# Calculate the price at time t=0

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?

• I don't get the question. Which process is defined by $e^{\beta t}S_t^3$? – Sanjay Sep 28 '19 at 15:33

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 $$K$$.