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29 0.0001 0.001 0.01 0.1 f FIGURE 3.9 The absolute value of the kernel of the differential operator (full line), in a logarithmic scale. The dotted line shows a simple EMA with range , demonstrating the much faster decay of the differential kernel. interval bracketing the time t  r has to be searched for. This return definition corresponds to a differential operator kernel made of two S functions (or to the limit x\, %i > 0 of the kernel in Figure 3.7). The quantity x(t)  x(t  ) can be quite noisy, so a further EMA might be taken to smooth it. In this case, the resulting effective differential operator kernel has two discontinuities, at 0 and at r, and decays exponentiallythat is, much slower than the kernel of [ ; ]. Thus it is cleaner and more efficient to compute returns with the A operator of Equation 3.61. Another quantity commonly used in finance is x  EMA[r; x], often called a momentum or an oscillator. This is also a differential with the kernel 5(0  exp(r/r)/r, with a S function aw = 0. A simple drawing shows that the kernel of Equation 3.61 produces a much less noisy differential. Other appropriate kernels can be designed, depending on the application. In general, there is a tradeoff between the averaging property of the kernel and a short response to shocks of the original time series. 3.3.10 Derivative and Derivative The derivative operator behaves exactly as the differential operator, except for the normalization D[r; t\ = 1. This derivative can be iterated in order to construct higher order derivatives: D[x\ = (3.62) D2[r] = D[t;D[t\] (3.63)
rr 0.04 0.03 0.021 0.01 0.004 0.01 0.02 0.03 0.04 I1 ,111111111 1 10 27.10 28.10 29.10 30.10 Date 31.10 1.11 2.11 FIGURE 3.10 A comparison between the differential computed using the formula in Equation 3.61 with = 24hr (full line) and the pointwise return x(t)  x(t  24h) (dotted line). The time lag of approximately 4hr between the curves is essentially due to the extent of both the positive part of the kernel (0 < t < 0.5) and the tail of the negative part (t > 1.5). The range of the secondorder derivative operator is 2 . More generally, the th order derivative operator Dn, constructed by iterating the derivative operator n times, has a range . As defined, the derivative operator has the dimension of an inverse time. It is easier to work with dimensionless operators and this is done by measuring r in some units. One year provides a convenient unit, corresponding to an annualized return when D[r]x is computed. The choice of unit is denoted by /ly, meaning that is measured in years, yet other units may also be used. For a random diffusion process, a more meaningful normalization for the derivative is to take D[r] = A[r] r/ly. For a space of processes as in Section 3.3.3, such that Equation 3.38 holds, the basic scaling behavior with r is eliminated, namely E [(D[t]z)2] = cr2. More generally, we can define a derivative as D[r,y] = A[r] (r/lyF In particular =0 = 0.5 = 1 differential stochastic diffusion process the usual derivative (3.64) (3.65) An empirical probability density function for the derivative is displayed in Figure 3.11. We clearly see that the main part of the scaling with is removed when using the yderivative with  0.5.
0.3 0.2 0.1 0.0 0.1 0.2 0.3 Derivative FIGURE 3.11 The annualized derivative D[r, = 0.5; x] for USDCHF from January I, 1988 to November I, 1998. The shortest time intervals r correspond to the most leptokurtic curves. In order to discard the daily and weekly seasonality, the time scale used is the business time i? as explained in Chapter 6 and in Dacorogna et al. (1993). The data were sampled every 2 hr (in #time) to construct the curves. The Gaussian probability density function added for comparison has a standard deviation of ct = 0.07, similar to that of the other curves. 3.3.11 Volatility The most common computation of realized or historical volatility is given by Equation 3.8 in Section 3.2.4, based on regularly spaced (e.g., daily) observations. Realized volatility can also be defined and measured with the help of convolution operators. Volatility is a measure widely used for random processes, quantifying the size and intensity of movements, namely the current "width" of the probability distribution P(Az) of the process increment Az, where A is a difference operator yet to be chosen. Often the volatility of market prices is computed, but volatility is a general operator that can be applied to any time series. There are many ways to turn this idea into a definition, and there is no unique, universally accepted definition of volatility in finance. In our new context, we can reformulate the
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