期权,期货及其衍生品第22弹.ppt

1、Chapter 22Estimating Volatilities and Correlations,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,1,Standard Approach to Estimating Volatility (page 498),Define sn as the volatility per day between day n-1 and day n, as estimated at end of day n-1 Define Si as the

2、value of market variable at end of day i Define ui= ln(Si/Si-1),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,2,Simplifications Usually Made in Risk Management (page 499),Define ui as (Si Si-1)/Si-1 Assume that the mean value of ui is zero Replace m1 by m This giv

3、es,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,3,Weighting Scheme,Instead of assigning equal weights to the observations we can set,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,4,ARCH(m) Model (page 500),In an ARCH(m) model w

4、e also assign some weight to the long-run variance rate, VL:,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,5,EWMA Model,In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time This leads

5、to,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,6,To Show that Weights Decline Exponentially,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,7,Attractions of EWMA,Relatively little data needs to be stored We need only remember th

6、e current estimate of the variance rate and the most recent observation on the market variable Tracks volatility changes 0.94 is a popular choice for l,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,8,GARCH (1,1) page 502,In GARCH (1,1) we assign some weight to the

7、 long-run average variance rate Since weights must sum to 1 g + a + b =1,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,9,GARCH (1,1) continued,Setting w = gV the GARCH (1,1) model is and,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull

8、2012,10,Example (Example 22.2, page 502),Suppose The long-run variance rate is 0.0002 so that the long-run volatility per day is 1.4%,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,11,Example continued,Suppose that the current estimate of the volatility is 1.6% per

9、 day and the most recent percentage change in the market variable is 1%. The new variance rate is The new volatility is 1.53% per day,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,12,GARCH (p,q),Options, Futures, and Other Derivatives, 8th Edition, Copyright John

10、C. Hull 2012,13,Maximum Likelihood Methods,In maximum likelihood methods we choose parameters that maximize the likelihood of the observations occurring,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,14,Example 1,We observe that a certain event happens one time in

11、ten trials. What is our estimate of the proportion of the time, p, that it happens? The probability of the event happening on one particular trial and not on the others is We maximize this to obtain a maximum likelihood estimate. Result: p = 0.1 (as expected),Options, Futures, and Other Derivatives,

12、 8th Edition, Copyright John C. Hull 2012,15,Example 2,Estimate the variance of observations from a normal distribution with mean zero,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,16,Application to GARCH,We choose parameters that maximize,Options, Futures, and Ot

13、her Derivatives, 8th Edition, Copyright John C. Hull 2012,17,S&P 500 Excel Application,Start with trial values of w, a, and b Update variances Calculate Use solver to search for values of w, a, and b that maximize this objective function Important note: set up spreadsheet so that you are searching f

14、or three numbers that are the same order of magnitude,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,18,S&P 500 Excel Application (Table 22.1),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,19,The Results (Figure 22.2, page 507),O

15、ptions, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,20,Variance Targeting,One way of implementing GARCH(1,1) that increases stability is by using variance targeting We set the long-run average volatility equal to the sample variance Only two other parameters then have to

16、 be estimated,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,21,How Good is the Model?,The Ljung-Box statistic tests for autocorrelation We compare the autocorrelation of the ui2 with the autocorrelation of the ui2/si2,Options, Futures, and Other Derivatives, 8th E

17、dition, Copyright John C. Hull 2012,22,Forecasting Future Volatility (equation 22.13, page 480),A few lines of algebra shows that The variance rate for an option expiring on day m is,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,23,Forecasting Future Volatility co

18、ntinued (equation 21.14, page 481),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,24,S&P Example,w = 0.000001366, a = 0.083394, b = 0.910116,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,25,Volatility Term Structures (Table 22.4)

19、,GARCH (1,1) suggests that, when calculating vega, we should shift the long maturity volatilities less than the short maturity volatilities When instantaneous volatility changes by Ds(0), volatility for T-day option changes by,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. H

20、ull 2012,26,Results for S&P 500,When instantaneous volatility changes by 1%,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,27,Correlations and Covariances (page 512-515),Define xi=(XiXi-1)/Xi-1 and yi=(YiYi-1)/Yi-1 Also sx,n: daily vol of X calculated on day n1 sy,

21、n: daily vol of Y calculated on day n1 covn: covariance calculated on day n1 The correlation is covn/(sx,n sy,n),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,28,Updating Correlations,We can use similar models to those for volatilities Under EWMA covn = l covn-1+(1-l)xn-1yn-1,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,29,Positive Finite Definite Condition,A variance-covariance matrix, W, is internally consistent if the positive s