Ch14_ValueatRis(金融工程学,华东师大)k.ppt

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1、Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.1,Value at Risk,Chapter 14,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.2,The Question Being Asked in Value at Ri

2、sk(VaR),“What loss level is such that we are X%confident it will not be exceeded in N business days?”,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.3,Meaning is Probability,(1-)%,%,Z,Options,Futures,and Other Derivatives,4th editi

3、on 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.4,VaR and Regulatory Capital,Regulators require banks to keep capital for market risk equal to the average of VaR estimates for past 60 trading days using X=99 and N=10,times a multiplication factor.(Usually the multiplication fact

4、or equals 3),Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.5,Advantages of VaR,It captures an important aspect of riskin a single numberIt is easy to understandIt asks the simple question:“How bad can things get?”,Options,Futures,

5、and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.6,Daily Volatilities,In option pricing we express volatility as volatility per yearIn VaR calculations we express volatility as volatility per day,Options,Futures,and Other Derivatives,4th edition 200

6、0 by John C.HullTang Yincai,2003,Shanghai Normal University,14.7,Daily Volatility(continued),Strictly speaking we should define sday as the standard deviation of the continuously compounded return in one dayIn practice we assume that it is the standard deviation of the proportional change in one day

7、,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.8,IBM Example(p.343),We have a position worth$10 million in IBM sharesThe volatility of IBM is 2%per day(about 32%per year)We use N=10 and X=99,Options,Futures,and Other Derivatives,4

8、th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.9,IBM Example(continued),The standard deviation of the change in the portfolio in 1 day is$200,000The standard deviation of the change in 10 days is,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang

9、Yincai,2003,Shanghai Normal University,14.10,IBM Example(continued),We assume that the expected change in the value of the portfolio is zero(This is OK for short time periods)We assume that the change in the value of the portfolio is normally distributedSince N(0.01)=-2.33,the VaR is,Options,Futures

10、,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.11,AT&T Example,Consider a position of$5 million in AT&TThe daily volatility of AT&T is 1%(approx 16%per year)The STD per 10 days isThe VaR is,Options,Futures,and Other Derivatives,4th edition 2000 b

11、y John C.HullTang Yincai,2003,Shanghai Normal University,14.12,Portfolio(p.344),Now consider a portfolio consisting of both IBM and AT&TSuppose that the correlation between the returns is 0.7,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal Univer

12、sity,14.13,STD of Portfolio,A standard result in statistics states thatIn this case sx=632,456 and sY=158,114 and r=0.7.The standard deviation of the change in the portfolio value is therefore 751,665,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Norm

13、al University,14.14,VaR for Portfolio,The VaR for the portfolio isThe benefits of diversification are(1,473,621+368,405)-1,751,379=$90,647What is the incremental effect of the AT&T holding on VaR?,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal U

14、niversity,14.15,The Linear Model,We assumeThe change in the value of a portfolio is linearly related to the change in the value of market variablesThe changes in the values of the market variables are normally distributed,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yinc

15、ai,2003,Shanghai Normal University,14.16,The General Linear Model continued(Equation 14.5),Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.17,Handling Interest Rates,We do not want to define every interest rate as a different market

16、 variableAn approach is to use the duration relationship DP=-DPDy so that sP=DPysy,where sy is the volatility of yield changes and sP is as before the standard deviation of the change in the portfolio value,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shangha

17、i Normal University,14.18,Alternative:Cash Flow Mapping(p.347),We choose as market variables zero-coupon bond prices with standard maturities(1mm,3mm,6mm,1yr,2yr,5yr,7yr,10yr,30yr)Suppose that the 5yr rate is 6%and the 7yr rate is 7%and we will receive a cash flow of$10,000 in 6.5 years.The volatili

18、ties per day of the 5yr and 7yr bonds are 0.50%and 0.58%respectively,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.19,Cash Flow Mapping(continued),We interpolate between the 5yr rate of 6%and the 7yr rate of 7%to get a 6.5yr rate

19、of 6.75%The PV of the$10,000 cash flow is,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.20,Cash Flow Mapping(continued),We interpolate between the 0.5%volatility for the 5yr bond price and the 0.58%volatility for the 7yr bond pric

20、e to get 0.56%as the volatility for the 6.5yr bondWe allocate a of the PV to the 5yr bond and(1-a)of the PV to the 7yr bond,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.21,Cash Flow Mapping(continued),Suppose that the correlation

21、 between movement in the 5yr and 7yr bond prices is 0.6To match variancesThis gives a=0.074,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.22,Cash Flow Mapping(continued),The cash flow of 10,000 in 6.5 years is replaced by in 5 yea

22、rs and byin 7 years.This cash flow mapping preserves value and variance,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.23,When Linear Model Can be Used,Portfolio of stocksPortfolio of bondsForward contract on foreign currencyIntere

23、st-rate swap,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.24,The Linear Model and Options(p.350),Consider a portfolio of options dependent on a single stock price,S.Defineand,Options,Futures,and Other Derivatives,4th edition 2000

24、 by John C.HullTang Yincai,2003,Shanghai Normal University,14.25,Linear Model and Options(continued),As an approximationSimilar when there are many underlying market variableswhere di is the delta of the portfolio with respect to the ith asset,Options,Futures,and Other Derivatives,4th edition 2000 b

25、y John C.HullTang Yincai,2003,Shanghai Normal University,14.26,Example,Consider an investment in options on IBM and AT&T.Suppose the stock prices are 120 and 30 respectively and the deltas of the portfolio with respect to the two stock prices are 1,000 and 20,000 respectivelyAs an approximationwhere

26、 Dx1 and Dx2 are the proportional changes in the two stock prices,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.27,Skewness,The linear model fails to capture skewness in the probability distribution of the portfolio value.,Options

27、,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.28,Quadratic Model(p.352),For a portfolio dependent on a single stock price(by Taylor expansion)this becomesWhere the proportional change,Options,Futures,and Other Derivatives,4th edition 200

28、0 by John C.HullTang Yincai,2003,Shanghai Normal University,14.29,Moments of DP(p.354),Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.30,Quadratic Model(continued),With many market variables and each instrument dependent on only on

29、ewhere di and gi are the delta and gamma of the portfolio with respect to the ith variable,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.31,Quadratic Model(continued),When the change in the portfolio value has the formwe can calcu

30、late the moments of DP analytically if the Dxi are assumed to be normal,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.32,Quadratic Model(continued),Once we have done this we can use the Cornish Fisher expansion to calculate fracti

31、les of the distribution of DP,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.33,Monte Carlo Simulation(p.355),The stages are as followsValue portfolio todaySample once from the multivariate distributions of the Dxi Use the Dxi to d

32、etermine market variables at end of one dayRevalue the portfolio at the end of day,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.34,Monte Carlo Simulation,Calculate DPRepeat many times to build up a probability distribution for DP

33、VaR is the appropriate fractile of the distribution times square root of NFor example,with 1,000 trial the 1 percentile is the 10th worst case.,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.35,Speeding Up Monte Carlo,Use the quadr

34、atic approximation to calculate DP,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.36,Historical Simulation(p.356),Create a database of the daily movements in all market variables.The first simulation trial assumes that the percenta

35、ge changes in all market variables are as on the first dayThe second simulation trial assumes that the percentage changes in all market variables are as on the second dayand so on,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.37,S

36、tress Testing(p.357),This involves testing how well a portfolio performs under some of the most extreme market moves seen in the last 10 to 20 years,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.38,Back-Testing(p.357),Tests how we

37、ll VaR estimates would have performed in the pastWe could ask the question:How often was the loss greater than the 99%/10 day VaR?,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.39,Principal Components Analysis(p.357),Suppose that

38、a portfolio depends on a number of related variables(eg interest rates)We define factors as a scenarios where there is a certain movement in each market variable(The movements are known as factor loadings)The observations on the variables can often be largely explained by two or three factors,Option

39、s,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.40,Results for Interest Rates(Table 14.4),The first factor is a roughly parallel shift(83.1%of variation explained)The second factor is a twist(10%of variation explained)The third factor is a bowing(2.8%of variation explained),Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,2003,Shanghai Normal University,14.41,Assignments,14.1 14.5,14.10 14.14Assignment Questions,