Multi-asset Black–Scholes model as a variable second class constrained dynamical system
Introduction
Since 1973, with the initial articles by Black and Scholes [1], and Merton [2], the Black–Scholes (B–S) model is being extensively used in financial engineering to price a derivative on equity. After that, different generalizations of the initial model were made, some of which included the concept of the volatility smile [3], [4], [5], [6], stochastic volatility models [7], [8], [9] and the incorporation of jumps, which gives rise to integrodifferential equations [10] for the price of the option. Another class of generalization that implies the consideration of many assets is called the multi-asset Black–Scholes model [3], [11]. In this case, the option price satisfies a diffusion equation in many variables. The numerical implementation of the solution can be very difficult for assets [12], [13], [14].
On the other hand, path integrals techniques have been applied to study the Black–Scholes model in its different forms [15], [16], [17], [18], [19], [20], [21]. Also, there are developments that try to understand the Black–Scholes equation as a quantum mechanical Schrödinger equation [22], [23], [24]. Finally, the Black–Scholes model, in the stochastic volatility setting, results in being a constrained dynamical system similar to those that appear in high-energy physics [25], [26].
In this paper, we want to study the multi-asset Black–Scholes model from a structural point of view. In that sense, we allow for the correlation parameters to take values not only between , but also outside this region in order to discuss convergence properties.
The principal goal of our study is to show that the multi-asset Black–Scholes model is a constrained dynamical system with a mathematical structure very close to the most important physical theories such as gravitation, electromagnetism, strong or weak force. In these last cases, the theory has a fixed number of constraints. An interesting and beautiful characteristic of the multi-asset Black–Scholes model instead, is that it has a variable constraint number, which depends on the particular region of the Kummer surface the correlation parameters lie. In a sense, the geometry of the Kummer surface completely determines the possible dynamical scenarios of this financial system.
In order to achieve this result, we started by writing the multi-asset Black–Scholes equation as a quantum mechanical Schrödinger equation and then by considering the underlying classical mechanics of the quantum model, it is found that there exists a surface in the correlation parameter space (the so called Kummer surface ) where the determinant of the correlation matrix is zero and where constraints are generated. Thus, the Kummer surface divides the parameter space into three regions: the inside, the Kummer surface itself and the outside part of it. The form propagator of the Black–Scholes equation (which is necessary to price an option on the assets set) depends on which of these regions we evaluate the correlation parameters. In the inside part (where no constraints appear) the usual expression of the propagator is found [3], but outside the propagator becomes divergent. On the Kummer surface, the propagator acquires a variable form, depending on the number of constraints that appear in a specific sector of the Kummer surface. To study the Black–Scholes model in an appropriated form, our analysis rests heavily on the Dirac’s method of constrained systems [27], [28] and on the constrained path integral representation of the quantum propagator [29].
Due to the fact that the Dirac’s method is specifically used in the high-energy physics literature, we have tried to make the paper self-content, so we dedicate some sections to this particular issue in order to:
- (1)
Establish a communication bridge between readers coming from the financial side and physicists.
- (2)
That financial readers appreciate the power of these methods and ideas, and they can apply them in future investigations, generating a valuable multidisciplinary area of research.
Our paper goes as follows:
In Section 2, we review the multi-asset model and obtain the multi-assets Black–Scholes equation.
For readers coming from the financial area the quantum and classical behaviors are resumed in Section 3. Section 3.1 considers the Hamiltonian and space phase point of view, whereas Section 3.2 discusses the Lagrangian approach. Also, we discuss here the relation between these approaches and the way in which constraints appear in the theory.
Section 4 analyses the constrained systems. It starts by considering a simple problem in finance in Section 4.1 and studies its properties in the phase space. In Section 4.2 we give a short review to the Dirac’s method and the corresponding classifications of constraints. Section 4.3 briefly discusses the principal constrained models in physics and emphasize them as a dynamical system with a fixed number of restrictions in the phase space. Section 4.4 presents the usual and Euclidean quantum mechanical framework and its classical counterpart. This section must be obligatory for financial readers to understand the Dirac’s method and the rest of the paper.
Section 5 is where our paper really starts. Here we interpret and write the multi-asset Black–Scholes as a dimensional one particle Euclidean Schrödinger equation.
Section 6 talks about the classical behavior associated with the Euclidean quantum model and the reason that the constraints appear in the high correlation limit case .
Part 7 reviews the Kummer surface and its properties shortly.
In Section 8 we examine the case of assets in detail. First, we study the geometry of the Kummer surface in this case and then we compute the propagator of the bidimensional Black–Scholes equation inside, on and outside the Kummer surface .
The same detailed study is done in the case assets in Section 9, where the geometric structure is a little more complex.
Section 10 resumes some characteristics of the general case of assets and gives the form of the propagator for an arbitrary sector of constraints on the Kummer surface.
Finally, in Section 11 we give the conclusions of our work.
Section snippets
The multi-asset Black–Scholes model
Consider a portfolio consisting of one option and underlying assets. Let be the price processes for the assets; where each asset satisfies the usual dynamic and the Wiener processes are correlated according to where is the symmetric matrix with So we have
If the price process for the option is , the value of the portfolio is given
Hamiltonian quantum and classical mechanics
In physics, the quantum dynamical behavior is defined by the Hamiltonian operator. For the simple case of a non-relativistic one-dimensional particle subjected to external potential , the Hamiltonian operator reads where is the momentum operator. The wave function at time is (given that the wave function at is ) which can be written as a convolution according to where the propagator admits
An example from finance
In finance, it is usual to find dynamical systems which have constraints. For example, in optimal control financial applications (see for example Ref. [30]) we started with a problem that consists in the optimization of a cost functional where represents a state variable (for example, production of a certain article) and is a control variable (such as the marketing cost). The state variable must satisfy the market dynamic The problem consists on how to
The multi-asset Black–Scholes equation as a Schrödinger equation
Here, some transformations, which map the multi-asset option pricing equation in a dimensional one particle Euclidean Schrödinger equation, are developed. If one makes the change of variables in (14), one can map this equation to At least if one defines as then satisfies the equation Now, by defining the variables the above equation can be written as
Classical equations of motion
If one takes the classical limit in (132), the Hamiltonian operator goes to the classical Hamiltonian function and the classical Euclidean Hamiltonian equations are read (see Ref. [26] for details) For the classical Hamiltonian (133), Eqs. (134), (135) give the following equations These systems of equations can be readily integrated to give where are
The Kummer surface
Now it is clear that the origin of constraints in the phase space is associated to the non-invertibility of the correlation matrix. Thus, the region of the correlation parameter space where the determinant of the matrix is zero plays a fundamental role in the model. In mathematical literature this region is called the Kummer surface [33], [34], [35], [36].
To explore the properties of this surface is convenient to parametrize the correlation matrix (3) by its triangular upper (or lower)
The geometry of the case
Here the two assets case will be analyzed. The correlation matrix is In this case the vector is one dimensional. For this parameterization, the determinant of the matrix is This determinant becomes zero when Then the Kummer surface consists of a discrete set of points: The diagonal form of the is Fig. 1 shows the eigenvalues and as a function of . Note that is always positive and
The geometry of the case
Now, the three assets case will be analyzed in detail. In fact, the matrix is where we write the vector as . For this parameterization the determinant of the matrix is This determinant becomes zero when or Fig. 2 shows the Kummer surface given by , the Kummer surface given by and the complete Kummer surface for the
The higher dimensional case
For the higher dimensional case , the correlation matrix (203) is completely characterized by the dimensional vector (152) that takes values on a hyperdimensional cube of length 2 centering in the origin. In this cube, the dimensional Kummer surface is defined by the equation
The geometric visualization becomes more difficult, so here we give some brief comments on the extremal vertices of . Table 2 shows the total
Conclusions and further research
In this paper, we have explored the multi-asset Black–Scholes equation from a structural point of view. By taking a physicists optic, we have interpreted this equation as a dimensional one particle Euclidean Schrödinger equation. We have analyzed this model in a quantum and classical framework. The study of the classical equations of motion, which underlie the quantum multi-assets Black–Scholes model, implies that over the Kummer surface the canonical momentum, in general, cannot be
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2017, Physica A: Statistical Mechanics and its ApplicationsCitation Excerpt :Some developments have also been used to try to understand the Black–Scholes equation as a quantum mechanical Schrödinger equation [8–10]. In the last few years, constrained systems techniques, through Dirac’s method [11,12] have been used to explain some features of stochastic volatility models [13,14] and the multi-asset Black–Scholes equation [15]. In this paper, we apply these same constrained methods to understand (from a physical point of view) a particular issue: the dynamic optimization problem.