Advanced econ.

1. (5 points) You are working in the product development department of an insurance company that sells equity-linked annuities. Your manager assigns you to investigate exotic options. (a) (1 point) Describe three common differences between exotic options and standard options. Assume that X" and Y, are stochastic processes satisfying the following stochastic differential equations: dX, = /X,di +X, (a dw()+ Baw(2), X, =1 dY, = /,Y,di + Y, (B aw," + yaw(3), Y. =1 where /, My, a, P, Y are positive constant parameters, W," and WE are two independent standard Wiener processes. Let B = ( a + B ) W."+ (8 +y ) W(2) V(a + B ) + ( B+7 ) S, = VX,Y, (b) (1.5 points) Derive the stochastic differential equation of S, . (c) (1.5 points) Show that B, is a standard Wiener process. (d) (/ point) Derive the condition on the parameters under which S, is a martingale with respect to the filtration generated by B