An analytic multi-currency model with stochastic volatility and stochastic interest rates

We introduce a tractable multi-currency model with stochastic volatility and correlated stochastic interest rates that takes into account the smile in the FX market and the evolution of yield curves. The pricing of vanilla options on FX rates can be performed effciently through the FFT methodology thanks to the affinity of the model Our framework is also able to describe many non trivial links between FX rates and interest rates: a second calibration exercise highlights the ability of the model to fit simultaneously FX implied volatilities while being coherent with interest rate products

The explicit Laplace transform for the Wishart process

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru. We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.Comment: Accepted on: Journal of Applied Probability 51(3), 201

A flexible matrix Libor model with smiles

We present a flexible approach for the valuation of interest rate derivatives based on Affine Processes. We extend the methodology proposed in Keller-Ressel et al. (2009) by changing the choice of the state space. We provide semi-closed-form solutions for the pricing of caps and floors. We then show that it is possible to price swaptions in a multifactor setting with a good degree of analytical tractability. This is done via the Edgeworth expansion approach developed in Collin-Dufresne and Goldstein (2002). A numerical exercise illustrates the flexibility of Wishart Libor model in describing the movements of the implied volatility surface

Smiles all around: FX joint calibration in a multi-Heston model

We introduce a novel multi-factor Heston-based stochastic volatility model, which is able to reproduce consistently typical multi-dimensional FX vanilla markets, while retaining the (semi)-analytical tractability typical of affine models and relying on a reasonable number of parameters. A successful joint calibration to real market data is presented together with various in- and out-of-sample calibration exercises to highlight the robustness of the parameters estimation. The proposed model preserves the natural inversion and triangulation symmetries of FX spot rates and its functional form, irrespective of choice of the risk-free currency. That is, all currencies are treated in the same way.Comment: Journal of Banking and Finance. Accepte

The Wishart short rate model

We consider a short rate model, driven by a stochastic process on the cone of positive semidefinite matrices. We derive sufficient conditions ensuring that the model replicates normal, inverse or humped yield curves

Cross-Currency Heath-Jarrow-Morton Framework in the Multiple-Curve Setting

We provide a general HJM framework for forward contracts written on abstract market indices with arbitrary fixing and payment adjustments. We allow for indices on any asset class, featuring collateralization in arbitrary currency denominations. The framework is pivotal for describing portfolios of interest rate products which are denominated in multiple currencies. The benchmark transition has created significant discrepancies among the market conventions of different currency areas: our framework simultaneously covers forward-looking risky IBOR rates, such as EURIBOR, and backward-looking rates based on overnight rates, such as SOFR. In view of this, we provide a thorough study of cross-currency markets in the presence of collateral, where the cash flows of the contract and the margin account can be denominated in arbitrary combinations of currencies. We finally consider cross-currency swap contracts as an example of a contract simultaneously depending on all the risk factors that we describe within our framework

Multi-Layer Deep xVA: Structural Credit Models, Measure Changes and Convergence Analysis

We propose a structural default model for portfolio-wide valuation adjustments (xVAs) and represent it as a system of coupled backward stochastic differential equations. The framework is divided into four layers, each capturing a key component: (i) clean values, (ii) initial margin and Collateral Valuation Adjustment (ColVA), (iii) Credit/Debit Valuation Adjustments (CVA/DVA) together with Margin Valuation Adjustment (MVA), and (iv) Funding Valuation Adjustment (FVA). Since these layers depend on one another through collateral and default effects, a naive Monte Carlo approach would require deeply nested simulations, making the problem computationally intractable. To address this challenge, we use an iterative deep BSDE approach, handling each layer sequentially so that earlier outputs serve as inputs to the subsequent layers. Initial margin is computed via deep quantile regression to reflect margin requirements over the Margin Period of Risk. We also adopt a change-of-measure method that highlights rare but significant defaults of the bank or counterparty, ensuring that these events are accurately captured in the training process. We further extend Han and Long's (2020) a posteriori error analysis to BSDEs on bounded domains. Due to the random exit from the domain, we obtain an order of convergence of O(h^(1/4-eps)) rather than the usual O(h^(1/2)). Numerical experiments illustrate that this method drastically reduces computational demands and successfully scales to high-dimensional, non-symmetric portfolios. The results confirm its effectiveness and accuracy, offering a practical alternative to nested Monte Carlo simulations in multi-counterparty xVA analyses