衍生工具与风险管理第3章课件

1、D. M. Chance,An 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

2、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 F

3、utures, December, 2001, p. 66,Important Concepts in Chapter 3,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, risk-free rate and volatility on an option price Difference between pr

4、ices of European and American options Put-call parity,D. M. Chance,2,An Introduction to Derivatives and Risk Management, 6th ed.,Basic Notation and Terminology,Symbols S0 (stock price) X (exercise price) T (time to expiration = (days until expiration)/365) r (see below) ST (stock price at expiration

5、) C(S0,T,X), P(S0,T,X),D. M. Chance,3,An Introduction to Derivatives and Risk Management, 6th ed.,Basic Notation and Terminology (continued),Computation of risk-free rate Date: 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 =

6、 4.41 T-bill price = 100 - 4.41(7/360) = 99.91425 T-bill yield = (100/99.91425)(365/7) - 1 = .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, p. 58 for prices of AOL options,D. M. Chance,4,An Introduction to Derivative

7、s and Risk Management, 6th ed.,Principles of Call Option Pricing,The Minimum Value of a Call C(S0,T,X) 0 (for any call) For American calls: Ca(S0,T,X) Max(0,S0 - X) Concept of intrinsic value: Max(0,S0 - X) Proof of intrinsic value rule for AOL calls Concept of time value See Table 3.2, p. 59 for ti

8、me values of AOL calls See Figure 3.1, p. 60 for minimum values of calls,D. M. Chance,5,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Call Option Pricing (continued),The Maximum Value of a Call C(S0,T,X) S0 Intuition See Figure 3.2, p. 61, which adds this to Figure 3.1 Th

9、e Value of a Call at Expiration C(ST,0,X) = Max(0,ST - X) Proof/intuition For American and European options See Figure 3.3, p. 63,D. M. Chance,6,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Call Option Pricing (continued),The Effect of Time to Expiration Two American cal

10、ls 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, p. 64 and Table 3.2, p. 59 Cannot be proven (yet) for European calls,D. M. Chance,7

11、,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Call Option Pricing (continued),The Effect of Exercise Price The 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. Se

12、e Table 3.3, p. 65. Portfolio A has non-negative payoff; therefore, Ce(S0,T,X1) Ce(S0,T,X2) Intuition: show what happens if not true Prices of AOL options conform,D. M. Chance,8,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Call Option Pricing (continued),The Effect of Ex

13、ercise Price (continued) Limits on the Difference in Premiums Again, note Table 3.3, p. 65. 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, p. 67. Prices of AOL options conform,D. M. Chance,9,An I

14、ntroduction to Derivatives and Risk Management, 6th ed.,Principles of Call Option Pricing (continued),The Lower Bound of a European Call Construct portfolios A and B. See Table 3.5, p. 68. B dominates A. This implies that (after rearranging) Ce(S0,T,X) Max0,S0 - X(1+r)-T This is the lower bound for

15、a European call See Figure 3.5, p. 69 for the price curve for European calls Dividend adjustment: subtract present value of dividends from S; adjusted stock price is S For foreign currency calls, Ce(S0,T,X) Max0,S0(1+)-T - X(1+r)-T,D. M. Chance,10,An Introduction to Derivatives and Risk Management,

16、6th ed.,Principles of Call Option Pricing (continued),American Call Versus European Call Ca(S0,T,X) Ce(S0,T,X) But S0 - X(1+r)-T S0 - X prior to expiration so Ca(S0,T,X) Max(0,S0 - X(1+r)-T) Look at Table 3.6, p. 70 for lower bounds of AOL calls If there are no dividends on the stock, an American ca

17、ll will never be exercised early. It will always be better to sell the call in the market. Intuition,D. M. Chance,11,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Call Option Pricing (continued),The Early Exercise of American Calls on Dividend-Paying Stocks If a stock pay

18、s 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 The Effect of Interest Rates The Effect of Stock Volatility,D. M. Chance,12,An Introduction to Derivatives an

19、d Risk Management, 6th ed.,Principles of Put Option Pricing,The Minimum Value of a Put P(S0,T,X) 0 (for any put) For American puts: Pa(S0,T,X) Max(0,X - S0) Concept of intrinsic value: Max(0,X - S0) Proof of intrinsic value rule for AOL puts See Figure 3.6, p. 74 for minimum values of puts Concept o

20、f time value See Table 3.7, p. 75 for time values of AOL puts,D. M. Chance,13,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Put Option Pricing (continued),The Maximum Value of a Put Pe(S0,T,X) X(1+r)-T Pa(S0,T,X) X Intuition See Figure 3.7, p. 76, which adds this to Figur

21、e 3.6 The 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, p. 77,D. M. Chance,14,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Put Option Pricing (continued),The Effect of Time to Expiration Two Ameri

22、can puts differing only by time to expiration, T1 and T2 where T1 T2. Pa(S0,T2,X) Pa(S0,T1,X) Proof/intuition See Figure 3.9, p. 78 and Table 3.7, p. 75 Cannot be proven for European puts,D. M. Chance,15,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Put Option Pricing (co

23、ntinued),The Effect of Exercise Price The 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, p. 79. Portfolio A has non-negative payoff; therefore, Pe(S0,T,X2) Pe(S0,T,X1) Intuition: show wh

24、at happens if not true Prices of AOL options conform,D. M. Chance,16,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Put Option Pricing (continued),The Effect of Exercise Price (continued) Limits on the Difference in Premiums Again, note Table 3.8, p. 79. We must have (X2 -

25、 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, p. 81. Prices of AOL options conform,D. M. Chance,17,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Put Option Pricing (continued),The Lowe

26、r Bound of a European Put Construct portfolios A and B. See Table 3.10, p. 81. A dominates B. This implies that (after rearranging) Pe(S0,T,X) Max(0,X(1+r)-T - S0) This is the lower bound for a European put See Figure 3.10, p. 82 for the price curve for European puts Dividend adjustment: subtract pr

27、esent value of dividends from S to obtain S,D. M. Chance,18,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Put Option Pricing (continued),American Put Versus European Put Pa(S0,T,X) Pe(S0,T,X) The Early Exercise of American Puts There is always a sufficiently low stock pri

28、ce that will make it optimal to exercise an American put early. Dividends on the stock reduce the likelihood of early exercise.,D. M. Chance,19,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Put Option Pricing (continued),Put-Call Parity Form portfolios A and B where the o

29、ptions are European. See Table 3.11, p. 84. The portfolios have the same outcomes at the options expiration. Thus, it must be true that 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 interpret

30、ations.,D. M. Chance,20,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Put Option Pricing (continued),Put-Call parity for American options can be stated only as inequalities: See Table 3.12, p. 86 for put-call parity for AOL options See Figure 3.11, p. 87 for linkages betw

31、een underlying asset, risk-free bond, call, and put through put-call parity.,D. M. Chance,21,An Introduction to Derivatives and Risk Management, 6th ed.,Principles of Put Option Pricing (continued),The Effect of Interest Rates The Effect of Stock Volatility,SummarySee Table 3.13, p. 90.,Appendix 3:

32、The Dynamics of Option Boundary Conditions: A Learning Exercise,D. M. Chance,22,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide),D. M. Chance,23,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide 5),(Return to text slide 7),D. M. Chanc

33、e,24,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide),D. M. Chance,25,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide),D. M. Chance,26,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide),D. M. Chance,2

34、7,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide),D. M. Chance,28,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide 9),(Return to text slide 8),D. M. Chance,29,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to te

35、xt slide),D. M. Chance,30,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide),D. M. Chance,31,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide),D. M. Chance,32,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide),D. M. Chance,33,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide),D. M. Chance,34,An Introduction to Derivatives and Risk Management, 6th ed.,(Return to text slide 15),(Return to text slide 13),D.