April 29th, 2012, 10:57 am
I am assuming spread = outperformance option.So payoff is Max[A-B,0]simplify this to B x Max[A/B-1,0]Now treat A/B as a single asset which will have a volatility ofsigmaA/B^2 = sigmaA^2 + sigmaB^2 - 2 correlation(A,B) sigmaA sigmaB (From Ito's)Max[A/B-1,0] is your black scholes formula with asset as A/B (let's say time t) instead of a stock, but the price is in B's unit...which I can convert to dollars by using B(t)(which is $ price) x Price of blackscholes payoff of Max[A/B-1,0] Similarly think of hedging in two steps1. Hedge with A in terms of B2. Hedge B in terms of dollar (or any currency) - (similar to a quanto actually....)Quantities might be different but finally A * change in price / change in A + B * change in price / change in B = Priceabove equality should hold. Or intuitively think in these termsA is at 130 and B = 2 or very small valuethen your price is Max[130 - 20,0] is mostly dependent on volatility of A and not on volatility of B. And if B's volatility is very small it might behave like a call option ITM with strike 20. Of course this is a very very crude example. But it's another way of seeing why quantity of A,B might be different in hedging.