Expectation under Q

Yossarian22 · November 17th, 2007, 4:24 am

Can someone explain to me the logic of the second line?How can a random payout at time t2 equal the stock price process at some earlier time?

lesliejinyu · November 17th, 2007, 6:06 am

In the second line, the first equality follows from the law of iterated conditioning. I can not explain the second equality before I know what H and F are? Where do you get the formulae? If F_{t1+} is a sigma-field, i.e. information set, then I do not see any reason we can take expectation of an information set. Could you elaborate a bit on the notation?

Yossarian22 · November 17th, 2007, 5:59 pm

F is not a filtration its the fund valueH = random payout at time tF_{tk^{-}} represents the value of the fund immdiately before reset and F_{tk^{+}} immediately after reset.F_{tk^{+}} = F_{tk^{-}} + H_{tk}.Its a reset option and I read the originally posted equation in an appendix, there is no explaination with it . Yps I know the collector has a closed form soln for one reset, Whaley et al.

lesliejinyu · November 17th, 2007, 6:22 pm

What (discounted) value process is a Q-martingale in this case? I think the second line uses some martingale property such as the discounted value process of the fund is a martingale. Does this help? Why don't you attach the paper here?

Yossarian22 · November 17th, 2007, 9:42 pm

The way I see it is that there is a random process which is the Fund F and there is a random variable H, which is the payoff. Under Q they are both martingales. That means that H can be transformed from a random variable to a random process. Its just not clear to me why the discounted value of H at t2 given the Fund value at time t1 is equal to the discounted value of the Fund at time t1. Unfortunately I can't post the original document as its in a book. Also there is no other explaination given as its in the appendix. Y

lesliejinyu · November 18th, 2007, 6:41 am

to my knowledge, discounted F and H processes are not Q-Martingales but the discounted total gain process is a Q-martingale, i.e. dG = dF + Hdt has a drift rate r under Q measure. (Think of a dividend paying stock price process, under Q it has a drift r-y, where y is the dividend yield and the actual payout is y*S. Substituting back we have dS + y*S dt has a drift rate r.)