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Applications of Eigenfunction Expansions in Continuous‐Time Finance

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  • Alan L. Lewis

Abstract

We provide exact solutions for two closely related valuation problems in continuous‐time finance. The first problem is to value generalized European‐style options on stocks that pay dividends at a constant dollar rate. The second problem is to find the yield curve associated with the economy of R. C. Merton's “An Asymptotic Theory of Growth Under Uncertainty.” In Merton's economic growth model, the interest rate process has a volatility linear in the rate level and a linear/quadratic drift. Both problems are solved by an eigenfunction expansion technique. The main technical difficulty is handling the problem of payoff functions that are not square‐integrable with respect to the natural weight function of the models.

Suggested Citation

  • Alan L. Lewis, 1998. "Applications of Eigenfunction Expansions in Continuous‐Time Finance," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 349-383, October.
    • Handle: RePEc:bla:mathfi:v:8:y:1998:i:4:p:349-383
    DOI: 10.1111/1467-9965.00059
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    Cited by:

    1. Dmitry Davydov & Vadim Linetsky, 2003. "Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach," Operations Research, INFORMS, vol. 51(2), pages 185-209, April.
    2. Hyungbin Park, 2015. "Sensitivity Analysis of Long-Term Cash Flows," Papers 1511.03744, arXiv.org, revised Sep 2018.
    3. Likuan Qin & Vadim Linetsky, 2014. "Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery and Long-Term Pricing," Papers 1411.3075, arXiv.org, revised Sep 2015.
    4. Petteri Piiroinen & Lassi Roininen & Tobias Schoden & Martin Simon, 2018. "Asset Price Bubbles: An Option-based Indicator," Papers 1805.07403, arXiv.org, revised Jul 2018.
    5. Decamps, Marc & De Schepper, Ann & Goovaerts, Marc, 2009. "Spectral decomposition of optimal asset-liability management," Journal of Economic Dynamics and Control, Elsevier, vol. 33(3), pages 710-724, March.
    6. Alan L. Lewis, 2001. "A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes," Related articles explevy, Finance Press.
    7. Sebastian F. Tudor & Rupak Chatterjee & Igor Tydniouk, 2021. "On a new parametrization class of solvable diffusion models and transition probability kernels," Quantitative Finance, Taylor & Francis Journals, vol. 21(10), pages 1773-1790, October.
    8. Damien Ackerer & Damir Filipović, 2020. "Option pricing with orthogonal polynomial expansions," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 47-84, January.
    9. Likuan Qin & Vadim Linetsky, 2016. "Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery, and Long-Term Pricing," Operations Research, INFORMS, vol. 64(1), pages 99-117, February.
    10. Runhuan Feng & Pingping Jiang & Hans Volkmer, 2020. "Geometric Brownian motion with affine drift and its time-integral," Papers 2012.09661, arXiv.org.
    11. J. Li & A. Metzler & R. M. Reesor, 2017. "A structural framework for modelling contingent capital," Quantitative Finance, Taylor & Francis Journals, vol. 17(7), pages 1071-1088, July.
    12. Feng, Runhuan & Jiang, Pingping & Volkmer, Hans, 2021. "Geometric Brownian motion with affine drift and its time-integral," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    13. Alan L. Lewis, 2018. "Exact Solutions for a GBM-type Stochastic Volatility Model having a Stationary Distribution," Papers 1809.08635, arXiv.org, revised May 2019.
    14. Vadim Linetsky, 2004. "Spectral Expansions for Asian (Average Price) Options," Operations Research, INFORMS, vol. 52(6), pages 856-867, December.
    15. Lim, Dongjae & Li, Lingfei & Linetsky, Vadim, 2012. "Evaluating callable and putable bonds: An eigenfunction expansion approach," Journal of Economic Dynamics and Control, Elsevier, vol. 36(12), pages 1888-1908.

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