Stéphane Crépey
LaMME, Univ. Evry, CNRS, Université Paris-Saclay
Paris IHP, January 26 and February 2, 9, and 16, 2018


Since the crisis, derivative dealers charge to their clients various add-ons, dubbed X-valuation adjustments (XVAs), meant to account for counterparty risk and its capital and funding implications. XVAs deeply affect the derivative pricing task by making it global, nonlinear, and entity dependent. However, before the technical implications, the fundamental points are to understand what deserves to be priced and what does not, and to establish, not only the pricing, but also the corresponding collateralization, dividend, and accounting policy of a bank. If banks cannot replicate jump-to-default related cash ows, deals trigger wealth transfers from bank shareholders to creditors and shareholders need to set capital at risk. On this basis, we devise a theory of XVAs, whereby so-called contra-liabilities and cost of capital are sourced from bank clients at trade inceptions, on top of the fair valuation of counterparty risk, in order to compensate shareholders for wealth transfer and risk on their capital.

The resulting all-inclusive XVA add-on, to be sourced from clients incrementally at every new deal, reads (CVA + FVA + MVA + KVA), where C sits for credit, F for funding, M for (initial) margin, and where the KVA is a cost of capital risk premium. This formula corresponds to the cost of the possibility for the bank to go into run-off, while staying in line with shareholder interest, from any point in time onward if wished. Moreover, economic capital (EC) can be used as a funding source by banks, at a risk-free cost instead of the additional credit spread of the bank in the case of unsecured borrowing. This intertwining of EC and FVA leads to an anticipated BSDE (backward stochastic differential equation « of the McKean type ») for the FVA, with coeffcient entailing a conditional risk measure of the one-year-ahead increment of the martingale part of the FVA itself.

Our XVA equations are solved by projection on a reduced ltration myopic to the default of the bank, the latter being assumed to be an invariance time as per Crépey and Song (2017). This assumption, which covers mainstream immersion setups (but not only), expresses the consistency of valuation across different trading desks with different focuses within the bank: the XVA desks versus the different business desks.

Finally, we present a nested Monte Carlo approach implemented on graphics processing units (GPU) to XVA computations. The overall XVA suite involves five compound layers of nested dependence. Higher layers are launched first and trigger nested simulations on-the-fly whenever required in order to compute a metric from a lower layer. With GPUs, error controlled nested Monte Carlo XVA computations are within reach. This is illustrated on XVA computations involving equities, interest rate, and credit derivatives, for both bilateral and central clearing XVA metrics.

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