ABSTRACT
We have seen that one powerful method of pricing options in the BS-framework, on a dividend paying asset, is via the pair of formulae (1.51), repeated below for convenience:
V (x, t) = e−rτ EQ{F (XT ) | Ft}; XT = x e((r−q)− 1
Essentially, this is the mathematical statement of the FTAP applied to the BS economy. The function F (XT ) is the payoff of the derivative at expiry T , and the expression for XT is the random asset price at time T , given its current value x at time t < T , as seen under the EMM, Q. This random price depends only on a single Gaussian random variable Z, although for each t there will be a different such Z.
