# Pricing valuation adjustments by correlation expansion

# Pricing valuation adjustments by correlation expansion

16.00h

Faculty of Mathematics and Computer Science, UB (Hist. Building).

*Abstract*

We consider firstly the problem of computing the Credit Value Adjustment (CVA) of a European option in a default intensity setting and in presence of the so-called Wrong Way Risk (WWR): that is, a decrease/increase in the credit quality of the counterparty produces a higher exposure in the portfolio of the derivative’s holder. This effect may be modeled by the correlation between the stochastic factors driving our market model. We consider a method, introduced in the papers [F. Antonelli, S. Scarlatti, Finance and Stochastics, 13, (2009)] and [F. Antonelli, A. Ramponi, S. Scarlatti, Review of Deriv. Research, 13, (2010)], which expands theoretically the solution of the PDE system in a Taylor’s series with respect to the correlation parameters.

Indeed, under quite general hypotheses, it is possible to verify that the solution to the PDE is regular with respect to the correlation parameters and therefore it can be expanded in series around the zero value for all of them. The coefficients of the series are characterized, by using Duhamel’s principle, as solutions to a chain of PDE problems and they are therefore identified by means of Feynman-Kac formulas and expressed as expectations, that turn to be easier to compute or to approximate.

Finally, we show that under appropriate conditions, the method can be extended to include several XVA’s, such as bilateral CVA, DVA (Debt Value Adjustment), FVA (Funding Value Adjustment) and LVA (Liquidity Value Adjustment) due to collateralization. In fact, we remark that the adjusted value of a defaultable claim (with default risk of both parties) that takes into account the funding and collateralization costs verifies a (possibly nonlinear) BSDE and that, under some hypothesis, it may be approximated by using the correlation expansion method.

## Lecturer

16.00h

Faculty of Mathematics and Computer Science, UB (Hist. Building).

*Abstract*

We consider firstly the problem of computing the Credit Value Adjustment (CVA) of a European option in a default intensity setting and in presence of the so-called Wrong Way Risk (WWR): that is, a decrease/increase in the credit quality of the counterparty produces a higher exposure in the portfolio of the derivative’s holder. This effect may be modeled by the correlation between the stochastic factors driving our market model. We consider a method, introduced in the papers [F. Antonelli, S. Scarlatti, Finance and Stochastics, 13, (2009)] and [F. Antonelli, A. Ramponi, S. Scarlatti, Review of Deriv. Research, 13, (2010)], which expands theoretically the solution of the PDE system in a Taylor’s series with respect to the correlation parameters.

Indeed, under quite general hypotheses, it is possible to verify that the solution to the PDE is regular with respect to the correlation parameters and therefore it can be expanded in series around the zero value for all of them. The coefficients of the series are characterized, by using Duhamel’s principle, as solutions to a chain of PDE problems and they are therefore identified by means of Feynman-Kac formulas and expressed as expectations, that turn to be easier to compute or to approximate.

Finally, we show that under appropriate conditions, the method can be extended to include several XVA’s, such as bilateral CVA, DVA (Debt Value Adjustment), FVA (Funding Value Adjustment) and LVA (Liquidity Value Adjustment) due to collateralization. In fact, we remark that the adjusted value of a defaultable claim (with default risk of both parties) that takes into account the funding and collateralization costs verifies a (possibly nonlinear) BSDE and that, under some hypothesis, it may be approximated by using the correlation expansion method.