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Assume the risk-free bond Bt and the stock St follow the dynamics of the Black & Scholes model (with interest rate r, stock drift $\mu$ and volatility $\sigma$).

Calculate the price at time $t = 0$ of a derivative with maturity T and payoff $(S^3_t-K)^+$. I know I need to use the Black Scholes formula for price of a call to find the price of the derivative but the formula also contains $N(d_1)$ and $N(d_2)$ so how would this get affected?

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  • $\begingroup$ I don't get the question. Which process is defined by $e^{\beta t}S_t^3$? $\endgroup$ – Sanjay Sep 28 '19 at 15:33
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I don't understand the question but I can try. I think the problem is to find the price of a contingent claim that has payoff $(S_T^3 - K)^+$. The well-known pricing formula is: \begin{equation} \pi(t)=\mathbb{E}^\mathbb{Q}[e^{-r(T-t)}(S_T^3 - K)^+|\mathcal{F}_t] \end{equation} Now put $Y=S^3$, by using Ito's Lemma \begin{equation} dY(t)=dS^3(t)=3S^2(t)dS(t) + \frac126S(t)\sigma^2S^2(t)dt \end{equation} In Black-Scholes model \begin{equation} dS(t)=\mu S(t) dt + \sigma S(t) dW(t) \end{equation} So we have: \begin{equation} dY(t)=3\mu S^3dt + 3\sigma^2S^3dt + 3\sigma S^3dW=(3\mu + 3\sigma^2)Ydt + 3\sigma YdW \end{equation} Now we define \begin{align} \tilde{\mu}&=3\mu + 3\sigma^2 \\ \tilde{\sigma}&=3\sigma \end{align} Now suppose $Y$ is a new stock with drift $\tilde{\mu}$ and volatility $\tilde{\sigma}$ and just substitute in the Black-Scholes formula for an european option with underlying $Y$ and strike $K$.

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  • $\begingroup$ Hi sorry, I edited the quesiton now.@ab94 $\endgroup$ – Anon Sep 28 '19 at 21:42

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