back start next
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [ 98 ] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134]
98 returns i(Axj; Ayj; At) = ( ,.  (Ax,.m) ,. /  ( ,. ))" (10.3) where (AJf,m> = V and < ,, > = £ (10.4) j=l 7=1 The most obvious choice for a is 0.5, though this can be investigated as a way to magnify or demagnify the weight given to farther outlying return values. A value of 0.5 is used in all cases described in this discussion. Equation 10.3 formulates covolatility around the mean rather than around zero and it therefore follows that = 0 for the case of returns derived from two linearly interpolated prices existing outside of our region of interest, At. These covolatility estimates can be inserted as weights in all the sums computed to obtain the variances and covariance of the correlation calculation: Q(Axit Ay,, to,) = Zjif I (Axi  < ))( ;  {Ay))coi ] s/Zfi? [ (Ax,  (Ax))2w, ]7Efiff [ (Ay;  (Ay»% ] (10.5) Note that Ax, and Ay, from Equation 10.5 are the same values as used in Equation 10.1, as they are logarithmic returns taken over the same time period, A;. These coarse return values can then be defined as the sum of the fine return values Ax, = Axi.mj (10.6) The sample means (Ax) and (Ay) have to be reconsidered in Equation 10.5. In the special case of risk assessment, we can still replace them by zero. Otherwise, we prefer that they are calculated again in a weighted fashion so that returns are considered only when observations over intervals of size At exist. Rather than keeping Equation 10.2, we define covolatility weighted mean values for both time series: YTI, {Ax cot) TjiAytcot) <)= / and (Ay)2 )Jl (10.7)
In this way, the means are calculated over the identically weighted data sample also used for the rest of the correlation calculation. The weights adjust for periods of lower or higher activity. Equation 10.3 is formulated in such a way that o>; = 0 for the case of returns interpolated over a data gapthat is, a tick interval that fully contains the analyzed interval of size At. Data gaps have no influence on the means, and the sums of Equations 10.5 and 10.7 are not updated there. The covolatility adjusted measure of correlation described by Equation 10.5 also retains the desirable characteristics of the original, standard linear correlation coefficient; it is scale free, and completely different measurements are directly comparable. In addition, this alternative method is only slightly more complicated to implement than the standard linear correlation coefficient and can easily be implemented on a computer. As will be applied later, this correlation measure easily fits into the framework of autocorrelation analysis. Given a time series of correlations gt, it can be correlated with a copy of itself but with different time lags (r) between the two, as shown in Equation 10.8: R(q(Axj, Ay,,<w,), r) = E!U+i(g»  (oi))(oir  (02)) [L?=r+i(or  (oi))2L"=r+i(err  (02»2],/2 for r > 0, where 1 " 1 " <ei> =  ) 0, and (q2) =  T q,r ( .9) r=r+l r=r + l For the discussions that follow, we measure correlation using the covolatility adjusted method described by Equation 10.5, unless otherwise stated, and always with m = 6 and a = 0.5 (see Equation 10.3). Any subsequent use of the commonly recognized linear correlation coefficient (Equation 10.1) will be referred to as the "standard" method. 10.3.2 Monte Carlo and Empirical Tests Various tests were performed on the covolatility adjusted correlation measure in order to test its behavior when applied to time series with differing frequencies and data gaps. A first test used synthetic Monte Carlo data to illustrate the effectiveness of the method. Two separate, uncorrelated, normally distributed, i.i.d. random time series, / and Bj, were produced, each with zero mean, standard deviation a = 0.01 and size m = 10, 000. A third series, C,, can then be formed as a linear combination of the previous two: qi, = *aj"=i + (i  ) =1 (10.10)
TABLE 10.1 Results of a Monte Carlo simulation of correlations. Comparing the covolatility adjusted linear correlation q to the standard linear correlation q, both applied to synthetic time series. The series £>, is like Cj, but regularly spaced sections of the data are replaced by linearly interpolated data. Details are described in the text. Note the similarity of the q(Aj, Cj) and q(Aj, £>,) columns. Multiplier Equation 10.10  Q(AjXi) Equation 10.1  Q(Aj,Dj) Equation 10.1  QiAj.Dj) Equation 10.5   0.00  0.00  0.00   0.12  0.10  0.12   0.23  0.15  0.22   0.38  0.28  0.38   0.52  0.40  0.51   0.69  0.51  0.69   0.83  0.62  0.82   0.92  0.67  0.91   0.97  0.72  0.95   0.99  0.74  0.97   1.00  0.74  0.99 
where the constant is selected such that 0 < < 1. In this way, the new series Cj has a controllable correlation to the original data series A,. The synthetic returns C, were then cumulated to synthetic prices P,, with starting value Pi = 10 and sample size m + 1 = 10, 001: pm+l e\n(Pl ,)+Ci 1 (10]1) The pure cumulation of C, leads to synthetic logarithmic prices that are transformed to synthetic nonlogarithmic prices by the exponential function. Repeated data sections, each consisting of 50 price observations, were then deleted in the time series P; and replaced by prices linearly interpolated from the prices bracketing the deleted sections. The distance between these artificial data gaps also consisted of 50 observations, creating an alternating series of original data patches followed by data gaps filled with linearly interpolated prices. Finally, the first differences of this altered price series were taken to build a new series of returns, Dj. Equation 10.5 was then used to measure the correlation between one of the original return distributions, A,, and the manipulated return distribution, D,, given various values of the constant multiplier k. Results are shown in comparison to the standard linear correlation calculation in Table 10.1. A comparison of columns two (q(A, C)) and four (&(A, D)) shows that the covolatility adjusted correlation measure described by Equation 10.5 successfully approximates the original standard linear correlation between distributions A and before some data patches were replaced by linearly interpolated values. Any
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [ 98 ] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134]
