Multi-asset Black–Scholes model as a variable second class constrained dynamical system

https://doi.org/10.1016/j.physa.2016.03.063Get rights and content

Highlights

  • Multiasset Black–Scholes model.

  • Quantum and Classical methods

  • Dirac method.

  • Propagators and constrained path integrals.

Abstract

In this paper, we study the multi-asset Black–Scholes model from a structural point of view. For this, we interpret the multi-asset Black–Scholes equation as a multidimensional Schrödinger one particle equation. The analysis of the classical Hamiltonian and Lagrangian mechanics associated with this quantum model implies that, in this system, the canonical momentums cannot always be written in terms of the velocities. This feature is a typical characteristic of the constrained system that appears in the high-energy physics. To study this model in the proper form, one must apply Dirac’s method for constrained systems. The results of the Dirac’s analysis indicate that in the correlation parameters space of the multi-assets model, there exists a surface (called the Kummer surface ΣK, where the determinant of the correlation matrix is null) on which the constraint number can vary. We study in detail the cases with N=2 and N=3 assets. For these cases, we calculate the propagator of the multi-asset Black–Scholes equation and show that inside the Kummer ΣK surface the propagator is well defined, but outside ΣK the propagator diverges and the option price is not well defined. On ΣK the propagator is obtained as a constrained path integral and their form depends on which region of the Kummer surface the correlation parameters lie. Thus, the multi-asset Black–Scholes model is an example of a variable constrained dynamical system, and it is a new and beautiful property that had not been previously observed.

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 N>3 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 ρij to take values not only between 1ρij1, 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 ρij 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 ΣK) where the determinant of the correlation matrix ρ is zero and where constraints are generated. Thus, the Kummer surface ΣK 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 N 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 ΣK 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 N 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 ρij=±1.

Part 7 reviews the Kummer surface and its properties shortly.

In Section  8 we examine the case of N=2 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 ΣK.

The same detailed study is done in the case N=3 assets in Section  9, where the geometric structure is a little more complex.

Section  10 resumes some characteristics of the general case of N assets and gives the form of the propagator for an arbitrary sector of NB 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 N underlying assets. Let Si be the price processes for the assets; i=1N where each asset satisfies the usual dynamic dSi=αiSidτ+σiSidWii=1N and the N Wiener processes Wi are correlated according to dWidWj=ρijdτ where ρ is the symmetric matrix (ρij)=(1ρ12ρ13ρ14ρ1Nρ121ρ23ρ24ρ2Nρ1Nρ2Nρ3Nρ4N1) with 1ρij1i,j=1N. So we have dSidSj=σiσjSiSjρijdτ.

If the price process for the option is Π=Π(S1,S2,Sn,τ), the value V 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 U(x), the Hamiltonian operator reads Ȟ=12mP̌x2+U(x)=ħ22m2x2+U(x) where P̌x=iħx is the momentum operator. The wave function at time t is (given that the wave function at t=0 is Φ0) Φ(x,t)=eiħȞtΦ0(x) which can be written as a convolution according to Φ(x,t)=K(x,t|x0)Φ0(x)dx where the propagator K 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 A[x,u]=t0t1F(x,u,t)dt where x represents a state variable (for example, production of a certain article) and u is a control variable (such as the marketing cost). The state variable must satisfy the market dynamic ẋ=f(x,u,t). 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 N dimensional one particle Euclidean Schrödinger equation, are developed. If one makes the change of variables xi=ln(Si)(r12σi2)τ in (14), one can map this equation to Πτ+12i,jσiσjρij2ΠxixjrΠ=0. At least if one defines Ψ as Π(x,τ)=er(Tτ)Ψ(x,τ) then Ψ satisfies the equation Ψτ+12i,jσiσjρij2Ψxixj=0. Now, by defining the variables χi=xiσi the above equation can be written as Ψτ+12i,jρij2Ψχiχj

Classical equations of motion

If one takes the classical limit in (132), the Hamiltonian operator Ȟ goes to the classical Hamiltonian function H(χ,Pχ)=12i,j=1NρijPχiPχj and the classical Euclidean Hamiltonian equations are read (see Ref.  [26] for details) iχ̇i={χi,H}=HPχiiṖχi={Pχi,H}=Hχi. For the classical Hamiltonian (133), Eqs. (134), (135) give the following equations iχ̇k=j=1NρkjPχjk=1NiṖχk=0k=1N. These systems of equations can be readily integrated to give χk=(j=1NρkjCj)tk=1NPχk=Ckk=1N where Ck 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 ρ matrix is zero plays a fundamental role in the model. In mathematical literature this region is called the Kummer surface ΣK   [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 2×2 case

Here the two assets case will be analyzed. The correlation matrix is ρ=(1ρ12ρ121)=(1xx1). In this case the vector a=(x,0,0,0) is one dimensional. For this parameterization, the determinant of the ρ matrix is det(ρ)=1x2. This determinant becomes zero when x=±1. Then the Kummer surface ΣK consists of a discrete set of points: ΣK={1,1}. The diagonal form of the ρ is D=(λ1(x)00λ2(x))=(1+|x|001|x|).Fig. 1 shows the eigenvalues λ1 and λ2 as a function of x. Note that λ1 is always positive and

The geometry of the 3×3 case

Now, the three assets case will be analyzed in detail. In fact, the ρ matrix is (ρij)=(1ρ12ρ13ρ121ρ23ρ13ρ231)=(1xyx1zyz1) where we write the vector a=(ρ12,ρ13,ρ23) as a=(x,y,z). For this parameterization the determinant of the ρ matrix is det(ρ)=2xyzx2y2z2+1. This determinant becomes zero when z=z+(x,y)=xy+x2y2x2y2+1 or z=z(x,y)=xyx2y2x2y2+1. ​Fig. 2 shows the Kummer surface ΣK+ given by z=z+(x,y), the Kummer surface ΣK given by z=z(x,y) and the complete Kummer surface ΣK for the ρ

The higher dimensional case

For the higher dimensional case N>3, the correlation matrix (203) is completely characterized by the M=N(N1)2 dimensional vector (152) that takes values on a hyperdimensional cube of length 2 centering in the origin. In this cube, the (M1) dimensional Kummer surface ΣK is defined by the equation det(1ρ12ρ13ρ14ρ1Nρ121ρ23ρ24ρ2Nρ1Nρ2Nρ3Nρ4N1)=0.

The geometric visualization becomes more difficult, so here we give some brief comments on the extremal vertices of ΣK. 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 N 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 ΣK the canonical momentum, in general, cannot be

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      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.

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