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Mathematical Finance
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Mathematical Finance

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 August 2017) | Viewed by 16715

Special Issue Editor

Prof. Dr. Indranil SenGupta

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, Hunter College, City University of New York (CUNY), NY 10065, USA
Interests: mathematical finance; stochastic processes; machine learning, deep learning, and neural network; information theory

Special Issue Information

Dear Colleagues,

Mathematical modeling in finance has continued to enjoy a strong interaction with research in stochastic processes. This interaction has gone both ways: mathematical finance has provided a natural set of applications for the vast panoply of tools and results developed in the theory of stochastic processes, and rekindled interest in some portions of this theory, such as expansion of filtrations, martingale representation theory, extensions of the Malliavin calculus, Levy processes, and their connection with partial integro-differential equations. However, mathematical finance has also been a source of new research questions and challenges that have generated new motivations and an impetus for research in stochastic processes. Examples may be found in the theory of Backward Stochastic differential equations, the theory of nonlinear filtration-consistent expectations and the nascent theory of optimal martingale transport, in which research agendas have been strongly motivated, if not directly inspired, by mathematical finance. The goal of this Special Issue is to explore such interactions between stochastic processes and mathematical finance.

Dr. Indranil SenGupta
Guest Editor

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Keywords

  • Stochastic processes
  • Mathematical finance
  • Integro-differential equations
  • Option pricing and stochastic volatility

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Published Papers (3 papers)

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Research

14 pages, 794 KiB  
Article
Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics
by Andronikos Paliathanasis, Richard M. Morris and Peter G. L. Leach
Mathematics 2016, 4(2), 34; https://doi.org/10.3390/math4020034 - 13 May 2016
Cited by 4 | Viewed by 4095
Abstract
We analyse two classes of ( 1 + 2 ) evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study [...] Read more.
We analyse two classes of ( 1 + 2 ) evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the ( 1 + 2 ) Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a ( 1 + 1 ) equation, the resulting equation is of maximal symmetry and so equivalent to the ( 1 + 1 ) Classical Heat Equation. Full article
(This article belongs to the Special Issue Mathematical Finance)
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14 pages, 832 KiB  
Article
Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility
by Andronikos Paliathanasis, K. Krishnakumar, K.M. Tamizhmani and Peter G.L. Leach
Mathematics 2016, 4(2), 28; https://doi.org/10.3390/math4020028 - 3 May 2016
Cited by 15 | Viewed by 6048
Abstract
We perform a classification of the Lie point symmetries for the Black-Scholes-Merton Model for European options with stochastic volatility, σ, in which the last is defined by a stochastic differential equation with an Orstein-Uhlenbeck term. In this model, the value of the [...] Read more.
We perform a classification of the Lie point symmetries for the Black-Scholes-Merton Model for European options with stochastic volatility, σ, in which the last is defined by a stochastic differential equation with an Orstein-Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, S, and a new variable, y. We find that for arbitrary functional form of the volatility, σ ( y ) , the (1 + 2) evolution equation always admits two Lie point symmetries in addition to the automatic linear symmetry and the infinite number of solution symmetries. However, when σ ( y ) = σ 0 and as the price of the option depends upon the second Brownian motion in which the volatility is defined, the (1 + 2) evolution is not reduced to the Black-Scholes-Merton Equation, the model admits five Lie point symmetries in addition to the linear symmetry and the infinite number of solution symmetries. We apply the zeroth-order invariants of the Lie symmetries and we reduce the (1 + 2) evolution equation to a linear second-order ordinary differential equation. Finally, we study two models of special interest, the Heston model and the Stein-Stein model. Full article
(This article belongs to the Special Issue Mathematical Finance)
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18 pages, 832 KiB  
Article
Barrier Option Under Lévy Model : A PIDE and Mellin Transform Approach
by Sudip Ratan Chandra and Diganta Mukherjee
Mathematics 2016, 4(1), 2; https://doi.org/10.3390/math4010002 - 4 Jan 2016
Cited by 2 | Viewed by 5445
Abstract
We propose a stochastic model to develop a partial integro-differential equation (PIDE) for pricing and pricing expression for fixed type single Barrier options based on the Itô-Lévy calculus with the help of Mellin transform. The stock price is driven by a class of [...] Read more.
We propose a stochastic model to develop a partial integro-differential equation (PIDE) for pricing and pricing expression for fixed type single Barrier options based on the Itô-Lévy calculus with the help of Mellin transform. The stock price is driven by a class of infinite activity Lévy processes leading to the market inherently incomplete, and dynamic hedging is no longer risk free. We first develop a PIDE for fixed type Barrier options, and apply the Mellin transform to derive a pricing expression. Our main contribution is to develop a PIDE with its closed form pricing expression for the contract. The procedure is easy to implement for all class of Lévy processes numerically. Finally, the algorithm for computing numerically is presented with results for a set of Lévy processes. Full article
(This article belongs to the Special Issue Mathematical Finance)
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