[Paper Review] On the Parameter Estimation in the Schwartz-Smiths Two-Factor Model

Karol Binkowski, Peilun He+2 more|arXiv (Cornell University)|Aug 4, 2021

Stochastic processes and financial applications16 references20 citations

TL;DR

This paper proposes a joint parameter estimation method for the Schwartz-Smith two-factor model using the Kalman Filter, addressing the parameter identification problem in the likelihood function by introducing an additional constraint. The approach yields consistent conditional Maximum Likelihood Estimators (MLEs) for the model parameters and unobserved state variables, validated through simulation studies showing robustness and convergence.

ABSTRACT

The two unobservable state variables representing the short and long term factors introduced by Schwartz and Smith in [16] for risk-neutral pricing of futures contracts are modelled as two correlated Ornstein-Uhlenbeck processes. The Kalman Filter (KF) method has been implemented to estimate the short and long term factors jointly with un- known model parameters. The parameter identification problem arising within the likelihood function in the KF has been addressed by introduc- ing an additional constraint. The obtained model parameter estimates are the conditional Maximum Likelihood Estimators (MLEs) evaluated within the KF. Consistency of the conditional MLEs is studied. The methodology has been tested on simulated data.

Motivation & Objective

• To address the parameter identification problem in the Kalman Filter likelihood function when estimating parameters of the two-factor model.
• To develop a method for jointly estimating the unobserved short- and long-term factors ($\chi_t$, $\xi_t$) and model parameters ($\kappa$, $\gamma$, $\mu_\xi$, $\sigma_\chi$, $\sigma_\xi$, $\rho_{\chi\xi}$).
• To ensure consistency of the conditional Maximum Likelihood Estimators (MLEs) within the Kalman Filter framework.
• To validate the proposed estimation methodology using simulated data under the risk-neutral two-factor model.

Proposed method

• Model the logarithm of the spot price as $\log(S_t) = \chi_t + \xi_t$, where $\chi_t$ and $\xi_t$ are correlated Ornstein-Uhlenbeck processes.
• Formulate the system as a partially observed linear state-space model with unobserved state vector $x_t = (\chi_t, \xi_t)^\top$.
• Implement the Kalman Filter algorithm to estimate the state variables and model parameters simultaneously via conditional MLE.
• Introduce an additional constraint in the likelihood function to resolve the parameter identification problem arising from the non-identifiability of the variance-covariance structure.
• Use grid search for initial parameter exploration to improve convergence and stability of the MLE estimation.
• Derive analytical expressions for the conditional mean and covariance of the state vector under the risk-neutral measure, based on the bivariate O-U process dynamics.

Experimental results

Research questions

• RQ1 How can the Kalman Filter be adapted to jointly estimate the unobserved state variables and model parameters in the Schwartz-Smith two-factor model?
• RQ2 What constraints are necessary to resolve the parameter identification problem in the likelihood function of the Kalman Filter for this model?
• RQ3 Are the resulting conditional Maximum Likelihood Estimators (MLEs) for the model parameters and states consistent?
• RQ4 How robust is the joint estimation procedure under varying parameter configurations and sample sizes in simulation?
• RQ5 What is the performance of the grid-search initialization in achieving convergence and accurate estimation of the MLEs?

Key findings

• The proposed method successfully resolves the parameter identification problem in the Kalman Filter likelihood function through an additional constraint, enabling joint estimation of model parameters and unobserved states.
• The conditional MLEs for the model parameters and state vector $x_t = (\chi_t, \xi_t)^\top$ are shown to be consistent under the proposed framework.
• Simulation results demonstrate robust performance of the grid-search initialization in achieving stable and accurate parameter estimates.
• The estimated parameters converge to their true values as sample size increases, confirming the consistency of the estimators.
• The Kalman Filter with constrained likelihood estimation effectively reproduces the term structure of commodity futures prices in simulated data.