Example 1.2.1. Consider the particular three-period model with So = 4, u = 2, and d = 1. We have the binomial "tree" of possible stock prices shown in Figure 1.2.2. Sz(HHH) = 32 S2(HH) = 16 S (H) = 8 S3(HHT) = S(HTH) S3(THH) = 8 So = 4 S2(HT) = S(TH) = 4 S(T) = 2 Sz(HTT) = S(THT) = S3(TTH) = 2 S2(TT) = 1 S(TTT) = .50 Problem 1. Consider a three-period model of Example 1.2.1 on p. 9 with So = 4, u = 2, d = { and interest rate r= , so that pr= . (i) Show that k Sk (1+r) -, k = 0,1,2, 3, is a martingale under the risk-neutral measure. (ii) Let f = f(s) be an arbitrary function of one variable. Find a function g (in terms of the other input parameters and f), such that x [F(x+1)] = g(S) and state such a g=g(s) explicitly. (iii) Is a Markov process under the risk-neutral measure? Justify your answer. Example 1.2.1. Consider the particular three-period model with So = 4, u = 2, and d = 1. We have the binomial "tree" of possible stock prices shown in Figure 1.2.2. Sz(HHH) = 32 S2(HH) = 16 S (H) = 8 S3(HHT) = S(HTH) S3(THH) = 8 So = 4 S2(HT) = S(TH) = 4 S(T) = 2 Sz(HTT) = S(THT) = S3(TTH) = 2 S2(TT) = 1 S(TTT) = .50 Problem 1. Consider a three-period model of Example 1.2.1 on p. 9 with So = 4, u = 2, d = { and interest rate r= , so that pr= . (i) Show that k Sk (1+r) -, k = 0,1,2, 3, is a martingale under the risk-neutral measure. (ii) Let f = f(s) be an arbitrary function of one variable. Find a function g (in terms of the other input parameters and f), such that x [F(x+1)] = g(S) and state such a g=g(s) explicitly. (iii) Is a Markov process under the risk-neutral measure? Justify your