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Closed-form approximations of moments and densities of continuous–time Markov models

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Listed:
  • Kristensen, Dennis
  • Lee, Young Jun
  • Mele, Antonio

Abstract

This paper develops power series expansions of a general class of moment functions, including transition densities and option prices, of continuous-time Markov processes, including jump–diffusions. The proposed expansions extend the ones in Kristensen and Mele (2011) to cover general Markov processes, and nest transition density and option price expansions recently developed in the literature, thereby connecting seemingly different ideas in a unified framework. We show how the general expansion can be implemented for fully general jump–diffusion models. We provide a new theory for the validity of the expansions which shows that series expansions are not guaranteed to converge as more terms are added in general once the time span of interest gets larger than some model–specific threshold. Thus, these methods should be used with caution when applied to problems with a larger time span of interest, such as long-term options or data observed at a low frequency. At the same time, the numerical studies in this paper demonstrate good performance of the proposed implementation in practice when applied to pricing options with time to maturity below three months. Thus, our expansions are particularly well suited for pricing ultra-short-term (such as “zero–day”) options.

Suggested Citation

  • Kristensen, Dennis & Lee, Young Jun & Mele, Antonio, 2024. "Closed-form approximations of moments and densities of continuous–time Markov models," Journal of Economic Dynamics and Control, Elsevier, vol. 168(C).
  • Handle: RePEc:eee:dyncon:v:168:y:2024:i:c:s0165188924001404
    DOI: 10.1016/j.jedc.2024.104948
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    References listed on IDEAS

    as
    1. Kristensen, Dennis & Shin, Yongseok, 2012. "Estimation of dynamic models with nonparametric simulated maximum likelihood," Journal of Econometrics, Elsevier, vol. 167(1), pages 76-94.
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    15. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
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    18. Yang, Nian & Chen, Nan & Wan, Xiangwei, 2019. "A new delta expansion for multivariate diffusions via the Itô-Taylor expansion," Journal of Econometrics, Elsevier, vol. 209(2), pages 256-288.
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    More about this item

    Keywords

    Continuous-time models; Jump-diffusion; Transition density; Stochastic volatility; Closed-form approximations; Maximum-likelihood estimation; Option pricing;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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