Rodney Hoskinson, Associate Director, Quantitative Support (Strategic Trading and Funding) at ANZ Banking Group presented on approximation methods for KVA under the final Basel III framework at QuantMinds International in Lisbon, here he explores the topic further.
Banks wrestling with the challenging and contentious calculations for the lifetime cost of servicing capital for derivatives (capital valuation adjustment, or KVA) incorporating regulatory capital requirements need to consider a pathwise calculation of capital under the (December 2017) final Basel III Standardised Approach for Credit Valuation Adjustment (SA-CVA) in their Monte Carlo simulation for valuation adjustments (XVA). Although this is a future regulatory requirement, arguably it should be included in forward-looking KVA calculations extending over future decades even now.
The SA-CVA is a sensitivity and weights-based calculation, analogous to the ISDA Standardised Initial Margin Method (SIMM) calculation. While SIMM can be seen as a first-order (delta/vega) approximation to a Value at Risk of 10-day Mark-to-Market variation, the SA-CVA can analogously be seen as a delta/vega approximation to the Expected Shortfall of CVA over a liquidity risk horizon. CVA sensitivities to market data are required.
KVA for SA-CVA is a similar problem to that of MVA but is computationally harder, as it requires CVA sensitivities as well as hedge Mark-to-Market sensitivities.
Much has been written in working papers and presented at conferences on dynamic initial margin for the purposes of calculating lifetime cost of funding initial margin (so-called Margin Valuation Adjustment or MVA). By contrast very little has been said about KVA for the SA-CVA. KVA for SA-CVA is a similar problem to that of MVA but is computationally harder, as it requires CVA sensitivities as well as hedge Mark-to-Market sensitivities. This creates a nested Monte Carlo problem as CVA itself is usually calculated by Monte Carlo methods. Brute force methods (with or without Adjoint Algorithmic Differentiation (AAD)) for calculating CVA sensitivities are even less appealing than they are for Mark-to-Market sensitivities as required for SIMM MVA. Just spot (time zero) CVA sensitivites are computationally demanding for banks.
One approach to this problem worthy of further investigation is to decompose the CVA sensitivity Jacobian matrix into a product of a Jacobian of CVA-to-model sensitivities and a Jacobian of model-to-market sensitivities. An approximation based on freezing the former at time zero and evolving the latter (ideally using AAD methods) would take the nested Monte Carlo calculation out of the equation.
In my QuantMinds International presentation, a simpler approach based on a new expected shortfall regression method was illustrated. This consists of two steps:
- First, obtain a pathwise CVA approximation by ordinary least squares regression.
- Secondly, get the pathwise Expected Shortfall of CVA over an appropriate risk horizon by Expected Shortfall regression using the pathwise CVA.
Details of the method and illustrative capital profiles and KVA examples are included in my presentation (here). The examples are based on interest rate and credit models with rich features I developed, including stochastic volatility and jumps in the interest rate curve, interest rate/credit dependence, and explicit credit grade evolution. However, the approach can be used with any set of interest rate, credit and other models used for XVA risk factor evolution. Ideally the models would include stochastic credit, in order to capture both market and credit evolution as intended by SA-CVA. The method can then allow for an approximate attribution of the capital profile into credit, market, and credit/market cross-gamma risks.
Even for banks who ultimately develop more elaborate SA-CVA KVA approximations, this simple method will provide a useful point of reference. For those banks without an AAD capability it may be one of few viable methods for approaching the calculation.