European Call Option and the Hamilton-Jacobi Type in Classical and Fractional Forms

José Rodrigo González Granada, Luis Fernando Plaza Gálvez, Javier Alexander Tenorio Quiñones

Abstract

This study aims to investigate the European Call Option in a fractional way and the reaction of the Hamilton-Jacobi differential equation with the European Call Option. We explore the applications of Mellin transforms as a method for solving such fractional differential equations. We can address initial value problems defined for arbitrary orders of differentiation and integration in fractional calculus that allows us to understand the solution's evolution, encompassing the values established in classical analysis. The theory of options permits the connection of physics and finance, particularly through dynamic systems and their relationship with the Hamilton-Jacobi equation. The European Call Option model exhibits energy in its Hamiltonian, leading to investigations of the Lagrangian and its potential functions within the European Call Option model. Caputo’s and Riemann-Liouville's fractional derivatives are crucial for differential equations with integer and non-integer initial conditions, respectively. This new perspective on calculus challenges traditional analysis, explores new horizons, and demands a redefinition of classical approaches. Understanding the efficacy of integro-differential operators is pivotal for advancing research, particularly in interpreting financial phenomena using classical calculus. The Mittag-Leffler function proves valuable for algebraic purposes and inverse transformations such as Mellin.

 

Keywords: European call option, the Hamilton-Jacobi equation, the Mittag-Leffler function, the Mellin transform, the Riemann-Liouville fractional derivative.

 

https://doi.org/10.55463/issn.1674-2974.50.7.13


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References


DAMODARAN A. Investment Valuation: Tools and Techniques for Determining the Value of Any Asset. 3rd ed. Wiley, 2002.

ARNOLD V.I. Mathematical Methods of Classical Mechanics. 2nd ed. Springer. 1989.

BAAQUIE B.E., CORIANO S., and SRIKANT M. Hamiltonian and potentials in derivative pricing models: exact results and lattice simulations. Physica A: Statistical Mechanics and its Applications, 2004, 334(3-4): 531-557.

BLACK F., and SCHOLES M. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 1973, 81(3), 637-654.

DEVILLE R. On the range of the derivative of a smooth function and applications. Journal of the Royal Academy of Exact, Physical and Natural Sciences. Series A. Mathematics, 2006, 100(1-2): 63-74.

GONZALEZ J.R., PLAZA L.F., and VASYUNKINA O. From Black-Scholes to Hamilton-Jacobi. Contemporary Engineering Sciences, 2018, 11(90): 4455-4463.

EL HADDAD E.M. Viscosity solutions of Hamilton-Jacobi equations in infinite dimension. Stationary case. Mathematical Publications of the Department of Mathematics of the Autonomous University of Barcelona, 1995, 39: 173-185.

HULL J.C. Options, Futures and Other Derivatives. 6th ed. Prentice Hall, 2006.

IBRAGIMOV N.H., and GAZIZOV R.K. Lie Symmetry analysis of differential equations in finance. Nonlinear Dynamics, 1998, 17: 387-407.

PLAZA L.F. Analytical-numerical study of the differential equation underlying the Black-Scholes model. Master's thesis. Technological University of Pereira, 2009.

RUND H. The Hamilton-Jacobi theory in the calculus of variations: Its role in mathematics and physics. D. Van Nostrand Company Ltd., 1966.

ZHANG H., LIU F., TURNER I., and YANG Q. Numerical solution of the time fractional Black-Scholes model governing European options. Computers & Mathematics with Applications, 2016, 71: 1772-1783.

PODLUBNY I. Fractional Differential Equations. Academic Press, 1999.

SAADATMANDI A., and DEHGHAN M.A. New operational matrix for solving fractional-order differential equations. Computers & Mathematics with Applications, 2010, 59, 1326-1336.

MITTAG-LEFFLER G.M. On the new function Eα(x). Reports of the Academy of Sciences, 1903, 137: 554-558.

OWOYEMI A., SUMIATI I., RUSYAMAN E., and FIRMAN S. Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation. International Journal of Quantitative Research and Modeling, 2020, 1(4): 194-207.

EDEKI S.O., RAJARAMA J., CHAKRAVERTY S., and BALEANU D. Coupled transform method for time-space fractional Black-Scholes option pricing model. Alexandria Engineering Journal, 2020, 59: 3239-3246.

TENORIO Q.A. Study of integral transforms as a method for solving fractional differential equations and their applications. Master's thesis. Technological University of Pereira, 2023.


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