Lognormal expectation problem

Question

A stock's price follows a lognormal distribution. To simulate its price over 10 years, scenarios are generated. In each scenario, the stock price at time t is generated by generating a standard normal random variable Z. Then $S_t$ is set to equal $S_{t-1}e^{.1+.2Z}$. Find the expected value of $S_{10}/S_{0}$.

This how I tried this and I can't find my mistake.

$S_{10} = S_{9}e^{.1+.2Z} = ... = S_{0}e^{1+2Z}$. So this implys that $S_{10}/S_{0} = e^{1+2Z}$. If $Z \sim n(0,1)$ this means that $2Z \sim n(0,2^2).$

$E(S_{10}/S_{0})= E(e^{1+2Z}) = e * E(e^{2Z})$. But $e^{2Z} \sim LN(0, 2^2)$ where $LN$ is the lognormal distribution. So this means $e* E(e^{2Z}) = e \cdot e^{0+.5(2^2)}=e*e^{2}= e^3.$

But this isnt the answer in the solution manual. Here is their solution https://i.stack.imgur.com/Oecpe.jpg

Im hoping someone could point out my mistake, thanks!!

Answer 1

The standard normal random variables are independent and are not the same random variable. In short, $2Z$ and $0.2Z_1 + 0.2Z_2+\ldots_+0.2Z_{10}$ (10 times) are different.

///////////////////////////////

Remark: If $Z_1,\ldots, Z_{10}$ are i.i.d. standard normal random variables, let $X=0.2Z_1+\ldots+0.2Z_{10}$, then $X$ is normally distributed with mean = $0$ and variance = $0.2^2\times 1 + \ldots + 0.2^2\times 1 = 0.4$.

//////////////////////////////

Therefore $\frac{S_{10}}{S_0} = \exp\left(1+X)\right)$ and hence $E(\frac{S_{10}}{S_0})= \exp(1+0+\frac{1}{2}\times 0.4)=\exp(1.2)$.

Answer 2

In fact, the underlying picture is something like that: Let $(\Omega,\mathcal{F},P)$ be a probability space and let $\{\mathcal{F}_{t}\mid t\geq0\}$ be filtration. Let $\{W_{t}\mid t\geq0\}$ be a standard Wiener process adapted to the filtration $\{\mathcal{F}_{t}\mid t\geq0\}$. Suppose that the stock price process $\{S_{t}\mid t\geq0\}$ satisfies the SDE: $$ dS_{t}=S_{t}\left(\alpha dt+\sigma dW_{t}\right), $$ with initial condition $S_{0}>0$, where $\alpha-\frac{1}{2}\sigma^{2}=0.1$ and $\sigma=0.2$.

Then, for $t\geq1$, we have $S_{t}=S_{t-1}\exp\left((\alpha-\frac{1}{2}\sigma^{2})+\sigma\left(W_{t}-W_{t-1}\right)\right),$ where $W_{t}-W_{t-1}$ distributed normally with mean = 0 and variance = 1. Denote $Z_{k}=W_{k}-W_{k-1}$, for $k=1,\ldots,10$. Then $Z_{1},\ldots,Z_{10}$ are i.i.d. standard normal random variables.