the exact expected result as follows:

1. Log Drift: The drift of the logarithm of S is given by μ - σ² / 2. Here, 0 - (0.5)² / 2 = -0.125.
2. Mean: The expected value is E[S] = S0 * e^(μ * T). Here, 1 * e^(0 * 1) = 1.0.
3. Median: The median is given by S0 * e^((μ - σ² / 2) * T). Here, 1 * e^(-0.125 * 1) ≈ 0.8824969025845955.
4. Variance: The variance is (S0²) * e^(2 * μ * T) * (e^(σ² * T) - 1). Here, 1 * e^(0) * (e^(0.25) - 1) ≈ 0.2840254166877415.

Experiment

Explore the effects of drift and volatility on the lognormal distribution of the terminal level. Open the Geometric Brownian Ito Drift module to interact with the parameters.

1. Set Drift to 0.5 and observe how the Mean and Median shift to higher values. The Mean will be greater than the Median due to the right-skewness of the lognormal distribution.
2. Increase the Volatility to 1. Notice how the gap between the Mean and Median widens significantly. High volatility pulls the median down relative to the mean, reflecting the negative -σ²/2 term in the log drift.
3. Change the Normal coordinate to 1 or -1 to see how a typical 1-standard-deviation shock affects the Terminal level relative to the median.

Limitations

This model assumes that the drift and volatility parameters (μ and σ) are constant over time. In real-world scenarios, such as financial markets, volatility often changes over time (volatility clustering) and may depend on the level of the variable itself (local volatility). Additionally, the model assumes that logarithmic returns are normally distributed, which implies that extreme events (tail risks) are extremely rare. Empirical data often show "fat tails" where extreme events occur more frequently than predicted by the lognormal distribution.

This educational visualization focuses on the analytical properties of the terminal distribution. A fixed z * √t curve does not represent a true Brownian sample path, as real paths are nowhere differentiable and require independent increments. This module is an ideal continuous stochastic model and does not constitute investment recommendations.

References

• MIT OpenCourseWare: Engineering Economy Module (Fall 2009), Lecture 3

  Discusses Geometric Brownian Motion and its properties in the context of modeling stochastic processes. This provides a deep explanation of how drift, volatility and time contribute to the continuous diffusion and distribution limits observed in the simulation.