A Second Look at Post Crisis Pricing of Derivatives - Part I: A Note on Money Accounts and Collateral

TL;DR

This paper critically reviews post-crisis derivative pricing approaches, identifies conceptual inconsistencies, and derives arbitrage-free pricing formulas for default-risky derivatives with collateral.

Contribution

It provides a rigorous derivation of no-arbitrage pricing formulas that corrects previous flawed approaches based on risk-free money accounts.

Findings

Identifies inconsistencies in existing post-crisis pricing models.

Derives arbitrage-free pricing formulas for default-risky derivatives.

Highlights issues with semi-replication and risk-free account assumptions.

Abstract

The paper reviews origins of the approach to pricing derivatives post-crisis by following three papers that have received wide acceptance from practitioners as the theoretical foundations for it - [Piterbarg 2010], [Burgard and Kjaer 2010] and [Burgard and Kjaer 2013]. The review reveals several conceptual and technical inconsistencies with the approaches taken in these papers. In particular, a key component of the approach - prescription of cost components to a risk-free money account, generates derivative prices that are not cleared by the markets that trade the derivative and its underlying securities. It also introduces several risk-free positions (accounts) that accrue at persistently non-zero spreads with respect to each other and the risk free rate. In the case of derivatives with counterparty default risk [Burgard and Kjaer 2013] introduces an approach referred to as semi-replication, which through the choice of cost components in the money account results in derivative prices that carry arbitrage opportunities in the form of holding portfolio of counterparty's bonds versus a derivative position with it. This paper derives no-arbitrage expressions for default-risky derivative contracts with and without collateral, avoiding these inconsistencies.