华东师范大学：《金融工程》英文版 Chapter 14 Value at risk

14.1 Value at risk Chapter 14 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

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

14.2 The question Being asked in Value at Risk (Var) What loss level is such that we are x%o confident it will not be exceeded in n business days? Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

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?

14.3 Meaning is Probability Y-N(0,o2) Pr(r<pm=a po=n(a) Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 14.3 Meaning is Probability 2 (0, ) Pr( ) ( ) * Y N Y p p N        = = (1-) % % Z

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, C 2003, Shanghai Normal University

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)

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, C 2003, Shanghai Normal University

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?

14.6 DailyⅤ volatilities In option pricing we express volatility as volatility per year In VaR calculations we express volatility as volatility per day year √252 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

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   day year = 252

14.7 Daily volatility(continued) Strictly speaking we should define o day 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, C 2003, Shanghai Normal University

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 day 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

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 m=10and￥99 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

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

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 2000010=S632456 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

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 200,000 10 = $632,456

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)=-233, the Var is 2.33×632456=S1,473,.621 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

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 2.33 632,456 = $1,473,621