Handbook of Financial Econometrics: Applications

by Yacine Ait-Sahalia (Editor), Lars Peter Hansen (Editor)

Overview

Applied financial econometrics subjects are featured in this second volume, with papers that survey important research even as they make unique empirical contributions to the literature. These subjects are familiar: portfolio choice, trading volume, the risk-return tradeoff, option pricing, bond yields, and the management, supervision, and measurement of extreme and infrequent risks. Yet their treatments are exceptional, drawing on current data and evidence to reflect recent events and scholarship. A landmark in its coverage, this volume should propel financial econometric research for years. - Presents a broad survey of current research - Contributors are leading econometricians - Offers a clarity of method and explanation unavailable in other financial econometrics collections

Product Details

ISBN-13:	9780444535498
Publisher:	Elsevier Science
Publication date:	10/21/2009
Series:	Handbooks in Finance , #2
Sold by:	Barnes & Noble
Format:	eBook
Pages:	384
File size:	5 MB

About the Author

Lars Peter Hansen is David Rockefeller Distinguished Service Professor at the University of Chicago, and is an internationally known leader in economic dynamics. Hansen guides the scholarly direction of the Becker Friedman Institute and chairs the Institute Research Council. He was one of the forces behind the 2008 creation of the Milton Friedman Institute, the predecessor of the Becker Friedman Institute, and served as its founding director. He was one of three in 2013 to be awarded The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel along with Eugene F. Fama and Robert J. Shiller "for their empirical analysis of asset prices." Hansen’s work explores formal implications of dynamic economic models in which decision makers face uncertain environments. The main theme of his research has been to devise and apply econometric methods that are consistent with the probabilistic framework of the economic models under investigation. His work has implications for consumption, savings investment, and asset pricing. Hansen's early research in econometrics was aimed at developing time series statistical methods to investigate one part of an economic model without having to fully specify and estimate all of the model ingredients. The applications he explored with several coauthors included systems that are rich enough to support models of asset valuation and to identify and clarify empirical puzzles, where real-world financial and economic data were at odds with prevailing academic models. He continues to explore, analyze, and interpret implications of dynamic economic models in environments with uncertainty from a time-series perspective. His recent research explores ways to quantify intertemporal risk-return tradeoffs and ways to model economic behavior when decision makers are uncertain about how to forecast future economic events. Hansen won the 2010 BBVA Foundation Frontiers of Knowledge Award in the Economics, Finance and Management “for making fundamental contributions to our understanding of how economic actors cope with risky and changing environments.” He also received the CME Group-MSRI Prize in Innovative Quantitative Applications in 2008 and the Erwin Plein Nemmers Prize in Economics from Northwestern University in 2006. Hansen is a fellow of the National Academy of Sciences and the American Finance Association. He also is a member of the American Academy of Arts and Sciences and past president of the Econometric Society. Hansen is the editor of two Elsevier publications – Handbook of Financial Econometrics, Volume 1, Tools; and Handbook of Financial Econometrics, Volume 2, Applications.

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Handbook of Financial Econometrics, Vol 2

Applications

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North-Holland

Copyright © 2010 Elsevier B.V.
All right reserved.
ISBN: 978-0-444-53549-8

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Chapter One

MCMC Methods for Continuous-Time Financial Econometrics

Contents

1. Introduction 2

2. Overview of Bayesian Inference and MCMC 5 2.1. MCMC Simulation and Estimation 5 2.2. Bayesian Inference 6

3. MCMC: Methods and Theory 9 3.1. Clifford-Hammersley Theorem 9 3.2. Gibbs Sampling 10 3.3. Metropolis-Hastings 12 3.4. Convergence Theory 14 3.5. MCMC Algorithms: Issues and Practical Recommendations 20

4. Bayesian Inference and Asset Pricing Models 24 4.1. States Variables and Prices 24 4.2. Time-Discretization: Computing p(Y|X,Θ) and p(X|Θ) 27 4.3. Parameter Distribution 30

5. Asset Pricing Applications 31 5.1. Equity Asset Pricing Models 32 5.2. Term Structure Models 54 5.3. Regime Switching Models 63

6. Conclusions and Future Directions 65 Acknowledgments 66 References 66

Abstract

This chapter develops Markov Chain Monte Carlo (MCMC) methods for Bayesian inference in continuous-time asset pricing models. The Bayesian solution to the inference problem is the distribution of parameters and latent variables conditional on observed data, and MCMC methods provide a tool for exploring these high-dimensional, complex distributions. We first provide a description of the foundations and mechanics of MCMC algorithms. This includes a discussion of the Clifford–Hammersley theorem, the Gibbs sampler, the Metropolis–Hastings algorithm, and theoretical convergence properties of MCMC algorithms. We next provide a tutorial on building MCMC algorithms for a range of continuous-time asset pricing models. We include detailed examples for equity price models, option pricing models, term structure models, and regime-switching models.

Keywords: continuous-time; Markov Chain Monte Carlo; financial econometrics; Bayesian inference; derivative pricing; volatility; jump diffusions; stochastic volatility; option pricing

1. INTRODUCTION

Dynamic asset pricing theory uses arbitrage and equilibrium arguments to derive the functional relationship between asset prices and the fundamentals of the economy: state variables, structural parameters, and market prices of risk. Continuous-time models are the centerpiece of this approach because of their analytical tractability. In many cases, these models lead to closed form solutions or easy to solve differential equations for objects of interest such as prices or optimal portfolio weights. The models are also appealing from an empirical perspective: through a judicious choice of the drift, diffusion, jump intensity, and jump distribution, these models accommodate a wide range of dynamics for state variables and prices.

Empirical analysis of dynamic asset pricing models tackles the inverse problem:extracting information about latent state variables, structural parameters, and market prices of risk from observed prices. The Bayesian solution to the inference problem is the distribution of the parameters, Θ, and state variables, X, conditional on observed prices, Y. This posterior distribution, p(Θ, X|Y), combines the information in the model and the observed prices and is the key to inference on parameters and state variables.

This chapter describes Markov Chain Monte Carlo (MCMC) methods for exploring the posterior distributions generated by continuous-time asset pricing models. MCMC samples from these high-dimensional, complex distributions by generating a Markov Chain over [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], whose equilibrium distribution is p(Θ, X|Y). The Monte Carlo method uses these samples for numerical integration for parameter estimation, state estimation, and model comparison.

Characterizing p(Θ, X|Y) in continuous-time asset pricing models is difficult for a variety of reasons. First, prices are observed discretely while the theoretical models specify that prices and state variables evolve continuously in time. Second, in many cases, the state variables are latent from the researcher's perspective. Third, p([THEA], X|Y) is typically of very high dimension and thus standard sampling methods commonly fail. Fourth, many continuous-time models of interest generate transition distributions for prices and state variables that are nonnormal and nonstandard, complicating standard estimation methods such as MLE. Finally, in term structure and option pricing models, parameters enter nonlinearly or even in a nonanalytic form as the implicit solution to ordinary or partial differential equations. We show that MCMC methods tackle all of these issues.

To frame the issues involved, it is useful to consider the following example: Suppose an asset price, St , and its stochastic variance, V_t, jointly solve:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where W^s_t (P) and W^v_t (P) are Brownian motions under P, N_t(P) counts the observed number of jump times, τ_j, prior to time t realized under P,μ_t is the equity risk premium, Z_j(P) are the jump sizes with a given predictable distribution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] under P, r_t is the spot interest rate, and μ^P_t is the expected jump size conditional on information available at time t. For simplicity, assume both the spot interest rate and equity premium are constants, although this is easily relaxed. Researchers also often observe derivative prices, such as options. To price these derivatives, it is common to assert that in the absence of arbitrage, there exists a probability measure, Q, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where W^s_t (Q), N_t(Q), and W^v_t (Q) are defined under Q. The parameters κ^Q_v and θ^Q_v capture the diffusive "price of volatility risk," and μ^Q_t is expected jump sizes under Q. Under Q, the price of a call option on S_t maturing at time T, struck at K, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

where Θ = (Θ^P, Θ^Q) are the structural and risk neutral parameters. The state variables, X, consist of the volatilities, the jump times, and jump sizes.

The goal of empirical asset pricing is to learn about the risk neutral and objective parameters, the state variables, namely, volatility, jump times, and jump sizes, and the model specification from the observed equity returns and option prices. In the case of the parameters, the marginal posterior distribution p(Θ|Y) characterizes the sample information about the objective and risk-neutral parameters and quantifies the estimation risk: the uncertainty inherent in estimating parameters. For the state variables, the marginal distribution, p(X|Y), combines the model and data to provide a consistent approach for separating out the effects of jumps from stochastic volatility. This is important for empirical problems such as option pricing or portfolio applications that require volatility estimates. Classical methods are difficult to apply in this model as the parameters and volatility enter in a nonanalytic manner in the option pricing formula, volatility, jump times, and jump sizes are latent, and the transition density for observed prices is not known.

To design MCMC algorithms for exploring p([TEHA], X|Y),we first follow Duffie (1996) and interpret asset pricing models as state space models. This interpretation is convenient for constructing MCMC algorithms as it highlights the modular nature of asset pricing models. The observation equation is the distribution of the observed asset prices conditional on the state variables and parameters, while the evolution equation consists of the dynamics of state variables conditional on the parameters. In the example mentioned earlier, (1.1) and (1.3) form the observation equations and (1.2) is the evolution equation. Viewed in this manner, all asset pricing models take the general form of nonlinear, non-Gaussian state space models.

MCMC methods are particularly well suited for continuous-time finance applications for several reasons.

1. Continuous-time asset models specify that prices and state variables solve parameterized stochastic differential equations (SDEs), which are built from Brownian motions, Poisson processes, and other i.i.d. shocks whose distributions are easy to characterize. When discretized at any finite time-interval, the models take the form of familiar time series models with normal, discrete mixtures of normals or scale mixtures of normals error distributions. This implies that the standard tools of Bayesian inference directly apply to these models. We will also later discuss the accuracy of discrete-time variants of continuous-time models.

2. MCMC is a unified estimation procedure, simultaneously estimating both parameters and latent variables. MCMC directly computes the distribution of the latent variables and parameters given the observed data. This is a stark alternative, the usual approach in the literature of applying approximate filters or noisy latent variable proxies. This allows the researcher, for example, to separate out the effects of jumps and stochastic volatility in models of interest rates or equity prices using discretely observed data.

3. MCMC methods allow the researcher to quantify estimation and model risk. Estimation risk is the inherent uncertainty present in estimating parameters or state variables, while model risk is the uncertainty over model specification. Increasingly in practical problems, estimation risk is a serious issue whose impact must be quantified. In the case of option pricing and optimal portfolio problems, Merton (1980) argues that the "most important direction is to develop accurate variance estimation models which take into account of the errors in variance estimates" (p. 355).

4. MCMC is based on conditional simulation, therefore avoiding any optimization or unconditional simulation. From a practical perspective, MCMC estimation is typically extremely fast in terms of computing time. This has many advantages, one of which is that it allows the researcher to perform simulation studies to study the algorithms accuracy for estimating parameters or state variables, a feature not shared by many other methods.

The rest of the chapter is outlined as follows. Section 2 provides a brief, nontechnical overview of Bayesian inference and MCMC methods. Section 3 describes the mechanics of MCMC algorithms, provides an overview of the limiting properties of MCMC algorithms, and provides practical recommendations for implementing MCMC algorithms. Section 4 discusses the generic problem of Bayesian inference in continuous-time models. Section 5 provides a tutorial on MCMC methods, building algorithms for equity price, option price, term structure, and regime switching models. Section 6 concludes and provides directions for future research.

2. OVERVIEW OF BAYESIAN INFERENCE AND MCMC

This section provides a brief, nontechnical overview of MCMC and Bayesian methods. We first describe the mechanics of MCMC simulation, and then we show how to use MCMC methods to compute objects of interest in Bayesian inference.

2.1. MCMC Simulation and Estimation

MCMC generates random samples from a given target distribution, in our case, the distribution of parameters and state variables given the observed prices, p(Θ, X|Y). One way to motivate the construction of MCMC algorithms is via a result commonly known as the Clifford-Hammersley theorem. The theorem states that a joint distribution can be characterized by its so-called complete conditional distributions. Specifically, the theorem implies that p(X|Θ, Y) and p(Θ|X, Y) completely characterize the joint distribution p(Θ, X|Y).

(Continues...)

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Excerpted from Handbook of Financial Econometrics, Vol 2 Copyright © 2010 by Elsevier B.V. . Excerpted by permission of North-Holland. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents

1. MCMC Methods for Continuous-Time Financial Econometrics- Michael Johannes, Nicholas Polson 2. The Analysis of the Cross Section of Security Returns- Ravi Jagannathan, Giorgios Skoulakis, Zhenyu Wang 3. Option Pricing Bounds and Statistical Uncertainty- Per A. Mykland 4. Inference for Stochastic Processes- Jean Jacod 5. Stock market Trading Volume- Andrew W. Lo, Jiang Wang

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Presents broad and eclectic surveys of applied financial econometric subjects.

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"With contributions from many (if not most) of the world's leading scholars in financial econometrics, these volumes summarize the key advances in this field over the past two decades." —Darrell Duffie, Stanford University

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