RiskBMMR

Description

RiskBMMR generates a Brownian motion with mean reversion process with long-term mean parameter , volatility parameter , speed of reversion parameter , and value at time 0.

Unlike standard Brownian Motion, a mean reverting model will tend toward a long-run equilibrium mean. When the series is above this level, it will tend to decrease, and vice versa. The parameter governs the speed of this reversion, with larger levels implying a quicker reversion.

This process was originally proposed by Vasicek in 1977 as a model for interest rates. It is typically not a good model for stock prices because there is typically no reason to believe that stock prices revert to some long-term mean. However, interest rates cannot rise indefinitely because of economic forces; they tend to revert back to some long-term mean value.

Examples

RiskBMMR(0.01, 0.05, 0.2, 0.015) generates a mean reverting Brownian motion process with long-term mean 1%, volatility 5%, speed of reversion rate 0.2, and value 1.5% at time 0.

RiskBMMR(C10, C11, C12, C13) generates a mean reverting Brownian motion process with parameters taken from cells C10 to C13.

Technical Details

Define = sample from a Normal(0,1) distribution

Then for any , ,

The discrete equivalent of this is

The conditional mean and variance of given , are

as and

as