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

A Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: the Exponent Expansion

In this paper we discuss a closed-form approximation of the likelihood functions of an arbitrary diffusion process. The approximation is based on an exponential ansatz of the transition probability for a finite time step $\Delta t$, and a series expansion of the deviation of its logarithm from that of a Gaussian distribution. Through this procedure, dubbed {\em exponent expansion}, the transition probability is obtained as a power series in $\Delta t$. This becomes asymptotically exact if an increasing number of terms is included, and provides remarkably accurate results even when truncated to the first few (say 3) terms. The coefficients of such expansion can be determined straightforwardly through a recursion, and involve simple one-dimensional integrals. We present several examples of financial interest, and we compare our results with the state-of-the-art approximation of discretely sampled diffusions [A\"it-Sahalia, {\it Journal of Finance} {\bf 54}, 1361 (1999)]. We find that the exponent expansion provides a similar accuracy in most of the cases, but a better behavior in the low-volatility regime. Furthermore the implementation of the present approach turns out to be simpler. Within the functional integration framework the exponent expansion allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. This is illustrated with the application to simple path-dependent interest rate derivatives. Finally we discuss how these results can also be used to increase the efficiency of numerical (both deterministic and stochastic) approaches to derivative pricing.Comment: 28 pages, 7 figure

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

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

Estimation in Models of the Instantaneous Short Term Interest Rate By Use of a Dynamic Bayesian Algorithm

This paper considers the estimation in models of the instantaneous short interest rate from a new perspective. Rather than using discretely compounded market rates as a proxy for the instantaneous short rate of interest, we set up the stochastic dynamics for the discretely compounded market observed rates and propose a dynamic Bayesian estimation algorithm (i.e. a filtering algorithm) for a time-discretised version of the resulting interest rate dynamics. The filter solution is computed via a further spatial discretization (quantization) and the convergence of the latter to its continuous counterpart is discussed in detail. The method is applied to simulated data and is found to give a reasonable estimate of the conditional density function and to be not too demanding computationally.

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

Divergences Test Statistics for Discretely Observed Diffusion Processes

In this paper we propose the use of $\phi$-divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process \de X_t = b(X_t, \theta)\de t + \sigma(X_t, \theta)\de W_t, from discrete observations $\{X_{t_i}, i=0, ..., n\}$ with $t_i = i\Delta_n$, $i=0, 1, >..., n$, under the asymptotic scheme $\Delta_n\to0$, $n\Delta_n\to\infty$ and $n\Delta_n^2\to 0$. The class of $\phi$-divergences is wide and includes several special members like Kullback-Leibler, R\'enyi, power and $\alpha$-divergences. We derive the asymptotic distribution of the test statistics based on $\phi$-divergences. The limiting law takes different forms depending on the regularity of $\phi$. These convergence differ from the classical results for independent and identically distributed random variables. Numerical analysis is used to show the small sample properties of the test statistics in terms of estimated level and power of the test