Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 1 Chapter 3: Principles of Option Pricing Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties. Kate Jennings Moral Hazard, Fourth Estate, 2002, p. 8 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th 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 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th 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) © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 4 Basic Notation and Terminology (continued) n Computation of risk-free rate (r) 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 = 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 for prices of DCRB options Table 3.1Table 3.1 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 5 Principles of Call Option Pricing n 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 DCRB calls u Concept of time value F See Table 3.2 for time values of DCRB calls Table 3.2Table 3.2 u See Figure 3.1 for minimum values of calls Figure 3.1Figure 3.1 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 6 Principles of Call Option Pricing (continued) n 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, which adds this to Figure 3.1Figure 3.2 n 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 Figure 3.3Figure 3.3 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 7 Principles of Call Option Pricing (continued) n 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 and Table 3.2 Figure 3.4Table 3.2Figure 3.4Table 3.2 u Cannot be proven (yet) for European calls © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 8 Principles of Call Option Pricing (continued) n Effect of Exercise Price u 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. Table 3.3Table 3.3 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 DCRB options conform © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 9 Principles of Call Option Pricing (continued) n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have Table 3.3Table 3.3 (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. Prices of DCRB options conform Table 3.4Table 3.4 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 10 Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. Table 3.5Table 3.5 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 for the price curve for European calls Figure 3.5Figure 3.5 u Dividend adjustment: subtract present value of dividends from S 0 ; adjusted stock price is S 0 ´ 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 ] © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th 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 for lower bounds of DCRB calls Table 3.6Table 3.6 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 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 12 Principles of Call Option Pricing (continued) n 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 Effect of Interest Rates n Effect of Stock Volatility © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 13 Principles of Put Option Pricing n 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 DCRB puts u See Figure 3.6 for minimum values of puts Figure 3.6Figure 3.6 u Concept of time value F See Table 3.7 for time values of DCRB puts Table 3.7Table 3.7 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 14 Principles of Put Option Pricing (continued) n 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, which adds this to Figure 3.6Figure 3.7 n 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 Figure 3.8Figure 3.8 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 15 Principles of Put Option Pricing (continued) n 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 and Table 3.7 Figure 3.9Table 3.7Figure 3.9Table 3.7 u Cannot be proven for European puts © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 16 Principles of Put Option Pricing (continued) n Effect of Exercise Price u 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. Table 3.8Table 3.8 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 DCRB options conform © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 17 Principles of Put Option Pricing (continued) n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have Table 3.8Table 3.8 (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. Prices of DCRB options conform Table 3.9Table 3.9 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 18 Principles of Put Option Pricing (continued) n Lower Bound of a European Put u Construct portfolios A and B. See Table Table 3.10Table 3.10 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 for the price curve for European puts Figure 3.10Figure 3.10 u Dividend adjustment: subtract present value of dividends from S to obtain S´ © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th 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 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. © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th 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 Table 3.11Table 3.11 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. © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th 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 for put-call parity for DCRB options Table 3.12Table 3.12 u See Figure 3.11 for linkages between underlying asset, risk-free bond, call, and put through put-call parity. Figure 3.11Figure 3.11 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 9th ed.Ch. 3: 22 Principles of Put Option Pricing (continued) n The Effect of Interest Rates n The Effect of Stock Volatility See Table Summary See Table 3.13.Table 3.13Table 3.13 Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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