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Markov-Madulated Shifted Wishart Process with Applications in Financial Mathematics

Azadie Faraz, Hossein | 2025

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 58531 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical science
  6. Advisor(s): Mahdavi Amiri, Nezaoddin; Arian, Hamid Reza
  7. Abstract:
  8. Wishart processes are popular models for stochastic correlation and volatility of financial assets due to their convenient analytical properties. However, they cannot reproduce the sudden and sustained surges in correlation and volatility observed during market crises. Moreover, the Wishart process (like the CIR model) does not impose a positive lower bound on variances, allowing them to approach zero. This thesis introduces a new stochastic process, called the Markov-Modulated Shifted Wishart (MMSW) process, to address these shortcomings. The MMSW process extends the Wishart model by adding a constant “shift” matrix to keep variances away from zero and by using a Markov chain to switch model parameters between normal and crisis regimes. The special case without regime switching, i.e., the Shifted Wishart (SW) process, is also studied. Analytical expressions for key quantities, including characteristic functions of the process, are derived under the proposed models, and methods for parameter estimation (such as a continuum-of-moment approach) are applied. Two financial applications of these models are then examined: derivative pricing and portfolio management. For correlation-dependent derivatives such as spread options, option prices under the MMSW and SW models are computed via Fourier transform and compared with those under the standard Wishart model. The results indicate that incorporating Markov regime shifts in the covariance model can significantly impact spread option prices (by several percentage points), yielding a more realistic pricing behavior compared to the classical Wishart model. In the portfolio management problem, the optimal asset allocation under a constant relative risk aversion (CRRA) utility is derived, assuming asset covariance dynamics follow the MMSW process. The obtained closed-form solution provides a flexible investment strategy capable of adapting to sudden market stress. This strategy alters the portfolio composition significantly during crisis periods (with weight differences of about 30% to 90% compared to the standard Wishart model) while preserving diversification benefits in normal periods. The contributions of this research include developing the SW and MMSW processes as a flexible framework for modeling asset covariance with regime shifts and a positive variance floor, and demonstrating the practical advantages of these processes in improving the accuracy of derivative pricing and the efficiency of asset allocation strategies
  9. Keywords:
  10. Option Pricing ; Portfolio Optimization ; Shifted Wishart Process ; Markov-Modulated Shifted Wishart Process ; Regime Change in Financial Markets ; Mathematical Finance

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