Arbitrage-free XVA

Arbitrage-free pricing of XVA

Arbitrage-Free Pricing of XVA – Part II: PDE Representation and Numerical Analysis

We study the semilinear partial differential equation (PDE) associated with the non-linear BSDE characterizing buyer’s and seller’s XVA in a framework that allows for asymmetries in funding, repo and collateral rates, as well as for early contract termination due to counterparty credit risk. We show the existence of a unique classical solution to the PDE by first proving the existence and uniqu...

Arbitrage-Free Pricing of XVA – Part I: Framework and Explicit Examples

We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on noarbitrage arguments, we derive a nonlinear backward stochastic differential equation (BSDE) associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buye...

From arbitrage to arbitrage-free implied volatilities

We propose a method for determining an arbitrage-free density implied by the Hagan formula. (We use the wording “Hagan formula” as an abbreviation of the Hagan– Kumar–Leśniewski–Woodward model.) Our method is based on the stochastic collocation method. The principle is to determine a few collocation points on the implied survival distribution function and project them onto the polynomial of an ...

Arbitrage-free SVI volatility surfaces

In this article, we show how to calibrate the widely-used SVI parameterization of the implied volatility smile in such a way as to guarantee the absence of static arbitrage. In particular, we exhibit a large class of arbitrage-free SVI volatility surfaces with a simple closed-form representation. We demonstrate the high quality of typical SVI fits with a numerical example using recent SPX optio...