Computing hitting time densities for CIR and OU diffusions: applications to mean- reverting models

This paper provides explicit analytical characterizations for first hitting time densities for Cox–Ingersoll–Ross (CIR) and Ornstein–Uhlenbeck (OU) diffusions in terms of relevant Sturm–Liouville eigenfunction expansions. Starting with Vasicek (1977) and Cox, Ingersoll and Ross (1985), the Gaussian Ornstein– Uhlenbeck and Feller’s (1951) square-root diffusions are among the most commonly used stochastic processes in finance. Mean-reverting OU and CIR processes are used to model interest rates (eg, Gorovoi and Linetsky, 2004), credit spreads (eg, Duffie and Singleton, 2003), stochastic volatility (eg, Heston, 1993), commodity convenience yields (eg, Schwartz, 1997), market capitalization of growth stocks (Kou and Kou, 2002) and other mean-reverting financial variables. Following the spectral expansion approach to diffusions (McKean, 1956; Itô and McKean, 1976, Section 4.11; Kent, 1980, 1982; Davydov and Linetsky, 2003; Linetsky, 2001, 2004a,b,c), we explicitly compute eigenfunction expansions for hitting time distributions for CIR and OU diffusions in terms of special functions (confluent hypergeometric and Hermite functions, respectively), give large-n asymptotics in terms of elementary functions, and provide several examples of explicit calculations in interest rate, credit risk and stochastic volatility modeling applications implemented in the Mathematica package. For the CIR diffusion, the Laplace transform of the first hitting time is known (Giorno et al., 1986; Göing-Jaeschke and Yor, 2003b; Leblanc and Scaillet, 1998). To recover the density, previous studies have used numerical Laplace transform