Infinitesimal Operator Based Methods For Continuous-Time Finance Models
Continuous time Markov processes, including diffusion, jump-diffusion and Levy jump-diffusion models, have become an essential tool of modern finance over the past three decades. Nowadays, they are widely used in modeling dynamics of, for instance, interest rates, stock prices, exchange rates and option prices. However, data are always recorded at discrete points in time, e.g., monthly, weekly, and daily, although these models are formulated in continuous time. This feature makes most econometric inferential procedures developed for discrete time econometrics unsuitable for continuous time models and complicates the econometric analysis considerably. For example, estimators obtained by applying discrete time econometric methods to the discretized version of continuous time models are not consistent for a fixed sampling interval. More seriously, although the maximum likelihood method is a very appealing econometric procedure due to its nice properties like efficiency, the transition density and hence likelihood function of most continuous time Markov models have no analytic expressions. This poses a serious impediment for the implementation of likelihood procedures. Many approaches have been proposed to deal with this problem but they either incur substantive computation burdens especially for multivariate cases or involve complicated approximation formulas with limited applicability. Consequently, there is a strong need for convenient econometric methodologies designed for continuous time mod- els given discrete sampled data. Unlike the transition density, the infinitesimal operator, as an important mathematical tool in probability theory, enjoys the nice property of being a closed-form expression of drift, diffusion and jump terms of the process. As a result, no approximated formulas or simulation based implementations are needed. Furthermore, it is equivalent to the transition density in characterizing the complete dynamics of the processes. Based on this convenient infinitesimal operator, this dissertation proposes a sequence of econometric procedures for continuous time Markov models with applications to affine jump diffusion (AJD) term structure models of interest rates. It is divided into four chapters. In the first chapter, "Infinitesimal Operator Based Estimation for Continuous Time Markov Processes", I propose an estimation method based on the infinitesimal operator for general multivariate continuous-time Markov processes, which cover diffusion, jump-diffusion and Levy-driven jump models as special cases. A conditional moment restriction is first obtained via the infinitesimal operator based identification of the process. Then an empirical likelihood type estimator is constructed by a kernel smoothing approach. Unlike the transition density which is rarely available in closed-form, the infinitesimal operator has an analytic form for all continuous time Markov models. As a result, different from the maximum likelihood estimator (MLE) which involves either numerical or simulated transition densities, the proposed estimator can be conveniently implemented by plugging in parametric components of the models. Furthermore, I prove that the proposed estimator attains the semi-parametric efficiency bound for conditional moment restrictions models of Markov processes and hence is asymptotically efficient. Simulation studies show that the proposed estimator has good finite sample performances comparable to the MLE. In the empirical application, I estimate Levy jump diffusion models for daily Euro/Dollar (2000-2010) and Yen/Dollar (1990-2000) rates. Results show that Levy jumps are important components in exchange rate dynamics and Poissontype jump diffusion models cannot capture them. In the second chapter, "Expectation Puzzles, Time Varying Conditional Volatility, and Jumps in Affine Term Structure Models", I study how jumps in interest rates, which are well documented in the literature, affect the term structure dynamics of the LIBOR-Swap curve in a multivariate AJD model. The motivation is that affine diffusion (AD) term structure models, as the major framework for interest rate dynamics, face two empirical challenges: first, they ignore well-documented jumps in interest rates as the state variables follow affine diffusions; second, they fail to capture simultaneously time variations in risk premiums implied by the violations of the "expectation hypothesis" and time variations in volatilities which are critical for pricing fixed-income derivatives. In this paper, I develop a multivariate AJD term structure model that overcomes these two challenges. Using LIBOR-Swap yields from 1990 to 2008, I estimate three-factor AJD models with infinitesimal operator methods and examine the contributions of jumps to term structure dynamics. I find that jumps are state dependent and negative. The risk premium is positive for jump size risk and negative for jump time risk, while the total jump risk premium is positive. Jump risk premiums lead to flexible time-varying market prices of risks without restricting time variations in conditional volatilities. As a result, two models in the three-factor AJD class capture time variations in both the risk premium and conditional volatility of LIBOR-Swap yields simultaneously. In the third chapter (part of this chapter has been published as Song (2011) in Journal of Econometrics, 162-2, 189-212.), "A Martingale Approach for Testing Diffusion Models Based on Infinitesimal Operator", I develop an omnibus specification test for diffusion models based on the infinitesimal operator instead of the transition density extensively used in literature. The infinitesimal operator based identification of the diffusion process is equivalent to a "martingale hypothesis" for the processes obtained by a transformation of the original diffusion model. My test procedure is then constructed by checking the "martingale hypothesis" via a multivariate generalized spectral derivative based approach which delivers an N(0,1) asymptotical null distribution for the test statistic. The infinitesimal operator of the diffusion process enjoys the nice property of being a closed-form function of drift and diffusion terms. Consequently, my test procedure covers both univariate and multivariate diffusion models in a unified framework and is particularly convenient for the multivariate case. Moreover, different transformed martingale processes contain separate information about the drift and diffusion specifications and about their interactions. This motivates me to propose a separate inferential test procedure to explore the sources of rejection when a parametric form is rejected. Simulation studies show that the proposed tests have reasonable size and excellent power performances. An empirical application of my test procedure using Eurodollar interest rates finds that most popular short-rate models are rejected and the drift mis-specification plays an important role in such rejections. In the fourth chapter, "Estimating Semi-Parametric Diffusion Models with Unrestricted Volatility via Infinitesimal Operator", two generalized method of moments estimators are proposed for the drift parameters in both univariate and multivariate semi-parametric diffusion models with unrestricted volatility based on the infinitesimal operator. The first estimator is obtained by integrating out the diffusion function via the quadratic variation (co-variation), which is estimated by the realized volatility (covariance) in a first step using high frequency data. The second is constructed based on the separate identification condition and is actually applicable for a general instantaneous conditional mean model in continuous time, which covers the stochastic volatility and jump diffusion models as special cases. Simulation studies show that they possess fairly good finite sample performances.
Infinitesimal Operator; Continuous Time Finance; Markov
Jarrow, Robert A.
Kiefer, Nicholas Maximillian
Ph.D. of Economics
Doctor of Philosophy
dissertation or thesis