#10. Problems 1 through 10, part a). Fi8000 Practice Set #1 Check Solutions 1. Payoff. Payoff #8 Payoff S

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1 Problems 1 through 1, part a). #1 #2 # #4 #5 # #7 #8 # #1-1 Fi8 Practice et #1 Check olutions 1

2 Problem b) Profitable Range c) Maximum Profit c) Maximum Loss 1 < $22.8 $12.8 Unlimited 2 > $22.6 $7.4 $2.6 3 > $13.1 $6.9 $ < $7. and > $33. Unlimited $8. 5 > $24.3 Unlimited $ > $3.2 Unlimited $1.2 7 < $22.8 $12.8 Unlimited 8 < $12. and > $38. Unlimited $13. 9 $12.2 < < $27.8 $2.8 $2.2 1 < $22. $2. $ Long Put, X=2; hort Bond, FV= Long Put, X=15; Long Call, X=25; Long Bond, FV= hort Put, X=2; hort Call, X=25; Long Bond, FV= Long Put, X=2; hort Put, X=1; Long Call, X=2; hort Call, X= hort Put, X=2; Long Call, X=25; hort Call, X=3; Long Bond, FV= hort Put, X=3; Lon Put, X=15; hort Call, X=3; Long Bond, FV= Note that the payoff to Call A is always greater than or equal to the payoff to Call B. ince Call A is cheaper, we buy Call A and write (sell) Call B, pocketing $1.25 today. At expiration, we use any payoff from Call A to payoff anything due on Call B. At worst, we get zero at expiration and at best we get $1 at expiration. hus, we have an arbitrage opportunity. he table shows the transactions and cash flows. oday (time ) Buy a call, X= Write a call, X= Net cash flow Expiration (time 1) Position < 5 = 5 5 < < 6 = 6 > 6 Long call, X= hort call, X=6 6 Net cash flow to 18. his is similar to the last problem. he payoff to Put Z is always greater than or equal to the payoff to Put Y. ince Put Z is cheaper, we buy Put Z and write (sell) Put Y, pocketing $.75 today. At expiration, we use any payoff from Put Z to payoff anything due on Put Y. At worst, we get zero at expiration and at best we get $1 at expiration. hus, we have an arbitrage opportunity. he table shows the transactions and cash flows. oday (time ) Buy a put, X=4-2.5 Write a put, X= Net cash flow +.75 Fi8 Practice et #1 Check olutions 2

3 Expiration (time 1) Position < 3 = 3 3 < < 4 = 4 > 4 Long put, X= hort put, X=3 3 Net cash flow + 1 to 19. We can now make general statements based on 17. and 18. Call options (on the same underlying and the same expiration date) with lower strike prices must have premiums greater than or equal to the premiums of call options with higher strike prices. o, C(X1) C(X2). Put options (on the same underlying and the same expiration date) with lower strike prices must have premiums less than or equal to the premiums of put options with higher strike prices. o, P(X1) P(X2). 2. According to put-call parity, X P = C +. Using the interest rate and prices for the call and, we can replicate a long position in a put at a cost of negative $2.68! he price of a put (whose minimum payoff is zero) must be at least zero. o, we can get paid for buying portfolio that replicates a long position in the put. his must be an arbitrage opportunity since we never have a negative cash flow and we earn a profit. he table shows the transactions and cash flows. oday (time ) Buy a call, X=9-3.5 Buy a bond, FV= hort sell the Net cash flow Expiration (time 1) Position < 9 = 9 > 9 Long call, X=9 9 Long bond, FV= hort Net cash flow to +9 X 21. According to put-call parity, P = C +. Using the interest rate and prices for the call and, we can replicate a long position in a put at a cost of $6.7. his is cheaper than we could buy the real put. o, buy the portfolio that replicates the put and sell the real put. he table shows the transactions and cash flows to answer part a).note that the transactions in the table are the same no matter what you replicate (call vs. put). b) $6.7 c) $7.93 d) $76.7 e).% oday (time ) Buy a call, X=8-6. Buy a bond, FV= hort sell the Write a put, X= Net cash flow Fi8 Practice et #1 Check olutions 3

4 Expiration (time 1) Position < 8 = 8 > 8 Long call, X=8 8 Long bond, FV= hort hort put, X=8 8 Net cash flow 22. he deviation from put-call parity should be fairly small. If it is within $.5 or so, then transaction costs are very likely to wipe out any arbitrage profit. Also, the borrowing rate might be higher for us than the - bill rate. o, try the arbitrage strategy assuming that you must pay a premium in excess of 1% over the - bill rate. Again, this along with transaction costs should render any attempt at the arbitrage strategy costly (i.e., no arbitrage). Also, be sure you have not chosen a dividend paying. What about early exercise? hese are American options. Put-call parity only applies to European options. 23. We can replicate the by rearranging the put-call parity formula as = C + P. o, we X replicate the by buying a call, buying a bond, and selling a put. Note that we have two strike prices, so we can get two synthetic values: X1 6 = C( X1) + P( X1) = X 2 65 = C( X 2) + P( X 2) = = 2 = and. ince the replicating portfolio using X2 is cheaper, we will buy this portfolio and sell the other portfolio that uses X1. Every time we do this, we create a riskless position and pocket the difference between the costs of the two portfolios. We must now show the cash flows at the relevant dates (time is today and time 1 is at expiration) to show that this is an arbitrage strategy. We enter into the positions today and check the value of those positions (assuming we get out of the positions) at expiration. Because we are using two strike prices here, we must check the value of our positions in 5 states of the world at expiration. oday (time ) Buy a call, X=65-6. Buy bond, FV= Write a put, X= Write a call, X=6. ell a bond, FV= Buy a put, X=6-3.5 Net cash flow +.28 Expiration (time 1) Position < 6 = 6 6 < < 65 = 65 > 65 Long call, X=65 65 Long bond, FV= hort put, X= hort call, X= hort bond, FV= Long put, X=6 6 Net cash flow Fi8 Practice et #1 Check olutions 4

5 24. As with the previous problem, we can rearrange the put-call parity formula to replicate the riskfree asset (i.e., the bond). o, X (1+ r) = P + C. Note that the right hand side of this equation is the portfolio that replicates the bond that pays a return or r. o, to earn this rate we would buy the put and and write a call. o determine what rate this portfolio will give us, we solve for the riskfree rate, X r = P + C 1 1. olving for the rate using the options with a strike price of X1, we find that r1=8.6957%. Using X2, we get r2=1%. Which rate should we buy? When we buy (i.e., invest), we want the highest rate of return. herefore, we should buy the portfolio using X2, since it gives us a riskfree return of 1%. We want to sell the portfolio that uses X1, since we will essentially borrow that money at a rate of %. he challenge here is to make sure that we never have a negative cash flow. If we were to naïvely buy the portfolio using X1 and sell the portfolio using X2, then we would end up with negative cash flows. his would violate the conditions for an arbitrage strategy. ince we are buying and selling bonds (really we are buying and selling portfolios that replicate bonds), let s make sure that we buy and sell bonds so that our total face value (i.e., payment at maturity) is the same between what we buy and sell. ince the X1 bond (portfolio using X1) has a face value of $1 and the X2 bond has a face value of $11, we can use 11 of the X1 bond and 1 of the X2 bond so that the total of both (i.e., the total face value) is $1,1 at maturity or expiration. Again, we must show the transactions and cash flows to open the positions today and the payoffs or values of these positions at expiration. Note that I have left the short 11 shares and long 1 shares as separate transactions in order to make this more transparent. Of course, we could just use short 1 share, leaving the result unchanged. oday (time ) Write 11 puts, X= ell short 11 shares of Buy 11 calls, X=1-275 Write 1 calls, X= Buy 1 shares of - 15 Buy 1 puts, X=11-1 Net cash flow +12 Expiration (time 1) Position < 1 = 1 1 < < 11 = 11 > 11 hort 11 puts, X= hort 11 shares Long 11 calls, X= hort 1 calls, X= Long 1 shares Long 1 puts, X= Net cash flow Fi8 Practice et #1 Check olutions 5

6 25. a) $5.1542; N=.8866, B= b) $.2311; N=-.1134, B= a) $3.6165; N=.6465, B= b) $.6934; N=-.3535, B=18.72 c) he call premium increases and put premium decreases as the price of the underlying asset increases. 27. a) $1.5947; N=.2743, B= b) $1.4793; N=-.7257, B=39.94 c) he call premium decreases and the put premium increases as the strike price increases. 28. a) $5.832; N=.7775, B= b) $.971; N=-.2225, B=12.7 c) he call and put premiums increase with an increase in volatility. 29. a) $5.9436; N=.8866, B=-41.5 b) $.1134; N=-.1134, B=6.12 c) he call premium increases and the put premium decreases with an increase in the riskfree rate. 3. a) and c) $7.564; Cu=$9.1631, Cd=$ b) and d) $.2842; Pu=$, Pd=$ e) he put and call premiums increase with an increase in time to expiration. Note: this is true, in general, for all call options, but not for put options. 31. a) $3.456; Cu=$4.8546, Cd=$ b) $1.367; Pu=$.4993, Pd=$3.815 c) $3.456; Cu=$4.8546, Cd=$ d) $1.8347; Pu=$.4993, Pd=$ e) he is no value to early exercise for an American call, but there is value to the early exercise option for some put options. 32. ince the market price for the real call is $4.75 and the price of the synthetic call (replicating portfolio) is $5.1542, we should buy the real call and short sell the replicating portfolio. Below are the transactions and cash flows that occur today (time ) and at expiration (time 1). We show that we satisfy the three conditions of an arbitrage opportunity. ime ime 1 CF Position CF (up) CF (down) hort sell.8866 shares hort.8866 shares of +$ of -$ $ Invest $ in the riskfree bond -$ Long riskfree bond +$ $ Buy one real call -$4.75 Long one call +$7.24 $. Net cash flow +$.442 New cash flow $. $. 33. Now the market price for the real call is higher than the price of the synthetic call. o, we should buy the replicating portfolio and sell the real call. ime ime 1 CF Position CF (up) CF (down) Buy.8866 shares of Long.8866 shares of -$ $ $ Borrow $ in the riskfree bond +$ hort riskfree bond -$ $ Write a real call +$5.375 hort one call -$7.24 $. Net cash flow +$.228 New cash flow $. $. Fi8 Practice et #1 Check olutions 6

7 34. he market price for the real put is higher than the price of the synthetic put. o, we should buy the replicating portfolio and sell the real put. ime ime 1 CF Position CF (up) CF (down) hort sell.1134 shares hort.1134 shares of +$6.96 of -$ $ Invest $6.248 in the riskfree bond -$6.248 Long riskfree bond +$ $6.494 Write a real put -$.5 hort one put $. -$.9259 Net cash flow +$.2688 New cash flow $. $. 35. he market price for the real call is higher than the price of the synthetic call. o, we should buy the replicating portfolio and sell the real call. Because we have multiple time periods, we must rebalance our portfolio at time 1 in order to replicate the payoffs of the call. Note that the rebalancing transactions cost us nothing. In other words, the replicating portfolio is self-financing. ime CF Buy.5945 shares of -$ Borrow $ against the riskfree bond +$ Write a real call +$4. Net cash flow +$.544 ime 1 ( goes up to $57.24) Current position Desired position Rebalancing transaction CF Long.5945 shares of Long.7732 shares of Buy.1787 shares of -$1.24 Owe $ Borrow $ Borrow $1.24 more +$1.24 hort one call hort one call - $. Net cash flow $. ime 1 ( goes down to $49.7) Current position Desired position Rebalancing transaction CF Long.5945 shares of Long shares of ell shares of +$ Owe $ Owe nothing Repay loan -$ hort one call hort one call - $. Net cash flow $. ime 2 ( price starts from $57.24 at time 1) Position CF (=$ ) CF (=$53.) Long.7732 shares of +$47.8 +$4.988 Owe $ $ $4.988 hort one call -$ $. Net cash flow $. $. ime 2 ( price starts from $49.7 at time 1) Position CF (=$53.) CF (=$45.439) Long shares $. $. Owe nothing $. $. hort one call $. $. Net cash flow $. $. Fi8 Practice et #1 Check olutions 7

8 Problems 36 through 4. You should use the Black-choles spreadsheet that I created. Problem 41. Create your own spreadsheet and solve for the volatility. Recall that the option premiums increase as the volatility increases. Problem 42. It should be the case that the volatility implied (calculated) from the Black-chole model (again, use a spreadsheet that you ve created) for the higher risk s should be higher. Fi8 Practice et #1 Check olutions 8