We can transform the log-optimal pricing formula into a risk-neutral pricing equation. From the log-optimal pricing equation we have

\[P=\mathrm{E}\left(\frac{d}{R^{*}}\right)\]

where $R^{*}$ is the return on the log-optimal portfolio. We can then define a new expectation operation $\hat{\mathrm{E}}$ by

\[\hat{\mathrm{E}}(x)=\mathrm{E}\left(\frac{R x}{R^{*}}\right)\]

This can be regarded as the expectation of an artificial probability. Note that the usual rules of expectation hold. Namely:

(a) If $x$ is certain, then $\hat{\mathrm{E}}(x)=x$. This is because $\mathrm{E}\left(1 / R^{*}\right)=1 / R$.(b) For any random variables $x$ and $y$, there holds $\hat{\mathrm{E}}(a x+b y)=a \hat{\mathrm{E}}(x)+b \hat{\mathrm{E}}(y)$.

(c) For any nonnegative random variable $x$, there holds $\hat{\mathrm{E}}(x) \geq 0$.

Using this new expectation operation, with the implied artificial probabilities, show that the price of any security $d$ is

\[P=\frac{\hat{\mathrm{E}}(d)}{R}\]