73323393-19-Estimating-Vol-and-Corr - E stitnating...

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Estitnating Volatilities and Correlations In this chapter we explain how historical data can be used to produce estimates of the current and future levels of volatilities and correlations. The chapter is relevant both to the calculation of value at risk using the model-building approach and to the valuation of derivatives. When calculating value at risk, we are most interested in the current levels of volatilities and correlations because we are assessing possible changes in the value of a portfolio over a very short period of time. When valuing derivatives, forecasts of volatilities and correlations over the whole life of the derivative are usually required. The chapter considers models with imposing names such as exponentially weighted moving average (EWMA), autoregressive conditional heteroscedasticity (ARCH), and generalized autoregressive conditional heteroscedasticity (GARCH). The distinctive feature of the models is that they recognize that volatilities and correlations are not constant. During some periods, a particular volatility or correlation may be relatively low, whereas during other periods it may be relatively high. The models attempt to keep track of the variation~ in the volatility or correlation through time. 19.1 ESTIMATING VOLATILITY Define (J1l as the volatility of a market variable on day n, as estimated at the end of day 12 - L The square of the volatility, (J~, on day 12 is the variance rate. We described the standard approach to estimating from historical data in Section 13.4. Suppose that the value of the market variable at the end of day i is Si' The variable lli is defined as the continuously compounded return during day i (between the end of day i-I the end of day i): = In-'- Si-I An unbiased estimate of the variance rate per day, observations on the using the most recent I1l (19.1) 461
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462 CHAPTER 19 where it is the mean of the lliS: 1 //I =- Lllll-i 111 i=l For the purposes ofmonitoring daily volatility, the formula in equation (19.1) is usually changed in a number of ways: 1. lli is defined as the percentage change in the market variable between the end of day i-I and the end ofday i, so that:! (19.2) 2. is assumed to be zero? 3. 111 - 1 is replaced by 111. 3 These three changes make very little difference to the estimates that are calculated, but they allow us to simplify the formula for the variance rate to 1 1 ,,1 a;; = - £...J ll;;-i 111 i=! where is given by equation (19.2).4 (19.3) Weighting Schemes Equation (19.3) gives equal weight to ll~-l> ll~-2"'" ll~_I/1' Our objective is to estimate the current level of volatility, all' It therefore makes sense to give more weight to recent data. A model that does this is (19.4) The variable (Xi is the amount of weight given to the observation i days ago. The (X's are positive. If we choose them so that (Xi < (Xj when i > j, less weight is given to older observations. The weights must sum to unity, so we have I This is consistent with the point made in Section 18.3 about the way that volatility is defined for the purposes of VaR calculations.
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73323393-19-Estimating-Vol-and-Corr - E stitnating...

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