# CVA and vulnerable options in stochastic volatility models

## TL;DR

This paper develops a general formula to evaluate the Credit Value Adjustment (CVA) for vulnerable options within stochastic volatility models, accounting for correlation with default risk, and validates it through numerical analysis.

## Contribution

It introduces a unified representation formula for CVA in stochastic volatility models with default correlation, applicable to models like SABR, Hull-White, and Heston.

## Key findings

- The formula explicitly captures the impact of correlation on CVA.
- Numerical results show good agreement with Monte Carlo simulations.
- The approach enhances accuracy in CVA evaluation for models with default risk.

## Abstract

In this work we want to provide a general principle to evaluate the CVA (Credit Value Adjustment) for a vulnerable option, that is an option subject to some default event, concerning the solvability of the issuer. CVA is needed to evaluate correctly the contract and it is particularly important in presence of WWR (Wrong Way Risk), when a credit deterioration determines an increase of the claim's price. In particular, we are interested in evaluating the CVA in stochastic volatility models for the underlying's price (which often fit quite well the market's prices) when admitting correlation with the default event. By cunningly using Ito's calculus, we provide a general representation formula applicable to some popular models such as SABR, Hull \& White and Heston, which explicitly shows the correction in CVA due to the processes correlation. Later, we specialize this formula and construct its approximation for the three selected models. Lastly, we run a numerical study to test the formula's accuracy, comparing our results with Monte Carlo simulations.