Answer Happy

3.1. A stock in the three-period binomial model satisfies So 4, S₁ (H) = 8, S₁ (T) 2, and r = 0.25. You wish to price an up-and-out call with barrier value 15 and strike price 5. This call is priced as a standard European call, except that the option dissolves (leaving the holder of the option with nothing) if the stock price ever meets or exceeds 15. = = Work out the value tree for this option and determine whether or not the pricess (Vo, V₁, V2, V3) is a Markov process in the risk-neutral measure. Here v = 1/(1+r) is the one-period discount factor for the risk-free rate.
3.2 In the same three-period model as the first problem, you have an asset that pays out Cn Sn in periods n = : 0, 1, 2, 3. We are going to price this asset and determine what sort of (stochastic) difference equation it satisfies. = (a) For n = 0, 1, 2, 3, work out the value tree for the asset that pays out Cn at time n. This will mean writing out four separate trees, with 0, 1, 2, and 3 periods respectively. You can do this numerically using the data from the prior part, or you may give answers symbolically in terms of the stock price process or other variables in the model. (b) Add the terms in these trees state-by-state, to get a single three-period tree. Denote that resulting stochastic process as (Yo, Y₁, Y2, Y3). This is supposed to be the value process of the cash flow introduced at the start of this problem. (c) Verify that (vYn) is not a Martingale process in the risk-neutral measure. (d) Verify that this does satisfies the following recursion Y3 C3 and Yn - Cn = v(pYn+1(H) + ãYn+1(T)) for n = 0,1,2 (e) Suppose I lumped together all of the replicating portfolios for each of the securities given in part (a) into a single portfolio. Do you think that this would give a replicating portfolio for the overall cash flow? How do you think this portfolio relates to the process given in part (b)? Note: you will receive full credit for this part for writing something reasonable and relevant to the problem, so you do not need to supply a full mathematical analysis here.