Abstract
In this paper we implement the method of Feynman path integral for the analysis of option pricing for certain Lévy process driven financial markets. For such markets, we find closed form solutions of transition probability density functions of option pricing in terms of various special functions. Asymptotic analysis of transition probability density functions is provided. We also find expressions for transition probability density functions in terms of various special functions for certain Lévy process driven market where the interest rate is stochastic.
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References
Baaquie, B.E.: Quantum finance: Path Integrals and Hamiltonians for Options and Interest Rates. Cambridge University Press, New York (2004)
Baaquie, B.E.: A path integral approach to option pricing with stochastic volatility: some exact results. Journal de Physique I 7(12), 1733–1753 (1997)
Blazhyevskyi, L.F., Yanishevsky, V.S.: The path integral representation kernel of evolution operator in Merton–Garman model. Condens. Matter Phys. 14(2), 116 (2011). 23001:
Bormetti, G., Montagna, G., Moreni, N., Nicrosini, O.: Pricing exotic options in a path integral approach. Quant. Finance 6(1), 55–66 (2006)
Cont, R., Voltchkova, E.: Integro-differential equations for option prices in exponential Lévy model. Finance Stoch. 9, 299–325 (2005)
Debnath, L., Mikusinski, P.: Introduction to Hilbert Spaces with Applications, 3rd edn. Academic Press, New York (2005)
Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Clarendon Press, Oxford (1982)
Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill Book Company, New York (1965)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, Corrected and Enlarged Edition. Academic Press, New York (1980)
Jacob, N.: Pseudo-Differential Operators and Markov Processes, 1st edn. Imperial College Press, London (2001)
Kakushadze, Z.: Path integral and asset pricing. Quant. Finance 15(11), 1759–1771 (2015)
Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets, 3rd edn. World Scientific Publishing Co., Inc., River Edge (2004)
Linetsky, V.: The path integral approach to financial modeling and options pricing. Comput. Econ. 11, 129–163 (1998)
Roepstorff, G.: Path Integral Approach to Quantum Physics. Springer, Berlin (1994)
Schulman, L.S.: Techniques and Applications of Path Integration. Wiley, New York (1981)
SenGupta, I.: Option pricing with transaction costs and stochastic interest rate. Appl. Math. Finance 21(5), 399–416 (2014)
Watson, G.N.: The harmonic functions associated with the parabolic cylinder. Proc. Lond. Math. Soc. s2–17(1), 116–148 (1918)
Whittaker, E.T.: On the functions associated with the parabolic cylinder in harmonic analysis. Proc. Lond. Math. Soc. 35, 417–427 (1902)
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Issaka, A., SenGupta, I. Feynman path integrals and asymptotic expansions for transition probability densities of some Lévy driven financial markets. J. Appl. Math. Comput. 54, 159–182 (2017). https://doi.org/10.1007/s12190-016-1002-2
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DOI: https://doi.org/10.1007/s12190-016-1002-2
Keywords
- Lévy density
- Mathematical finance
- Parabolic cylinder functions
- Incomplete gamma functions
- Watson’s lemma