How to compute the dynamic of stock using Geometric Brownian Motion?

Question

I have been given the following question:

Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$

For the first part of the question, I have got this answer: $$dS_t = \mu S_tdt + \sigma S_t dWt$$

Is it correct?

And for the second part, I know that the price $f(t,S_t)$ follows the process $$df = (\frac{\partial f}{\partial t}+\mu S_t \frac{\partial f}{\partial S_t}+\frac{1}{2} \sigma ^2S_t\frac{\partial^2f}{\partial S_t^2})dt +\sigma S_t dWt$$

I am having trouble finding the answer using this process and given the information.

Any help is appreciated.

Answer 1

The above equation should correctly read as follows:

$df=\big(\frac{\partial f}{\partial t}+\mu S_t \frac{\partial f}{\partial S_t}+\frac{1}{2}\sigma^2 S_t^2\frac{\partial^2 f}{\partial S_t^2}\big)+\sigma S_t \frac{\partial f}{\partial S_t}dW$

Using:

(a) $\frac{\partial f}{\partial t}=S_t^2f$

(b) $\frac{\partial f}{\partial S_t}=2S_ttf$

(c) $\frac{\partial^2 f}{\partial S_t^2}=2tf+4S_t^2t^2f$

The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:

$\frac{df}{f}=dt \big(S_t^2+2 \mu S_t^2t+\sigma^2S_t^2t+2\sigma^2S_t^4t^2 \big)+2S_t^2t\sigma dW$