北京大学：《金融学概论》课程教学资源（作业习题）第四次作业答案

第四次作业答案 a) Buy IBX stock in Tokyo and simultaneously sell them in NY, and your arbitrage profit is $2 per share b) The prices will convers them, roughly $0.35 m this ng exactly equal, there can remain a 1% discrepancy between c)Instead of the prices becom 2 a)Money market hedge: borrow the dollar now, convert the dollar into the sterling and deposit the sterling. The future dollar cost is fixed Forward market hedge: buy(long)f forward contract b) The one-year f forward rate is 1+0.08 S 1+s=1.50× =14464$/￡ 1+0.12 c)If the market f forward rate is $1.55/f, there is an arbitrage opportunity. Assuming the contract size is fl 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 Cash Flows at t=0 Cash Flows at t=1 Sell￡ forward short 0 +$155and-￡1 Borrow dollar s1.50/(1+0.12) $150×108/1.12 Convert dollar into pound[-1.50/(1+0.12)and f1/(1+0. 12) Deposit pound ￡11(+0.12) +￡1 155-14464=5010361 See lecture notes and Textbook C Gary Xu AcF2 14 Princip les of finance

© Gary Xu AcF214 Principles of Finance 1 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 $0.35 in this case. 2. a) Money market hedge: borrow the dollar now, convert the dollar into the sterling and deposit the sterling. The future dollar cost is fixed. Forward market hedge: buy (long) £ forward contract b) The one-year £ forward rate is: 1.4464$ / £ 1 0.12 1 0.08 1.50 1 1 £ $ 0,1 0 = + + =  + + = r r F S c) 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 $1.50 /(1+ 0.12) −$1.50/(1+ 0.12)and £1/(1+ 0.12) − £1/(1+ 0.12) −$1.501.08/1.12 +£1 0 1.55−1.4464 = $0.1036 3 . See Lecture Notes and Textbook

erminal payoffs ong put Short put Profit/los Long Put Short put X c+e"=2+32/√1+0.10=$32.51 You may want to convert the interest rate with annual compound ing into the one with continuous compounding first) p+S=4+29=$33 Therefore,(p+s)-(c+ xe") 33-3251=$049 The arbitrage involves selling the put option and the underlying share short and buying the call options and lend ing 32//1.10(30.51)for six months. The details of transactions and the resulting cash flows are as follo C Gary Xu AcF2 14 Princip les of finance 2

© Gary Xu AcF214 Principles of Finance 2 4. Terminal payoffs: Profit/Loss: 5. + = 2 + 32 / 1+ 0.10 = $32.51 −r c Xe (You may want to convert the interest rate with annual compounding into the one with continuous compounding first) p + S = 4 + 29 = $33 Therefore, ( + ) − ( + ) = 33− 32.51 = $0.49 −r p s c Xe The arbitrage involves selling the put option and the underlying share short and buying the call options and lending 32 / 1.10(= 30.51) for six months. The details of transactions and the resulting cash flows are as follows: Long Put X Short Put X Long Put X Short Put X T S T S T S T S

t=6 months Arbitrage transactions Cash Flows at t=0 Sr32 ell the put short s(32-S) 0 Sell the share short +$29 20 0 ( Lend32/110(=301)-32/10(=3051)+$32 +S32 0 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 $o (if the stock price is $48) Construct a portfolio consisting of A shares and Bo borrowing or lending. The payoff of the portfolio replicates the payoff the call option, therefore △53+Be12=4 △48+B 0 Solving the above two equations gives △=0.8,andB=-37.7654 The value of the portfolio today 0.8×50-37.7654=2235 To avoid arbitrage, the value of the call option must be $2. 235 Risk-neutral valuation 48 Down factor: d=-=0.96 Risk-neutral probability of an up movement 0.568l 1.06-0.96 The value of the put option is given by 0.5681×4+0 =$2235 C Gary Xu AcF2 14 Princip les of finance

© Gary Xu AcF214 Principles of Finance 3 Arbitrage Transactions Cash Flows at t = 0 t = 6 months ST  32 ST  32 Sell the put short +$4 ( ) − $ 32 − ST 0 Sell the share short + $29 − $ST − $ST Buy the call long −$2.0 0 $( − 32) ST Lend $32 / 1.10(= 30.51) − 32 / 1.10(= 30.51) + $32 + $32 +$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 B0 borrowing or lending. The payoff of the portfolio replicates the payoff the call option, therefore 48 0 53 4 12 2 0.10 0 12 2 0.10 0  + =  + =   B e B e Solving the above two equations gives  = 0.8, and B0 = −37.7654 The value of the portfolio today is 0.850−37.7654 = 2.235 To avoid arbitrage, the value of the call option must be $2.235. Risk-neutral Valuation Up factor: 1.06 50 53 u = = Down factor: 0.96 50 48 d = = Risk-neutral probability of an up movement: 0.5681 1.06 0.96 0.96 12 2 0.10 = − − =  e  The value of the put option is given by $2.235 0.5681 4 0 12 2 0.10 =  +  e

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 A shares and Bo borrowing or lending. The payoff of the portfolio replicates the payoff the put option, therefore △45+B0(1+002)=0 △35+B(1+002)=5 Solving the equations gi Bo The value of the portfolio today is (-0.5)×40+220588=$20588 Therefore, the value of the put option is $2.0588 Risk-neutral valuation Up factor,, 45 1.125 Down factor: d =35=0.875 Risk-neutral probability of an up movement 102-0.875 1.125-09750.58 The value of the put option is given by 0.58×0+042×5 1.02 =$2.0588 a)S=52,X=50,r=12%,G=0.30,=0.25 h(52/50)+(012+032)×025 030√0.25 =0.5365 d2=d1-o√r=0.5365-0.30√0.25=0.3865 N(05365)=0.7042,N(0.3865)=06504 C Gary Xu AcF2 14 Princip les of Finance

© Gary Xu AcF214 Principles of Finance 4 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 B0 borrowing or lending. The payoff of the portfolio replicates the payoff the put option, therefore ( ) 35 (1 0.02) 5 45 1 0.02 0 0 0  + + =  + + = B B Solving the equations gives  = −0.5, and B0 = 22.0588 The value of the portfolio today is (− 0.5)40 + 22.0588 = $2.0588 Therefore, the value of the put option is $2.0588. Risk-neutral Valuation Up factor: 1.125 40 45 u = = Down factor: 0.875 40 35 d = = Risk-neutral probability of an up movement: 0.58 1.125 0.975 1.02 0.875 = − −  = The value of the put option is given by $2.0588 1.02 0.58 0 0.42 5 ＝  +  8. a) S = 52, X = 50,r = 12%, = 0.30, = 0.25 ( ) ( ) 0.5365 0.30 0.25 ln 52 / 50 0.12 0.3 0.25 2 1 = + +  d = d2 = d1 −  = 0.5365 − 0.30 0.25 = 0.3865 N(0.5365) = 0.7042, N(0.3865) = 0.6504

The price of the European call is c=52×0.7042-50×e-012025×06504=506 b) The initial replicating portfolio consists of N(d,)(long)shares and borrowing of Xe" N(d, ) In this case, the replicating portfolio includes 0.7042 shares and borrowing of $31.56 a) The product provides a Six-month return equal to max(0, 0. 4R), where r is the return on FTSE 100 index. Suppose So is the current value of the index and Sr 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 Amx(0,04*S Sa=044 S maxOS-So) This 0.AA of the European call options on the index with the strike price of So b) with the usual notions, the value of the option offered 0.4A s(Soe - q N(d, )-Soe- N(d2)) 044(eN(d1)-eN(d2) In this case,r=008,q=0.03,=0.25,7=0.50 (008-0.03+0252/2)×0.50 0.2298 025×√0.50 d2=d1-a√T=00530 N(d1)=0.5909 N(d2)=0.5212 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: 2 In(4 6.6% 0.9675A The return of 6.6% per annum with continuous compound ing is lower than the riskfree rate of interest C Gary Xu AcF2 14 Princip les of finance

© Gary Xu AcF214 Principles of Finance 5 The price of the European call is: 52 0.7042 50 0.6504 5.06 0.12 0.25 =  −   = −  c e b) The initial replicating portfolio consists of ( ) N d1 (long) shares and borrowing of ( ). Xe N d2 −r 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 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: max( 0, ) 0.4 max( 0,0.4* ) 0 0 0 0 S S S A S S S A T T = − − This is 0 0.4 S A of the European call options on the index with the strike price of S0. b) With the usual notions, the value of the option offered is: 0.4 ( ( ) ( )) ( ( ) ( )) 0.4 1 2 0 1 0 2 0 A e N d e N d S e N d S e N d S A qT rT qT rT − − − − = − − In this case, r = 0.08,q = 0.03, = 0.25,T = 0.50 0.0530 0.2298 0.25 0.50 (0.08 0.03 0.25 / 2) 0.50 2 1 2 1 = − = =  − +  = d d T d  ( ) 0.5212 ( ) 0.5909 2 1 = = N d N d 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: ) 6.6% 0.9675 2ln( = A A The return of 6.6% per annum with continuous compounding is lower than the riskfree rate of interest

C Gary Xu AcF2 14 Princip les of finance 6

© Gary Xu AcF214 Principles of Finance 6