Breaking down XVAs

In the post-crisis world, it has become a prevalent practise to include certain costs in the pricing of OTC derivatives that in many cases have previously been ignored, the so-called valuation adjustments or “XVAs”. The crisis revealed that counterparty credit risks associated with OTC derivatives can be very substantial (e.g., CVA losses caused by US monoliners) and ought to be mitigated. Market participants incur costs associated with hedging counterparty credit risks through their CVA management activities. Other costs include capital, funding and liquidity costs.

XVAs are counterparty or netting set level metrics, and necessitate a combined assessment of a full range of trades, typically across a broad range of asset classes. For general OTC derivatives netting sets, the calculation of the valuation adjustments is highly complex, and requires the generation of exposure profiles over the lifetime of the trades. The complexity of the modelling is due to the fact that exposures are aggregated across a whole portfolio of trades (both long and short) covered by a master netting agreement, which requires a simultaneous projection of the mark-to-market values of all trades within the netting set.

From an accounting perspective, CVA, DVA and FVA provisions are often only set aside for a reduced set of netting sets that are uncollateralised or have a high margin threshold. Under the CVA rules introduced by Basel III, however, CVA risk needs to be assessed for all bilateral netting sets, even those that are closely collateralised with a low threshold.

Computational complexity

The estimation of the netting set level exposure profiles is typically performed using a Monte Carlo simulation. Within each individual Monte Carlo path, one generates a joint realisation of all risk factors that drive the market value of the netting set. For this “realisation of the future”, one then performs a revaluation of all trades in order to obtain the corresponding exposure.

A large OTC derivative netting set can easily have in excess of 100 risk drivers, spanning different interest rates curves, foreign exchange rates, inflation curves, credit spreads, equity indices, etc. Estimating the valuation adjustment for a particular netting set is thus computationally very expensive.

XVA sensitivities

Aside from the XVA metric in itself, a quantity of interest is the sensitivity of the XVA to the netting set’s key risk drivers. Indeed, in recent years, OTC valuation adjustments have become traded quantities (especially CVA). Managing a CVA desk hence not only requires a market value, but also creates a need for calculating CVA Greeks (mainly deltas and vegas), in order to adequately hedge the risks.

While the core mandate of a CVA desk is to manage the volatility of accounting CVA, the introduction of capital requirements for CVA risk under Basel III made banks focus on additionally managing CVA from a regulatory capital perspective. In fact, the standardised approach for CVA risk introduced in Basel IV (SA-CVA) explicitly requires the calculation of CVA sensitivities for all risk factors.

As previously mentioned, an XVA calculation in itself is computationally very expensive. Calculating sensitivities of XVAs to the relevant risk factors adds a new dimension to the computational costs. Indeed, if one would apply “brute force” to calculate first order XVA Greeks using finite differences, one would have to re-run the full XVA simulation hundreds of times (once for each risk factor). In most practical cases, this tends to be computationally unfeasible. Recent developments in quantitative finance, however, show that the calculation of XVA Greeks can be made computationally tractable by making use of algorithmic differentiation (AD).

Cost allocations and pre-trade assessments

In larger organisations, the allocation of XVAs to desks or trades is a prerequisite for efficient management of OTC derivatives trading businesses. Understanding how individual trades affect the XVAs is a key component in pre-trade assessments. Furthermore, a fair allocation of costs along an institution’s business hierarchy ensures that everyone in the business is incentivise to deploy the firm’s capital in the most efficient way.

Since the valuation adjustments are calculated at an aggregate level across all trades in a netting set, sophisticated methods have to be applied in order to break down the netting set-level XVAs into contributions by single desks or trades.

Our solution: sensitivity-based XVA allocations

In this article, we introduce an intuitive and fair XVA allocation technique based on risk factor sensitivities. The approach can be integrated as a stand-alone addition to any XVA calculation infrastructure, leveraging an existing XVA sensitivity framework (e.g., the CVA sensitivities used in the capital calculations for the Basel IV standardised approach for CVA risk^[1]). We further discuss how XVA sensitivities can be calculated efficiently using algorithmic differentiation techniques.

The article concludes by providing practical example, in which CVA is allocated for a simple portfolio of equity derivatives. With this example we illustrate computational gains that can be made by applying algorithmic differentiation.

An efficient calculation of XVA sensitivities provides several useful applications, and can significantly improve OTC derivatives portfolio management.

_{[1] Basel Committee. (2017) “Basel III: Finalising post-crisis reforms”.}