Expectation from Musiela's Book

DCH · June 23rd, 2010, 3:54 pm

Hi Folks,I'm reading the section on Lookback options from Martingale Methods in Financial Modelling by Marek Musiela and Marek Rutkowski, and have got stuck on one expectation:This is on p. 240 of my copy (2nd Ed, 3rd Printing). I just can't see how this works - can anyone explain, step by step, how the last expression follows ?Aside: Seems that \mathbb and \mathds are not supported in the equation editor here - I hope the meaning is clear nonetheless. "1" in the first expression is the indicator function.Many thanks,David.

Alan · June 23rd, 2010, 4:34 pm

I haven't checked it all the way through, but what you should try is:Write the lhs expectation asint(z,infty) (e^(-y) - e^(-z)) d/dy P(M < y) dy.Then, just integrate by parts, and of course use P(M < y) = 1 - P(M > y)

DCH · June 25th, 2010, 2:34 pm

Thanks for the reply, Alan.I think this is in the right direction, but I'm not sure your integral gives what we want. int(z,infty) e^-y = e^-z so we lose any integral of e^-y which appears on the RHS of Musiela's expression.

Alan · June 26th, 2010, 1:42 pm

int u dv = u v - int v du; u = e^(-y); dv = [d/dy P(M < y)] dy

DCH · June 29th, 2010, 1:42 pm

Forgive me, Alan. What you say is an absolute load of correctness! It comes out in about 6 or 7 lines.You would think authors would put occasional hints in about how expressions come about, or perhaps I'm just more stupid than everyone else ;-)Thanks,David.