ch16期權期貨與衍生證券（第五版）

1、Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.1,Value at Risk Chapter 16,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.2,The Question Being

2、Asked in 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, 5th edition © 2002 by John C. Hull,16.3,VaR and Regulatory Capita

3、l,Regulators base the capital they require banks to keep on VaRThe market-risk capital is k times the 10-day 99% VaR where k is at least 3.0,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C.

4、Hull,16.4,VaR vs. C-VaR (See Figures 16.1 and 16.2),VaR is the loss level that will not be exceeded with a specified probabilityC-VaR is the expected loss given that the loss is greater than the VaR levelAlthough C-Va

5、R is theoretically more appealing, it is not widely used,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.5,Advantages of VaR,It captures an important aspect of riskin a single numbe

6、rIt is easy to understandIt asks the simple question: “How bad can things get?”,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.6,Time Horizon,Instead of calculating the 10-day, 99%

7、 VaR directly analysts usually calculate a 1-day 99% VaR and assumeThis is exactly true when portfolio changes on successive days come from independent identically distributed normal distributions,Options, Futures, an

8、d Other Derivatives, 5th edition © 2002 by John C. Hull,16.7,Historical Simulation (See Table 16.1 and 16.2),Create a database of the daily movements in all market variables.The first simulation trial assumes tha

9、t the percentage 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

10、Other Derivatives, 5th edition © 2002 by John C. Hull,16.8,Historical Simulation continued,Suppose we use m days of historical dataLet vi be the value of a variable on day iThere are m-1 simulation trialsThe ith

11、 trial assumes that the value of the market variable tomorrow (i.e., on day m+1) is,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.9,The Model-Building Approach,The main alternative

12、to historical simulation is to make assumptions about the probability distributions of return on the market variables and calculate the probability distribution of the change in the value of the portfolio analyticallyTh

13、is is known as the model building approach or the variance-covariance approach,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.10,Daily Volatilities,In option pricing we express volat

14、ility as volatility per yearIn VaR calculations we express volatility as volatility per day,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.11,Daily Volatility continued,Strictly spe

15、aking 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 percentage change in one day,Options, Futures, and Oth

16、er Derivatives, 5th edition © 2002 by John C. Hull,16.12,Microsoft Example,We have a position worth $10 million in Microsoft sharesThe volatility of Microsoft is 2% per day (about 32% per year)We use N=10 and X=9

17、9,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.13,Microsoft Example continued,The standard deviation of the change in the portfolio in 1 day is $200,000The standard deviation of t

18、he change in 10 days is,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.14,Microsoft Example continued,We assume that the expected change in the value of the portfolio is zero (This i

19、s OK for short time periods)We assume that the change in the value of the portfolio is normally distributedSince N(–2.33)=0.01, the VaR is,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hu

20、ll,16.15,AT&T Example,Consider a position of $5 million in AT&TThe daily volatility of AT&T is 1% (approx 16% per year)The S.D per 10 days isThe VaR is,Options, Futures, and Other Derivatives, 5th edition

21、 © 2002 by John C. Hull,16.16,Portfolio,Now consider a portfolio consisting of both Microsoft and AT&TSuppose that the correlation between the returns is 0.3,Options, Futures, and Other Derivatives, 5th editio

22、n © 2002 by John C. Hull,16.17,S.D. of Portfolio,A standard result in statistics states thatIn this case sX = 200,000 and sY = 50,000 and r = 0.3. The standard deviation of the change in the portfolio value in o

23、ne day is therefore 220,227,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.18,VaR for Portfolio,The 10-day 99% VaR for the portfolio isThe benefits of diversification are(1,473,62

24、1+368,405)–1,622,657=$219,369What is the incremental effect of the AT&T holding on VaR?,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.19,The Linear Model,We assumeThe daily ch

25、ange in the value of a portfolio is linearly related to the daily returns from market variablesThe returns from the market variables are normally distributed,Options, Futures, and Other Derivatives, 5th edition © 2

26、002 by John C. Hull,16.20,The General Linear Model continued (equations 16.1 and 16.2),,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.21,Handling Interest Rates: Cash Flow Mapping,

27、We choose as market variables 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

28、years.The volatilities per day of the 5yr and 7yr bonds are 0.50% and 0.58% respectively,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.22,Example continued,We interpolate between t

29、he 5yr rate of 6% and the 7yr rate of 7% to get a 6.5yr rate of 6.75%The PV of the $10,000 cash flow is,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.23,Example continued,We interp

30、olate between the 0.5% volatility for the 5yr bond price and the 0.58% volatility for the 7yr bond price 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 t

31、he 7yr bond,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.24,Example continued,Suppose that the correlation between movement in the 5yr and 7yr bond prices is 0.6To match variances

32、This gives a=0.074,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.25,Example continued,The value of 6,540 received in 6.5 yearsin 5 years and byin 7 years.This cash flow ma

33、pping preserves value and variance,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.26,When Linear Model Can be Used,Portfolio of stocksPortfolio of bondsForward contract on foreign

34、currencyInterest-rate swap,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.27,The Linear Model and Options,Consider a portfolio of options dependent on a single stock price, S. Defin

35、eand,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.28,Linear Model and Options continued (equations 16.3 and 16.4),As an approximationSimilar when there are many underlying ma

36、rket variableswhere Di is the delta of the portfolio with respect to the ith asset,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.29,Example,Consider an investment in options on

37、Microsoft 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 dx1 and dx2

38、are the percentage changes in the two stock prices,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.30,Skewness (See Figures 16.3, 16.4 , and 16.5),The linear model fails to capture s

39、kewness in the probability distribution of the portfolio value.,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.31,Quadratic Model,For a portfolio dependent on a single stock price it

40、 is approximately true thatthis becomes,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.32,Moments of dP for one market variable,,Options, Futures, and Other Derivatives, 5th edit

41、ion © 2002 by John C. Hull,16.33,Quadratic Model continued,With many market variables and each instrument dependent on only onewhere Di and Gi are the delta and gamma of the portfolio with respect to the ith va

42、riable,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.34,Quadratic Model continued,When the change in the portfolio value has the formwe can calculate the moments of DP analytica

43、lly if the dxi are assumed to be normal,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.35,Quadratic Model continued,Once we have done this we can use the Cornish Fisher expansion to

44、calculate fractiles of the distribution of dP,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.36,Monte Carlo Simulation,To calculate VaR using M.C. simulation weValue portfolio today

45、Sample once from the multivariate distributions of the dxi Use the dxi to determine market variables at end of one dayRevalue the portfolio at the end of day,Options, Futures, and Other Derivatives, 5th edition ©

46、 2002 by John C. Hull,16.37,Monte Carlo Simulation,Calculate dPRepeat many times to build up a probability distribution for dPVaR is the appropriate fractile of the distribution times square root of NFor example, wit

47、h 1,000 trial the 1 percentile is the 10th worst case.,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.38,Speeding Up Monte Carlo,Use the quadratic approximation to calculate dP,Optio

48、ns, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.39,Stress Testing,This involves testing how well a portfolio performs under some of the most extreme market moves seen in the last 10 to 20

49、years,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.40,Back-Testing,Tests how well VaR estimates would have performed in the pastWe could ask the question: How often was the actual

50、 10-day loss greater than the 99%/10 day VaR?,Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull,16.41,Principal Components Analysis for Interest Rates,The first factor is a roughly parall