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An Integrated Model for Hybrid Securities

by Sanjiv R. Das of Santa Clara University, and
Rangarajan K. Sundaram of New York University

September 2007

Abstract: We develop a model for pricing securities whose value may depend simultaneously on equity, interestrate, and default risks. The framework may also be used to extract probabilities of default (PD) functions from market data. Our approach is entirely based on observables such as equity prices and interest rates, rather than on unobservable processes such as firm value. The model stitches together in an arbitrage-free setting a constant elasticity of variance (CEV) equity model (to represent the behavior of equity prices prior to default), a default intensity process, and a Heath-Jarrow-Morton (HJM) model for the evolution of riskless interest rates. The model captures several stylized features such as a negative relation between equity prices and equity volatility, a negative relation between default intensity and equity prices, and a positive relationship between default intensity and equity volatility. We embed the model on a discrete-time, recombining lattice, making implementation feasible with polynomial complexity. We demonstrate the simplicity of calibrating the model to market data and of using it to extract default information. The framework is extensible to handling correlated default risk and may be used to value distressed convertible bonds, debt-equity swaps, and credit portfolio products such as collateralized debt obligations (CDOs). Applied to the CDX INDU (credit default index-industrials) Index, we find the S&P 500 index explains credit premia.

Keywords: securities, hybrid models, default, credit risk.

Published in: Management Science, Vol. 53, No. 9, (September 2007), pp. 1439-1451.

Previously titled: "A Simple Model for Pricing Securities with Equity, Interest-Rate, and Default Risk" --and-- "A Simple Unified Model for Pricing Derivative Securities with Equity, Interest-rate, and Default Risk"

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