Interest Rates and The Credit Crunch: New Formulas and Market Models

Abstract

We start by describing the major changes that occurred in the quotes of market rates after the 2007 subprime mortgage crisis. We comment on their lost analogies and consistencies, and hint on a possible, simple way to formally reconcile them. We then show how to price interest rate swaps under the new market practice of using different curves for generating future LIBOR rates and for discounting cash flows. Straightforward modifications of the market formulas for caps and swaptions will also be derived.

Finally, we will introduce a new LIBOR market model, which will be based on modeling the joint evolution of FRA rates and forward rates belonging to the discount curve. We will start by analyzing the basic lognormal case and then add stochastic volatility. The dynamics of FRA rates under different measures will be obtained and closed form formulas for caplets and swaptions derived in the lognormal and Heston (1993) cases.

Alternative, and more recent, formulations of multi-curve LIBOR market models can be downloaded at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1563685 http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1583081

Keywords: credit crunch, credit, liquidity, market rates, forward curve, discount curve, bootstrapping, FRAs, swaps, caps, swaptions, LIBOR market models, measure changes, stochastic volatility, Heston volatility, closed form formulas

JEL Classification: E45, G13