binomial representation for pricing stock options

Exercise on Binomial Representation March 21, 2021 A stock process S follows a binomial tree. Let S; = stock price at time i. Si is a random variable with two possible values SH and Swith corresponding risk neutral probabilities qi and 1 qi. Let Fi be the stock history up to time i, X a claim on S at time T and let Ni = EQ(X|F) Then N; is also a random variable with possible values NP and N! a) Express Si as a linear combination of S and S b) Express N; as a linear combination of N and N c)What are the coefficients of the hedge portfolio Ci+1 and Vi+1 at time i? Recall that these are chosen so that N; = 0i+1S; + Vi+1 It follows that Ni+1 = 0i+1 Si+1 + Vi+1. Why? Justify this by using the Binomial Representation Theorem The next hedge portfolio Pi+2 Si+1 + $i+2 has the same value Ni+1 What property of the hedge portfolio does this express? Which of the fol- lowing variables are known at time i? Si, Si+1,0i+1, Vi+1 ,Ni, Ni+1 n! and N! were so named because they correspond to SP and S! Find an example of a claim X where sometimes N! <>