Example: The most popular way to model single stocks is a Geometric Brownian motion (GBM): $$dS_t = \mu S_t dt + S \sigma dW_t$$
- Determine $\mathbb{E}[S_t]$
- Determine $\text{var}[S_t]$
Tip: The MGF $\varphi(t) = \mathbb{E}[\exp(tX)]$ when $X \sim N(a,b^2)$ is given by $\exp(at+ \frac{b^2t^2}{2})$. Use the result for $X = \log(S)$ using Ito's lemma.
I am trying to understand the steps they take to solve the questions but I don't quite understand what they are doing and how they are using Ito's lemma or knowledge about GBM.
- Using Ito's lemma we know that the solution of the SDE describing the share price $S$ as a GBM is: $$S_t = S_0\exp(X)$$ with $$X = (\mu-\frac{\sigma^2}{2})t + \sigma W_t.$$
Hence: $\mathbb{E}[S_t] = S_0\mathbb{E}[\exp(X)]$ with $X \sim N((\mu-\frac{\sigma^2}{2}t, \sigma^2t).$ Using definition of the MGF of $X$: $\mathbb{E}[S_t] = S_0 \varphi_X(1)$ with \begin{align*} \varphi_X(1) &= \exp((\mu-\frac{\sigma^2}{2})t + \frac{\sigma^2t^2}{2})\\ &= \exp(\mu t) \end{align*} Therefore $\mathbb{E}[S_t] = S_0 \exp(\mu t)$.
- For the variation they use $\mathbb{E}[S_t^2] = S_0^2 \mathbb{E}[\exp(2X)]$ which uses what they did in 1. but I also don't understand it there. The rest of this part I do understand.
So what is $\varphi_X(2)$ worth ? $$\varphi_X(2)=exp(2.(\mu-\sigma^2/2)t+2^2.\sigma^2.t/2)=exp(2.(\mu)+\sigma^2.t(2-1))=exp(2t.(\mu)+\sigma^2.t)= exp(2.a +4.b^2)$$ and as $a=\mu.t$ (see below) you get $$b=\sqrt{(2t.\mu+\sigma^2.t -2a)}/2=\frac{\sigma\sqrt{t}}{2}$$
BTW : $$\varphi_X(1)=exp(1.(\mu-\sigma^2).t+1^2.\sigma^2.t/2)=exp((\mu-\sigma^2/2).t+\sigma^2.t/2))=exp(\mu.t)$$ so as $a=\mu.t$