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Heat Kernel Interest Rate Models With Time-Inhomogeneous Markov Processes

Author

Listed:
  • JIRÔ AKAHORI

    (Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu 52-8577, Shiga, Japan)

  • ANDREA MACRINA

    (Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK;
    Institute of Economic Research, Kyoto University Yoshida-Honmachi, Sakyo-Ku, Kyoto 606-8501, Japan)

Abstract

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.

Suggested Citation

  • Jirô Akahori & Andrea Macrina, 2012. "Heat Kernel Interest Rate Models With Time-Inhomogeneous Markov Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(01), pages 1-15.
    • Handle: RePEc:wsi:ijtafx:v:15:y:2012:i:01:n:s0219024911006553
    DOI: 10.1142/S0219024911006553
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    References listed on IDEAS

    as
    1. Robert A. Jarrow, 2009. "Credit Risk Models," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 37-68, November.
    2. Andrea Macrina & Priyanka A. Parbhoo, 2010. "Security Pricing with Information-Sensitive Discounting," World Scientific Book Chapters, in: Masaaki Kijima & Chiaki Hara & Keiichi Tanaka & Yukio Muromachi (ed.), Recent Advances In Financial Engineering 2009, chapter 6, pages 157-180, World Scientific Publishing Co. Pte. Ltd..
    3. Jiro Akahori & Yuji Hishida & Josef Teichmann & Takahiro Tsuchiya, 2009. "A Heat Kernel Approach to Interest Rate Models," Papers 0910.5033, arXiv.org.
    4. Joanne Kennedy & Phil Hunt & Antoon Pelsser, 2000. "Markov-functional interest rate models," Finance and Stochastics, Springer, vol. 4(4), pages 391-408.
    5. L. C. G. Rogers, 1997. "The Potential Approach to the Term Structure of Interest Rates and Foreign Exchange Rates," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 157-176, April.
    6. Andrea Macrina & Priyanka A. Parbhoo, 2010. "Security Pricing with Information-Sensitive Discounting," Papers 1001.3570, arXiv.org, revised Jun 2010.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Andrea Macrina & Priyanka A. Parbhoo, 2011. "Randomised Mixture Models for Pricing Kernels," Papers 1112.2059, arXiv.org.
    2. Andrea Macrina, 2012. "Heat Kernel Framework for Asset Pricing in Finite Time," Papers 1211.0856, arXiv.org, revised Sep 2013.
    3. Andrea Macrina & Obeid Mahomed, 2018. "Consistent Valuation Across Curves Using Pricing Kernels," Papers 1801.04994, arXiv.org, revised Feb 2018.
    4. Andrea Macrina & David Skovmand, 2020. "Rational Savings Account Models for Backward-Looking Interest Rate Benchmarks," Risks, MDPI, vol. 8(1), pages 1-18, March.
    5. Andrea Macrina & Priyanka Parbhoo, 2014. "Randomised Mixture Models for Pricing Kernels," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 21(4), pages 281-315, November.
    6. Andrea Macrina & Priyanka A. Parbhoo, 2010. "Security Pricing with Information-Sensitive Discounting," Papers 1001.3570, arXiv.org, revised Jun 2010.
    7. Andrea Macrina & Priyanka A. Parbhoo, 2010. "Securities Pricing with Information-Sensitive Discounting," KIER Working Papers 695, Kyoto University, Institute of Economic Research.
    8. Andrea Macrina & Obeid Mahomed, 2018. "Consistent Valuation Across Curves Using Pricing Kernels," Risks, MDPI, vol. 6(1), pages 1-39, March.

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    More about this item

    Keywords

    Time-inhomogeneous Markov processes; Lévy processes; heat kernels; pricing kernels; information-based pricing; interest rate models; fixed-income assets;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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