C T P T S T

Transcription

1 Fi8 Valuation of Financial Assets pring emester 21 Dr. Isabel Tkatch Assistant Professor of Finance Today Review of the Definitions Arbitrage Restrictions on Options Prices The Put-Call Parity European Call and Put Options American vs. European Options Monotonicity of the Option Price Convexity of the Option Price A Call Option Buying a Call Payoff Diagram 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: C T = Max { T, } tock price = T Payoff = Max{ T -, } C T T A Put Option Buying a Put Payoff Diagram A European put option gives the buyer of the option the right to sell the 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 T = Max { - T, } tock price = T Payoff = Max{- T, } P T T 1

2 The Put Call Parity trategy I Portfolio Payoff Compare the payoffs of the following strategies: trategy I: Buy one call option (strike=, expiration= T) Buy one risk-free bond (face value=, maturity= T, return= rf) trategy II Buy one share of stock Buy one put option (strike=, expiration= T) tock price Buy Call Buy Bond All (Portfolio) trategy II Portfolio Payoff The Put Call Parity tock price Buy tock Buy Put All (Portfolio) If two portfolios have the same payoffs in every possible state and date in the future, their prices must be equal: C+ PV( ) = + P 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 profit Arbitrage a Technical Definition Let CF tj be the cash flow of an investment strategy at time t and state j. If the 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 tj for every t and j. (ii) at least one cash flow is strictly positive - there exists a pair ( t, j ) for which CF tj >. 2

3 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 h expires in one year and has an exercise price of $4, is $7.25; The price of a put option on that stock, which expires in one year and has an exercise price of $4, is $6.5; The annual risk free rate is 6%. Arbitrage Example In this case we must check whether the put call parity holds. ince we can see that this parity relation is violated, we will show that there is an arbitrage opportunity. $4 C + = $ = $ T 1 (1 + rf ) (1 +.6) + P = $39 + $6.5 = $45.5 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; 2. For each asset, use the sign as an indicator of the appropriate investment in the asset: If the sign is negative then the cash flow at time t= is negative (which means that you buy the stock, bond or option). If the sign is positive reverse the position. Arbitrage Example In this case we move all terms to the LH: T (1 + rf ) ie.. + P C > (1 + rf ) T ( + P) C+ = $45.5 $ = $.514 > Arbitrage Example In this case we should: 1. ell (short) one share of stock 2. Write one put option 3. Buy one call option 4. Buy a zero coupon risk-free bond (lend) Arbitrage Example Time: t = hort stock Write put Buy call Buy bond tate: T < = 4 T > = 4 CF CF T1 CF T2 3

4 Arbitrage Example Arbitrage Example Time: t = Time: t = tate: T < = 4 T > = 4 tate: T < = 4 T > = 4 hort stock +=$39 hort stock +=$39 - T - T Write put +P=$6.5 Write put +P=$6.5 -(- T ) Buy call -C=( C=(-$7.25) Buy call -C=( C=(-$7.25) ( T -) Buy bond -/(1+rf)=( /(1+rf)=(-$37.736) Buy bond -/(1+rf)=( /(1+rf)=(-$37.74) +P-C-/(1+rf) = P C - /(1+rf) =.514 > - T -(- T ) + = - T +( T -) + = Arbitrage 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 h expires in one year and has an exercise price of $4, is $7.25; The price of a put option on that stock, which expires in one year and has an exercise price of $4, is $6.5; The annual risk free rate is 6%. Arbitrage In this case the put call parity relation is violated again, and there is an arbitrage profit opportunity. $4 C + = $ = $ T 1 (1 + rf ) (1+.6) + P= $37 + $6.5 = $43.5 Arbitrage In this case we get C+ + P = T (1 + rf ) = > ie.. C+ P> (1 + rf ) T ( ) $ $43.5 $1.486 Arbitrage Time: t = Long stock Buy put Write call ell bond tate: T < = 4 T > = 4 -=$37 + T + T -P=( P=(-$6.5) (- T ) +C=$7.25 -( T -) +/(1+rf)=$ P + C + /(1+rf) = > T + (- T ) - = T -( T -) - = 4

5 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{ -, } < C EU < Max{ -, } < C EU < < C : the owner has a right but not an obligation. - < C : arbitrage proof. C < : you will not pay more than $, the market price of the stock, for an option to buy that stock for $. Buying the stock itself is always an alternative to buying the call option. C - Example: The current stock price is $83 The stock will not pay dividends in the next six months A call option on that stock is traded for $3 The exercise price is $8 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? Time: t = hort stock Buy call Buy bond tate: T < = 8 T = 8 +=$83 - T - T -C=( C=(-$3) ( T -) -/(1+rf)=( /(1+rf)=(-$76.19) C-/(1+rf) = $3.81 > - T + > - T +( T -)+ = The call option price is bounded Max{ -, } < C EU < The Call option price is monotonically increasing in the stock price If then C() The call option price is a convex function of the stock price (see sketch). 5

6 Exercise Prices C - Assume there are two European call options on the same stock, with the same expiration date T, that have different exercise prices 1 < 2. C() Then, C( 1 ) > C( 2 ) I.e., the price of the call option is monotonically decreasing in the exercise price (if then C() ). Example how 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. 1 = $4 and 2 = $5 C 1 = $3 and C 2 = $4 Time: t = trategy: tate: T < 1 = 4 1 < T < 2 T > 2 = 5 Buy Call 1 -C 1 =(-$3) ( T - 1 ) ( T - 1 ) ell Call 2 +C 2 =$4 -( T - 2 ) -C 1 + C 2 = $1 > = ( T - 1 ) = T -4 > 2-1 = 5-4 > Option Price Convexity Assume there are three European call options on the same stock, with the same expiration date T, that have different exercise prices 1 < 2 < 3. If 2 = α 1 +(1 (1- α) ) 3 Then C( 2 ) < αc( 1 ) + (1- α)c( 3 ) I.e., the price of a call option is a convex function of the exercise price. Example how 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, there is an opportunity to make arbitrage profits. 1 = $4, 2 = $5 and 3 = $6 C 1 = $4.6, C 2 = $4 and C 3 = $3 6

7 Time: t = Buy Call 1 (one unit) ell Call 2 (two units) Buy Call 3 (one unit) tate: T < 1 = 4 1 < T < 2 2 < T < 3 T > 3 = 6 -C 1 = ( T - 1 ) ( T - 1 ) ( T - 1 ) 1 T 1 T 1 T 1 2C 2 = 8-2*( T - 2 ) -2*( T - 2 ) -C 3 = -3 ( T - 3 ) C1-2C2+C3 =.4 > = = T - 1 = T - 4 > = T = 6 - T > =1-4-6 = The call option price is bounded Max{ -, } < C EU < The Call option price is monotonically decreasing in the exercise price If then C() The call option price is a convex function of the exercise price. C C - - C() Assumptions: 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: C(European) = C(American) Hint: compare the payoff from immediate exercise to the lower bound of the European call option price. If C Eu < C Am then you can make arbitrage profits, but the strategy is dynamic and involves transactions in the present and in a future date t < 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 ( = 1) but the American option costs more than the European (C Eu = 5 < 6 = C Am ). Assume that the buyer of the American call option considers to exercise after 6 months (t = 1). how that if the semi-annual interest rate is rf = 5% then there is an opportunity to make arbitrage profits. 7

8 ince the no-arbitrage restriction is C Eu = C Am but the market prices are C Eu = $5 < $6 = C Am, we can make arbitrage profits if we buy the cheap option (C Eu = $5) and sell (write) the expensive one (C Am = $6). If the buyer of the American call option decides to exercise before expiration (date t < T), we should remember that C Am = - < - < C Eu and (1)sell the stock, (2)buy a bond. If the buyer of the American call option decides to exercise only on expiration date, then the future CFs of the American and European call options will offset each other. If the American Call is not Exercised before Expiration Time: t = t = 2 =T Buy Eu Call (date t=) ell Am Call (date t=) tate: T < = 1 T > = 1 -C EU EU = -5 ( T -) C AM = 6 -( T -) C AM -C EU = 6-5 > = = If the American Call is Exercised before Expiration on date t < T Time: t = t = 1 t = 2 =T Buy Eu Call (date t=) ell Am Call (date t=) ell tock (date t=1) Buy Bond (date t=1) tate: t < =1 t > =1 T < =1 T > =1 -C EU = -5 ( T -) C AM = 6 -( t -) C AM -C EU = 6-5 > = = t - T - T - FV() = FV() - T > - T > FV() = FV() - > Application ay only a European option is traded in the market and on date t=1 you really wish you could exercise since the price is much lower than the strike (say 1 =15). Describe a strategy that will be as good as (if not better than) exercising a call option before expiration. Intuition short stock and long bond will generate at least the same payoff as exercising an American call option. We can use this strategy to exercise the European call option. Note: if you own an American call option, the same intuition implies that you should guarantee the profit rather than exercise. The payoff of adding short stock and long bond to your American call is better than exercising before expiration. Exercise a European Call before Expiration Time: t = -1 t = t = 1 =T Buy Eu Call (date t=-1) ell tock (date t=) Buy Bond (date t=) -C EU t > 15 > 1 EU = -5 -C EU =(- 5) T < 8 < 1 T > 13 > 1 ( T -)= T > 17 > 1 ( T -)= t =15 - T =-8 - T =-13 - T =-17 -= =-1 FV()=15 t -= 15-1= 1= 5 = FV() - T = 15-8 = 25 > FV()=15 = FV() - = 15-1 = 5 > FV()=15 = FV() - = 15-1 = 5 > Assumptions: 1. A European put 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{ -, } < P EU < 8

9 Max{ -, } < P EU < < P : the owner has a right but not an obligation. - < P : arbitrage proof. P < : the highest profit from the put option is realized when the stock price is zero. That profit is $ and it is realized on date T, which is equivalent to PV($) today. Example: The current stock price is $75 The stock will not pay dividends in the next six months A put option on that stock is traded for $1 The exercise price is $8 The expiration of the option is in 6 months The 6 months risk free rate is 5% Is there an opportunity to make arbitrage profits? Time: t = Buy stock Buy put ell bond tate: T < = 8 T = 8 -=( =(-$75) + T + T -P=( P=(-$1) (- T ) +/(1+rf)=$ P+ P+/(1+rf) = $.19 > + T +(- T )- = T - > The put price is bounded Max{ -, } < P EU < The put option price is monotonically decreasing in the stock price If then P() The put option price is a convex function of the stock price (see sketch). P P - - P() 9

10 Exercise Prices Assume there are two European put options on the same stock, with the same expiration date T, that have different exercise prices 1 < 2. Then, P( 1 ) < P( 2 ) I.e., the price of the put option is monotonically increasing in the exercise price (if then P() ). Example how 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. 1 = $4 and 2 = $5 P 1 = $4 and P 2 = $3 Time: t = trategy: tate: T < 1 = 4 1 < T < 2 T > 2 = 5 ell Put 1 P 1 =$4 -( 1 - T ) ( 1 = $4) Buy Put 2 ( 2 = $5) P 1 -P 2 = $1 > -P 2 =(-$3) ( 2 - T ) ( 2 - T ) 2-1 = 5-4 > ( 2 - T ) = 5- T > = Option Price Convexity Assume there are three European put options on the same stock, with the same expiration date T, that have different exercise prices 1 < 2 < 3. If 2 = α 1 +(1 (1- α) ) 3 Then P( 2 ) < αp( 1 ) + (1- α)p( 3 ) I.e., the price of a put option is a convex function of the exercise price. Example how 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 follows, there is an opportunity to make arbitrage profits. 1 = $4, 2 = $5 and 3 = $6 P 1 = $3, P 2 = $4 and P 3 = $4.6 Time: t = Buy put 1 (one unit) ell put 2 (two units) Buy put 3 (one unit) tate: T < 1 = 4 1 < T < 2 2 < T < 3 T > 3 = 6 -P 1 = -3 ( 1 T ) 1 1 T 2P 2 = 8-2*( 2 T ) -2*( 2 T ) -P 3 = ( 3 T ) ( 3 T ) ( 3 T ) P1-2P2+P3 =.4 > = = = T = T 4 > = 3 - T = 6 - T > = 1

11 The put price is bounded Max{ -, } < P EU < The put option price is monotonically increasing in the exercise price P - If then P() The put option price is a convex function of the exercise price. FV() P P() - Assumptions: 1. Two Put options European and American 2. The underlying asset is a stock that pays no dividends before expiration 3. The stock is traded d 4. A risk free bond is traded FV() For a put option, regardless of dividends: P(European) P(American) Note: in this case an example is enough. Determinants of the Values of Call and Put Options Variable stock price exercise price σ stock price volatility T time to expiration r risk-free interest rate Div dividend payouts C Call Value Decrease Decrease P Put Value Decrease Decrease Practice Problems BKM Ch. 21: 7th Ed.: End of chapter - 1, 2 Example 21.1 and concept check Q #4 8th Ed.: End of chapter - 1, 6 Example 21.1 and concept check Q #4 Practice set: