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Let $S$ be the value of a stock that evolves according to $dS$=$\mu$$Sdt$ + $\sigma$$SdB$. A contract has a payoff at expiration of $V_T$=$(S_T)^3$. What is the value of $V_0$ of the contract at $t=0$?

Express your answer in terms of $r, T, \sigma, \mu$ and $S_0$

I tried to tackle this problem using Ito's Lemma but it doesn't seem to give me the right answer. Is there any way I can use $r$ as the risk-neutral rate?

Hector Lombard
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  • Hint: write down the explicit solution of $S_T$. It is a exponential (geometric Brownian motion). Then calculate $E[e^{-rT}S_T^3],.$ – Kurt G. Aug 24 '21 at 16:59
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    How do I get the explicit solution? Is it through Ito's Lemma or is there some other special method? – Hector Lombard Aug 24 '21 at 18:38
  • The explicit solution of $dS=\mu S,dt+\sigma SdB$ is well-known: $S_T=S_0\exp(\sigma B_T-\sigma^2T/2+\mu T),.$ It can be verified using Ito's lemma. – Kurt G. Aug 24 '21 at 18:50

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