Abstract
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.
TopIntroduction
In several problems of financial engineering, such as options and commodities trading, forecasting of options’ values, estimation of financial distress and credit risk assessment, validation of option pricing models, etc. one comes against Partial Differential Equations (PDEs). Moreover, in problems of control of financial systems where the aim is to stabilize financial processes which are described by PDE models, one has to harness again the complex PDE dynamics through the application of an external input. In the recent years differential flatness theory has emerged as an approach to the control and stabilization of systems described by PDE dynamics (Rudolph, 2003), (Rigatos, 2015). This research work focuses on differential flatness theory for the control and stabilization of single asset and multi-asset option price dynamics, described by PDE models. It is shown how the differential flatness approach achieves, stabilization of distributed parameter financial systems (that is systems modelled by PDEs) and how it enables convergence to specific financial performance indexes (Rigatos, 2014a; Rigatos, 2014b; Rigatos, 2014c; Rigatos, 2015a; Rigatos, 2015b; Rigatos, 2015c).
The Black-Scholes PDE is the principal financial model used in this study. It is demonstrated how with the use of 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 proven that differential flatness properties hold (Rigatos, 2011; Rigatos, 2013; Rigatos, 2015). This permits to arrive at a solution for the associated control problem and to ascertain stabilization of the options’ dynamics. By proving that it is feasible to control the single-asset (equivalently multi-asset) Black-Scholes PDE it is also concluded that through a selected trading policy, the price of options can be made to converge and stabilize at specific reference values.
Key Terms in this Chapter
Pricing: The procedure of defining the price of traded assets, services or commodities. Spot pricing is an on-site agreement about the price of the traded resource based on offer and demand. There can be more elaborated pricing schemes taking place in longer time intervals (trading of financial options or commodities) which apart from offer and demand are also dependent on supporting services and procedures for the trading transaction (e.g. ability to store and transport goods, availability of equipment, credibility of the trading parts etc.).
Distributed Parameter System: A dynamical system that evolves not only in time but also in space. Otherwise stated the system exhibits spatiotemporal dynamics along the time axis and along one or more spatial axes. Systems described by partial differential equations are distributed parameter ones.
Feedback Control: The action of applying an external excitation to a dynamical system which is dependent on the value of the system’s state vector and on the deviation of this state vector from a reference value that is called setpoint.
Black-Scholes PDE: A diffusion partial differential equation which describes the dynamics of option prices. It computes the distribution of the option price as a function of time and an underlying asset variable. It can be dependent on one single asset or on multiple assets.
Options: Financial derivatives which generate a secondary value for traded assets (commodities, goods, services etc.). Options are related to long term agreements about the exploitation of the traded assets so they do not reflect only the spot price of the assets but they also show the ability to accomplish the contract term’s.
Differential Flatness Theory: A primary research direction in the area of nonlinear dynamical systems control. It considers that instead of describing the system’s dynamics through its entire state vector, one can use for this purpose specific algebraic variables which are called flat outputs and which are dependent only on certain elements of the state vector. Differential flatness theory enables to succeed global linearization for complicated nonlinear dynamics and in this manner to solve the associated control and state estimation problems.
Boundary Control: An approach to the control of partial differential equations in which the control action is exerted to the PDE through its boundary conditions. This is different to distributed or pointwise control of PDEs, in which the control action is exerted at several points of the system’s state space.
Asset: A parameter in financial models which denotes a possession or a resource which can be evaluated and traded. Assets can take the form of financial derivatives (such as options and futures) or commodities or the form of equipment or other proprietary holdings.
Lyapunov Function: This is an energy function of the system which depends on quadratic terms of the system’s state vector error. It takes positive values apart from the equilibrium where it becomes zero. A system is stabilized when the associated Lyapunov function becomes zero.
Stability: A property of a dynamical system denoting that the state vector of the system converges to a specific point in the state space, which is called equilibrium or to a bounded region in the state-space which is called domain of attraction and remains there.