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1、Chapter 5: Option Pricing Models: The Black-Scholes ModelWhen I first saw the formula I knew enough about it to know that this is the answer. This solved the ancient problem of risk and return in the stock market. It was recognized by the profession for what it was as a real tour de force. Merton Mi
2、ller Trillion Dollar Bet, PBS, February, 2000D. M. ChanceCh. 5: 1An Introduction to Derivatives and Risk Management, 6th ed.Important Concepts in Chapter 5nThe Black-Scholes option pricing modelnThe relationship of the models inputs to the option pricenHow to adjust the model to accommodate dividend
3、s and put optionsnThe concepts of historical and implied volatilitynHedging an option positionD. M. Chance2An Introduction to Derivatives and Risk Management, 6th ed.Origins of the Black-Scholes FormulanBrownian motion and the works of Einstein, Bachelier, Wiener, ItnBlack, Scholes, Merton and the 1
4、997 Nobel PrizeD. M. Chance3An Introduction to Derivatives and Risk Management, 6th ed.The Black-Scholes Model as the Limit of the Binomial ModelnRecall the binomial model and the notion of a dynamic risk -free hedge in which no arbitrage opportunities are available.nConsider the AOL June 125 call o
5、ption. Figure 5.1, p. 131 shows the model price for an increasing number of time steps.nThe binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.D. M. Chance4An Introduction to Derivatives and Risk Management, 6th ed.The Assumptions of the
6、 ModelnStock Prices Behave Randomly and Evolve According to a Lognormal Distribution. uSee Figure 5.2a, p. 134, 5.2b, p. 135 and 5.3, p. 136 for a look at the notion of randomness.uA lognormal distribution means that the log (continuously compounded) return is normally distributed. See Figure 5.4, p
7、. 137.nThe Risk-Free Rate and Volatility of the Log Return on the Stock are Constant Throughout the Options LifenThere Are No Taxes or Transaction CostsnThe Stock Pays No DividendsnThe Options are EuropeanD. M. Chance5An Introduction to Derivatives and Risk Management, 6th ed.A Nobel FormulanThe Bla
8、ck-Scholes model gives the correct formula for a European call under these assumptions.nThe model is derived with complex mathematics but is easily understandable. The formula isD. M. Chance6An Introduction to Derivatives and Risk Management, 6th ed.A Nobel Formula (continued)uwhereFN(d1), N(d2) = c
9、umulative normal probabilityF = annualized standard deviation (volatility) of the continuously compounded return on the stockFrc = continuously compounded risk-free rateD. M. Chance7An Introduction to Derivatives and Risk Management, 6th ed.A Nobel Formula (continued)nA Digression on Using the Norma
10、l DistributionuThe familiar normal, bell-shaped curve (Figure 5.5, p. 139)uSee Table 5.1, p. 140 for determining the normal probability for d1 and d2. This gives you N(d1) and N(d2).D. M. Chance8An Introduction to Derivatives and Risk Management, 6th ed.A Nobel Formula (continued)nA Numerical Exampl
11、euPrice the AOL June 125 calluS0 = 125.9375, X = 125, rc = ln(1.0456) = .0446, T = .0959, = .83.uSee Table 5.2, p. 141 for calculations. C = $13.21.uFamiliarize yourself with the accompanying softwareFExcel: bsbin3.xls. See Software Demonstration 5.1. Note the use of Excels =normsdist() function.FWi
12、ndows: bsbwin2.2.exe. See Appendix 5.B. D. M. Chance9An Introduction to Derivatives and Risk Management, 6th ed.A Nobel Formula (continued)nCharacteristics of the Black-Scholes FormulauInterpretation of the FormulaFThe concept of risk neutrality, risk neutral probability, and its role in pricing opt
13、ionsFThe option price is the discounted expected payoff, Max(0,ST - X). We need the expected value of ST - X for those cases where ST X.D. M. Chance10An Introduction to Derivatives and Risk Management, 6th ed.A Nobel Formula (continued)nCharacteristics of the Black-Scholes Formula (continued)uInterp
14、retation of the Formula (continued)FThe first term of the formula is the expected value of the stock price given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.FThe second term is the expected value of the payment o
15、f the exercise price at expiration.D. M. Chance11An Introduction to Derivatives and Risk Management, 6th ed.A Nobel Formula (continued)nCharacteristics of the Black-Scholes Formula (continued)uThe Black-Scholes Formula and the Lower Bound of a European CallFRecall from Chapter 3 that the lower bound
16、 would beFThe Black-Scholes formula always exceeds this value as seen by letting S0 be very high and then let it approach zero.D. M. Chance12An Introduction to Derivatives and Risk Management, 6th ed.A Nobel Formula (continued)nCharacteristics of the Black-Scholes Formula (continued)uThe Formula When T = 0FAt expiration, the formula must converge to the intrinsic value.FIt does but requires taking limits since otherwise it would be
