# Expectation of the negative exponential utility function for a Grossman and Miller model

I have the standard $$3$$-period Grossman and Miller model with $$2$$ outside traders and $$M$$ market makers.

I'm told:

• $$W_t^{(1)}, W_t^{(2)}, W_t^{(m)}$$ is the wealth of the first outside trader, second outside trader and market maker at time $$t$$.
• $$B_t^{(1)}, B_t^{(2)}, B_t^{(m)}$$ is the cash of the first outside trader, second outside trader and market maker at time $$t$$.
• $$x_t^{(1)},x_t^{(2)},x_t^{(m)}$$ is the shares of the security of the first outside trader, second outside trade and market maker at time $$t$$.
• $$\tilde{p}_t$$ is the unknown security price at time $$t$$ that follows a normal distribution.

Let $$x$$ be the number of share holdings, $$\tilde{p} \sim N(\mu, \sigma^2)$$ be the security price and $$W=x \cdot \tilde{p}$$ be the wealth. Show that $$E(U(W)) = -e^{(((\lambda^2\sigma^2)/2)x^2)-\lambda\mu x}$$

So I know that the negative exponential utility function is: $$U(W)=-e^{-\lambda W}$$, in a previous part of this question I formulated the utility optimization problem for the first outside trader, second outside trader and market maker, but I'm not sure if these help (if they will be of use I can upload them).

This is where I'm stuck, I'm not sure how I can find $$E(U(W))$$ given what I know and what is provided in the question.

Any help would be greatly appreciated, even if it's only a hint to get me started.

### Deriving the expected utility comes from an application of the moment generating function of a Normal random variable:

Let $$U(W)=-e^{-\lambda W}$$ and $$W=x \cdot \bar{p}$$ where $$\bar{p} \sim N(\mu, \sigma^2)$$ be given as above and then further note that $$W$$ is Normally distributed with $$W \sim N(\mu \cdot x, x^2 \sigma^2)$$.

See that:

\begin{align} \mathbb{E}\left[U(W)\right] &= \mathbb{E}\left[-e^{-\lambda W}\right]\\ &=-\mathbb{E}\left[e^{-\lambda W}\right], \end{align}

is very similar to the m.g.f. of a Normal random variable.

Remember that the moment generating function of a Normal random variable with distribution $$x \sim N(\mu, \sigma^2)$$ is given by (see here or here for formula):

$$\mathbb{E}\left[e^{tx}\right] = e^{t\mu+\frac{\sigma^2t^2}{2}}$$

Continuing from above, you can apply the moment generating function to the expected utility, which yields the desired solution:

\begin{align} \mathbb{E}\left[U(W)\right] &= \mathbb{E}\left[-e^{-\lambda W}\right]\\ &=-\mathbb{E}\left[e^{-\lambda W}\right]\\ &=-e^{-\mu x\lambda + \frac{x^2\sigma^2\lambda^2}{2}}\\ &=-e^{\frac{\lambda^2\sigma^2x^2}{2} - \lambda\mu x}, \end{align} where I have realigned everything in the last equation, in order to fit the solution in your question.

• @CharlieP Hi. Feel free to comment on my solution. Did my answer help or is it not what you're looking for? I am always willing to explain more in detail, if that is what you need.
– Pleb
Commented Jan 13, 2022 at 7:18
• This is perfect, thanks a lot Commented Jan 13, 2022 at 18:41
• Glad I could help :-)
– Pleb
Commented Jan 13, 2022 at 18:51