On stiffness in affine asset pricing models

On stiffness in affine asset pricing models

25 Pages · 2007 · 387 KB · English

School of Economics and Social Sciences, Singapore Management University,. 90 Stamford Road, Singapore 178903; email: [email protected] Economic and econometric analysis of continuous-time affine asset pricing models often necessitates solving systems of ordinary differential equations.

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Journal of Computational Finance (99–123) Volume 10/Number 3, Spring 2007 On stiffness in af?ne asset pricing models Shirley J Huang Lee Kong Chian School of Business, Singapore Management University, 50 Stamford Road, Singapore 178899; email: [email protected] Jun Yu School of Economics and Social Sciences, Singapore Management University, 90 Stamford Road, Singapore 178903; email: [email protected] Economic and econometric analysis of continuoustime af?ne asset pricing models often necessitates solving systems of ordinary differential equations (ODEs) numerically Explicit Runge?Kutta (ERK) methods have been suggested to solve these ODEs both in the theoretical ?nance literature and in the ?nancial econometrics literature In this paper we show that under many empirically relevant circumstances the ODEs involve stiffness, a phenomenon which leads to some practical dif?culties for numerical methods with a ?nite region of absolute stability, including the whole class of ERK methods The dif?culties are highlighted in the present paper in the context of pricing zerocoupon bonds as well as econometric estimation of dynamic term structure models via the empirical characteristic function To overcome the numerical dif?culties, we propose to use implicit numerical methods for the ODEs The performance of these implicit methods relative to certain widely used ERK methods are examined in the context of bond pricing and parameter estimation The results show that the implicit methods greatly improve the numerical ef?ciency 1 INTRODUCTION “···around 1960, things became completely different and everyone became aware that world was full of stiff problems” Dahlquist (1985) When valuing nancial assets in a continuoustime, arbitragefree framework, one often needs to nd the numerical solution to a partial differential equation (PDE) (examples include the Feynman–Kac PDE for bond prices; see Dufe Both authors gratefully acknowledge nancial support from the Research Ofce at Singapore Management University We would like to thank the anonymous referee for constructive comments that have substantially improved the article We also wish to thank John Butcher, David Chen, Peter Phillips, Dai Min and participants in the Econometric Society World Congress in London, 2005 International Symposium on Econometrics Development in Beijing, the Inaugural Saw Centre for Financial StudiesConference on Quantitative Finance and the Singapore Econometric Study Group Meeting for comments on an earlier version of the paper All computations were performed using MATLABr12 and Compact Visual FORTRAN 66a 99 100 S J Huang and J Yu (2001) for details and references) Given the fact that in many practically rel evant cases solving the PDE is computationally demanding and even becomes impractical when the number of states is modestly large, considerable attention has been paid to the class of afne asset pricing models where the riskneutral drift and volatility functions of the process for the state variables are afne Under the afne specication, many asset prices have either completely or nearly closed form expressions Important examples from the rst category include Black and Scholes (1973) for pricing equity options, Vasicek (1977) and Cox, Ingersoll, and Ross ((1985); hereafter CIR) for pricing bonds and bond options, and Heston (1993) for pricing equity and currency options Important examples from the second category include Dufe and Kan (1996) for pricing bonds, Chacko and Das (2002) for pricing interest derivatives, Bates (1996) for pricing currency options, and Dufeet al(2000) for a treatment of very general pricing relations The solutions have nearly closedform expressions in the sense that the PDE is decomposed into a system of ordinary differential equations (ODEs) and hence only a system of ODEs, as opposed to a PDE, has to be solved numerically Such decomposition greatly facilitates the numerical implementation of pricing (Piazzesi (2003)) Computational burdens are even heavier for econometric analysis of continuoustime dynamic asset pricing models based on discretely sampled observations The reasons for this are: (i) the implied transition density of discretely sampled observations are solutions to PDEs which have to be solved numerically at every data point and at each iteration in the numerical optimizations (examples include the apparent need to solve the Kolmogorov forward and backward equations for the transition density in the maximum likelihood (ML) approach; see Lo (1988) for details

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