ch17期權(quán)期貨與衍生證券（第五版）

1、Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.1,Estimating Volatilities and Correlations Chapter 17,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hu

2、ll,17.2,Standard Approach to Estimating Volatility,Define sn as the volatility per day between day n-1 and day n, as estimated at end of day n-1Define Si as the value of market variable at end of day iDefine ui= ln(Si/

3、Si-1),Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.3,Simplifications Usually Made,Define ui as (Si-Si-1)/Si-1Assume that the mean value of ui is zeroReplace m-1 by mThis gives,

4、Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.4,Weighting Scheme,Instead of assigning equal weights to the observations we can set,Options, Futures, and Other Derivatives, 5th editi

5、on © 2002 by John C. Hull,17.5,ARCH(m) Model,In an ARCH(m) model we also assign some weight to the long-run variance rate, VL:,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.6,

6、EWMA Model,In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through timeThis leads to,Options, Futures, and Other Derivatives, 5th edition © 20

7、02 by John C. Hull,17.7,Attractions of EWMA,Relatively little data needs to be storedWe need only remember the current estimate of the variance rate and the most recent observation on the market variableTracks volatil

8、ity changesRiskMetrics uses l = 0.94 for daily volatility forecasting,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.8,GARCH (1,1),In GARCH (1,1) we assign some weight to the long-r

9、un average variance rateSince weights must sum to 1g + a + b =1,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.9,GARCH (1,1) continued,Setting w = gV the GARCH (1,1) model is

10、and,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.10,Example,SupposeThe long-run variance rate is 0.0002 so that the long-run volatility per day is 1.4%,Options, Futures, and O

11、ther Derivatives, 5th edition © 2002 by John C. Hull,17.11,Example continued,Suppose that the current estimate of the volatility is 1.6% per day and the most recent percentage change in the market variable is 1%.T

12、he new variance rate isThe new volatility is 1.53% per day,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.12,GARCH (p,q),,Options, Futures, and Other Derivatives, 5th edition 

13、9; 2002 by John C. Hull,17.13,Other Models,We can design GARCH models so that the weight given to ui2 depends on whether ui is positive or negativeWe do not have to assume that the conditional distribution is normal,Op

14、tions, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.14,Variance Targeting,One way of implementing GARCH(1,1) that increases stability is by using variance targetingWe set the long-run aver

15、age volatility equal to the sample varianceOnly two other parameters then have to be estimated,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.15,Maximum Likelihood Methods,In maximu

16、m likelihood methods we choose parameters that maximize the likelihood of the observations occurring,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.16,Example 1,We observe that a cer

17、tain event happens one time in ten trials. What is our estimate of the proportion of the time, p, that it happens?The probability of the outcome isWe maximize this to obtain a maximum likelihood estimate: p=0.1,Option

18、s, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.17,Example 2,Estimate the variance of observations from a normal distribution with mean zero,Options, Futures, and Other Derivatives, 5th edi

19、tion © 2002 by John C. Hull,17.18,Application to GARCH,We choose parameters that maximize,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.19,How Good is the Model?,The Ljung-Box

20、 statistic tests for autocorrelationWe compare the autocorrelation of theui2 with the autocorrelation of the ui2/si2,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.20,Forecasting

21、Future Volatility,A few lines of algebra shows thatThe variance rate for an option expiring on day m is,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.21,Volatility Term Structur

22、es,The GARCH (1,1) model allows us to predict volatility term structures changesIt suggests that, when calculating vega, we should shift the long maturity volatilities less than the short maturity volatilities,Options,

23、Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.22,Correlations,Define ui=(Ui-Ui-1)/Ui-1 and vi=(Vi-Vi-1)/Vi-1Alsosu,n: daily vol of U calculated on day n-1sv,n: daily vol of V calculated o

24、n day n-1covn: covariance calculated on day n-1The correlation is covn/(su,n sv,n),Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.23,Correlations continued,Under GARCH (1,1)covn

25、 = w + a un-1vn-1+b covn-1,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,17.24,Positive Finite Definite Condition,A variance-covariance matrix, W, is internally consistent if the posit