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Heat Kernel Framework for Asset Pricing in Finite Time

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  • Andrea Macrina

Abstract

A heat kernel approach is proposed for the development of a general, flexible, and mathematically tractable asset pricing framework in finite time. The pricing kernel, giving rise to the price system in an incomplete market, is modelled by weighted heat kernels which are driven by multivariate Markov processes and which provide enough degrees of freedom in order to calibrate to relevant data, e.g. to the term structure of bond prices. It is shown how, for a class of models, the prices of bonds, caplets, and swaptions can be computed in closed form. The dynamical equations for the price processes are derived, and explicit formulae are obtained for the short rate of interest, the risk premium, and for the stochastic volatility of prices. Several of the closed-form asset price models presented in this paper are driven by combinations of Markovian jump processes with different probability laws. Such models provide a rich basis for consistent applications in several sectors of a financial market including equity, fixed-income, commodities, and insurance. The flexible, multidimensional and multivariate structure, on which the asset price models are constructed, lends itself well to the transparent modelling of dependence across asset classes. As an illustration, the impact on prices by spiralling debt, a typical feature of a financial crisis, is modelled explicitly, and contagion effects are readily observed in the dynamics of asset returns.

Suggested Citation

  • Andrea Macrina, 2012. "Heat Kernel Framework for Asset Pricing in Finite Time," Papers 1211.0856, arXiv.org, revised Sep 2013.
    • Handle: RePEc:arx:papers:1211.0856
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    References listed on IDEAS

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    1. Back, Kerry, 2010. "Asset Pricing and Portfolio Choice Theory," OUP Catalogue, Oxford University Press, number 9780195380613.
    2. Edward Hoyle & Lane P. Hughston & Andrea Macrina, 2010. "Stable-1/2 Bridges and Insurance," Papers 1005.0496, arXiv.org, revised Apr 2014.
    3. Jirô Akahori & Andrea Macrina, 2022. "Heat Kernel Interest Rate Models With Time-Inhomogeneous Markov Processes," World Scientific Book Chapters, in: Dorje Brody & Lane Hughston & Andrea Macrina (ed.), Financial Informatics An Information-Based Approach to Asset Pricing, chapter 9, pages 179-193, World Scientific Publishing Co. Pte. Ltd..
    4. Jiro Akahori & Yuji Hishida & Josef Teichmann & Takahiro Tsuchiya, 2009. "A Heat Kernel Approach to Interest Rate Models," Papers 0910.5033, arXiv.org.
    5. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    6. Farshid Jamshidian, 1996. "Bond, futures and option evaluation in the quadratic interest rate model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(2), pages 93-115.
    7. Dorje C. Brody & Lane P. Hughston & Andrea Macrina, 2008. "Information-Based Asset Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(01), pages 107-142.
    8. 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.
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