Assign2 - STAT/ACTSC 446/846 Assignment #2 (due October 28,...

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STAT/ACTSC 446/846 Assignment #2 (due October 28, 2009) (1-3) Problems from the book “Financial Economics . ..”: 5.2 (do this by using the fundamental theorem of asset pricing, not graphically as suggested), 5.5, and 5.8. (4) First we introduce definition of a predictable process. Suppose that we are given a filtration {F n } n 0 . A stochastic process { X n } n 1 is called {F n } n 0 - predictable if X n is F n - 1 - measurable for all n 1. Prove that a predictable martingale is constant. (5) Suppose that { X n } n 0 is adapted to the filtration {F n } n 0 and that { φ n } n 1 is {F n } n 0 - predictable. Define a new process Z n = Z 0 + n - 1 X j =0 φ j +1 ( X j +1 - X j ) , where Z 0 is a constant. Show that if { X n } n 0 is a martingale with respect to the filtration {F n } n 0 then so is { Z n } n 1 . (6) (This is Problem 6.5 from the “Financial Economics” book) A European derivative pays the square of the asset price in 3 month’s time. The current price is 20, and the asset price has a
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