Value at Risk Chapter 20 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 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?” Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 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 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 VaR vs Expected Shortfall (See Figures 20.1 and 20.2, page 431) VaR is the loss level that will not be exceeded with a specified probability Expected shortfall is the expected loss given that the loss is greater than the VaR level Although expected shortfall is theoretically more appealing than VaR, it is not widely used Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 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?” Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 Historical Simulation 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 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 Historical Simulation continued Suppose we use 501 days of historical data Let vi be the value of a market variable on day i There are 500 simulation trials The ith trial assumes that the value of the market variable tomorrow is v500 vi vi −1 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 Historical Simulation continued The portfolio’s value tomorrow is calculated for each simulation trial The loss between today and tomorrow is then calculated for each trial (gains are negative losses) The losses are ranked and the one-day 99% VaR is set equal to the th worst loss Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 The Model-Building Approach The main alternative to historical simulation is to make assumptions about the probability distributions of return on the market variables This is known as the model building approach or the variance-covariance approach Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 Daily Volatilities In option pricing we express volatility as volatility per year In VaR calculations we express volatility as volatility per day σ day = σ year 252 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 10 But the distribution of the daily return on an option is not normal (See Figure 20.4, page 444) Positive Gamma Negative Gamma Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 29 Translation of Asset Price Change to Price Change for Long Call (Figure 20.5, page 445) Long Call Asset Price Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 30 Translation of Asset Price Change to Price Change for Short Call (Figure 20.6, page 445) Asset Price Short Call Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 31 Quadratic Model For a portfolio dependent on a single stock price where γ is the gamma of the portfolio This becomes ∆P = δ∆S + γ (∆S ) 2 ∆P = Sδ ∆x + S γ (∆x) 2 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 32 Use of Quadratic Model Analytic results are not as readily available Monte Carlo simulation can be used in conjunction with the quadratic model (This avoids the need to revalue the portfolio for each simulation trial) The quadratic model is also sometimes used in conjunction with historical simulation Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 33 Estimating Volatility for Model Building Approach (equation 20.6) Define σn as the volatility per day between day n-1 and day n, as estimated at end of day n1 Define Si as the value of market variable at end of day i Define ui= ln(Si/Si-1) The usual estimate of volatility from m observations is: m σ = ( u − u ) ∑ n −i m − i =1 n m u = ∑ u n −i m i =1 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 34 Simplifications (equations 20.7 and 20.8) Define ui as (Si–Si-1)/Si-1 Assume that the mean value of ui is zero Replace m–1 by m This gives m σ = ∑i =1 un −i m n Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 35 Weighting Scheme Instead of assigning equal weights to the observations we can set σ = ∑i =1 α i u m n n −i where m ∑α i =1 i =1 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 36 EWMA Model (equation 20.10) In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time This leads to σ = λσ n n −1 + (1 − λ )u n −1 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 37 Attractions of EWMA Relatively little data needs to be stored We need only remember the current estimate of the variance rate and the most recent observation on the market variable Tracks volatility changes RiskMetrics uses λ = 0.94 for daily volatility forecasting Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 38 Correlations Define ui=(Ui-Ui-1)/Ui-1 and vi=(Vi-Vi-1)/Vi-1 Also σu,n: daily vol of U calculated on day n-1 σv,n: daily vol of V calculated on day n-1 covn: covariance calculated on day n-1 covn = ρn σu,n σv,n where ρn is the correlation between U and V Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 39 Correlations continued (equation 20.12) Using the EWMA covn = λcovn-1+(1-λ)un-1vn-1 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 40 Model Building vs Historical Simulation Approaches Model building approach has the disadvantage that it assumes that market variables have a multivariate normal distribution Historical simulation is computationally slower and cannot easily incorporate volatility updating schemes Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 41 Back-Testing Tests how well VaR estimates would have performed in the past We could ask the question: How often was the loss greater than the VaR level Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 42 Stress Testing This involves testing how well a portfolio would perform under some of the most extreme market moves seen in the last 10 to 20 years Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 2010 43 ... S Define and ∆P δ= ∆S ∆S ∆x = S Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 201 0 26 Linear Model and Options continued (equations 20. 3 and 20. 4) As... standard deviation of the change in In this case σX = 200 ,000 and σY = 50,000 the portfolio value in one day is therefore 220, 227 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, ... $368,405 Fundamentals of Futures and Options Markets, 7th Ed, Ch 20, Copyright © John C Hull 201 0 15 Portfolio (See Example 20. 1) Now consider a portfolio consisting of both Microsoft and AT&T