Risk Neutral Pricing and Financial Mathematics: A Primer

Here, we have defined the Radon–Nikodym derivative for our change of measure:

$\frac{\mathrm{d}\mathrm{\mathbb{Q}}}{\mathrm{d}\mathrm{\mathbb{P}}}\left(x\right)=\xi \left(x\right)={\mathrm{e}}^{-\mathit{\mu }x+\frac{{\mathit{\mu }}^2}{2}}$ (6.7)

(6.7)

This change of measure allows us to take a random variable that had a nonzero mean with respect to one probability space and convert the probability space so that the random variable has a zero mean with respect to the new probability space, while leaving its variance unchanged. After illustrating and generalizing this change, we will perform a similar change of measure on a Brownian motion process.

Illustration: Change of Normal Density

Suppose that we were to begin with a random variable X with mean μ under probability measure , and ...