Geodesic
Stochastic approximation schemes for economic capital and risk margin computations
David Barrera1 ; Stéphane Crépey2 ; Babacar Diallo2, 3 ; Gersende Fort4 ; Emmanuel Gobet1 ; Uladzislau Stazhynski1
1 Centre de Mathématiques Appliqués (CMAP), Ecole Polytechnique and CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau Cedex, France
2 LaMME, Univ. Evry, CNRS, Université Paris-Saclay, 91037, Evry, France
3 Quantitative Research GMD/GMT Crédit Agricole CIB, 92160 Montrouge, France
4 CNRS, Institut de Mathématiques de Toulouse (IMT), 31062 Toulouse Cedex 9, France
ESAIM. Proceedings, Tome 65 (2019), pp. 182-218.

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We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja [17] or by regression à la Broadie, Du, and Moallemi [10]. Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in [17] and [10]. Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking parlance. This is illustrated numerically by a KVA case study implemented on GPUs.
DOI : 10.1051/proc/201965182

David Barrera 1 ; Stéphane Crépey 2 ; Babacar Diallo 2, 3 ; Gersende Fort 4 ; Emmanuel Gobet 1 ; Uladzislau Stazhynski 1

1 Centre de Mathématiques Appliqués (CMAP), Ecole Polytechnique and CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau Cedex, France
2 LaMME, Univ. Evry, CNRS, Université Paris-Saclay, 91037, Evry, France
3 Quantitative Research GMD/GMT Crédit Agricole CIB, 92160 Montrouge, France
4 CNRS, Institut de Mathématiques de Toulouse (IMT), 31062 Toulouse Cedex 9, France 
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     title = {Stochastic approximation schemes for economic capital and risk margin computations},
     journal = {ESAIM. Proceedings},
     pages = {182--218},
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     doi = {10.1051/proc/201965182},
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David Barrera; Stéphane Crépey; Babacar Diallo; Gersende Fort; Emmanuel Gobet; Uladzislau Stazhynski. Stochastic approximation schemes for economic capital and risk margin computations. ESAIM. Proceedings, Tome 65 (2019), pp. 182-218. doi : 10.1051/proc/201965182. http://geodesic.mathdoc.fr/articles/10.1051/proc/201965182/

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