Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2015 / 1 / e / Solution

For the forward-start call option, the strike is fixed by the intermediate stock price. The four path payoffs areThe same final price eight has different payoffs on the two middle paths, so one must retain the history at expiry. At time one, risk-neutral pricing gives $C_1^u=(3/10)(2\cdot 4)=12/5$ and $C_1^d=(3/10)(2\cdot 2)=6/5$. ThusEquivalently, its value at the strike-fixing date is $S_1/5$ by homogeneity, so the martingale property of discounted stock prices gives the initial value $S_0/5$.