Relating path statistics to terminal distributions for a class of positive martingales.

Abstract: We consider a class of strictly positive martingales satisfying \$dS(t) = A(t)S(t)dW(t)\$ where \$W\$ is a Brownian motion and the process A is a stochastic process independent of W.  We derive various results of the form \$E(f(S(T))) = E(F[S])\$, where \$E\$ denotes expectation, \$F[S]\$ is a functional of the path of \$S\$ (e.g., possibly depending jointly on the running maximum, running minimum and quadratic variation of \$S\$) and the function \$f\$ is determined from \$F\$.  From a financial standpoint, these results allow us to determine the value of a path-dependent claim on a stock \$S\$ relative to value of a path-independent European claim. This is joint work with Peter Carr and Roger Lee.