Monte Carlo Greeks: Smoothed Perturbation Analysis help

[Keval]

Hi everybody,

I am trying to derive a gamma estimator using smoothed perturbation analysis (SPA)(Smooth Infinitesimal Perturbation Analysis, smooth pathwise derivative), as discussed briefly in Glassermans [2004] book, page 399-400. I have uploaded the pages I am concerned with.

I am trying to apply it to basket options, in particular a geometric average basket option ( in contrast to arithmetic average).

Essentially the estimator derived is for the delta for a digital call option. But it also works for the Gamma of a European call, with the strike(call option)=discontinuity(digital option).

Looking at photos 1 and 2:
I would have thought that the delta estimator of a european call option is the digital call payoff multiplied by dS(T)/dS(0)=S(T)/S(0) (using chain rule argument, as seen in photo 3)(but Glasserman seems to miss this bit out).
. And then to find the gamma through SPA, you take the second derivative dS(T-epsilon)/dS(0) * d/dS(T-epsilon). But I want to to know formally why you use the digital call payoff, in contrast to the pathwise delta estimator for a european call option when takign the second derivative to find the gamma estimator.

This is important as you have a random variable on the outside of the indicator function, which makes the derivation a fair bit more involved. I have done some long derivations, but none of them are correct( I have tested them out).

Any help will be greatly appreciated.

Thank You.

 

[Keval]

If anybody wants to know, I have solved the computational issue( I am getting the gammas, and stuff correctly, but I still don't understand why my derivation works.