Total value adjustment in a multicurrency framework with stochastic exchange rates and mean-reversion spreads

Íñigo Arregui, Mirco Martini, Roberta Simonella and Carlos Vázquez

Abstract

By using portfolio replication and dynamic hedging techniques, we deduce different models for pricing financial derivatives in multicurrency markets and in the presence of counterparty credit risk, as well as the associated valuation adjustments. For this purpose, we consider that the foreign exchange rates between the different currencies follow stochastic dynamics, while the credit spread is governed by mean-reversion dynamics. We derive linear and nonlinear models in terms of partial differential equations, expectations and backward stochastic differential equations. For their numerical solution, we propose Monte Carlo-based numerical schemes (eventually combined with a multilevel Picard iteration method), as well as deep neural networks. Finally, we present and solve some realistic tests.