 Solution

For a call option on a power payoff, the replicating asset $V$ from part (c) has $V_T=S_T^3$, volatility $3\sigma$, and risk-neutral drift $r$. The new payoff is therefore a European call option on $V$, with strike $K$. Applying the given Black-Scholes formula givesHere $V_t=S_t^3{e}^{(2r+3{\sigma }^2)\tau }$. Equivalently $D_1=[\log (S_t^3/K)+(3r+15{\sigma }^2/2)\tau ]/(3\sigma \sqrt{\tau })$. It is the price process $V_t$, rather than the raw power $S_t^3$, that serves as the self-financing underlying for this application.