第四次作业答案 1. Buy IBX stock in Tokyo and simultaneously sell them in NY, and your arbitrage profit is $2 per share. The prices will converge. Instead of the prices becoming exactly equal, there can remain a 1% discrepancy between them, roughly $0.35 in this case. 2. Money market hedge: borrow the dollar now, convert the dollar into the sterling and deposit the sterling. The future dollar cost is fixed. Forward market hedge: buy (long) ￡ forward contract The one-year ￡ forward rate is:  If the market ￡ forward rate is $1.55/￡, there is an arbitrage opportunity. Assuming the contract size is ￡1 million, then the arbitrageur should borrow the dollars, convert into the pounds and invest in pounds, and sell them at the market forward rate. The details and cash flows (in millions) of the transactions are as follows: Arbitrage Transactions Cash Flows at t = 0 Cash Flows at t = 1   Sell ￡ forward short 0 +$1.55 and (￡1   Borrow dollar Convert dollar into pound Deposit pound     +￡1    0    3 . See Lecture Notes and Textbook 4. Terminal payoffs: Profit/Loss: 5.  (You may want to convert the interest rate with annual compounding into the one with continuous compounding first)  Therefore,  The arbitrage involves selling the put option and the underlying share short and buying the call options and lending for six months. The details of transactions and the resulting cash flows are as follows: Arbitrage Transactions  Cash Flows at t = 0 t = 6 months       Sell the put short   0  Sell the share short     Buy the call long  0   Lend       +$0.49 0 0   6. No Arbitrage Valution At maturity (two months) the payoff of the call option with strike price of $49 will be either $4 (if the stock price is $53) or $0 (if the stock price is $48). Construct a portfolio consisting of ( shares and  borrowing or lending. The payoff of the portfolio replicates the payoff the call option, therefore  Solving the above two equations gives  The value of the portfolio today is  To avoid arbitrage, the value of the call option must be $2.235. Risk-neutral Valuation Up factor:  Down factor:  Risk-neutral probability of an up movement:  The value of the put option is given by  7. No-arbitrage valuation At maturity (three months) the payoff of the European put option with strike price of $40 will be either 0 (if the stock price is $45) or $5 (if the stock price is $45). Construct a portfolio consisting of ( shares and  borrowing or lending. The payoff of the portfolio replicates the payoff the put option, therefore  Solving the equations gives  The value of the portfolio today is  Therefore, the value of the put option is $2.0588. Risk-neutral Valuation Up factor:  Down factor:  Risk-neutral probability of an up movement:  The value of the put option is given by  8. a)     The price of the European call is:  b) The initial replicating portfolio consists of (long) shares and borrowing of  In this case, the replicating portfolio includes 0.7042 shares and borrowing of $31.56. 9. The product provides a six-month return equal to max (0, 0.4R), where R is the return on FTSE 100 index. Suppose S0 is the current value of the index and ST is the value of the index in six months. When an amount A is invested, the return received at the end of six months is:  This is  of the European call options on the index with the strike price of S0. With the usual notions, the value of the option offered is:  In this case,    The value of the call option is 0.0325A Initial investment: A-0.0325A=0.9675A At six months: A Therefore return with continuous compounding is:  The return of 6.6% per annum with continuous compounding is lower than the riskfree rate of interest.