September 20th, 2010, 3:48 am
isn't that simply a "forward-starting" call option?as long as the strike is going to be proportional to the underlying (in your case, it is equal to the underlying), then you can use the fact that an option is linearly homogeneous with respect to the spot and the strike. at the time t when the strike is set, the call price will be:C(t)=C(S(t),K=a*S(t),T-t,r,q,vol)=S(t)*C(1,a,T-t,r,q,vol)=S(t)*X where in your case t=T/2, a=1. So at time t the option is worth X shares of stock, then today it must be worth X*S(0)*exp(-q*t), where X=C(1,a,T-t,r,q,vol)you don't really need GBM for this argument - an option should be linearly homogeneous with respect to the spot and strike for other underlying diffusion processes, otherwise our model will give us different prices in different currency denominations