Swap Pricing with Two-Sided Default Risk in a Rating-Based Model

by Brian Huge of the University of Copenhagen, and
David Lando of the University of Copenhagen

January 1999

Introduction: This paper analyzes the pricing of defaultable securities in rating based models where the default of more than one agent is involved. We extend the model of Duffie and Huang (1996) to a framework which explicitly takes into account the rating of each party. Although our method is by no means restricted to swap con-tracts we will use as our illustrative example a plain vanilla interest rate swap. Our extension allows us to investigate the effects on swap spreads of early termination provisions, i.e., credit triggers, which are linked to the ratings of the contracting parties. Clearly, a credit trigger will make each counterparty look less risky, as illustrated for example in Wakeman (1996), simply because the trigger eliminates those defaults that occur after a sequence of downgrades. How much this affects swap spreads can be studied using the technique presented in this paper. We also consider the following questions:

• How does the degree of rating asymmetry affect swap spreads?
• How does the swap spread vary with rating when the two parties have the same rating?
• How important is a stochastic specification of the transition intensities (as in Lando (1994, 1998))?

We will work primarily with the continuous-time Markov model as in Jarrow, Lando and Turnbull[11] and Lando[15][REF] to model the credit risk of both counterparties. For recovery in default we will however use the idea of Duffie and Singleton[5] of using a fractional recovery of predefault market value.

The modeling of default risk in this paper is 'intensity-based' or 'reduced form' as in the papers by, among others, Artzner and Delbaen[2][REF], Duffie and Singleton[4], [5], Duffie, Schroder and Skiadas[7], Jarrow, Lando and Turnbull[11], Jarrow and Turnbull[12] , Lando[15][REF], Lando[17], Madan and Unal[19], Schönbucher[21]. For a survey on the various approaches to modeling default risk, see for example Lando[16][REF]. The approach taken in this paper is to consider ratings as a significant factor in the modeling of the default intensity. Rating-based models of default risk are popular for modeling defaultable bonds since they use readily observable data which enable financial institution to control and model credit risk without having to carefully monitor each counterparty. There is currently a rising interest in models based on ratings starting with Jarrow, Lando and Turnbull[11] and Lando[15][REF]. Recent contributions include Kijima[14][REF] and Li[18][REF]. Although there may be analytical solutions to prices of corporate bonds under various assumptions it is typically difficult to have analytical solutions to swap spreads in general and therefore a model based on ratings must have a numerical implementation to be of practical use. This paper provides such an implementation.

Papers dealing with one-sided default risk in swaps include Abken[1][REF], Artzner and Delbaen[2][REF], Cooper and Mello[1991], Rendleman[20][REF] and Turnbull[22][REF].

Rendleman[20][REF] also considers two-sided risk in an analysis based on the asset values of two firms entering into the contract.

An elegant way of studying two-sided default risk in a reduced-form setting is presented in Duffie and Huang[6] and our paper can be viewed as an extension of that model to a case where rating information can be taken explicitly into account. A key feature of this added jump risk component is the finite state space for the Markov chain governing default risk which implies that the solution equations become a system of quasi-linear PDEs. We pay special attention to an ADI-method which is well-suited for this problem which initially seems large due to the fact that both a spot rate and a two-dimensional rating process are involved. We also show in this paper a derivation of the valuation PDEs which does not build on recursive methods.

Other work dealing with two sided default risk includes Hübner[10], who uses an approximation to obtain swap spreads analytically, and Jarrow and Turnbull[13][REF] who present a discrete-time implementation which like the above mentioned approaches does not model ratings changes before default. Dufresne and Solnik[8] also consider swap spreads but argue that default risk of the swap contract is not a significant factor in determining swap spreads. In our paper, default risk is assumed to be important for the contract we study and the method is illustrated on swap contracts. The issue of whether some other source of risk or method of indexing the contract is driving the spreads, is not considered here.

Published in: European Financial Review, Vol. 3, No. 3, (January 1999), pp. 239-268.

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