第四次作业答案1.a) Buy IBX stock in Tokyo and simultaneously sell them in NY, and your arbitrage profit is $2 per share.b) The prices will converge.c) Instead of the prices becoming exactly equal, there can remain a 1% discrepancy between them, roughly $0.35 in this case.2.a) 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 contractb) The one-year £ forward rate is:£/$4464.112.0108.0150.111£$01,0=++⨯=++=r r S Fc) 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:3 .See Lecture Notes and Textbook4.Terminal payoffs:Profit/Loss:5. 51.32$10.01/322=++=+-τr Xe c(You may want to convert the interest rate with annual compounding into the one with continuous compounding first)33$294=+=+S pTherefore, ()()49.0$51.3233=-=+-+-τr Xe c s pThe arbitrage involves selling the put option and the underlying share short and buying the call options and lending ()51.3010.1/32=for six months. The details of transactions and the resulting cash flows are as follows:Long PutShort Put Long PutShort Put T S T S T S T SArbitrage Transactions Cash Flows at t = 0 t = 6 months32<T S 32>T SSell the put short 4$+ ()T S --32$ 0 Sell the share short 29$+ T S $- T S $-Buy the call long 0.2$- 0 ()32$-T S Lend ()51.3010.1/32$=()51.3010.1/32=-32$+ 32$+ +$0.49 0 06. No Arbitrage ValutionAt 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 0B borrowing or lending. The payoff of the portfolio replicates the payoff the call option, therefore 04845312210.0012210.00=+∆=+∆⨯⨯eB e BSolving the above two equations gives7654.37 and ,8.00-==∆BThe value of the portfolio today is235.27654.37508.0=-⨯To avoid arbitrage, the value of the call option must be $2.235.Risk-neutral ValuationUp factor: 06.15053==u Down factor: 96.05048==dRisk-neutral probability of an up movement:5681.096.006.196.012210.0=--=⨯e π The value of the put option is given by 235.2$045681.012210.0=+⨯⨯e7. No-arbitrage valuationAt 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 0B borrowing or lending. The payoff of the portfolio replicates the payoff the put option, therefore()()502.0135002.014500=++∆=++∆B BSolving the equations gives0588.22 and ,5.00=-=∆BThe value of the portfolio today is()0588.2$0588.22405.0=+⨯-Therefore, the value of the put option is $2.0588.Risk-neutral ValuationUp factor: 125.14045==u Down factor: 875.04035==dRisk-neutral probability of an up movement:58.0975.0125.1875.002.1=--=πThe value of the put option is given by 0588.2$02.1542.0058.0＝⨯+⨯8.a) 25.0 ,0.30 ,%12 ,50 ,52=====τσr X S()()5365.025.030.025.03.012.050/52ln 21=⨯++=d3865.025.030.05365.012=-=-=τσd d()()6504.03865.0 ,7042.05365.0==N NThe price of the European call is:06.56504.0507042.05225.012.0=⨯⨯-⨯=⨯-e cb) The initial replicating portfolio consists of ()1d N (long) shares and borrowing of ().2d N Xe r τ- In this case, the replicating portfolio includes 0.7042 shares and borrowing of $31.56.9.a) The product provides a six-month return equal to max (0, 0.4R), where R is the return on FTSE 100 index. Suppose S 0 is the current value of the index and S T 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:),0max(4.0)*4.0,0max(0000S S S A S S S A T T -=- This is 04.0S A of the European call options on the index with the strike price of S 0. b) With the usual notions, the value of the option offered is:))()((4.0))()((4.02120100d N e d N e A d N e S d N e S S A rT qT rT qT -----=-In this case, 50.0,25.0,03.0,08.0====T q r σ0530.02298.050.025.050.0)2/25.003.008.0(1221=-==⨯⨯+-=T d d d σ5212.0)(5909.0)(21==d N d N The value of the call option is 0.0325AInitial investment: A-0.0325A=0.9675AAt six months: ATherefore return with continuous compounding is: %6.6)9675.0ln(2=AA The return of 6.6% per annum with continuous compounding is lower than the riskfree rate of interest.。