Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / B / 5

Put $\sigma =\sqrt{{\sigma }_1^2+{\sigma }_2^2}>0$. The linear combination $W_3$ starts at zero and has continuous paths. An increment over an interval of length $h$ is a linear combination of independent centred normal random variables, so is centred normal with varianceIncrements on disjoint intervals are independent, since they depend on disjoint increments of the two independent Brownian motions. These are the defining properties, so $W_3$ is a standard Brownian motion.

Cubing the supplied geometric Brownian motion solution gives $S_t^3=s^3\exp ((3\mu -3{\sigma }^2/2)t+3\sigma W_3(t))$. The moment-generating function of a normal distribution yields $\mathbb{E}[e^{a{W}_3(t)}]=e^{a^{2}t/2}$. Therefore

For this stock-dependent payoff, replace the stock drift by $r$ under a risk-neutral measure. Conditional lognormal moments then give, with $\tau =T-t$,This is the Power payoff in the Black-Scholes model. The delta hedge is ${\Delta }_t=\partial V/\partial S=3V_t/S_t$ shares, so the stock holding is worth $3V_t$. The rest is a bank position worth $-2V_t$, or $-2V_t/B_t$ units of the bank asset. This gives a replicating portfolio; its self-financing identity follows from Itô's formula and the Black-Scholes pricing equation, and its terminal value is $S_T^3$.

The full two-Brownian filtration contains an additional orthogonal noise direction, but this particular payoff is replicated using the traded stock and bank asset, so that extra noise does not affect its price.

Apply Itô's formula to $V_t=A(t)S_t^3$, where $A^{\prime }(t)=-(2r+3{\sigma }^2)A(t)$. Since $d{S}_t/S_t=\mu dt+\sigma d{W}_3(t)$, the drift and diffusion areThusUnder the risk-neutral measure its drift becomes $3r-2r=r$, as required for a traded, non-dividend-paying replicating asset.

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.