D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 1 Chapter 3: Principles of Option Pricing Asking a fund manager about arbitrage opportunities is akin to asking a fisherman where his favorite hole is. He will be glad to tell you a fish story from long ago, but he will not tell you where he caught the trout that in our analogy can be translated into millions of dollars, lest there will be hundreds of fishermen in his spot pulling in their own trout and reducing the inefficiency that made that arbitrage opportunity profitable in the first place. Daniel P. Collins Futures, December, 2001, p. 66

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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)

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 58 for prices of AOL options Table 3.1, p. 58Table 3.1, p. 58

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 59 for time values of AOL calls Table 3.2, p. 59Table 3.2, p. 59 u See Figure 3.1, p. 60 for minimum values of calls Figure 3.1, p. 60Figure 3.1, p. 60

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 61, which adds this to Figure 3.1Figure 3.2, p. 61 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. 63 Figure 3.3, p. 63Figure 3.3, p. 63

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 64 and Table 3.2, p. 59 Figure 3.4, p. 64Table 3.2, p. 59Figure 3.4, p. 64Table 3.2, p. 59 u Cannot be proven (yet) for European calls

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 65. Table 3.3, p. 65Table 3.3, p. 65 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

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 65. We must have Table 3.3, p. 65Table 3.3, p. 65 (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. 67. Prices of AOL options conform Table 3.4, p. 67Table 3.4, p. 67

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 68. Table 3.5, p. 68Table 3.5, p. 68 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. 69 for the price curve for European calls Figure 3.5, p. 69Figure 3.5, p. 69 u Dividend adjustment: subtract present value of dividends from S; adjusted stock price is S´ u For foreign currency calls,  C e (S 0,T,X) Max[0,S 0 (1+  ) -T - X(1+r) -T ]  C e (S 0,T,X)  Max[0,S 0 (1+  ) -T - X(1+r) -T ]

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 70 for lower bounds of AOL calls Table 3.6, p. 70Table 3.6, p. 70 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

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. (For a currency, the foreign interest can induce early exercise.) u Intuition n The Effect of Interest Rates n The Effect of Stock Volatility

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 74 for minimum values of puts Figure 3.6, p. 74Figure 3.6, p. 74 u Concept of time value F See Table 3.7, p. 75 for time values of AOL puts Table 3.7, p. 75Table 3.7, p. 75

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 76, which adds this to Figure 3.6Figure 3.7, p. 76 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. 77 Figure 3.8, p. 77Figure 3.8, p. 77

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 78 and Table 3.7, p. 75 Figure 3.9, p. 78Table 3.7, p. 75Figure 3.9, p. 78Table 3.7, p. 75 u Cannot be proven for European puts

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 79. Table 3.8, p. 79Table 3.8, p. 79 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

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 79. We must have Table 3.8, p. 79Table 3.8, p. 79 (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. 81. Prices of AOL options conform Table 3.9, p. 81Table 3.9, p. 81

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 81. Table 3.10, p. 81Table 3.10, p. 81 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. 82 for the price curve for European puts Figure 3.10, p. 82Figure 3.10, p. 82 u Dividend adjustment: subtract present value of dividends from S to obtain S´

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 84. Table 3.11, p. 84Table 3.11, p. 84 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.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 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. 86 for put-call parity for AOL options Table 3.12, p. 86Table 3.12, p. 86 u See Figure 3.11, p. 87 for linkages between underlying asset, risk-free bond, call, and put through put-call parity. Figure 3.11, p. 87Figure 3.11, p. 87

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 3: 22 Principles of Put Option Pricing (continued) n The Effect of Interest Rates n The Effect of Stock Volatility See Table 3.13, p. 90. Summary See Table 3.13, p. 90.Table 3.13, p. 90Table 3.13, p. 90 Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise

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