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Multivariate Time-Changed Brownian Motion: The Expectation–Maximization Estimation Method

In: HANDBOOK OF HEAVY-TAILED DISTRIBUTIONS IN ASSET MANAGEMENT AND RISK MANAGEMENT

Author

Listed:
  • Michele Leonardo Bianchi
  • Stoyan V Stoyanov
  • Gian Luca Tassinari
  • Frank J Fabozzi
  • Sergio M Focardi

Abstract

The main topics covered in this chapter are:a description of a method to estimate the parameters of models based on the multivariate time-changed Brownian motion;a review of the expectation–maximization (EM) maximum likelihood estimation (MLE) method to estimate the parameters of multivariate generalized hyperbolic distributions;an extension of the EM-based MLE algorithm to normal mean–variance mixture distributions in which only the characteristic function of the mixing distribution is known in closed form, while the density function is not;an error analysis of the estimation method applied to the multivariate normal tempered stable case;an empirical test showing the model performance on a five-and a 30-dimensional series of index (stock) returns;how to evaluate well-known risk measures (i.e., value-at-risk and average value-at-risk) under this framework;how to backtest the value-at-risk by taking into account the number of exceedances.

Suggested Citation

  • Michele Leonardo Bianchi & Stoyan V Stoyanov & Gian Luca Tassinari & Frank J Fabozzi & Sergio M Focardi, 2019. "Multivariate Time-Changed Brownian Motion: The Expectation–Maximization Estimation Method," World Scientific Book Chapters, in: HANDBOOK OF HEAVY-TAILED DISTRIBUTIONS IN ASSET MANAGEMENT AND RISK MANAGEMENT, chapter 8, pages 323-366, World Scientific Publishing Co. Pte. Ltd..
    • Handle: RePEc:wsi:wschap:9789813276208_0008
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    Keywords

    Heavy Tail Distributions; Fat Tail Distributions; Lévy Processes; Tempered Stable Distributions; Multivariate Time-changed Brownian Motion; Extreme Value Theory; Risk Management;
    All these keywords.

    JEL classification:

    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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