Valuing credit derivatives using Gaussian quadrature: A stochastic volatility framework

Abstract

AbstractThis article proposes semi‐closed‐form solutions to value derivatives on mean reverting assets. A very general mean reverting process for the state variable and two stochastic volatility processes, the square‐root process and the Ornstein‐Uhlenbeck process, are considered. For both models, semi‐closed‐form solutions for characteristic functions are derived and then inverted using the Gauss‐Laguerre quadrature rule to recover the cumulative probabilities. As benchmarks, European call options are valued within the following frameworks: Black and Scholes (1973) (represents constant volatility and no mean reversion), Longstaff and Schwartz (1995) (represents constant volatility and mean reversion), and Heston (1993) and Zhu (2000) (represent stochastic volatility and no mean reversion). These comparisons show that numerical prices converge rapidly to the exact price. When applied to the general models proposed (represent stochastic volatility and mean reversion), the Gauss‐Laguerre rule proves very efficient and very accurate. As applications, pricing formulas for credit spread options, caps, floors, and swaps are derived. It also is shown that even weak mean reversion can have a major impact on option prices. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:3–35, 2004