Ch14ValueatRisk(金融工程-华东师范大学汤银才).ppt

Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.1,Value at Risk,Chapter 14,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.2,The Question Being Asked in Value at Risk (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. Hull Tang Yincai, 2003, Shanghai Normal University,14.3,Meaning is Probability,(1-) %,%,Z,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang 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 factor equals 3),Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.5,Advantages of VaR,It captures an important aspect of risk in a single number It is easy to understand It asks the simple question: “How bad can things get?”,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.6,Daily Volatilities,In option pricing we express volatility as volatility per year In VaR calculations we express volatility as volatility per day,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang 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 day In practice we assume that it is the standard deviation of the proportional change in one day,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.8,IBM Example (p. 343),We have a position worth $10 million in IBM shares The volatility of IBM is 2% per day (about 32% per year) We use N=10 and X=99,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.9,IBM Example (continued),The standard deviation of the change in the portfolio in 1 day is $200,000 The standard deviation of the change in 10 days is,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang 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 distributed Since N(0.01)=-2.33, the VaR is,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.11,AT&T Example,Consider a position of $5 million in AT&T The daily volatility of AT&T is 1% (approx 16% per year) The STD per 10 days is The VaR is,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.12,Portfolio (p. 344),Now consider a portfolio consisting of both IBM and AT&T Suppose that the correlation between the returns is 0.7,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.13,STD of Portfolio,A standard result in statistics states that In 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. Hull Tang Yincai, 2003, Shanghai Normal University,14.14,VaR for Portfolio,The VaR for the portfolio is The benefits of diversification are (1,473,621+368,405)-1,751,379=$90,647 What is the incremental effect of the AT&T holding on VaR?,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.15,The Linear Model,We assume The change in the value of a portfolio is linearly related to the change in the value of market variables The changes in the values of the market variables are normally distributed,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 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. Hull Tang Yincai, 2003, Shanghai Normal University,14.17,Handling Interest Rates,We do not want to define every interest rate as a different market variable An 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. Hull Tang Yincai, 2003, Shanghai 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 volatilities 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. Hull Tang 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 of 6.75% The PV of the $10,000 cash flow is,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang 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 price to get 0.56% as the volatility for the 6.5yr bond We 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. Hull Tang Yincai, 2003, Shanghai Normal University,14.21,Cash Flow Mapping (continued),Suppose that the correlation between movement in the 5yr and 7yr bond prices is 0.6 To match variances This gives a=0.074,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang 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 years and by in 7 years. This cash flow mapping preserves value and variance,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.23,When Linear Model Can be Used,Portfolio of stocks Portfolio of bonds Forward contract on foreign currency Interest-rate swap,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang 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. Define and,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.25,Linear Model and Options (continued),As an approximation Similar when there are many underlying market variables where di is the delta of the portfolio with respect to the ith asset,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang 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 respectively As an approximation where Dx1 and Dx2 are the proportional changes in the two stock prices,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.27,Skewness,The linear model fails to capture skewness in the probability distribution of the portfolio value.,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.28,Quadratic Model (p. 352),For a portfolio dependent on a single stock price (by Taylor expansion) this becomes Where the proportional change,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.29,Moments of DP (p. 354),Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.30,Quadratic Model (continued),With many market variables and each instrument dependent on only one where 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. Hull Tang Yincai, 2003, Shanghai Normal University,14.31,Quadratic Model (continued),When the change in the portfolio value has the form we can calculate the moments of DP analytically if the Dxi are assumed to be normal,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.32,Quadratic Model (continued),Once we have done this we can use the Cornish Fisher expansion to calculate fractiles of the distribution of DP,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.33,Monte Carlo Simulation (p. 355),The stages are as follows Value portfolio today Sample once from the multivariate distributions of the Dxi Use the Dxi to determine market variables at end of one day Revalue the portfolio at the end of day,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.34,Monte Carlo Simulation,Calculate DP Repeat many times to build up a probability distribution for DP VaR is the appropriate fractile of the distribution times square root of N For example, with 1,000 trial the 1 percentile is the 10th worst case.,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.35,Speeding Up Monte Carlo,Use the quadratic approximation to calculate DP,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang 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 percentage changes in all market variables are as on the first day The second simulation trial assumes that the percentage changes in all market variables are as on the second day and so on,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,14.37,Stress 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. Hull Ta