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Please help me solve this question. Thank you.

Let the Geometric Brownian motion be: $$ \frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t} $$

$\Delta S$ = change in stock price (s)

$\mu$ = expected rate of return

$\sigma$ = volatility of shock

$epsilon$ has standard normal N(0,1) distribution

$\sigma \epsilon \sqrt{\Delta t}$ = stochastic companion

i) Derive the Ito process with a drift for the above ii) Given that the option price at time $t$ is $f(s,t)$, derive the process with Ito's lemma. Give an example.

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  • $\begingroup$ Welcome to Mathematics Stack Exchange! Users typically shun a straight transcription of homework problems. Please try to augment your question with you efforts so far. A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, use MathJax. $\endgroup$ – dantopa Jun 16 at 6:35
  • $\begingroup$ This depends very much on what level of rigour you're requiring. A 'formal' answer may be out of the scope of the course/notes you're reading. In this case the It\^o process would be the one satisfying the SDE $$ dS_t = \mu S_t dt + \sigma S_t dW_t, $$ i.e. $S_t = S_0 + \int_0^t \mu S_s ds + \int_0^t \sigma S_s dW_s$. $\endgroup$ – Panda Jun 16 at 9:38

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