Options and Futures
Options and Futures FIN 444
Cal State Fullerton
Popular in Course
Popular in Finance
This 8 page Class Notes was uploaded by Mr. April Weber on Wednesday September 30, 2015. The Class Notes belongs to FIN 444 at California State University - Fullerton taught by Tsong Lai in Fall. Since its upload, it has received 117 views. For similar materials see /class/217013/fin-444-california-state-university-fullerton in Finance at California State University - Fullerton.
Reviews for Options and Futures
Report this Material
What is Karma?
Karma is the currency of StudySoup.
Date Created: 09/30/15
CHAPTER 3 PRINCIPLES OF OPTION PRICING ENDOFCHAPTER QUESTIONS AND PROBLEMS 1 Basic Notation and Terminology The average of the bid and ask discounts is 822 Discount 7 82268360 7 15527 Price 7100 715527 7 984473 Yield 7 GOO984473 36568 7 1 7 00876 Note that even though the Tbill matured in 67 days we must use 68 days since that is the options time to expiration 2 Minimum Value of a Call This would create an arbitrage opportunity The call would be purchased and immediately exercised For example suppose S0 44 X 40 and the call price is 3 Then an investor would buy the call and immediately exercise it This would cost 3 for the call and 40 for the stock Then the stock would be immediately sold for 44 netting a riskfree profit of 1 In other words the investor could obtain a 44 stock for 43 Since everyone would do this it would drive the price of the call up to at least 4 If the call were European however immediate exercise would not be possible unless of course it was the expiration day so the European call could technically sell for less than the intrinsic value of the American call We saw though that the European call has a lower bound of the stock price minus the present value of the exercise price assuming no dividends Since this is greater than the intrinsic value the European call would sell for more than the intrinsic value Then at expiration it would sell for the intrinsic value 3 Principles of Call Option Pricing European call We know that its price cannot exceed S0 but must exceed Max0 S0 7 Xlr39T With an infinite time to expiration the present value of X is zero so the lower bound is So and since the upper bound is So the call price must be So American call We know that its price cannot exceed S0 but it must be at least as valuable as a European call Thus its value must also be So Note that if exercised early it would be worth only So 7 X so it will never be exercised early 4 Effect of Time to Expiration Ordinarily the option with the longer time to expiration would sell for more If both options were deep outofthemoney however they could both sell for essentially nothing The market would be expecting that both the shorter and longerlived options will expire outofthe money 5 Effect of Exercise Price If both options were deep outofthemoney they might have prices of zero As in the previous question the two options are expected to expire outofthemoney 6 Lower Bound of a European Call The call is underpriced so buy the call sell short the stock and buy riskfree bonds with face value of X The cash received from the stock is greater than the cost of the call and bonds Thus there is a positive cash flow up front The payoffs from the portfolio at expiration are as follows Value at Expiration Transaction 1ltX IgtX Long call 0 ST X Short stock 7ST ST Long bonds X X Total X 7 ST 0 Observe that in the case where ST lt X the payoff from this portfolio is X 7 ST which is positive In the other outcome the payoff is zero Thus this portfolio produces either a positive cash flow at expiration or Chapter 3 8 EndofChapter Solutions no cash flow at all Yet it also produces a positive cash ow at the start So there is no way to lose money and a guaranteed gain of money at the start with a possibility of more at expiration 7 Effect of Time to Expiration Time value is a measure of the amount of uncertainty in an option Uncertainty relates to whether the option will expire in or outofthemoney When the option is deep in themoney there is little uncertainty about the fact that the option will expire inthemoney The option will then begin to behave about like the stock When the option is deepoutthemoney there is also little uncertainty since it is likely to finish outofthemoney When the option is atthemoney there is considerable uncertainty about how it will finish 8 Effect of Exercise Price Assuming the stock pays no dividends there is no reason to exercise a call early this obviously presumes the call is American The tendency to believe that exercising an option because the stock can go up no further ignores the fact that an option can generally be sold Exercising an option throws away any chance that the stock can go up further If the stock falls the option holder would be hurt but if the option holder exercised and became a stock holder he would also be hurt by a falling stock price There is simply no reason to give away the time value that arises because of the possibility that the stock can always go further upward In simple mathematical terms exercising captures only the intrinsic value So 7X The call can always be sold for at least S0 7X l r 39T 9 Effect of Stock Volatility The paradox is resolved by recalling that if the option expires outofthe money it does not matter how far outofthemoney it is The loss to the option holder is limited to the premium paid For example suppose the stock price is 24 the exercise price is 20 and the call price is 6 Higher volatility increases the chance of greater gains to the holder of the call It also increases the chance of a larger stock price decrease If however the stock price does end up below 20 the investor39s loss is the same regardless of whether the stock price at expiration is 19 or 1 If the stock were purchased instead of the call the loss would obviously be greater if the stock price went to 1 than if it went to 19 For this reason holders of stocks dislike volatility while holders of calls like volatility A similar argument applies to puts 10 American Put Versus European Put The minimum value of an American put is Max0 X 7 So This is always higher than the lower bound of a European put X l r 39T 7 S0 except at expiration when the two are equal ll Effect of Interest Rates When buying a call option one hopes to exercise it at a later date Thus the exercise price will be paid out later If interest rates are higher additional interest can be earned on the money that will eventually be paid out as the exercise price When buying a put option one hopes to exercise it later thus receiving the exercise price If interest rates are higher the put is less valuable because the holder is foregoing interest by having to wait to exercise the put Higher interest rates make the present value of the exercise price be lower In the case of the call this is good because the call holder anticipates having to pay out the exercise price For the put holder this is bad because the put holder anticipates receiving the exercise price 12 PutCall Parity If the put price is higher than predicted by the model the put is overpriced Then the put should be sold The funds should be used to construct a portfolio consisting of a long call a short position in the stock and a long position in riskfree bonds with face value of X The payoffs at expiration of this strategy are shown below Value at Expiration Transaction 1ltX IgtX Short put X Sr 0 Long call 0 ST X Short stock ST ST Long bonds X Total 0 0 Chapter 3 9 EndofChapter Solutions Chapter 3 10 Thus this portfolio has no cash in ow or out ow at expiration but the sale of the put will produce more cash than the cost to buy the combination of long call short stock and long bonds that replicates the purchase of a put Early Exercise of American Puts An American call is exercised early only to capture a dividend When a stock goes exdividend the call will lose value as the stock drops This will cause a loss in value to the holder of the call The call holder knows this loss will be incurred as soon as the stock goes exdividend If the call were exercised just before the stock goes exdividend however the call holder would capture the stock and the dividend which might be enough to offset the otherwise loss in the value of the call For a put however dividends are not necessary to make the argument that it might be optimal to exercise early The holder of an American put faces a situation in which the gains are limited to the exercise price Since the stock price can go down only to zero early exercise of a put on a bankrupt firm would obviously be advisable But the firm does not have to go bankrupt If the stock price is low enough the gains from waiting for it to go lower are not worth the wait If dividends were added to the picture however they would discourage early exercise The more dividends paid the lower the stock price is driven and the more valuable it is to hold on to the put Principles of Call Option Pricing a July 160 Intrinsic value Max 0 16513 7160 513 Time value 67 513 087 Lower bound T 11365 00301 1 r39T 7 105163900301 7 09985 Lower bound Max0 16513 7 16009985 537 b October 155 Intrinsic value Max 0 16513 7155 1013 Time value 14 71013 387 Lower bound T 102365 2795 1 r39T 7 10588quot 795 71 7 09842 Lower bound Max0 16513 7 15509842 12579 c August 170 Intrinsic value Max0 16513 7 170 0 Time value 320 7 0 320 Lower bound T 46365 01260 1 r39T 7 105503901260 7 09933 Lower bound Max0 16513 7 17009933 0 All prices conform to the boundary conditions so there are no profitable opportunities Principles of Put Option Pricing a July 165 Intrinsic value Max0 165 7 16513 0 Time value 235 7 0 235 Lower bound Max0 16509985 7 16513 0 b August 160 EndofChapter Solutions Chapter 3 11 Intrinsic value Max0 160 7 16513 0 Time value 275 7 0 275 Lower bound Max0 16009933 7 16513 0 c October 170 Intrinsic value Max0 170 7 16513 487 Time value 97 487 413 Lower bound Max0 17009842 7 16513 2184 All prices conform to the boundary conditions so there are no profitable opportunities PutCall Parity In each case we compute the value ofP So and compare it to the value C X 1 r39T a July 155 P So 020 16513 16533 C X 1 r T 105 15509985 1652675 Difference 0625 b August 160 P So 275 16513 16788 CX 1 r 39T 810 16009933 167023 Difference 0857 c October 170 P So 9 16513 17413 CX 1 r T 6 17009842 173314 Difference 08160 These values are supposed to be zero If arbitrage could be executed at a cost less than the indicated difference it would be advisable to do so Consider the October 170 combination The difference of 08160 suggests that a portfolio of short put long call short stock and long riskfree bonds would generate a cash in ow of 08160 with no cash out ow at expiration assuming of course that there is no early exercise PutCall Parity In each case we compute the value of C X P So and C X1 r39T The values should line up in descending order a July 155 CX 105 155 1655 P So 020 16513 16533 C X1 r T 105 15509985 1652675 These align correctly b August 160 CX 810 160 16810 P So 275 16513 16788 CX1 r 39T 810 16009933 167028 These align correctly c October 170 CX6170176 P So 9 16513 17413 EndofChapter Solutions Chapter 3 12 C X1 r39T 6 17009842 173314 These align correctly Effect of Exercise Price The difference in premiums of American calls should not exceed the difference in exercise prices Difference in Difference in Pair Premiums Exercise Prices Aug155 160 37 5 Oct 160 165 3 5 Neither pair represents a violation Effect of Exercise Price The difference in premiums of American puts should not exceed the difference in exercise prices Difference in Difference in Pair Premiums Exercise Prices Aug 155 160 15 Oct 160 170 45 10 Neither pair represents a violation Principles of CallPut Option Pricing The time period is from December 9 to January 13 is 35 days So T 35365 00959 a Intrinsic Value Max0 So 7X 7 x0 4728 4 46 128 b Lower bound Max0 501 p39T 7 X1 r39T 7 Max0 4728103639 959 4 461071390 0959 142 c Time Value call price 7 intrinsic value 63 7 128 035 d Intrinsic Value Max0 X 7 SO 7 Max0 46 4 4728 0 e Lower bound Max0 X1 r39T 7 501 p T 7 Max0 4610713900959 4 47281036o 0959 7 0 Note this is the lower bound only for a European put f Time value put price 7 intrinsic value 014 7 0 014 g c7p 501p39T7X1r39T EndofChapter Solutions Chapter 3 13 The lefthand side is 163 The right hand side is 014 4728103639 0959 7 4610713900959 15615 Putcall parity does not hold however transaction costs could account for the difference Lower Bound of a European Call The most likely arbitrage opportunity is a violation of a lower boundary condition With dividends the lower boundary for a call is expressed as Cs0TX2Maxlos0 7D7X1VTTJ 1141140100471001005 1J 376 Note that the present value of the dividend is D because the dividend is paid in an instant Because the lower boundary is higher than the quoted call price of 35 there is an arbitrage opportunity One method to assess the appropriate trading strategy is to rearrange the boundary condition such that one side is greater than zero In this case the boundary is nonzero and it is violated therefore s0 7D7X1r 7 7CZS0TXgt 0 or in this case 1007171001005 1 735 026 gt 0 To generate 026 in cash flow today execute the trades suggested by their symbols short sell stock lend the present value of the strike price and buy the call option In the next instant the short seller must pay the dividend One way to demonstrate that this is an arbitrage is to create a cash flow table lt Therefore 026 per share is generated today without any chance of a future liability This is a money machine and clearly an arbitrage opportunity Notice that selling pressure on the stock will drive its price down and buying pressure on the call will drive its price up Hence this trading activity will eventually eliminate the arbitrage opportunity Lower Bound of a European Put The most likely arbitrage opportunity is a violation of a lower boundary condition With dividends the lower boundary for a put is expressed as PsOTX2 Max0X1rtT is D 114111401000005 1 7901 624 Again we see that the present value of the dividend is D because it occurs in an instant Because the lower boundary is higher than the quoted call price of 6 there is an arbitrage opportunity One method to assess the appropriate trading strategy is to rearrange the boundary condition such that one side is greater than zero In this case the boundary is nonzero and it is violated therefore X1r T is D7PZS0TXgt 0 EndofChapter Solutions or in this case 1001005 1 7901760 024 gt 0 To generate 024 in cash flow today execute the trades suggested by their symbols borrow the present value of the exercise price buy stock and buy the put option In the next instant the stock buyer will receive the dividend One way to demonstrate that this is an arbitrage is to create a cash flow table lt nonnegative due to Therefore 024 per share is generated today without any chance of a future liability This is a money machine and clearly an arbitrage opportunity Notice that buying pressure on the stock will drive its price up and buying pressure on the put will drive its price up Hence this trading activity will eventually eliminate the arbitrage opportunity 23 PutCall Parity Putcall parity is P SOT X 50 7 CCSOTX X1 r 392 so 50 7 CCSOTX 7 PCS0TX Xl r 39T So the equivalent of a position in a quotpointquot would be to buy a call sell a put and invest Xlr39T dollars in riskfree bonds paying X dollars at the end of the game This is demonstrated below where ST is the number of points at the end of the game Payoff at Expiration Transaction S lt X S gt X Point ST ST Buy call 0 ST 7 X Sell put X 7 ST 0 Buy bonds X X Total ST ST Notice that at the beginning of the game the call starts off outofthemoney and the put starts off inthe money This is because the point total is at its minimum and can only grow By the way the options have to be European or holders of American puts would exercise theirs immediately At any given time the above formula for S0 must equal the number of points Thus at the beginning of the game So 0 and C SOTX 7 PCSOTX 7x0 r39T 24 PutCall Parity Remember that put call parity says that P S0TX S0 C S0TX Xl r39T To isolate a short position in the stock we need to get 7 So on one side of the equation This would be PeS0TX 7 CeS0TX 7 Xl r39T 7 So So to do the equivalent of a short sale you would buy a put sell a call and borrow the present value of the exercise price The table below demonstrates this point Chapter 3 Endof Chapter Solutions Chapter 3 Transaction Short sale Long put Short call B orrow T otal Payoff at Expiration 1ltX Eli X isT isT X 7 ST 0 0 ST X i ST ST 15 Endof Chapter Solutions