Using the Replication with zero-cost tradeable hedges of Antonov et al. (2014) we generalize the Funding Invariance principle of Elouerkhaoui (2016) and classify systematically funding/hedging strategies. Other applications of the method include the gap adjustment GVA when close-out amount and PV are not discounted at the same rate. This allows a company to choose its PV discount rate freely while maintaining consistent pricing across foreign entities.

**Introduction to CUDA Programming in Finance (SSRN, 2015)**

Chapter 15 of A Practical Guide To Quantitative Portfolio Trading. Introduction to programming in CUDA intended for quants without prior knowledge of the subject. We describe the most important features of the language for the purpose of building financial applications, and we illustrate these concepts with 2 case studies, the Monte-Carlo pricing of hybrid exotics and the optimization of an implied volatility surface using the Differential Evolution algorithm.

We propose a Monte-Carlo calibration method for multi-currency Hybrid Local Volatility models a la Dupire. The algorithm follows a systematic approach to the evaluation of the bias due to the stochasticity of the interest rates and is applicable to all Markovian interest rate models. We explain it in details before demonstrating its excellent accuracy in several types of models including Black or SVI implied volatilities with Hull-White or CIR interest rates. We improve its performance by using parallel programming on GPUs with CUDA. We obtain gains of up to 28X in realistic configurations leading to calibration times below the second. CUDA sample code is provided.

We investigate the loss of efficiency of Sobol low-discrepancy sequence at high dimensions and the apparent improvement provided by the use of the Brownian bridge construction of Brownian motion paths. We show numerically that some often cited potential causes for these phenomena such as low quality of Sobol coordinates at high dimensions are not to blame and instead isolate a bias in Sobol sequence which we conjecture to be the main cause of the problem. We motivate this conjecture by an analysis of the equations defining both the Incremental and Brownian bridge constructions in a simplified setting, showing how the identified bias is removed by the use of the Brownian bridge. We further give numerical evidence that randomizing Sobol sequence can remove most of this bias and achieve a good convergence at high dimensions. We explain why this is particularly relevant for efficient inline implementations in massively parallel environments such as GPUs under the programming language CUDA. Tested products and models include vanilla and barrier options as well as TARN PRDCs in Local Volatility. We also provide proofs of the homogeneity properties of the Gaussian deviates derived from Sobol sequence for particular numbers of iterations.

We derive a set of sufficient conditions on the parametric forms of the Stochastic Volatility Inspired (SVI) implied volatility model parameters in order to satisfy the no-calendar spread arbitrage constraint while preserving the (necessary) condition of no-strike arbitrage. We propose a strategy to find solutions to these constraints and give one such example. We fit it to the market data of USD/JPY and show that the good fitting quality of SVI model is essentially preserved. We exhibit the Dupire local volatilities based on the original and term structure models and illustrate how the term structure of the implied volatility leads to a much smoother behavior in the time direction and more reasonable call calendar spread prices. We also show that the local volatility calibration remains robust even in the presence of arbitrage in the market data.

We describe several strategies for the calibration of one factor Hull-White model with constant or time-dependent mean reversion and volatility parameters to the interest rate vanillas. We propose an efficient approximation formula for the swaption implied volatility which enables us to estimate the mean reversion independently of the volatility. We give the closed-forms for exact pricing using explicit integrals of the model parameters and propose parametric forms for the mean reversion and volatility. We test their performance in terms of quality of fitting and stability w.r.t. market changes, and show that excellent fits can be obtained without suffering from instabilities. Furthermore, our calibration methods and parameter control techniques allow for an elegant interpretation of market moves, which we illustrate with an in-depth analysis of Lehman crisis in the fall of 2008.