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 [1], 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.@en

Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). The classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied here. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black–Scholes equation, whereas for the American option we solve the linear complementarity problem formulation. Two-asset exotic option values are also computed, since ANNs enable the accurate valuation of high-dimensional options. The resulting errors of the ANN approach are assessed by comparing to the analytic option values or to numerical reference solutions (for American options, computed by finite elements). In the short note, previously published, a brief introduction to this work was given, where some ideas to price vanilla options by ANNs were presented, and only European options were addressed. In the current work, the methodology is introduced in much more detail.

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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.

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