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Contingent claims valuation when the security price is a combination of an Ito process and a random point process

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  • Aase, Knut K.

Abstract

This paper develops several results in the modern theory of contingent claims valuation in a frictionless security market with continuous trading. The price model is a semi-martingale with a certain structure, making the return of the security a sum of an Ito-process and a random, marked point process. Dynamic equilibrium prices are known to be of this form in an ArrowDebreu economy, so there is no real limitation in our approach. This class of models is also advantageous from an applied point of view. Within this framework we investigate how the model behaves under the equivalent martingale measure in the P*-equilibrium economy, where discounted security prices are marginales. Here we present some new results showing how the marked point process affects prices of contingent claims in equilibrium. We derive a new class of option pricing formulas when the Ito process is a general Gaussian process, one formula for each positive L2[O, T]-function. We show that our general model is complete, although the set of equivalent martingale measures is not a singleton. We also demonstrate how to price contingent claims when the underlying process has after-effects in all of its parameters.

Suggested Citation

  • Aase, Knut K., 1988. "Contingent claims valuation when the security price is a combination of an Ito process and a random point process," Stochastic Processes and their Applications, Elsevier, vol. 28(2), pages 185-220, June.
    • Handle: RePEc:eee:spapps:v:28:y:1988:i:2:p:185-220
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    Citations

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    Cited by:

    1. Jean-Luc Prigent, 2001. "Option Pricing with a General Marked Point Process," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 50-66, February.
    2. Morten Christensen & Eckhard Platen, 2004. "A General Benchmark Model for Stochastic Jump Sizes," Research Paper Series 139, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Gerald Cheang & Carl Chiarella, 2011. "A Modern View on Merton's Jump-Diffusion Model," Research Paper Series 287, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Yang Wang & Baojun Bian & Jizhou Zhang, 2014. "Viscosity Solutions of Integro-Differential Equations and Passport Options in a Jump-Diffusion Model," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 122-144, April.
    5. Bardhan, Indrajit & Chao, Xiuli, 1996. "Stochastic multi-agent equilibria in economies with jump-diffusion uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 20(1-3), pages 361-384.
    6. Chou, Ching-Sung & Lin, Hsien-Jen, 2006. "Asian options with jumps," Statistics & Probability Letters, Elsevier, vol. 76(18), pages 1983-1993, December.
    7. Vajanne, Laura, . "The Exchange Rate Under Target Zones," ETLA A, The Research Institute of the Finnish Economy, number 16.
    8. Sergei Fedotov & Sergei Mikhailov, 1998. "Option Pricing Model for Incomplete Market," Papers cond-mat/9807397, arXiv.org, revised Aug 1998.
    9. C. Mancini, 2002. "The European options hedge perfectly in a Poisson-Gaussian stock market model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 9(2), pages 87-102.
    10. Oleg Szehr, 2021. "Hedging of Financial Derivative Contracts via Monte Carlo Tree Search," Papers 2102.06274, arXiv.org, revised Apr 2021.
    11. Bardhan, Indrajit & Chao, Xiuli, 1996. "On martingale measures when asset returns have unpredictable jumps," Stochastic Processes and their Applications, Elsevier, vol. 63(1), pages 35-54, October.
    12. Wee, In-Suk, 1999. "Stability for multidimensional jump-diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 193-209, April.
    13. Barbarin, Jérôme, 2008. "Heath-Jarrow-Morton modelling of longevity bonds and the risk minimization of life insurance portfolios," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 41-55, August.
    14. Sergei Fedotov & Sergei Mikhailov, 2001. "Option Pricing For Incomplete Markets Via Stochastic Optimization: Transaction Costs, Adaptive Control And Forecast," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 179-195.
    15. Anindya Goswami & Omkar Manjarekar & Anjana R, 2018. "Option Pricing in a Regime Switching Jump Diffusion Model," Papers 1811.11379, arXiv.org, revised Oct 2019.
    16. Gerald H.L. Cheang & Carl Chiarella, 2008. "Hedge Portfolios in Markets with Price Discontinuities," Research Paper Series 218, Quantitative Finance Research Centre, University of Technology, Sydney.
    17. Moller, Christian Max, 1995. "A counting process approach to stochastic interest," Insurance: Mathematics and Economics, Elsevier, vol. 17(2), pages 181-192, October.
    18. Olsen, Trond E. & Stensland, Gunnar, 1996. "On optimal control of income generating activities, and the value of flexible production design," International Review of Economics & Finance, Elsevier, vol. 5(4), pages 349-361.

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