The Fundamental Theorems of Asset Pricing in the Context of XVA, Funding Risk, and Collateral

Abstract

We present an approach to the fundamental theorems of asset pricing based on Black-Scholes hedging arguments and elementary linear algebra. The framework yields mainstream industry derivative pricing results, from simple Black-Scholes call option pricing to more recent results in counterparty credit risk pricing in the presence of funding costs and collateral agreements. In addition to 1) a new derivation of no-arbitrage pricing equations for credit derivatives---including Credit and Funding Valuation Adjustments (CVA and FVA)---for jump-diffusion markets and 2) explicit incorporation of collateral agreements into the fundamental theorems of asset pricing, the main contribution of the paper is, we believe, 3) the illumination of the nature of the fundamental theorems, and in particular the link between no arbitrage and the existence of a martingale measure, that is afforded by the perspective taken and the reliance only on elementary linear algebra.

Keywords: Fundamental Theorem of Asset Pricing, Derivative, FVA, XVA, Collateral, Credit Risk, Jump-Diffusion

JEL Classification: G13