Distributed Parameter Systems Control and Its Applications to Financial Engineering

2018 ◽  
pp. 15-35
Author(s):  
Gerasimos Rigatos ◽  
Pierluigi Siano
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Option Price ◽  
Distributed Parameter ◽  
Specific Reference ◽  
Differential Flatness ◽  
Differential Flatness Theory ◽  
Financial Model ◽  
Performance Indexes ◽  
Black Scholes

The chapter analyzes differential flatness theory for the control of single asset and multi-asset option price dynamics, described by PDE models. Through these control methods, stabilization of distributed parameter (PDE modelled) financial systems is achieved and convergence to specific financial performance indexes is made possible. The main financial model used in the chapter is the Black-Scholes PDE. By applying semi-discretization and a finite differences scheme the single-asset (equivalently multi-asset) Black-Scholes PDE is transformed into a state-space model consisting of ordinary nonlinear differential equations. For this set of differential equations it is shown that differential flatness properties hold. This enables to solve the associated control problem and to stabilize the options' dynamics. By showing the feasibility of control of the single-asset (equivalently multi-asset) Black-Scholes PDE it is proven that through selected purchases and sales during the trading procedure, the price of options can be made to converge and stabilize at specific reference values.

Distributed Parameter Systems Control and Its Applications to Financial Engineering

2019 ◽  
pp. 17-45
Author(s):  
Gerasimos Rigatos ◽  
Pierluigi Siano
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
State Space Model ◽  
Option Price ◽  
Distributed Parameter ◽  
Specific Reference ◽  
Differential Flatness ◽  
Differential Flatness Theory ◽  
Financial Model ◽  
Black Scholes

The chapter analyzes differential flatness theory for the control of single asset and multi-asset option price dynamics, described by PDE models. Through these control methods, stabilization of distributed parameter (PDE modelled) financial systems is achieved and convergence to specific financial performance indices are made possible. The main financial model used in the chapter is the Black-Scholes PDE. By applying semi-discretization and a finite differences scheme the single-asset (equivalently multi-asset) Black-Scholes PDE is transformed into a state-space model consisting of ordinary nonlinear differential equations. For this set of differential equations, it is shown that differential flatness properties hold. This enables one to solve the associated control problem and to stabilize the options' dynamics. By showing the feasibility of control of the single-asset (equivalently multi-asset) Black-Scholes PDE, it is proven that through selected purchases and sales during the trading procedure, the price of options can be made to converge and stabilize at specific reference values.

scholarly journals Jump Models with Delay—Option Pricing and Logarithmic Euler–Maruyama Scheme

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1932
Author(s):  
Nishant Agrawal ◽  
Yaozhong Hu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Financial Market ◽  
Fourier Transformation ◽  
Option Price ◽  
European Option ◽  
Transformation Technique ◽  
The Monte Carlo Method ◽  
Black Scholes ◽  
Convergence Of The Scheme

In this paper, we obtain the existence, uniqueness, and positivity of the solution to delayed stochastic differential equations with jumps. This equation is then applied to model the price movement of the risky asset in a financial market and the Black–Scholes formula for the price of European option is obtained together with the hedging portfolios. The option price is evaluated analytically at the last delayed period by using the Fourier transformation technique. However, in general, there is no analytical expression for the option price. To evaluate the price numerically, we then use the Monte-Carlo method. To this end, we need to simulate the delayed stochastic differential equations with jumps. We propose a logarithmic Euler–Maruyama scheme to approximate the equation and prove that all the approximations remain positive and the rate of convergence of the scheme is proved to be 0.5.

scholarly journals Radial Basis Function in Nonlinear Black-Scholes Option Pricing Equation with Transaction Cost

2019 ◽  
pp. 1-11
Author(s):  
Godwin Onwona-Agyeman ◽  
Francis T. Oduro
Keyword(s):  
Differential Equations ◽  
Radial Basis Function ◽  
Basis Function ◽  
Numerical Technique ◽  
Option Price ◽  
Meshfree Approximation ◽  
Black Scholes Model ◽  
Black Scholes ◽  
The World ◽  
Radial Basis

Differential equations play significant role in the world of finance since most problems in these areas are modeled by differential equations. Majority of these problems are sometimes nonlinear and are normally solved by the use of numerical methods. This work takes a critical look at Nonlinear Black-Scholes model with special reference to the model by Guy Barles and Halil Mete Soner. The resulting model is a nonlinear Black-Scholes equation in which the variable volatility is a function of the second derivative of the option price. The nonlinear equation is solved by a special class of numerical technique, called, the meshfree approximation using radial basis function. The numerical results are presented in diagrams and tables.

One Example of Comparison between the Efficiency of Two Approaches to Differential Equations Computational Solution

2018 ◽  
Vol 931 ◽  
pp. 170-173
Author(s):  
Nikolay G. Bandurin ◽  
Sergey Y. Kalashnikov
Keyword(s):  
Differential Equations ◽  
Difference Scheme ◽  
Ordinary Differential Equations ◽  
Nonlinear Differential Equations ◽  
Error Reduction ◽  
Specific Reference ◽  
Accuracy Order ◽  
Computational Solution

In this work we present main correlations of ordinary differential equations computational solution, based on the previously suggested interpolator procedure which differs from the known approaches with the application not to the sought-for function but to its derivative. With specific reference we show that this method of ordinary nonlinear differential equations system solving provides an error reduction by a huge ratio in comparison to the three-point difference scheme of the 5thaccuracy order.

Feedback control of the multi-asset Black–Scholes PDE using differential flatness theory

2016 ◽  
Vol 03 (02) ◽  
pp. 1650008 ◽  
Author(s):  
G. Rigatos ◽  
P. Siano
Keyword(s):  
Feedback Control ◽  
Differential Equations ◽  
State Space ◽  
State Space Model ◽  
The State ◽  
Differential Flatness ◽  
Control Input ◽  
Space Model ◽  
Virtual Control ◽  
Black Scholes

A method for feedback control of the multi-asset Black–Scholes PDE is developed. By applying semi-discretization and a finite differences scheme the multi-asset Black–Scholes PDE is transformed into a state-space model consisting of ordinary nonlinear differential equations. For this set of differential equations it is shown that differential flatness properties hold. This enables to solve the associated control problem and to succeed stabilization of the options’ dynamics. It is shown that the previous procedure results into a set of nonlinear ordinary differential equations (ODEs) and to an associated state equations model. For the local subsystems, into which a Black–Scholes PDE is decomposed, it becomes possible to apply boundary-based feedback control. The controller design proceeds by showing that the state-space model of the Black–Scholes PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the multi-asset Black–Scholes PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the Black–Scholes PDE system so as to assure that all its state variables will converge to the desirable setpoints.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

scholarly journals Solving nonlinear differential equations with differentiable quantum circuits

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
Oleksandr Kyriienko ◽  
Annie E. Paine ◽  
Vincent E. Elfving
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Quantum Circuits
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