Closed-Form Solution for Defaultable Bond Options under a Two-Factor Gaussian Model for Risky Rates Modeling

Abstract

In this article, the authors provide a closed-form solution for defaultable bond options under a two-factor Gaussian model for risky rates. The key feature of the proposed stochastic model is the introduction of two stochastic dynamics to address the behavior of both risk-free interest rates and credit spreads where the two sources of risk are correlated. Moreover, the model can match exactly the term structure of defaultable interest rates. In order to model credit-sensitive bonds, the authors assume an intensity-based reduced-form framework where the approach based on the recovery of market value is considered and where the recovery rate is one of the model’s input. Under this framework, the authors derive closed formulas for the price of European options having as their underlying both defaultable zero-coupon bonds and coupon-bearing bonds, which the authors assume are approximately log-normally distributed. Moreover, the options are assumed to be knocked out upon a default event of the bond, with zero value to the option holder. An empirical analysis is performed to illustrate the calibration process of the proposed model using financial market data to price call and put options across different credit ratings and different levels of the loss given default.

TOPICS: Factor-based models, derivatives, options

Key Findings

• • An option pricing model that considers the dynamic of risky rates under a two-factor Gaussian model where risky rates are the sum of risk-free rates and credit spreads is proposed. This distributional hypothesis is consistent with negative interest rates and benefits from analytical tractability.

• • Under this framework, closed formulas for the price of European options having both defaultable zero-coupon bonds and coupon-bearing bonds as their underlying are derived under an approximately log-normal distribution assumption.

• • A detailed two-step calibration procedure is provided. The first step estimates the parameters of the one-factor Hull–White model for the risk-free rates, and the second step estimates the parameters of the model’s credit component under a grid of correlation values.