Consider, now, the time series X ≡ (x1, x2, …, xN) consisting of observations of the sample path x(t) at the discrete times t = nΔt with n = 1, …, N. Then, using the Markov property of the Ornstein-Uhlenbeck process, the probability of the sample path is given by11
The probability P(λ, D|X) of the parameters, given the sample path, can now be computed using Bayes theorem, as
The denominator P(X) is an unimportant normalization, independent of the parameters, that we henceforth ignore. Since both k and γ must be positive, for stability and positivity of entropy production respectively, we use informative priors for λ and D, P(λ, D) = H(λ)H(D), where H is the Heaviside step function. This assigns zero probability weight for negative values of the parameters and equal probability weight for all positive values. The logarithm of the posterior probability, after using the explicit forms of P1|1 and P1, is
where we have defined the two quantities
The maximum a posteriori (MAP) estimate (λ*, D*) solves the stationary conditions ∂lnP(λ, D|X)/∂λ = 0 and ∂lnP(λ, D|X)/∂D = 0, while the error bars of this estimate are obtained from the Hessian matrix of second derivatives evaluated at the maximum3,12,13. The analytical solution of the stationary conditions, derived in the Supplementary Information, yields the MAP estimate to be
where both I2 and Δn are now evaluated at λ = λ* and the sum runs from n = 1, …, N − 1. These provide direct estimates of the parameters without the need for fitting, minimization, or Monte Carlo sampling.
The error bars are obtained from a Taylor expansion of the log posterior to quadratic order about the MAP value,
where Δλ = λ − λ* and ΔD = D − D* and ∑−1 is the matrix of second derivatives of the log posterior evaluated at the maximum. The elements , , of the covariance matrix ∑ are the Bayesian error bars; they determine the size and shape of the Bayesian credible region around the maximum13. Their unwieldy expressions are provided in the Appendix and are made use of when computing credible regions around the MAP estimates. We refer to this Bayesian estimation procedure as “Bayes I” below.
A second Bayesian procedure for directly estimating the trap stiffness results when X is interpreted not as a time series but as an exchangeable sequence, or, in physical terms, as repeated independent observations of the stationary distribution P1(x)12. In that case, the posterior probability, assuming an informative prior that constrains k to positive values, is
The MAP estimate and its error bar follow straightforwardly from the posterior distribution as
and, not unexpectedly, the standard error decreases as the number of observations increases. This procedure is independent of Δt and is equivalent to the equipartition method when the Heaviside prior is used for k. We refer to this procedure as “Bayes II” below.
The posterior probabilities in both methods can be written in terms of four functions of the data
which, therefore, are the sufficient statistics of the problem. The entire information in the time series X relevant to estimation is contained in these four statistics12. Their use reduces computational expense greatly, as only four numbers, rather than the entire time series, is needed for evaluating the posterior distributions.
The posterior distributions obtained above are for flat priors. Other choice of priors are possible. In particular, since both D and k are scale parameters a non-informative Jeffreys prior is appropriate3. Jeffreys has observed, however, that “An accurate statement of the prior probability is not necessary in a pure problem of estimation when the number of observations is large”3. The number of observations are in the tens of thousands in time series we study here and the posterior is dominated by the likelihoood rather than the prior. The prior, then, has an insignificant contribution to the posterior.
We note that the error bars obtained in both Bayes I and Bayes II refer to Bayesian credible intervals, which are relevant to the uncertainty in the parameter estimates, given the data set X. In contrast, conventional error bars refer to frequentist confidence intervals, which are relevant to the outcomes of hypothetical repetitions of measurement. In general, Bayesian credible intervals and frequentist confidence intervals are not identical and should not be compared as they provide answers to separate questions9.
A comparison of the estimates for the trap stiffness obtained from these independent procedures provides a check on the validity of the Ornstein-Uhlenbeck process as a data model. Any significant disagreement between the estimates from the two methods signals a breakdown of the applicability of the model and the assumptions implicit in it: overdamped dynamics, constant friction, harmonicity of the potential, and thermal equilibrium. This completes our description of the Bayesian procedures for estimating λ, D, and k.
Jump-Diffusion Term Structure and Ito Conditional Moment Generator
36 PagesPosted: 21 Sep 2001
Tsinghua University - PBC School of Finance
Date Written: April 2001
This paper implements a Multivariate Weighted Nonlinear Least Square estimator for a class of jump-diffusion interest rate processes (hereafter MWNLS-JD), which also admit closed-form solutions to bond prices under a no-arbitrage argument. The instantaneous interest rate is modeled as a mixture of a square-root diffusion process and a Poisson jump process. One can derive analytically the first four conditional moments, which form the basis of the MWNLS-JD estimator. A diagnostic conditional moment test can also be constructed from the fitted moment conditions. The market prices of diffusion and jump risks are calibrated by minimizing the pricing errors between a model-implied yield curve and a target yield curve. The time series estimation of the short-term interest rate suggests that the jump augmentation is highly significant and that the pure diffusion process is strongly rejected. The cross-sectional evidence indicates that the jump-diffusion yield curves are both more flexible in reducing pricing errors and are more consistent with the Martingale pricing principle.
Keywords: Jump-diffusion, term structure of interest rates, conditional moment generator, multivariate weighted nonlinear least square, market price of risk
JEL Classification: C51, C52, G12
Suggested Citation:Suggested Citation
Zhou, Hao, Jump-Diffusion Term Structure and Ito Conditional Moment Generator (April 2001). FEDS Working Paper No. 2001-28. Available at SSRN: https://ssrn.com/abstract=278440 or http://dx.doi.org/10.2139/ssrn.278440
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