Chance/Brooks Kate Jennings Moral Hazard, Fourth Estate, 2002, p. 8 CHAPTER 3: PRINCIPLES OF OPTION PRICING Ch. 3: 1 An Introduction to Derivatives and Risk Management, 9th ed. 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. © 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. An Introduction to Derivatives and Risk Management, 9th ed. Role of arbitrage in pricing options Minimum value, maximum value, value at expiration and lower bound of an option price Effect of exercise price, time to expiration, riskfree rate and volatility on an option price Difference between prices of European and American options Put-call parity Chance/Brooks IMPORTANT CONCEPTS IN CHAPTER 3 Ch. 3: 2 © 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. Symbols An Introduction to Derivatives and Risk Management, 9th ed. S0 (stock price) X (exercise price) T (time to expiration = (days until expiration)/365) r (see below) ST (stock price at expiration) C(S0,T,X), P(S0,T,X) Chance/Brooks BASIC NOTATION AND TERMINOLOGY Ch. 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. Computation of risk-free rate (r) May 14. Option expiration: May 21 T-bill bid discount = 4.45, ask discount = 4.37 Average T-bill discount = (4.45+4.37)/2 = 4.41 T-bill An Introduction to Derivatives and Risk Management, 9th ed. Date: Chance/Brooks BASIC NOTATION AND TERMINOLOGY (CONTINUED) price = 100 - 4.41(7/360) = 99.91425 T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 So 4.57 % is risk-free rate for options expiring May 21 Other risk-free rates: 4.56 (June 18), 4.63 (July 16) See Table 3.1 for prices of DCRB options Ch. 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. Minimum Value of a Call Concept of intrinsic value: Max(0,S0 - X) Proof of intrinsic value rule for DCRB calls Concept of time value Ca(S0,T,X) Max(0,S0 - X) See Table 3.2 for time values of DCRB calls See Figure 3.1 for minimum values of calls An Introduction to Derivatives and Risk Management, 9th ed. C(S0,T,X) 0 (for any call) For American calls: Chance/Brooks PRINCIPLES OF CALL OPTION PRICING Ch. 3: 5 © 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. Maximum Value of a Call Value of a Call at Expiration C(ST,0,X) = Max(0,ST - X) Proof/intuition The same for American and European options See Figure 3.3 An Introduction to Derivatives and Risk Management, 9th ed. C(S0,T,X) S0 Intuition See Figure 3.2, which adds this to Figure 3.1 Chance/Brooks PRINCIPLES OF CALL OPTION PRICING (CONTINUED) Ch. 3: 6 © 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. Effect of Time to Expiration Two American calls differing only by time to expiration, T1 and T2 where T1 < T2. Ca(S0,T2,X) Ca(S0,T1,X) Proof/intuition Deep in- and out-of-the-money Time value maximized when at-the-money Concept of time value decay See Figure 3.4 and Table 3.2 Cannot be proven (yet) for European calls An Introduction to Derivatives and Risk Management, 9th ed. Chance/Brooks PRINCIPLES OF CALL OPTION PRICING (CONTINUED) Ch. 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. Effect of Exercise Price Effect on Option Value Two European calls differing only by strikes of X1 and X2. Which is greater, Ce(S0,T,X1) or Ce(S0,T,X2)? Construct portfolios A and B. See Table 3.3. Portfolio A has non-negative payoff; therefore, Ce(S0,T,X1) Ce(S0,T,X2) Intuition: show what happens if not true Prices of DCRB options conform An Introduction to Derivatives and Risk Management, 9th ed. Chance/Brooks PRINCIPLES OF CALL OPTION PRICING (CONTINUED) Ch. 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. Effect of Exercise Price (continued) Limits on the Difference in Premiums Again, note Table 3.3. We must have (X2 - X1)(1+r)-T Ce(S0,T,X1) - Ce(S0,T,X2) X2 - X1 Ce(S0,T,X1) - Ce(S0,T,X2) X2 - X1 Ca(S0,T,X1) - Ca(S0,T,X2) Implications See Table 3.4. Prices of DCRB options conform An Introduction to Derivatives and Risk Management, 9th ed. Chance/Brooks PRINCIPLES OF CALL OPTION PRICING (CONTINUED) Ch. 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. Lower Bound of a European Call Ce(S0,T,X) Max[0,S0 - X(1+r)-T] This is the lower bound for a European call See Figure 3.5 for the price curve for European calls Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´ For foreign currency calls, Ce(S0,T,X) Max[0,S0(1+)-T - X(1+r)-T] An Introduction to Derivatives and Risk Management, 9th ed. Construct portfolios A and B. See Table 3.5. B dominates A. This implies that (after rearranging) Chance/Brooks PRINCIPLES OF CALL OPTION PRICING (CONTINUED) Ch. 3: 10 © 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. An Introduction to Derivatives and Risk Management, 9th ed. 23) Suppose the current stock price is $100, the exercise price is $100, the annually compounded interest rate is 5 percent, the stock pays a $1 dividend in the next instant, and the quoted call price is $3.50 for a one-year option. Identify the appropriate arbitrage opportunity Chance/Brooks PRINCIPLES OF CALL OPTION PRICING (CONTINUED) Ch. 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. American Call Versus European Call Ca(S0,T,X) Max(0,S0 - X(1+r)-T) Look at Table 3.6 for lower bounds of DCRB calls 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. Intuition An Introduction to Derivatives and Risk Management, 9th ed. Ca(S0,T,X) Ce(S0,T,X) But S0 - X(1+r)-T > S0 - X prior to expiration so Chance/Brooks PRINCIPLES OF CALL OPTION PRICING (CONTINUED) Ch. 3: 12 © 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. Early Exercise of American Calls on DividendPaying Stocks 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.) Intuition Effect of Interest Rates Effect of Stock Volatility An Introduction to Derivatives and Risk Management, 9th ed. If Chance/Brooks PRINCIPLES OF CALL OPTION PRICING (CONTINUED) Ch. 3: 13 © 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. Minimum Value of a Put Pa(S0,T,X) Max(0,X - S0) Concept of intrinsic value: Max(0,X - S0) Proof of intrinsic value rule for DCRB puts See Figure 3.6 for minimum values of puts Concept of time value See Table 3.7 for time values of DCRB puts An Introduction to Derivatives and Risk Management, 9th ed. P(S0,T,X) 0 (for any put) For American puts: Chance/Brooks PRINCIPLES OF PUT OPTION PRICING Ch. 3: 14 © 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. Maximum Value of a Put Value of a Put at Expiration P(ST,0,X) = Max(0,X - ST) Proof/intuition For American and European options See Figure 3.8 An Introduction to Derivatives and Risk Management, 9th ed. Pe(S0,T,X) X(1+r)-T Pa(S0,T,X) X Intuition See Figure 3.7, which adds this to Figure 3.6 Chance/Brooks PRINCIPLES OF PUT OPTION PRICING (CONTINUED) Ch. 3: 15 © 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. An Introduction to Derivatives and Risk Management, 9th ed. 18) Suppose a European put has an exercise price of $110 on February 5. The put expires in 45 days. Suppose the appropriate discount rate on Treasury bills maturing in 44 days is 7.615. What is the maximum value of the European put? If the put were instead an American put, what would be its maximum value? Chance/Brooks PRINCIPLES OF PUT OPTION PRICING (CONTINUED) Ch. 3: 16 © 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. Effect of Time to Expiration Proof/intuition See Figure 3.9 and Table 3.7 Cannot be proven for European puts An Introduction to Derivatives and Risk Management, 9th ed. Two American puts differing only by time to expiration, T1 and T2 where T1 < T2. Pa(S0,T2,X) Pa(S0,T1,X) Chance/Brooks PRINCIPLES OF PUT OPTION PRICING (CONTINUED) Ch. 3: 17 © 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. Effect of Exercise Price Effect on Option Value Two European puts differing only by X1 and X2. Which is greater, Pe(S0,T,X1) or Pe(S0,T,X2)? Construct portfolios A and B. See Table 3.8. Portfolio A has non-negative payoff; therefore, Pe(S0,T,X2) Pe(S0,T,X1) Intuition: show what happens if not true Prices of DCRB options conform An Introduction to Derivatives and Risk Management, 9th ed. Chance/Brooks PRINCIPLES OF PUT OPTION PRICING (CONTINUED) Ch. 3: 18 © 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. Effect of Exercise Price (continued) Limits on the Difference in Premiums An Introduction to Derivatives and Risk Management, 9th ed. Again, note Table 3.8. We must have (X2 - X1)(1+r)-T Pe(S0,T,X2) - Pe(S0,T,X1) X2 - X1 Pe(S0,T,X2) - Pe(S0,T,X1) X2 - X1 Pa(S0,T,X2) - Pa(S0,T,X1) Implications See Table 3.9. Prices of DCRB options conform Chance/Brooks PRINCIPLES OF PUT OPTION PRICING (CONTINUED) Ch. 3: 19 © 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. Lower Bound of a European Put Pe(S0,T,X) Max(0,X(1+r)-T - S0) This is the lower bound for a European put See Figure 3.10 for the price curve for European puts Dividend adjustment: subtract present value of dividends from S to obtain S´ An Introduction to Derivatives and Risk Management, 9th ed. Construct portfolios A and B. See Table 3.10. A dominates B. This implies that (after rearranging) Chance/Brooks PRINCIPLES OF PUT OPTION PRICING (CONTINUED) Ch. 3: 20 © 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. American Put Versus European Put Early Exercise of American Puts There is always a sufficiently low stock price that will make it optimal to exercise an American put early. Dividends on the stock reduce the likelihood of early exercise. An Introduction to Derivatives and Risk Management, 9th ed. Pa(S0,T,X) Pe(S0,T,X) Chance/Brooks PRINCIPLES OF PUT OPTION PRICING (CONTINUED) Ch. 3: 21 © 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. Put-Call Parity S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T This is called put-call parity. It is important to see the alternative ways the equation can be arranged and their interpretations. An Introduction to Derivatives and Risk Management, 9th ed. Form portfolios A and B where the options are European. See Table 3.11. The portfolios have the same outcomes at the options’ expiration. Thus, it must be true that Chance/Brooks PRINCIPLES OF PUT OPTION PRICING (CONTINUED) Ch. 3: 22 © 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. Put-Call parity for American options can be stated only as inequalities: C a (S , T, X) + X + å D j (1 + r) -t j j=1 ³ S0 + Pa (S'0 , T, X) ³ C a (S'0 , T, X) + X(1 + r) -T See Table 3.12 for put-call parity for DCRB options See Figure 3.11 for linkages between underlying asset, risk-free bond, call, and put through put-call parity. An Introduction to Derivatives and Risk Management, 9th ed. N ' 0 Chance/Brooks PRINCIPLES OF PUT OPTION PRICING (CONTINUED) Ch. 3: 23 © 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. Summary See Table 3.13. An Introduction to Derivatives and Risk Management, 9th ed. The Effect of Interest Rates The Effect of Stock Volatility Chance/Brooks PRINCIPLES OF PUT OPTION PRICING (CONTINUED) Ch. 3: 24 © 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. An Introduction to Derivatives and Risk Management, 9th ed. 2) Suppose someone offers you the following gamble: You pay $7 and toss a coin. If the coin comes up heads, he pays you $10, and if tails comes up, h e p a y s y o u $ 5 . Y o u i n tu r n g e t t h e id e a o f offering another person a coin toss in which he pays you $7 and tosses another coin. You tell him that if heads comes up, you will pay him $9 and if tails comes up, you will pay him $5. You think you see an opportunity to earn an arbitrage profit by engaging in both transactions at the same time. Why is this not an arbitrage opportunity? How could you make it one assuming that you could get two people to engage in these gambles? Chance/Brooks PRINCIPLES OF OPTION PRICING Ch. 3: 25 © 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. An Introduction to Derivatives and Risk Management, 9th ed. 10) Suppose a European put price exceeds the value predicted by put-call parity. How could an investor profit? Demonstrate that your strategy is correct by constructing a payoff table showing the outcomes at expiration. Chance/Brooks PRINCIPLES OF OPTION PRICING Ch. 3: 26 © 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. An Introduction to Derivatives and Risk Management, 9th ed. 25) On December 9 of a particular year, a January Swiss franc call option with an exercise price of 46 had a price of 1.63. The January 46 put was at 0.14. The spot rate was 47.28. All prices are in cents per Swiss franc. The option expired on January 13. The U.S. riskfree rate was 7.1 percent, and the Swiss risk-free rate was 3.6 percent. Do the following: a. Determine the intrinsic value of the call. b. Determine the lower bound of the call. c. Determine the time value of the call. d. Determine the intrinsic value of the put. e. Determine the lower bound of the put. f. Determine the time value of the put. g. Determine whether put-call parity holds. Chance/Brooks PRINCIPLES OF OPTION PRICING Ch. 3: 27 © 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. An Introduction to Derivatives and Risk Management, 9th ed. 26) Suppose Congress decides that investors should not profit when stock prices go down, so it outlaws short selling. Congress has not figured out options, however, so there are no restrictions on option trading. Explain how to accomplish the equivalent of a short sale by using options. Chance/Brooks PRINCIPLES OF OPTION PRICING Ch. 3: 28 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 29 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide 5) (Return to text slide 7) Ch. 3: 30 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 31 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 32 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 33 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 34 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide 8) (Return to text slide 9) Ch. 3: 35 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 36 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 37 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 38 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 39 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 40 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide 13) (Return to text slide 15) Ch. 3: 41 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 42 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 43 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 44 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide 16) (Return to text slide 17) Ch. 3: 45 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 46 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 47 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 48 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 49 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 50 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 51 © 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/Brooks An Introduction to Derivatives and Risk Management, 9th ed. (Return to text slide) Ch. 3: 52 © 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. 10 An Introduction to Derivatives and Risk Management, 9th ed. (Put-Call Parity) 18 (Maximum Value of a Put) 23 (Lower Bound of a European Call) 25 (Principles of Call/Put Option Pricing) 26 (Put-Call Parity) Chance/Brooks MY PROBLEMS Ch. 3: 53 © 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. 5 An Introduction to Derivatives and Risk Management, 9th ed. (Lower bound of European Call) 12 (Intrinsic Value, Time Value, Arbitrage) 15 (Put-Call Parity) 21 (Lower Bound of a European Put) 22 (Put-Call Parity) Chance/Brooks TA PROBLEMS Ch. 3: 54 © 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.