H OMEWORK 3 Homework 3 is due by Friday October 21st at 5:00 pm. (1) Consider the Binomial model with parameters u = 1.3, d = 0.9, r = 0.1, S0 = 113, p = 0.7, q = 0.3 and N = 5. Let P be the probability measure determined be the risk neutral measure defined in class. by the numbers p, q and let P Compute the following expressions e 5 |F3 ) E(S5 |F3 ), E(S Note: Note that in principle you need to compute 16 different quantities. If you obtain your answers using some code make sure you attach it. St Is the discounted stock price (1+r) t a P-martingale? If not show explicitly why this is not the case. St e Is the discounted stock price (1+r) t a P-martingale? If not show explicitly why this is not the case. (2) (Basic code for pricing and hedging) Write a code whose input parameters are u, d, S0 , N , r and the values of the payoff VN associated to a contract, and whose output is the price at time zero for the contract according to the binomial model. Attach your code. Write a code whose input parameters are u, d, S0 , N , r, the values of the payoff VN associated to a contract, and a string of Hs and T s of length N (or a string of ones and zeros) representing the realization of coin flips, and whose outputs are a sequence of numbers 0 , 1 , . . . , N and a sequence of numbers X0 , . . . , XN representing the units of stock and the value of a replicating portfolio for the payoff VN when the outputs of the coin flips are those contained in the input string. Attach your code. (3) (European derivatives) A European derivative on the stock price of a company is a contract whose payoff at time N is given by VN := f (SN ), where f is some function. Such payoff scheme is an example of a contract that is path-independent. We showed in class that the price at time zero to enter the contract is 1 e (f (SN )) V0 := E (1 + r)N e Use your answer to write an expres What is the distribution of SN under P? sion for V0 . Note: SN being a discrete random variable has a p.m.f. not a density! Explain why the code you created in Problem 2 is not the best way to compute the replicating portfolio for the payoff associated to a European derivative. Write down a code that works better in settings in which the payoff is of the form f (SN ). Hint: Use the fact that the payoff is path independent and hence only depends on the total number of Hs and Ts and not on the order in which the outcomes were observed. Does the payoff VN := max (St SN ) t=0,...,N correspond to the payoff of a European derivative? Explain. Note: The above payoff is the payoff associated to a contract known as lookback option. Use 1 2 your codes from question 2 to find the value at time zero of a lookback option when the parameters of the model are given by N = 7, S0 = 100, u = 1.3, d = 0.9, r = 0.1. Also, provide the replicating portfolio for the payoff VN if the realization of the coin flips is HT HHT T H (these are actual outcomes that I observed while I was writing this problem). (4) (Put option and put-call parity) A put option is a contract that gives to the owner the right to sell one unit of stock at time N at the price K (the strike of the option). We can think that the payoff at time N for a put option is: (K SN )+ Plot in the Cartesian plane the functions s 7 (K s)+ and s 7 (s K)+ . What is the function (s K)+ (K s)+ ? Use your answer to the previous question to write the value of a put option at time zero only in terms of: the price of a call option with the same strike at time zero, the strike K, S0 and the interest rate r. Explain your answer. (5) (Chooser option) A chooser option with strike K, maturity N and choosing time m ( 0 m N ) is a contract that allows the owner to decide at time m whether she wants her option to be paid as a call option or as a put option at time N . Find the price at time zero for a chooser option with strike K = 120, maturity N = 100 and choosing time m = 30, if the model parameters are u = 1.4, d = 0.9, r = 0.1, S0 = 100? You need to explain how you obtained your answer. (6) (Currency derivatives) XYZ is an international student in the US who regularly sends money to his 2 aunts living in Narnia. XYZ receives her stipend in dollars and uses part of it to buy lions (Narnia's most popular currency) that she then sends to her aunts. XYZ is worried about a possible political earthquake in the US that may affect its economy making the exchange rate of dollars per lion to skyrocket. XYZ decides to go to GB who then offers her the following deal: \"Let us fix an exchange rate K (dollars per lion) today. In N days you will have the right to purchase one lion for K dollars. To enter this contract you need to pay me G dollars today\". Let us denote by Et the exchange rate of dollars per lion at time t. Notice that XYZ only knows the value E0 . We assume that there are two risk free assets : one in dollars with constant interest rate rd (d standing for domestic) and one in lions with constant interest rate rf (f standing for foreign). Your goal is to determine the fair price G of the contract. Find a formula for G analogous to the risk neutral pricing formula we saw in class. NOTE: This problem can not be solved using directly the theory that we have developed in class. Notice in particular that there are two interest rates, one in dollars and one in lions. The problem is solved by finding a measure Q under which all discounted (in dollars) self-financing portfolios are Q-martingales. Indeed, think what XYZ can do: she can either put her dollars in a US bank or put dollars (in the form of lions) in a bank in Narnia. Her portfolio should consist of units of US risk-free asset and units of Narnia risk free asset. It may be a good idea to write the value of XYZ's portfolio in dollars; write down a formula for it. What are self-financing portfolios in this case? Try then to find a probability measure Q under which the discounted value of all self-financing portfolios are Q-martingales. You may assume that the exchange rate Et is described by a binomial model with parameters u, d, p, q ( the parameters p, q are irrelevant). You 3 will need to assume some conditions on the parameters for the Q to exist. What are these conditions? O PTIONAL P ROBLEMS (1) In page 23 of the lecture notes there are two optimization problem (one minimize KL divergence one maximize a certain expression) that are said to be equivalent. Prove that indeed any solution to one of the problems is a solution to the other. (2) Prove, using the definitions, all properties of conditional expectation listed in the lecture notes. (3) In class we showed that a P-martingale in our N -coin toss space M0 , . . . , MN satisfies E(Mt ) = M0 for all t. Is the converse statement true? Namely, is it always true that an adapted process whose expectation does not change in time is a P-martingale? If yes prove it, if not provide a counterexample. (4) Imagine that we model a stock price to evolve according to St+1 = t+1 St , but now we assume t+1 can take the values {a1 , . . . , ak } with probabilities p1 , . . . , pk . Assuming that you also have a risk free asset with interest rate r, can you find necessary and sufficient conditions on the parameters of the model for the absence of arbitrage in the market consisting of risky asset and risk free asset? In case the market is arbitrage free can you construct an equivalent measure under which the discounted stock price is a martingale