Fall 2023 MATH 5474 Financial Calculus Homework #3 Due: September 21, 2023, 10:30am Problem #1 (10 pts). Toss a coin repeatedly, with the probability of head on each toss is }, as is the probability of tail. Let X, = 1 if jth toss is a head and X, = -1 if jth toss is a tail. Define Mo = 0 and Me = XA ISnSN. (1) J=1 This process is called a symmetric random walk. It steps up one with each head, and steps down one with each tail. 1. Show that the symmetric random walk { Mo, M1, ... . My ) is a martingale. 2. Let & be a positive constant. Define T,, = k. for n = 0, 1. ... N. This is the so-called geometrical symmetric random walk. Show that {Go. GI. ..., Gx} is not a martingale. 3. Next we "discount" the geometrical symmetric random walk by defining S, = (TH ) Gn for n - 0. 1. . .. . N. Show that the discounted the geometrical symmetric random walk (50. S1, ... . Sy} is a mar- tingale