Copyright © 2001 by Harcourt, Inc. All rights reserved.1 Chapter 3: Principles of Option Pricing Order and simplification are the first steps toward mastery of a subject - the actual enemy is the unknown. Thomas Mann The Magic Mountain 1924

Copyright © 2001 by Harcourt, Inc. All rights reserved.2 Important Concepts in Chapter 3 n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate and volatility on an option price n Difference between prices of European and American options n Put-call parity

Copyright © 2001 by Harcourt, Inc. All rights reserved.3 Basic Notation and Terminology n Symbols u S 0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (see below) u S T (stock price at expiration) u C(S 0,T,X), P(S 0,T,X)

Copyright © 2001 by Harcourt, Inc. All rights reserved.4 Basic Notation and Terminology (continued) n Computation of risk-free rate u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = ( )/2 = 4.41 u T-bill price = (7/360) = u T-bill yield = (100/ ) (365/7) - 1 =.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16) n See Table 3.1, p. 74 for prices of America Online options

Copyright © 2001 by Harcourt, Inc. All rights reserved.5 Principles of Call Option Pricing n The Minimum Value of a Call  C(S 0,T,X)  C(S 0,T,X)  0 (for any call) u For American calls:  C a (S 0,T,X)  Max(0,S 0 - X) u Concept of intrinsic value: Max(0,S 0 - X) F Proof of intrinsic value rule for AOL calls u Concept of time value F See Table 3.2, p. 76 for time values of AOL calls u See Figure 3.1, p. 77 for minimum values of calls

Copyright © 2001 by Harcourt, Inc. All rights reserved.6 Principles of Call Option Pricing (continued) n The Maximum Value of a Call  C(S 0,T,X)  C(S 0,T,X)  S 0 u u Intuition u u See Figure 3.2, p. 78, which adds this to Figure 3.1 n The Value of a Call at Expiration u C(S T,0,X) = Max(0,S T - X) u Proof/intuition u For American and European options u See Figure 3.3, p. 80

Copyright © 2001 by Harcourt, Inc. All rights reserved.7 Principles of Call Option Pricing (continued) n The Effect of Time to Expiration u Two American calls differing only by time to expiration, T 1 and T 2 where T 1 < T 2.  C a (S 0,T 2,X) C a (S 0,T 1,X)  C a (S 0,T 2,X)  C a (S 0,T 1,X) F Proof/intuition u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4, p. 81 and Table 3.2, p. 82 u Cannot be proven (yet) for European calls

Copyright © 2001 by Harcourt, Inc. All rights reserved.8 Principles of Call Option Pricing (continued) n The Effect of Exercise Price u The Effect on Option Value F Two European calls differing only by strikes of X 1 and X 2. Which is greater, C e (S 0,T,X 1 ) or C e (S 0,T,X 2 )? F Construct portfolios A and B. See Table 3.3, p. 82. F Portfolio A has non-negative payoff; therefore, C e (S 0,T,X 1 )  C e (S 0,T,X 2 )C e (S 0,T,X 1 )  C e (S 0,T,X 2 ) Intuition: show what happens if not trueIntuition: show what happens if not true F Prices of AOL options conform

Copyright © 2001 by Harcourt, Inc. All rights reserved.9 Principles of Call Option Pricing (continued) n The Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3, p. 82. We must have (X 2 - X 1 )(1+r) -T C e (S 0,T,X 1 ) - C e (S 0,T,X 2 )(X 2 - X 1 )(1+r) -T  C e (S 0,T,X 1 ) - C e (S 0,T,X 2 ) X 2 - X 1 C e (S 0,T,X 1 ) - C e (S 0,T,X 2 )X 2 - X 1  C e (S 0,T,X 1 ) - C e (S 0,T,X 2 ) X 2 - X 1 C a (S 0,T,X 1 ) - C a (S 0,T,X 2 )X 2 - X 1  C a (S 0,T,X 1 ) - C a (S 0,T,X 2 ) ImplicationsImplications F See Table 3.4, p. 85. Prices of AOL options conform

Copyright © 2001 by Harcourt, Inc. All rights reserved.10 Principles of Call Option Pricing (continued) n The Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5, p. 86. u B dominates A. This implies that (after rearranging)  C e (S 0,T,X) Max(0,S 0 - X(1+r) -T )  C e (S 0,T,X)  Max(0,S 0 - X(1+r) -T ) F This is the lower bound for a European call F See Figure 3.5, p. 86 for the price curve for European calls u Dividend adjustment: subtract present value of dividends from S; adjusted stock price is S´

Copyright © 2001 by Harcourt, Inc. All rights reserved.11 Principles of Call Option Pricing (continued) n American Call Versus European Call  C a (S 0,T,X) C e (S 0,T,X)  C a (S 0,T,X)  C e (S 0,T,X) u But S 0 - X(1+r) -T > S 0 - X prior to expiration so  C a (S 0,T,X) Max(0,S 0 - X(1+r) -T )  C a (S 0,T,X)  Max(0,S 0 - X(1+r) -T ) F Look at Table 3.6, p. 88 for lower bounds of AOL calls u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market. F Intuition

Copyright © 2001 by Harcourt, Inc. All rights reserved.12 Principles of Call Option Pricing (continued) n The Early Exercise of American Calls on Dividend-Paying Stocks u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. u Intuition n The Effect of Interest Rates n The Effect of Volatility

Copyright © 2001 by Harcourt, Inc. All rights reserved.13 Principles of Put Option Pricing n The Minimum Value of a Put  P(S 0,T,X)  P(S 0,T,X)  0 (for any put) u For American puts:  P a (S 0,T,X)  Max(0,X - S 0 ) u Concept of intrinsic value: Max(0,X - S 0 ) F Proof of intrinsic value rule for AOL puts u See Figure 3.6, p. 92 for minimum values of puts u Concept of time value F See Table 3.7, p. 93 for time values of AOL puts

Copyright © 2001 by Harcourt, Inc. All rights reserved.14 Principles of Put Option Pricing (continued) n The Maximum Value of a Put  P e (S 0,T,X)  P e (S 0,T,X)  X(1+r) -T   P a (S 0,T,X)  X u u Intuition u u See Figure 3.7, p. 94, which adds this to Figure 3.6 n The Value of a Put at Expiration u P(S T,0,X) = Max(0,X - S T ) u Proof/intuition u For American and European options u See Figure 3.8, p. 95

Copyright © 2001 by Harcourt, Inc. All rights reserved.15 Principles of Put Option Pricing (continued) n The Effect of Time to Expiration u Two American puts differing only by time to expiration, T 1 and T 2 where T 1 < T 2.  P a (S 0,T 2,X) P a (S 0,T 1,X)  P a (S 0,T 2,X)  P a (S 0,T 1,X) F Proof/intuition u See Figure 3.9, p. 96 and Table 3.7, p. 93 u Cannot be proven for European puts

Copyright © 2001 by Harcourt, Inc. All rights reserved.16 Principles of Put Option Pricing (continued) n The Effect of Exercise Price u The Effect on Option Value F Two European puts differing only by X 1 and X 2. Which is greater, P e (S 0,T,X 1 ) or P e (S 0,T,X 2 )? F Construct portfolios A and B. See Table 3.8, p. 97. F Portfolio A has non-negative payoff; therefore, P e (S 0,T,X 2 ) P e (S 0,T,X 1 )P e (S 0,T,X 2 )  P e (S 0,T,X 1 ) Intuition: show what happens if not trueIntuition: show what happens if not true F Prices of AOL options conform

Copyright © 2001 by Harcourt, Inc. All rights reserved.17 Principles of Put Option Pricing (continued) n The Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8, p. 97. We must have (X 2 - X 1 )(1+r) -T P e (S 0,T,X 2 ) - P e (S 0,T,X 1 )(X 2 - X 1 )(1+r) -T  P e (S 0,T,X 2 ) - P e (S 0,T,X 1 ) X 2 - X 1 P e (S 0,T,X 2 ) - P e (S 0,T,X 1 )X 2 - X 1  P e (S 0,T,X 2 ) - P e (S 0,T,X 1 ) X 2 - X 1 P a (S 0,T,X 2 ) - P a (S 0,T,X 1 )X 2 - X 1  P a (S 0,T,X 2 ) - P a (S 0,T,X 1 ) ImplicationsImplications F See Table 3.9, p. 99. Prices of AOL options conform

Copyright © 2001 by Harcourt, Inc. All rights reserved.18 Principles of Put Option Pricing (continued) n The Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10, p u A dominates B. This implies that (after rearranging)  P e (S 0,T,X) Max(0,X(1+r) -T - S 0 )  P e (S 0,T,X)  Max(0,X(1+r) -T - S 0 ) F This is the lower bound for a European put F See Figure 3.10, p. 101 for the price curve for European puts u Dividend adjustment: subtract present value of dividends from S to obtain S´

Copyright © 2001 by Harcourt, Inc. All rights reserved.19 Principles of Put Option Pricing (continued) n American Put Versus European Put  P a (S 0,T,X) P e (S 0,T,X)  P a (S 0,T,X)  P e (S 0,T,X) n The Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.

Copyright © 2001 by Harcourt, Inc. All rights reserved.20 Principles of Put Option Pricing (continued) n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11, p u The portfolios have the same outcomes at the options’ expiration. Thus, it must be true that F S 0 + P e (S 0,T,X) = C e (S 0,T,X) + X(1+r) -T F This is called put-call parity. F It is important to see the alternative ways the equation can be arranged and their interpretations.

Copyright © 2001 by Harcourt, Inc. All rights reserved.21 Principles of Put Option Pricing (continued) u Put-Call parity for American options can be stated only as inequalities: u See Table 3.12, p. 105 for put-call parity for AOL options u See Figure 3.11, 106 for linkages between stock, risk- free bond, call, and put through put-call parity.

Copyright © 2001 by Harcourt, Inc. All rights reserved.22 Principles of Put Option Pricing (continued) n The Effect of Interest Rates n The Effect of Stock Volatility See Table 3.13, p Summary See Table 3.13, p. 107.