Accelerated computations of sensitivities for xVA*

ABSTRACT

Exposure simulations are fundamental to many xVA calculations and are a nested expectation problem where repeated portfolio valuations create a significant computational expense. Sensitivity calculations which require shocked and unshocked valuations in bump-and-revalue schemes exacerbate the computational load. A known reduction of the portfolio valuation cost is understood to be found in polynomial approximations, which we apply in this article to interest rate sensitivities of expected exposures. We consider a method based on the approximation of the shocked and unshocked valuation functions, as well as a novel approach in which the difference between these functions is approximated. Convergence results are shown, and we study the choice of interpolation nodes. Numerical experiments with interest rate derivatives are conducted to demonstrate the high accuracy and remarkable computational cost reduction. We further illustrate how the method can be extended to more general xVA models using the example of CVA with wrong-way risk.

KEYWORDS:

• xVA
• sensitivity
• exposure simulation
• portfolio approximation
• polynomial interpolation
• Monte Carlo

MSC CLASSIFICATIONS:

• 91G30
• 91G20

Acknowledgments

The authors thank an anonymous referee for valuable comments that helped improve this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In this article, ‘sensitivity’ refers to the effect on the expected exposure induced by small changes to the term structure of interest rates. It can be thought of as a particular type of derivative, which is approximated by difference quotients. Then, ‘shocked’ and ‘unshocked’ valuations refer to the function valuations with and without shifted parameters, respectively. A comprehensive description of the procedure is provided in Section 3.

2 We denote expectations conditional on the filtration ${\mathcal{F}}_t$ with a subscript ${\mathbb{E}}_t$.

3 The expected negative exposure is identical to (2.2(2.2) $\mathrm{EE}(t_0,t)={\mathbb{E}}_{t_0}^{\mathbb{Q}}[\frac{B(t_0)}{B(t)}{V}^{+}(t,X(t))]\approx \frac{1}{M}\sum _{j=1}^M\frac{B(t_0)}{B(t;{\omega }_j)}{V}^{+}(t,X(t;{\omega }_j)),$(2.2) ) with the positive exposure replaced by the negative exposure $V^{-}(t,X(t))$. Within this article, we consider only the positive exposure, by symmetry the results can be extended to negative exposure computations.

Additional information

Funding

This research is part of the ABC–EU–XVA project and has received funding from the European Union's Horizon 2020 Research and Innovation programme under the Marie Skłodowska–Curie grant agreement No. 813261.