Gaussian Estimation of Continuous Time Models of the Short Term Interest Rate

This paper proposes a Gaussian estimator for nonlinear continuous time models of the short term interest rate. The approach is based on a stopping time argument that produces a normalizing transformation facilitating the use of a Gaussian likelihood. A Monte Carlo study shows that the finite sample performance of the proposed procedure offers an improvement over the discrete approximation method proposed by Nowman (1997). An empirical application to U.S. and British interest rates is given.Gaussian estimation, nonlinear diffusion, normalizing transformation

Simulation-based Estimation of Contingent-claims Prices

A new methodology is proposed to estimate theoretical prices of financial contingent-claims whose values are dependent on some other underlying financial assets. In the literature the preferred choice of estimator is usually maximum likelihood (ML). ML has strong asymptotic justification but is not necessarily the best method in finite samples. The present paper proposes instead a simulation-based method that improves the finite sample performance of the ML estimator while maintaining its good asymptotic properties. The methods are implemented and evaluated here in the Black-Scholes option pricing model and in the Vasicek bond pricing model, but have wider applicability. Monte Carlo studies show that the proposed procedures achieve bias reductions over ML estimation in pricing contingent claims. The bias reductions are sometimes accompanied by reductions in variance, leading to significant overall gains in mean squared estimation error. Empirical applications to US treasury bills highlight the differences between the bond prices implied by the simulation-based approach and those delivered by ML. Some consequences for the statistical testing of contingent-claim pricing models are discussed.Bias reduction, Bond pricing, Indirect inference, Option pricing, Simulation-based estimation

A Two-Stage Realized Volatility Approach to the Estimation for Diffusion Processes from Discrete Observations

This paper motivates and introduces a two-stage method for estimating diffusion processes based on discretely sampled observations. In the first stage we make use of the feasible central limit theory for realized volatility, as recently developed in Barndorff-Nielsen and Shephard (2002), to provide a regression model for estimating the parameters in the diffusion function. In the second stage the in-fill likelihood function is derived by means of the Girsanov theorem and then used to estimate the parameters in the drift function. Consistency and asymptotic distribution theory for these estimates are established in various contexts. The finite sample performance of the proposed method is compared with that of the approximate maximum likelihood method of Ait-Sahalia (2002).Maximum likelihood, Girsnov theorem, Discrete sampling, Continuous record, Realized volatility

Jackknifing Bond Option Prices

In continuous time specifications, the prices of interest rate derivative securities depend crucially on the mean reversion parameter of the associated interest rate diffusion equation. This parameter is well known to be subject to estimation bias when standard methods like maximum likelihood (ML) are used. The estimation bias can be substantial even in very large samples and it translates into a bias in pricing bond options and other derivative securities that is important in practical work. The present paper proposes a very general method of bias reduction for pricing bond options that is based on Quenouille's (1956) jackknife. We show how the method can be applied directly to the options price itself as well as the coefficients in continuous time models. The method is implemented and evaluated here in the Cox, Ingersoll and Ross (1985) model, although it has much wider applicability. A Monte Carlo study shows that the proposed procedure achieves substantial bias reductions in pricing bond options with only mild increases in variance that do not compromise the overall gains in mean squared error. Our findings indicate that bias correction in estimation of the drift can be more important in pricing bond options than correct specification of the diffusion. Thus, even if ML or approximate ML can be used to estimate more complicated models, it still appears to be of equal or greater importance to correct for the effects on pricing bond options of bias in the estimation of the drift. An empirical application to U.S. interest rates highlights the differences between bond and option prices implied by the jackknife procedure and those implied by the standard approach. These differences are large and suggest that bias reduction in pricing options is important in practical applications.Bias Reduction, Option Pricing, Bond Pricing, Term Structure of Interest Rate, Re-sampling, Estimation of Continuous Time Models

Maximum Likelihood and Gaussian Estimation of Continuous Time Models in Finance

This paper overviews maximum likelihood and Gaussian methods of estimating continuous time models used in finance. Since the exact likelihood can be constructed only in special cases, much attention has been devoted to the development of methods designed to approximate the likelihood. These approaches range from crude Euler-type approximations and higher order stochastic Taylor series expansions to more complex polynomial-based expansions and infill approximations to the likelihood based on a continuous time data record. The methods are discussed, their properties are outlined and their relative finite sample performance compared in a simulation experiment with the nonlinear CIR diffusion model, which is popular in empirical finance. Bias correction methods are also considered and particular attention is given to jackknife and indirect inference estimators. The latter retains the good asymptotic properties of ML estimation while removing finite sample bias. This method demonstrates superior performance in finite samples.Maximum likelihood, Transition density, Discrete sampling, Continuous record, Realized volatility, Bias reduction, Jackknife, Indirect inference

Explosive Behavior in the 1990s Nasdaq : When Did Exuberance Escalate Asset Values?

A recursive test procedure is suggested that provides a mechanism for testing explosive behavior, date-stamping the origination and collapse of economic exuberance, and providing valid conOdence intervals for explosive growth rates. The method involves the recursive im- plementation of a right-side unit root test and a sup test, both of which are easy to use in practical applications, and some new limit theory for mildly explosive processes. The test procedure is shown to have discriminatory power in detecting periodically collapsing bubbles, thereby overcoming a weakness in earlier applications of unit root tests for economic bubbles. An empirical application to Nasdaq stock price index in the 1990s provides conOrmation of ex- plosiveness and date-stamps the origination of Onancial exuberance to mid -1995, prior to the famous remark in December 1996 by Alan Greenspan about irrational exuberance in Onancial market, thereby giving the remark empirical content.Explosive root, irrational exuberance, Mildly explosive process, Nasdaq bubble, periodically collapsing bubble, sup test, unit root test

Explosive Behavior in the 1990s Nasdaq: When Did Exuberance Escalate Asset Values?

A recursive test procedure is suggested that provides a mechanism for testing explosive behavior, date-stamping the origination and collapse of economic exuberance, and providing valid con?dence intervals for explosive growth rates. The method involves the recursive im- plementation of a right-side unit root test and a sup test, both of which are easy to use in practical applications, and some new limit theory for mildly explosive processes. The test procedure is shown to have discriminatory power in detecting periodically collapsing bubbles, thereby overcoming a weakness in earlier applications of unit root tests for economic bubbles. An empirical application to Nasdaq stock price index in the 1990s provides con?rmation of ex- plosiveness and date-stamps the origination of ?nancial exuberance to mid -1995, prior to the famous remark in December 1996 by Alan Greenspan about irrational exuberance in ?nancial market, thereby giving the remark empirical content.Explosive root, irrational exuberance, mildly explosive process, Nasdaq bubble, periodically collapsing bubble, sup test, unit root test

Explosive Behavior in the 1990s Nasdaq: When Did Exuberance Escalate Asset Values?

A recursive test procedure is suggested that provides a mechanism for testing explosive behavior, date-stamping the origination and collapse of economic exuberance, and providing valid confidence intervals for explosive growth rates. The method involves the recursive implementation of a right-side unit root test and a sup test, both of which are easy to use in practical applications, and some new limit theory for mildly explosive processes. The test procedure is shown to have discriminatory power in detecting periodically collapsing bubbles, thereby overcoming a weakness in earlier applications of unit root tests for economic bubbles. An empirical application to Nasdaq stock price index in the 1990s provides confirmation of explosiveness and date-stamps the origination of financial exuberance to mid -1995, prior to the famous remark in December 1996 by Alan Greenspan about irrational exuberance in financial markets, thereby giving the remark empirical content.Explosive root, Irrational exuberance, Mildly explosive process, Nasdaq bubble, Periodically collapsing bubble, Sup test, Unit root test

Specification Sensitivity in Right-Tailed Unit Root Testing for Explosive Behavior

Right-tailed unit root tests have proved promising for detecting exuberance in economic and financial activities. Like left-tailed tests, the limit theory and test performance are sensitive to the null hypothesis and the model specification used in parameter estimation. This paper aims to provide some empirical guidelines for the practical implementation of right-tailed unit root tests, focussing on the sup ADF test of Phillips, Wu and Yu (2011), which implements a right-tailed ADF test repeatedly on a sequence of forward sample recursions. We analyze and compare the limit theory of the sup ADF test under different hypotheses and model specifications. The size and power properties of the test under various scenarios are examined in simulations and some recommendations for empirical practice are given. Empirical applications to the Nasdaq and to Australian and New Zealand housing data illustrate these specification issues and reveal their practical importance in testing.Unit root test, Mildly explosive process, Recursive regression, Size and power