Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time. Application to Merton's optimal portfolio
By Guy Jumarie. Computers and Mathematics with Applications, 2010
325 citations recorded by the source corpus.
Abstract By using the new fractional Taylor’s series of fractional order f ( x + h ) = E α ( h α D x α ) f ( x ) where E α ( . ) denotes the Mittag–Leffler function, and D x α is the so-called modified Riemann–Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration, one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a (usual) Gaussian white noise. One can then derive two new families of fractional Black–Scholes equations, and one shows how one can obtain their solutions. Merton’s optimal portfolio is once more considered and some new results are contributed, with respect to the modeling on one hand, and to the solution on the other hand. Finally, one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics.
Subfields: Economics and Econometrics, Finance, Modeling and Simulation
View Paper: https://www.semanticscholar.org/paper/72e6790465ac0c74a80e36df5b75c53d2e7e3543
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Citations in the corpus
- Fractional Order Theory of Thermoelastic Diffusion (2011, methodology): “Ezzat [ 36 , 37 ] established a new model of fractional heat conduction equation by using the new Taylor series expansion of time-fractional order which developed by Jumarie [ 38 ].”
- Generalized thermoelasticity with memory-dependent derivatives involving two temperatures (2016, methodology): “So that, ∣∣Dω f (x, t )∣∣ ≤ ∣∣∣∣∂ f (x, t )∂t ∣∣∣∣ = ∣∣∣∣ limω→0 f (x, t + ω) − f (x, t )ω ∣∣∣∣ Ezzat [19] established a new model of fractional heat conduction equation by using the new Taylor series expansion of time-fractional order, developed by Jumarie [20] as: qi + τ α o α!”
- Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method (2013, methodology): “Jumarie [12, 13] derived the time- and space-fractional Black-Scholes equations and gave optimal fractional Merton’s portfolio.”