Applying the Gaussian Shift Theorem to Correlated Normal Variables

Summary

The note shows how to use the Gaussian shift theorem to evaluate an expectation involving two correlated standard normal variables. It first states the theorem: an exponential tilt inside an expectation can be converted into a shifted argument of the function, multiplied by a normalizing exponential factor.

To apply it in the bivariate case, the second normal variable is decomposed into a component proportional to the first and an independent Gaussian component. Independence factors the original expectation into two simpler expectations, to which the theorem can then be applied. This gives a general calculation route for Gaussian expectations that arise in quantitative finance, including option pricing. The note outlines the setup rather than carrying the calculation through to a final closed-form value, and assumes the stated standard-normal decomposition and correlation structure.

Key ideas

• The Gaussian shift theorem converts exponential weighting into a shifted function argument.
• A correlated bivariate normal can be represented using one variable and an independent normal component.
• The decomposition separates the expectation into factors through independence.
• The theorem can then be applied to each resulting Gaussian expectation.

Tags

# Showing the Gaussian shift theorem for bivariate case


# Showing the Gaussian shift theorem for bivariate case












I was reading about the Gaussian shift theorem in "An Introduction to Exotic Option Pricing" by Peter Buchen and came across a question that I can't seem to figure. In the book, he uses F(Z) (a measurable scalar function of Z, Z being Gaussian rv with a normal variate) but the function doesn't appear in the question and rather just uses Z1 and Z2.

where 1D is the univariate Gaussian distribution and GST is the Gaussian shift theorem

Any help would be much appreciated.

## Answer by Gordon (score 1)

https://quant.stackexchange.com/a/41627

The Gaussian Shift Theorem says that, for a standard Gaussian random variable $Z$, constant $c$, and function $F$, we have the expectation \begin{align*} E\left(e^{cZ} F(Z) \right) = e^{\frac{1}{2} c^2}E\big(F(Z+c) \big). \end{align*} Given the decomposition of $Z_2=\rho Z_1 + \sqrt{1-\rho^2} Z$, where $Z$ is independent of $Z_1$, \begin{align*} E\left(Z_1 e^{a Z_2} \right) &= E\left(Z_1 e^{a \rho Z_1 + a \sqrt{1-\rho^2}Z } \right)\\ &=E\left(Z_1 e^{a \rho Z_1}\right) E\left(e^{a \sqrt{1-\rho^2}Z } \right). \end{align*} Now, you can apply the Gaussian Shift Theorem to compute each of them.

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