Since the 2007/2008 financial crisis, the total value adjustment (XVA) should be included when pricing financial derivatives. In the present paper, the derivative values of European and American options have been priced where we take into account counterparty risk. Whereas European and American options considering counterparty risk have already been priced under Black-Scholes dynamics in [2], here the novel contribution is the introduction of stochastic volatility resulting in a Heston stochastic volatility type partial differential equation to be solved. We derive the partial differential equation modeling the XVA when stochastic volatility is assumed. For both European and American options, a linear and a nonlinear problem have been deduced. In order to obtain a numerical solution, suitable and appropriate boundary conditions have been considered. In addition, a method of characteristics for the time discretization combined with a finite element method in the spatial discretization has been implemented. The expected exposure and potential future exposure are also computed to compare the current model with the associated Black–Scholes model.

自2007/2008年金融危机以来，对金融衍生产品定价时应包括总价值调整（XVA）。在本文中，我们在考虑了交易对手风险的情况下对欧洲和美国期权的衍生价值进行了定价。考虑到交易对手风险的欧美期权已经在[2]中的Black-Scholes动力学模型下定价，这里的新贡献是引入了随机波动率，从而解决了Heston随机波动率类型偏微分方程。当假设随机波动率时，我们推导了对XVA建模的偏微分方程。对于欧洲和美国的选择，都推导出了线性和非线性问题。为了获得数值解，已经考虑了合适的边界条件。另外，已经实现了时间离散化的特征方法与空间离散化中的有限元方法相结合。还计算了预期的暴露量和潜在的未来暴露量，以将当前模型与关联的Black-Scholes模型进行比较。