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# VALUATION OF FIN ASSETS FI 8000

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This 11 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 Isabel Tkatch in Fall. Since its upload, it has received 33 views. For similar materials see /class/209803/fi-8000-georgia-state-university in Finance at Georgia State University.

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Date Created: 09/21/15

Fi8000 Valuation of Financial Assets Spring Semester 2009 Dr Isabel Tkatch Assistant Professor of Finance Today Review of the Definitions Arbitrage Restrictions on Options Prices The PutCall Parity European Call and Put Options American vs European Options to Monotonicity of the Option Price to Convexity of the Option Price A Call Option A European call option gives the buyer of the option the right to purchase the underlying asset at the exercise price on expiration date It is optimal to exercise the call option if the stock price exceeds the strike price 0 Max 87 X 0 Buying a Call Payoff Diagram Stock price Payoff ST MaxS X 0 0 0 5 0 10 0 15 0 20 0 25 5 30 10 A Put Option A European put option gives the buyer of the option the right to seIIthe underlying asset at the exercise price on expiration date It is optimal to exercise the put option if the stock price is below the strike price P Max X 8 0 Buying a Put Payoff Diagram Stock price Payoff m ST MaxXST0 i Pr 0 20 I 5 15 j 10 10 15 5 w 20 0 2 25 0 j 30 0 The Put Call Parity Compare the payoffs of the following strategies 9 Strategy I a Buy one call option str ke X expiration T a Buy one riskfree bond face value X maturity T return n 9 Strategy ll 0 Buy one share of stock a Buy one put option strike X expiration T Strategy Portfolio Payoff Stock Buy All Strategy Portfolio Payoff Stock Buy Buy A price Stock Put Puma 0 0 20 20 5 5 15 20 10 10 10 20 Jmnmnmun 15 15 5 20 20 20 0 20 25 25 0 25 30 30 0 30 The Put Call Parity If two portfolios have the same payoffs in every possible state and time in the future their prices must be equal CPVXSP Arbitrage the Law of One Price If two assets have the same payoffs in every possible state in the future and their prices are not equal there is an opportunity to make arbitrage profits We say that there exists an arbitrage opportunity if we identify that There is no initial investment There is no risk of loss There is a positive probability of pro t Arbitrage a Technical Definition Let CF be the cash flow of an investment strategy at timet and state j Ifthe following conditions are met this strategy generates an arbitrage profit i all the possible cash flows in every possible state and time are positive or zero CF 2 0 for every t and j ii at least one cash flow is strictly positive there exists a pair t j forwhich CF gt 0 Arbitrage Example Is there an arbitrage opportunity if we observe the following market prices The price of one share of stock is 39 The price of a call option on that stock which expires in one year and has an exercise price of 40 is 725 The price of a put option on that stock which expires in one year and has an exercise price of 40 is 650 The annual risk free rate is 6 Arbitrage Example In this case we must check whether the put ca parity holds Since we can see that this parity relation is violated we will show that there is an arbitrage opportunity CL 7254 0 44986 1rfT 10061 SP 39 650 455 Construction of an Arbitrage Transaction Constructing the arbitrage strategy 1 Move all the terms to one side of the equation so their sum will be positive N For each asset use the sign as an indicator of the appropriate investment in the asset lfthe sign is negative then the cash ow at time t0 is negative which means that you buythe stock bond or option lfthe sign is positive reverse the position Arbitrage Example In this case we move all terms to the LHS SP7C j4557449860514gt0 119 SP7C7 ix 7 Arbitrage Example In this case we should 1 Sell short one share of stock 2 Write one put option 3 Buy one call option 4 Buy a zero coupon riskfree bond lend Arbitrage Example t Arbitrage Example t Arbitrage Example t Arbitrage Example Continued Is there an arbitrage opportunity if we observe the following market prices The price of one share of stock is 37 The price of a call option on that stock which expires in one year and has an exercise price of 40 is 725 The price of a put option on that stock which expires in one year and has an exercise price of 40 is 650 The annual risk free rate is 6 Arbitrage Example Continued In this case the put call parity relation is violated again and there is an arbitrage profit opportunity mi 725 44986 1 rfT 1 0061 SP 3750435 Arbitrage Example Continued In this case we get ma 04594498674351486gt0 119 C7S7PgtO Arbitrage Example Continued a sip c Xlrf gt The Value of a Call Option Assumptions 1 A European Call option 2 The underlying asset is a stock that pays no dividends before expiration 3 The stock is traded 4 A risk free bond is traded Arbitrage restrictions Max SPVX 0 lt CEU lt S The Value of a Call Option MaxSPVX 0 lt CEU lt s C I the owner has a right but not an o 0 lt obligation SPVX lt C I arbitrage proof C lt S I you will not pay more than 8 the market price of the stock for an option to buy that stock for X Buying the stock itself is always an alternative to buying the call option The Value of a Call Option Example The current stock price is 83 0 The stock will not pay dividends in the next six A call option on that stock is traded for 3 0 The exercise price is 80 0 The expiration of the option is in 6 months The 6 months risk free rate is 5 Is there an opportunity to make an arbitrage profit Example Continued t a The Value of a Call Option 3 The call option price is bounded Max SPVX 0 lt CEU lt S 3 The Call option price is monotonically increasing in the stock price 8 If si then 08 3 The call option price is a convex function of the stock price 8 see sketch The Value of a Call Option y SPVX The Value of a Call Option Exercise Prices Assume there are two European call options on the same stock 8 with the same expiration date T that have different exercise prices X lt X2 Then CX4 gt CX2 e the price of the call option is monotonically decreasing in the exercise price if XT then CX1 Example Show that if there are two call options on the same stock that pays no dividends and both have the same expiration date but different exercise prices as follows there is an opportunity to make arbitrage profits x1 40 and X2 50 C1 3 and Oz 4 Example Continued t Option Price Convexity Assume there are three European call options on the same stock 8 with the same expiration date T that have different exercise prices X lt X2 lt X3 If X2dX lCl3 Then CX2 lt doom 1 acX3 e the price of a call option is a convex function of the exercise price Example Show that if there are three call options on the same stock that pays no dividends and all three have the same expiration date but different exercise prices as follows thew is an opportunity to make arbitrage profits x1 40 x2 50 and x3 60 0 46 02 4 and cs 3 Example Continued t 0 tT X1ltsltX2 x2ltsltX3 sgtx36u ST XO Sr xi Srrxi 0 arisrxz 0 0 The Value of a Call Option 3 The call option price is bounded MaxS PVX 0 lt CEU lt S Q The Call option price is monotonically decreasing in the exercise price X Ith then CXu The call option price is a convex function ofthe exercise price X The Value of a Call Option The Value of a Call Option The Value of a Call Option mm 1 Two Call options European and American 2 The underlying asset is a stock that pays no dividends before expiration 3 The stock is traded 4 A risk free bond is traded Arbitrage restriction CEuropean CAmerican m compare tne payoff rrom immediate exercise to tne lower bound ofthe European caii option price ifCE lt CM tnen you can make arbitrage profits but tne strategy is dynamic and inyoiyes transactions in tne present and in a future date t lt T Example There are two call options on the same stock that pays no dividends one is American and one is European Both have the same expiration date a year from now T 2 and exercise price X 100 but the American option costs more than the European CE 5 lt 6 Cm Assume that the buyer of the American call option considers to exercise after 6 months t 1 Show that if the semiannual interest rate is if 5 then there is an opportunity to make arbitrage profits Exam le Continued lfthe American Call is not Exercised p before Expiration Since theno arbitrage restrictiog is CEu CAm but the t0 t2 T et prices are Eu Am we can make arbitrage profits if we buy the cheap option C 5 a and sell write the expensive one CM 6 Eu S39lt X 100 S X 100 If the buyer ofthe American call option decides to 5 er exercise betore expiration datet lt I we should rememberthat CAquot S X lt SPVX lt CEquot and 1sell the stock 2buy a bond lfthe buyer ofthe American call option decides to 0 exercise only on expiration date then the future CFs of e American and European call options will cancel eac 0 er Si X lfthe American Call is Exercised before Expiration on datet lt T The value Of a Put Optlon lTlmea t0 ti t2T W3 7 7 7 7 1 A European put option Strategy 1 state a 5 ltgtltelEIEI s gtgtltelEIEI sltx emu SYgtgtltelEIEI 2 The underlying asset is a Stock that pays no mfg rcEuzs o o 0 Sex dividends before expiration SellAm Call 7 3 The stock is traded date 1 CW 6 0 78 7 X 0 0 4 A risk free bond is traded SEllStpck 8 VSY VSY SEEM Arbitrage restrictions 123 x mm mm CWCEU FVWSY FVWX Max PVX S 0 lt PEU lt PVX TutalCF 675gtE n n gtxrsgtn gtn The Value of a Put Option The Value of a Put Option Max PVXS O lt PEU lt PVX Example The current stock price is 75 0 lt P I the owner has a right but not an i The stock will not pay dividends in the next six obligation months PVXS lt P arbitrage proof A put option on that stock is traded for 1 i The exercise price is 80 P lt 39 the h39gheSt prom mime pm i The expiration ofthe option is in 6 months option is realized when the stock price is zero 0 That profit is sex and it is realized on date T The 6 months risk free rate IS 54 h39h39 39lttPVXtd W lo IS equwa en 0 as o ay lsthere an opportunity to make arbitrage profits Example Continued t The Value of a Put Option 3 The put price is bounded Max PVXS O lt PEU lt PVX The put option price is monotonically decreasing in the stock price 8 If ST then PSi The put option price is a convex function ofthe stock price 8 see sketch 0 The Value of a Put Option PVX PVXS The Value of a Put Option PVX PVXS Exercise Prices Assume there are two European put options on the same stock 8 with the same expiration date T that have different exercise prices X lt X2 Then PXi lt F X2 e the price of the put option is monotonically increasing in the exercise price if XT then PXT Example Show that if there are two put options on the same stock that pays no dividends and both have the same expiration date but different exercise prices as follows there is an opportunity to make arbitrage profits x1 40 and x2 50 P1 4 and P2 3 Example Continued Option Price Convexity Assume there are three European put options on the same stock 8 with the same expiration date T that have different exercise prices X1lt X2 lt X3 If X2GX1ICl3 Then F X2 lt GPX4 1 GF X3 e the price of a put option is a convex function of the exercise price Example Show that if there are three put options on the same stock that pays no dividends and all three have the same expiration date but different exercise prices as fwlows thew is an opportunity to make arbitrage profits x1 40 x2 50 and x3 60 P1 3 P2 4 and P3 46 The Value of a Put Option 3 The put price is bounded Max PVXS O lt PEU lt PVX The put option price is monotonically increasing in the exercise price X IfXT then PXT The put option price is a convex function ofthe exercise price X Example Continued tT ltx14u x1ltsltx2 x2ltsltx3 sgtx3 x4784 0 The Value of a Put Option PVXS The Value of a Put Option woosl PVXS The Value of a Put Option Assumptions 1 Two Put options European and American 2 The underlying asset is a stock that pays no dividends before expiration 5 me SIOCK IS traded 4 A risk free bond is traded For a put option regardless of dividends PEuropean S PAmerican Note ll i this case an example is enough Determinants of the Values of Call and Put Options Call Value Put Value stock price We stock price volatility timeto expiration nskrfree interest rate dividei id payouts Practice Problems BKM Ch 21 7th Ed End of chapter 1 2 Example 211 and concept check Q 4 8th Ed End of chapter 1 6 Example 211 and concept check Q 4 Practice set 17 24

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