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# VALUATION OF FIN ASSETS FI 8000

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This 7 page Class Notes was uploaded by Nellie Simonis on Monday September 21, 2015. The Class Notes belongs to FI 8000 at Georgia State University taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/209808/fi-8000-georgia-state-university in Finance at Georgia State University.

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Date Created: 09/21/15

Option Payoffs Problems 1 through 10 Assume that the stock is currently at 20 per share and options and bonds have the prices given in the table below Depending on the strike price 0 of the option or the face value FV of the bond do the followmg a Sketch the payoff and pro t diagrams for the strategy b Describe the pro table range in terms of the price of the underlying security c Describe the maximum potential pro m and losses X or FV all Premium Put Premium Bond Price 5 15 010 450 10 1 1 80 090 900 15 900 260 1360 20 690 500 1810 25 540 800 2260 30 430 1140 2710 35 340 1510 3170 1 Short one call X 1 0 Long one bond F V1 0 2 Long one caJLX20 Short one call X30 3 Long one share of stock short one callX20 4 Long one puLX15 Long one calX25 5 Short one putX15 Long one call X20 6 Long one puLX10 Short one put X20 Long one caJLX30 7 Short one put X 1 0 Long one put X 20 Short one call X 20 8 Long two calls X 25 Short one call X 1 0 Long one bond F V1 0 9 Long one puLX10 Short one put X15 Short one call X25 Long one call X30 10 Short one share of stock Short one put X 20 Long one bond F V25 Long one calLX25 Problems 11 through 16 Describe as I have in 110 the strategy depicted by each payoff diagram paw7 11 paw7 12 paw7 13 20 20 20 10 10 10 0 s s 0 3 10 20 10 20 20 10 10 10 paw7 14 paw7 15 paw7 16 20 20 20 10 10 10 0 S 0 S 0 A S 10 20 0 20 10 10 10 Fi8000 Practice Set 1 Arbitrage 17 Suppose that there are two call options written on the same underlying asset Call A has a strike price of r 00 r N N r N N N 50 and Call B has a strike price of 60 Call A s premium is 525 and Call B s premium is 650 Describe the arbitrage opportunity that takes advantage of these prices and prove that this is an arbitrage strategy H int draw the payoff diagrams Suppose that there are two put options written on the same underlying asset Put Y has a strike price of 30 and PutZ has a strike price of 40 Put Y s premium is 325 and Put Z s premium is 250 Describe the arbitrage opportunity that takes advantage of these prices and prove that this is an arbitrage strategy H int draw the payoff diagrams Suppose that you know that strike price X2 is greater than strike price X1 ie X1 lt X2 Generalizing the resulm from problems 17 and 18 what conditions must hold in equilibrium for the call option premiums in terms of X1 and X2 assuming the same underlying asset and time to expiration That is which call option should have the higher premium What about the put option premiums Suppose that a stock s current price is 88 and the riskfree interest rate is 10 Oneyear European call options with a strike price of 90 trade for 350 Is there an arbitrage opportunity here If so prove it Suppose that a stock39s current price is 78 Options written on the stock have a strike price of 80 and expire in 6 months The price of the call option is 6 and the price of the put option is 8 The riskfree interest rate is 5 per year a Show how you could take advantage of these prices to earn an arbitrage pro t b Taking the stock price call price and interest rate as given what is the equilibrium put premium c Taking the stock price put price and interest rate as given what is the equilibrium call premium d Taking the call price put price and interest rate as given what is the equilibrium stock price e Taking the stock price put price and call price as given what is the equilibrium riskfree rate Look up a stock in the Wall Street Journal that has options actively traded on it you should start by looking in the options section of the WSJ If possible find a stock that will not pay a dividend prior to the expiration of options and use options with at least 1 month to expiration a Using putcall parity find the value of a synthetic put b Using putcall parity find the value of asynthetic call c How far off are your premiums for the synthetic options versus the actual options Is there an arbitrage opportunity Why or why not Suppose that there are options written on ABC stock with two strike prices X1 60 and X265 The current premiums of the call options are CX110 and CX26 ie the premium on the call option with a strike price of X1 60 is 10 The current put premiums are PX1350 and PX2450 The rileEree interest rate is 6 All options expire in one year Show how you could take advantage of these prices to earn an arbitrage pro t H int Note that you cannot trade the ABC stock directly since you don t know what is price is There will be five possible outcomes at expiration Suppose that there are options written on XYZ stock with two strike prices X1 1 00 and X2110 One share of XYZ stock currently trades for 105 The current premiums of the call options are CXl 25 and CX2 15 ie the premium on the call option with a strike price of X1 1 00 is 25 The current put premiums are PX1 12 and PX2 10 Show how you could take advantage of these prices to earn an arbitrage pro t H int This problem is similar in nature to the previous problem What price don39t you know Fi8000 Practice Set 1 2 Binomial Option Pricing Model Problems 25 through 31 The de nitions of the variables are Scurrent stock priceXstrillte price of the options Urmultiplier for up movements in the stock price each period rinterest rate per period Tmber of periods until expiration Problem S X U r T 25 53 50 108 4 1 26 51 50 108 4 1 27 53 55 108 4 1 28 53 50 1 1 2 4 1 29 53 50 108 6 1 30 53 50 108 4 2 31 53 55 108 4 2 25 Use the singleperiod binomial option model to answer the following questions N 0 N u N 00 N O L 0 a What is the equilibrium call premium b What is the equilibrium put premium Use the singleperiod binomial option pricing model to answer the following questions a What is the equilibrium call premium b What is the equilibrium put premium c Compare your answers with 25 What can you conclude about how option prices change with a change in the price of the underlying stock Use the singleperiod binomial option pricing model to answer the following questions a What is the equiliban call premium b What is the equilibrium put premium c Compare your answers with 25 What can you conclude about how option prices change with a change in the strike price of the option Use the singleperiod binomial option model to answer the following questions a What is the equilibrium call premium b What is the equilibrium put premium c Compare your answers with 25 What can you conclude about how option prices change with a change in the volatility of the underlying stock Use the singleperiod binomial option pricing model to answer the following questions a What is the equilibrium call premium b What is the equilibrium put premium c Compare your answers with 25 What can you in the riskfree rate I 1 L 4hOW Op onr 39 1 39q 1 Use the twoperiod binomial option pricing model to answer the following questions a What is the equilibrium European call premium b What is the equilibrium European put premium c What is the equilibrium American call premium d What is the equilibrium American put premium e Compare your answers with 25 What can you conclude about how option prices change with a change inthe time to expiration Fi8000 Practice Set 1 3 L i L N L L L 4 L U Use the twoperiod binomial option model to answer the following questions a What is the equilibrium European call premium b What is the equilibrium European put premium c What is the equilibrium American call premium d What is the equilibrium American put premium e From your answers above what can you conclude about the value of the option to exercise early for an American call For an American put Suppose the market price for the call option in 25 is 475 Show an arbitrage strategy that takes advantage of this m1spnc1ng H int This is not a putcall parity question Do not assume that a put option exism The only asses in the market are a stock a bond riskless borrowing and lending and a call option Suppose the market price for the call option in 25 is 5375 Show an arbitrage strategy that takes advantage of this mispnc1ng H int This is not a putcall parity question Do not assume that a put option exism The only asses in the market are a stock a bond riskless borrowing and lending and a call option Suppose the market price for the put option in 25 is 050 Show an arbitrage strategy that takes advantage of this m1spnc1ng H int This is not a putcall parity question Do not assume that a call option exism The only asses in the market are a stock a bond riskless borrowing and lending and a put option Suppose the market price for the call option in 31 is 400 Show an arbitrage strategy that takes advantage of this mispricing H int This is not a putcall parity question Do not assume thata call option exism The only asses in the market are a stock a bond riskless borrowing and lending and a call option Also you must illustrate your cash ows at every date and in every possible state of the world at each date At time 1 there are two possible states At time 2 there are three possible states Be sure to describe how your portfolio is rebalanced in each state at time 1 BlackScholes Option Pricing Model Problems 36 through 42 View these problems in terms of the BlackScholes option model You may want to use the Black Scholes spreadsheet thatl have provided on the course web site You will need to program the Black Scholes model into your own spreadsheet in order to answer questions 38 and 39 36 L 0 Suppose the value of the underlying asset increases what impact does this have on a the call premium Why b the put premium Why Suppose the strike price of the option increases what impact does this have on a the call premium Why b the put premium Why Suppose the volatility of the underlying asset increases what impact does this have on a the call premium Why b the put premium Why Suppose the riskfree interest rate decreases what impact does this have on a the call premium Why b the put premium Why Fi8000 Practice Set 1 4 40 Suppose the time to expiration of an option decreases as would happen as the option would get closer to expiration while no other key variables change what impact does this have on a the call premium Why b the put premium Why 4 i Suppose that a stock currently trades for 76125 You know that the true standard deviation of the stock39s returns is exactly 415 The call option expires in 70 days and has a strike price of 75 The riskfree rate of return is 41 The market price of the call is currently 550 a Is the market39s assessment of the volatility of the underlying asset too high or too low Why b What is the market39s estimate of the volatility of the stock according to the market prices of the call and the stock H int You might need to use the 39Solver function in Microso Excel in order to 39back out the implied volatility 4 N Use the Wall StreetJournal or an internet quote service to obtain data for the following questions For the riskfree rate of return use the ask yield on the Tbill that matures in the same month or closest as the month of expiration for the options a Find an options contract on a 39Bluechip39 company such as Boeing or General Electric Using the market prices determine the implied volatility the sigma that the market is using for the call option that is closest to being atthemoney and that expires in one to three months Try not to use a firm that pays dividends H int You will need to use the 39Solver function in Microso Excel in order to 39back out39 the implied volatility b Find an options contract on an internet company Using the market prices determine the implied volatility the sigma that the market is using for the call option that is closest to being atthemoney and that expires in one to three months H int You will need to use the 39Solver function in Microso Excel in order to 39back out the implied volatility c Compare your answers to a and b above Is this what you would expect Why or why not Fi8000 Practice Set 1 5 Futures and Forward Contracts 1 N 9 Je V Follow the price of an SampP 500 Futures contract traded on the CME for 5 days Use the contract that calls for delivery in a month or so Assume that the initial margin is 20000 and the variationmargin is 15000 per contract Make a table to track the market value of the contract the daily profit loss your margin account balance and your 39cash in when there are margin calls Do this twice once for a long position in 5 contracts and the second time for a short position in two contracts What are the profits from the two strategies Suppose the current spot price of gold is 387 per ounce There is a oneyear forward contract for 1000 ounces ofgold The oneyear forward price is 460 per ounce The interest rate is 12 per year for borrowing and lending a Are these equilibrium prices If not show the arbitrage profit from trading one forward contract If so explain why b Solve for the equilibrium forward price taking the spot price as given c Solve forthe equilibrium spot price taking the forward price as given d Solve for the equilibrium cost of carry interest rate taking the forward and spot prices of gold as given Suppose the current spot price of gold is 205 per ounce There is a oneyear forward contract for 1000 ounces ofgold The oneyear forward price is 224 per ounce To borrow it will cost you 101 per year To lend you will earn an interest rate of87 per year a Are these equilibrium prices If not show the arbitrage profit from trading one forward contract If so explainw y b Solve for the equilibrium futures price bounds taking the spot price as given c Solve for the equilibrium spot price bounds taking the forward price as given d What happens to the equilibrium bounds for the forward price if the spread between the lending and borrowing rate increases For example suppose the lending rate is only 85 Are the equilibrium forward price bounds wider than in Suppose that when the stock market closes Friday Feb 18 the SampP 500 is valued at 135500 At the same time the Mar00 futures contract on the SampP 500 is trading at 136108 Assume that interest is compounded daily and that we have perfect markets same borrowing and lending rates etc Treat the futures contract as a forward contract ie ignore the implications of a margin account a Just before the stock market opens in the morning on Tuesday Feb 22 you see that the SampP 500 futures are trading at 135822 lfthe futures remain at this level what change do you expect to see in the stock market specifically in the SampP 500 index b Just before the stock market opens in the morning on Tuesday Feb 22 you see that the SampP 500 futures are trading at 137830 lfthe futures remain at this level what change do you expect to see in the stock market specifically in the SampP 500 index c What would 39fair value39 be for the futures contract In other words ifthe stock market were to open at exactly the same price on Tuesday morning as it closed on Friday afternoon what would the futures contract value be Suppose the current spot price ofgold has a bid price of 401 per ounce and an ask price of 403 There is a one year forward contract for 1000 ounces ofgold The oneyear forward price is bid at 431 per ounce and offered asked at 433 per ounce To borrow it will cost you 84 per year To lend you will earn an interest rate of83 per ear a Are these equilibrium prices If not show the arbitrage profit from trading one forward contract If so explain wh b Solve for the equilibrium futures price bounds taking the spot price as given c Solve for the equilibrium spot price bounds taking the forward price as given d Show that a decrease in the bidask spread in the spot market will decrease the bounds on the equilibrium forward price To do this assume that the bid price increases by 050 and the ask price decreases by 050 Practice Set 2 1 Foreign Exchange 6 Suppose the current spot exchange rate between a US and UK is 16746 USUK The spot exchange rate between US and German Mark DM is 05785 USDM In equilibrium what should the direct exchange rate be for UK to DM 7 Look up in the Wall Street Journal or possibly the AJC the exchange rate between the US and the following countries currency Canada France J apan and Switzerland Fill in the following table for direct conversion of currencies you should check that your obtained answers are very close to the cross rates reported in the Wall StreetJournal under 39Key Currency Cross Rates39 Country Canada France Japan Switzerland Canada 10000 CaCa CaFF Ca CaSF France FFCa 10000 FFFF FF FFSF Japan Ca FF 10000 SF Switzerland SFCa SFF F SF 10000 SF SF 8 Suppose that you have the following information i Perfect markets le no transactions costs and no restrictions on shortselling ii The spot exchange rate is 13425 US iii The 1year forward exchange rate is 12950 US iv The 1year return on denominated riskfree bonds in the US is 485 v The 1year return on denominated riskfree bonds in Japan is 430 Answer the following questions a Is there an arbitrage opportunity If so show how you would take advantage of it b Assuming that iiv are correct what is the equilibrium no arbitrage rate of return on denominated riskfree Japanese bonds c Assuming that iiii and v are correct what is the equilibrium no arbitrage rate of return on denominated riskfree US bonds d Assuming that i ii iv and v are correct what is the equilibrium no arbitrage forward price of US in terms of in terms of US e Assuming that i and iiiv are correct what is the equilibrium no arbitrage spot price of the US in terms of the in terms ofUS 9 Suppose that you have the following information i Perfect markets le no transactions costs and no restrictions on shortselling ii The spot exchange rates are 13550 US and 16750 USUK iii The 1year forward exchange rates are 13490 US and 1 6600 USUK iv The 1year return on denominated riskfree bonds in the UK is 595 v The 1year return on denominated riskfree bonds in Japan is 410 Is there an arbitrage opportunity lfso show how you would take advantage of it by showing the exact arbitrage transactions that you would use Hint You have no information on the direct conversion rates between and Therefore you may not assume any exchange rates other than those given Practice Set 2 2

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