金融衍生工具DerivativesExercises AnswersQuestion 1The basis strengthens unexpectedly. Which of the following is true (circle one)(a) A short hedger's position improves.(b) A short hedger's position worsens.(c) A short hedger's position sometimes worsens and sometimes improves.(d) A short hedger's position stays the same.Short Hedge•Suppose thatF1: Initial Futures PriceF2: Final Futures PriceS2: Final Asset Price•You hedge the future sale of an asset by entering into a short futures contract•Price Realized=S2+(F1 –F2)= F1+Basis3Question 2On March 1 the spot price of a commodity is $20 and the July futures price is $19. On June 1 the spot price is $24 and the July futures price is $23.50. A company entered into a futures contract on March 1 to hedge the purchase of the commodity on June 1. It closed out its position on June 1. What is the effective price paid by the company for the commodity?Long Hedge•Suppose thatF1: Initial Futures PriceF2: Final Futures PriceS2: Final Asset Price•You hedge the future purchase of an asset by entering into a long futures contract•Cost of Asset=S2–(F2–F1)= F1+ Basis =19+(24-23.5)=19.55Question 3The following futures prices were observed for gold at the end of five consecutive trading days:$900, $903, $912.5, $898, and $890.Mr Smith short one gold futures (contract size 100 oz.) at the end of Day 1. Suppose the initial margin requirement is $4,000 and the maintenance margin at the 75% level.1.Is Mr Smith bullish or bearish on gold?pute his cash flows and margin levels.3.At what price of gold will Mr Smith get a margin call?4.How much of a variation margin will Mr Smith have to secure?5.How much in total during the five days has Mr Smith earned?Margins and Marking-to-Market, in DollarsDATE FUTURESPRICE CASHFLOWBEGINNINGMARGINCASHWITHDRAWALENDINGMARGINDay 1900004,0004,000 Day 2903-3003,70003,700 Day 3912.5-9502,7501,2504,000 Day 489814505,450-1,4504,000 Day 58908004,800-8004,000•Since he sold gold futures, he is bearish.•The beginning margin is the margin level at the end of the trading day. •The ending margin is the after-cash flow adjustment based on the margin call or removal of excess margin.–There is no margin call until the margin reaches 75% of $4,000, or $3,000.Then, it has to be restored to the initial margin level of $4,000.–If the profit exceeds the initial margin level, this excess margin can be removed (as on Days 4 and 5).•Mr Smith will get a margin call when the account value falls to $3,000 or loses 4,000 –3,000 = $1,000. As Mr Smith loses $100 for each $1increase in gold price, he will lose $1,000 if gold increases by $10.•He will have to secure $1,000 in cash.•His profit during the five days is: 1450 + 800 –1250 = $1000Question 4•Suppose that a March put option to sell a share for $40 costs $3.5 and is held until March.a. Draw a diagram illustrating how the profit from a shortposition in the option depends on the stock price atmaturity of the option.b. Under what circumstances will the option beexercised? If the option is exercised, will the seller ofthe option make profit? What is the maximum loss the seller may suffer?Question 5•The current £/$US spot exchange rate is 0.680. If you invested one UK pound for 90 days in the domesticriskless asset you would earn £1.05, and if you invested one US dollar for 90 days in the US riskless asset you would earn 1.01 dollars (assume continuouscompounding). A broker offers you a 90-day forwardcontract to buy or sell one million US dollars at theexchange rate of 0.60 pounds/dollar.•Can you make an arbitrage profit? If so, explain.Answer:Yes, there is arbitrage profit.90 days Present value Present value basedon spot rate$USD 1 million 1/1.01=0.9901 million 0.9901 * 0.68= UKP 0.6733 M£Worth:0.6733*1.05 = UKP0.7070 M USD 1.1783 M UKP 0.6733 MUKP 0.6733 M1.Borrow USD 0.9901M, change to UKP 0.6733 M2.Hold for 90 days, 0.6733*1.05 =UKP0.7070 M3.Change to USD, 0.7070M/0.60 = USD 1.17834.Repayment of the loan. Profit = 1.1783 –1 = USD 0.1783 millions.Question 6Prove the result F2= F1e r(t2 –t1)where F2and F1are two forward prices on the same commodity with maturity dates t1and t2(t1< t2), and r is the risk-free interest rate.Answer:F(t,T) = S(t)e r(T –t)where S is today’s spot, S1is spot at time t1and S2is spot at time t2,F1= Se rt1=> S = F1e–rt1and F2= Se rt2=> S = F2e–rt2Equate the two results to get F1e–rt1= F2e–rt2Multiplying both sides by e rt2gives the result.Question 7This is February. Forward prices for April and June forward contracts for gold are $900 and $915 per ounce, respectively, and the riskless yearly rate is 5%. Can you make arbitrage profits? Explain. Assume that the time differencebetween the maturity of the two contracts is 60 days.Question 8The current value of the Dow Jones Industrial Average is 11,200.The dividend yield is 3.00 per annum, assuming continuouscompounding and a 365-day year.a) What is the 60-day Dow Jones Industrial Average futures price?The interest rate is 10%.b) The Dow slides down and a week later ends at 10,500. What isthe 53-day forward price? What is the value of the forwardcontract to the long? Assume that the forward contract’s payoffs are determined by multiplying the above results by $10.a)R=10%, the continuous compounded interest rate:e r= 1+10%r = ln(1.10) = 9.53%The 60-day forward price:F0= S0e(r–q )T= 11,200e(0.0953 –0.03)(60/365)= 11,321b) The 60-day forward price:F0= S0e(r–q )T= 10,500 e(0.0953 –0.03)(53/365)= 10,600the value of the forward contract to long:= (10,600–11,321) e-0.0953 *(53/365)= –711.1If the forward’s payoffs are determined by multiplying the payoff by $10, then the long has lost $7,111Question 9•Show that a newly written forward contract is equivalent to a portfolio consisting of one purchased European call option on the underlying asset and one written European put option on the underlying asset, both with a common expiration date equal to the delivery date, and both with a common striking price equal to the forward price, F.Question 10A company has a $36 million portfolio with a beta of 1.2. The S&P index is currently standing at 900. Futures contracts on $250 times the index can be traded. What trade is necessary to achieve the following. (Indicate the number of contracts that should be traded and whether the position is long or short.)•Eliminate all systematic risk in the portfolio •Reduce the beta to 0.9•Increase beta to 1.8Question 11•What is the lower bound for the price of a four-month call option on a non-dividend paying stock when thestock price is $28, the strike price is $25, and therisk-free interest rate is 8% per annum?•What is the lower bound for the price of a one-month European put option on a non-dividend paying stockwhen the stock price is $12, the strike price is $15,and the risk-free interest rate is 6% per annum?i) 66.32528)12/4(08.00=−=−−−eKeS rTii)93.215)12/1(06.00==−−−eS KerTQuestion 12An investor is looking at the options market, where the following prices are being quoted for American options (stock price = $30).MONTH & STRIKE CALL PRICE PUT PRICEMay 2542May 3013June 3012a) The investor is exploring the ways to make some arbitrage profits. Explain the different ways to make arbitrage profits based on the above information.b) Draw the payoff diagrams with options. In particular,i) a straddle (long call and long put with May 25)ii) a bull spread (long May 25 call and short May 30 call)Question 13The following prices are given for European options on a particular stock expiring in December.Dec 50 call = $7Dec 55 call = $3Dec 50 put = $4Dec 55 put = $8Draw the profit and loss diagram on the expiration date from the following positions:(a) A bull spread (long Dec 50 call and short Dec 55 call).(b) A straddle (long Dec 55 call and long Dec 55 put).(c) A strangle (long Dec 50 put and long Dec 55 call)Question 14A non-dividend paying stock sells for $7.59. What is the theoretical value of a European style, $8 call with 1 year until expiration, assuming interest rates of 12% (with continuous compounding) and annual volatility of 30%?)1( 0D Kec S p rT++=+−724.16.06.06.0)12/7(12.0)12/4(12.0)12/2(12.0=++=−−−e e e D The put-call parity relationship with dividends becomes:where D denotes the present value (PV) of dividend payments. We expect three dividend payments of $0.60 in 2, 4 and 7 months. Therefore:956.5724.1363)12/8(12.000=⇒++=−++=⇒++=+−−−p e S D Ke c p D Ke c S p rT rT Now we have everything we need in equation (1) to compute the value of the put option:Question 16Use the put-call parity relationship to derive, for a non-dividend-paying stock, the relationship between:•The delta of a European call and the delta of a European put.•The gamma of a European call and the gamma of a European put.•The vega of a European call and the vega of a European put.•The theta of a European call and the theta of a European put.Question 17• A non-dividend paying stock sells for $10/share. In three months it will be either $12/share or $8/share. The risk free interest rate is 5% per annum (with continuouscompounding).(a) What is the value of a put option if the strike price is $10using no-arbitrage valuation?(b) What is the call price with the same strike and maturity?(c) Suppose that the market value of the put option is $2.Explain whether you can make arbitrage profits.The riskless portfolio is: long 0.5 shares and long 1 put option. The value of the portfolio in 3 months is 12*0.5 = 6.The value of the portfolio today is 6*e – 0.05*0.25 = 5.92.The value of the put option = 5.92 – 10*0.5 = $0.92– r t- 0.05*0.25b) c = p + S0 - K e = 0.92 + 10 - 10 e = $1.04c)Sell the traded put for $2buy a synthetic put for $0.92 (short sell 0.5 stocks and lend $(0.5*10 + 0.92) = 5.92 at the interest rate of 5% per annum).Today: Net cash flow is 2 + 0.5*10 - 5.92 = + $1.08.At maturity:(i) If the stock price is 12,cash flow from the traded put is 0,cash flow from the synthetic put is –0.5*12 + 5.92 e 0.05*0.25 = 0. (ii) If the stock price is 8,cash flow from the traded put is –(10 – 8) = –2,cash flow from the synthetic put is –0.5*8 + 5.92 e 0.05*0.25 = 2.Net cash inflow is –2 + 2 = 0.Thus $1.08 is the arbitrage profit you can make today.Question 18Consider a European call option on a non-dividend-paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is six months.a)Calculate u, d, and p for a two-step tree.b)Value the option using a two-step tree.Question 19The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price change in one trading day?Question 20Explain the principle of risk-neutral valuation.。
