# Expected shortfall of stable distribution by Stoyanov

I've been working on calculating parametric ES assuming the returns follow Paretian stable law. Given the four parameters - $\alpha, \beta,\sigma,\mu$- Stoyanov introduces closed form solution of the problem (assuming returns as the r.v., not losses - hence VaR is the negative quantile and we are integrating over the left tail, as I understand it. - see below.

I have been struggling for two days with implementing this in R, but I cannot get the results they publish at the end of the paper - expect for a special symmetric case ($\beta=0$)

Any ideas where I could possibly go wrong? Here are my results: I use integrate in Rfor the integral and it seems to give reasonable values of the integral with absolute error around 1e-04.

Any help would be greatly appreciated.

> dat
alpha    myES  target relError integral myVaR absErrorIntegral subdivisions
1   1.04 160.837 157.204    0.023   -0.159 6.119            0.000            7
2   1.07  93.020  86.393    0.077   -0.166 5.755            0.000            6
3   1.10  66.317  58.302    0.137   -0.174 5.430            0.000            5
4   1.13  52.234  43.331    0.205   -0.184 5.140            0.000            4
5   1.16  43.650  34.082    0.281   -0.194 4.878            0.000            4
6   1.19  37.961  27.831    0.364   -0.205 4.642            0.000            4
7   1.22  33.996  23.344    0.456   -0.217 4.428            0.000            4
8   1.25  31.154  19.979    0.559   -0.231 4.234            0.000            3
9   1.28  29.094  17.369    0.675   -0.246 4.057            0.000            3
10  1.31  27.617  15.292    0.806   -0.264 3.895            0.000            3
11  1.34  26.599  13.604    0.955   -0.283 3.746            0.000            3
12  1.37  25.965  12.200    1.128   -0.305 3.610            0.000            2
13  1.40  25.672  11.030    1.327   -0.331 3.486            0.000            2
14  1.43  25.703  10.030    1.563   -0.360 3.371            0.000            2
15  1.46  26.058   9.170    1.842   -0.395 3.266            0.000            2
16  1.49  26.758   8.430    2.174   -0.436 3.169            0.000            2
17  1.52  27.840   7.771    2.583   -0.486 3.080            0.000            2
18  1.55  29.360   7.200    3.078   -0.546 2.998            0.000            2
19  1.58  31.399   6.680    3.701   -0.619 2.923            0.000            2
20  1.61  34.067   6.229    4.469   -0.710 2.854            0.000            2
21  1.64  37.506   5.817    5.448   -0.824 2.791            0.000            2
22  1.67  41.917   5.445    6.698   -0.967 2.733            0.000            2
23  1.70  47.576   5.100    8.329   -1.148 2.679            0.000            1
24  1.73  54.877   4.790   10.457   -1.383 2.630            0.000            2
25  1.76  64.422   4.500   13.316   -1.690 2.585            0.000            2
26  1.79  77.145   4.250   17.152   -2.103 2.544            0.000            2
27  1.82  94.616   4.010   22.595   -2.673 2.505            0.000            2
28  1.85 119.654   3.790   30.571   -3.497 2.470            0.000            3
29  1.88 157.937   3.580   43.116   -4.766 2.437            0.000            3
30  1.91 222.734   3.390   64.703   -6.928 2.406            0.000            3
31  1.94 353.958   3.220  108.925  -11.330 2.378            0.001            3
32  1.97 751.596   3.050  245.425  -24.725 2.351            0.001            4