《金融期货与期权》（英文版） Chapter 16 Value at Risk

16.1 Value at risk Chapter 16 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 16.1 Value at Risk Chapter 16

16.2 The Question Being Asked in Var What loss level is such that we are y% confident it will not be exceeded in n business days?” Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 16.2 The Question Being Asked in VaR “What loss level is such that we are X% confident it will not be exceeded in N business days?

16.3 VaR and regulatory capital Regulators base the capital they require banks to keep on VaR The 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. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 16.3 VaR and Regulatory Capital • Regulators base the capital they require banks to keep on VaR • The market-risk capital is k times the 10- day 99% VaR where k is at least 3.0

16.4 VaRⅴs.CVaR (See Figures 16.1 and 16.2) Var is the loss level that will not be exceeded with a specified probability C-VaR is the expected loss given that the loss is greater than the var level Although C-Var is theoretically more appealing, it is not widely used Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. 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 probability • C-VaR is the expected loss given that the loss is greater than the VaR level • Although C-VaR is theoretically more appealing, it is not widely used

16.5 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, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 16.5 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?

16.6 Time horizon Instead of calculating the 10-day, 99% VaR directly analysts usually calculate a 1-day 99% VaR and assume 10- day VaR=√10×1- day VaR This is exactly true when portfolio changes on successive days come from independent identically distributed normal distributions Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 16.6 Time Horizon • Instead of calculating the 10-day, 99% VaR directly analysts usually calculate a 1-day 99% VaR and assume • This is exactly true when portfolio changes on successive days come from independent identically distributed normal distributions 10 - day VaR = 10 1- day VaR

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 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, 5th edition 2002 by John C. Hull

Options, Futures, and 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 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

16.8 Historical simulation continued Suppose we use m days of historical data Let vi be the value of a variable on day i There are m-1 simulation trials The ith 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

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 16.8 Historical Simulation continued • Suppose we use m days of historical data • Let vi be the value of a variable on day i • There are m-1 simulation trials • The ith trial assumes that the value of the market variable tomorrow (i.e., on day m+1) is i−1 i m v v v

169 The Model-Building Approach The main alternative 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 analytically This is known as the model building approach or the variance-covariance approach Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 16.9 The Model-Building Approach • The main alternative 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 analytically • This is known as the model building approach or the variance-covariance approach

16.10 Daily 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, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 16.10 Daily Volatilities • In option pricing we express volatility as volatility per year • In VaR calculations we express volatility as volatility per day 252 year day   =