Self-Funding Portfolios and Collateralised Derivative Pricing

Abstract

A general framework is proposed for including the effects of collateral management in derivative pricing. Comprehensive yet simple formulas are obtained allowing the impact of different types of collateral agreements to be incorporated in valuations. The approach centers on analysing the self-financing condition on portfolios which explicitly include the market funding of derivative, collateral and hedging positions. Major simplifications, including a martingale pricing formula, are obtained when funding and collateral are in the form of cash, but the general framework aims at greater generality. The functional dependence of the collateral balance on the value of the derivative also plays a central role in the analysis. The “institutional” implications for liquidity management (pertaining to derivative desk funding, bank treasury “transfer pricing,” and the interplay with the repo market) are briefly considered.

Keywords: collateral, funding, Black-Scholes-Merton, martingale pricing, discounting

JEL Classification: G12