RiskGBM

Description

RiskGBM generates a geometric Brownian motion (GBM) process with location parameter , volatility parameter at time 0.

A geometric Brownian motion is a continuous-time stochastic process in which the logarithm of the series follows a Brownian motion, also called a Wiener process. In a financial context, the series is typically the price of a security, which is lognormally distributed. In this case, the “log return” of the series, essentially the change in the price, is normally distributed.

The RiskGBM function requires an exponential transformation and first order integrations.  This is done by using the RiskTSTransform and RiskTSIntegrate property functions.

Examples

RiskGBM(0.01, 0.05, RiskTSTransform(1,0), RiskTSIntegrate(1,1)) generates a GBM process with drift 1% and volatility 5%.

RiskGBM(C10, C11, RiskTSTransform(1,0), RiskTSIntegrate(1,1)) generates a GBM process with parameters taken from cells C10 and C11.

Technical Details

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

Then for any

, ,

The discrete equivalent of this is

The conditional mean and variance of given , are

and

If this is in a financial context and is the price of a security at time t, then the term inside the square brackets in the equation for , the “log return” of the security, is normally distributed with mean and variance .