July 6th, 2002, 5:18 am
there is any reasonable explanation for a negative vega of an Asian option, coming out from a Monte Carlo simulation pricing function?More generally, I' m wondering if there is a specific literature on the vega for Asian Option >>I vote for the vega of the (continuous) arithmetic average option, with a put or call payout, alwaysbeing positive (well non-negative to be really picky).I haven't done this carefully, but I believe here's how you prove it. You start withthe PDE developed by Roger's and Shi. For example, for the Asian call option, they show C = S f(x,t),where 0 = df/dt + (1/2)sig^2 x^2 (d^2 f/dx^2) - (r x + 1/T) df/dx with terminal conditionf(x,T) = -min[x,0]. Also x= (K-Sbar)/S, where K=Strike and Sbar is the average-to-date for a seasoned optionand 0 for a fresh option.Anyway, what you want to prove is that g(x,t) = df/dsig >= 0. To do that, you basically follow thetype of argument used in, for example, Romano & Touzi (1997, Math. Finance, 7, 399-412).Briefly, the method is to differentiate the PDE with respect to sig. This gives you a new PDEfor g. You write a Feynman-Kac style solution for that PDE, which shows that g(x,t) has the same sign as d^2f/dx^2.Then you go through the same exercise for h(x,t) = df/dx and k(x,t) = d^2f/dx^2, getting PDEs for them. It finallyboils down to d^2f/dx^2 being non-negative for all (x,t) if the payoff is convex (which it is for puts/calls).Hence, vega must be non-negative. Regards,Alan