The multi-period binomial model - This is a generalization of the one-period binomial model. - A series of times: 0,T,2T,,NT=T. - Two securities: 1. Stock. 2. Bond. - Interest rate ri during time interval [iT,(i+1)T] for the bond. - A binomial tree of possible states for the stock prices, that is for each stock price at time jT, there are two possible values it can take at the next time (j+1)T. The multi-period binomial model Figure: Iwo-period binomial model. - A European call option at time t is worth at least as much as a forward c(St,Tt,K)StKer(Tt); see slides from week 1. - Exercising an American option now will give StK. - If r>0 then StKer(Tt)>StK, and exercising the option now is suboptimal. Intuition: - If we exercise now, we have to pay the K. - If r>0 we would rather wait until maturity and pay K then. - The risk-neutral expectation of discounted stock remains the same. 2. This question concerns pricing an American call option on a dividend paying stock in a two-period multiplicative binomial model. The stock will pay a dividend at rate of at the end of the second period. That is if the price of the stock pre-dividend payment is S2, then post-dividend payment the stock price would be (1)S2 and the holder would receive S2 in cash. Assume also, that the dividend payment happens at time T=2T, but before the option holder can exercise his option. Namely, if the option holder has not exercised his option before time T, then exercising the call option at time T, will entitle him to payoff of ((1)S2K)+, that is gives him the right to buy the stock ex-dividends for the price of K. Let the multiplicative two-period binomial model have the initial stock price S0=4, with constant annual continuously compounded interest rate r, and the factors u=2,d=21, and time step T=1, with dividend payment rate of =41. a) (40 Points): Compute the price of an American call option with strike K=3, assuming r=0. State what is the optimal exercise time, and compute the replicating portfolio at each step, in all cases, and time steps, even if the option was not optimally exercised. b) Bonus (25 Points): Find the minimum interest rate r such that the price of the American option would be equal to the price of European option. In other words, find r such that the holder will be indifferent between exercising the option early, and waiting until maturity