A quantum model of option pricing: When Black–Scholes meets Schrödinger and its semi-classical limit

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Abstract

The Black–Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black–Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black–Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black–Scholes–Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrödinger equation in imaginary time for a particle of mass 1/σ2 with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black–Scholes model represent a particular case. Finally, since the Schrödinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black–Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrödinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing.

Introduction

Since this visionary statement of Kaldor in the early 1970s, many of the branches of economics have moved away from this narrow concept of equilibrium. Examples of this could be found in international economics, endogenous growth theory, game theory, labor economics, environmental economics, among others, see for a review Ref. [4]. The field of industrial organization has summarized this disequilibrium view in economics through a model of the firm, that considers the conjectural-variations approach, see Ref. [5]. This model allows economists to empirically estimate the structure of an industry, or in other words, its level of competition and hence efficiency. In this context, the equilibrium view of perfect markets is now an empirical issue instead of a dogmatic one, and more importantly the perfect market assumption is just one of the potential states of several others, like monopoly and oligopoly.

The main question that arises from this methodological change in economics is why the notion of equilibrium is still dominant in finance, specially regarding the recent financial crisis, where the assumptions of a perfect informed and competitive market were clearly and systematically violated during many years. Indeed, since the 80’s economists have realized that, in a real market, futures contracts are not always traded at the price predicted by the simple no-arbitrage relation. Strong empirical evidence have supported this point many times and in different settings, see for example: Refs. [6], [7], [8] among others. However, economists have tended to develop several alternative explanations for the variability of the arbitrage, such as: differential tax treatment for spots and futures [9], and marking-to-market requirements for futures, [10]. It was also noted that there are certain factors that influence the arbitrage strategies and slow down the market’s reaction on the arbitrage. The factors include constrained capital requirements [11], position limits, and transaction costs [12].

In this paper, we are not interested in explaining why these market imperfections are produced, but which are the effects of arbitrage on the option pricing dynamics. We know that the Black–Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black–Scholes model is violated, implying a non-risk-free portfolio. From Ref. [1] we know that an arbitrage environment is a necessary condition to embedding the Black–Scholes option pricing model in a more general quantum physics setting.

In this context, the main aim of this paper is to propose a new Black–Scholes–Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Ref. [2]. Thus, we will derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new model could be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black–Scholes model represents a particular case. The second aim of this research is to apply a semiclassical approximation of our Black–Scholes–Schrödinger model. Finally, given our semiclassical solution, we will test several arbitrage bubble functional forms, or in physical terms different potentials.

This paper is structured as follow. First, our disequilibrium (arbitrage) model is presented and discussed in the light of quantum physics. Second, the new Black–Scholes–Schrödinger model is developed and interpreted as a Schrödinger equation in imaginary time for a particle with a wave function in an external time dependent field force generated by the potential. This new model has, as particular case, the perfect market equilibrium of the Black–Scholes model. Third, since the Schrödinger equation is in place, we will apply semiclassical methods of common use in theoretical physics to find an approximate analytical solution of the Black–Scholes equation in the presence of market imperfections, as in the case of an arbitrage bubble. From this approximation, it is quite obvious that the perfect market equilibrium state postulated by the Black–Scholes model represents a particular case of a more general quantum model. As in the case of industrial organization mentioned above, we will have a parameter that will allow us to model the size (importance) of the disequilibrium in the market. Finally, some numerical illustrations of the potential of this Schrödinger equation analogy, the semiclassical approximations are compared for different arbitrage bubble forms (step, linear and parabolic) and with the exact solution of our general quantum model of option pricing.

Section snippets

The basic disequilibrium financial model1

The Black–Scholes (B–S) model gives the dynamic of the option prices in financial markets. This model has a long life in finance and has been widely used since the seventies. In its usual arbitrage free version, the B–S equation is given by πt+12σ2S22πS2+r(SπSπ)=0 where π=π(S,t) represents the option price as a function of the underlying asset price S and time t. Changing coordinates, as we will see later, the B–S equation can be mapped into the heat equation πt+12σ22πx2=0.

By doing a

The Black–Scholes model and the Schrödinger equation revisited

There has been only a few papers on the issue of applying more general elements of quantum physics on finance, specifically in applying a more general version of the Schrödinger differential equation. For some early and very important attempts see Refs. [20], [21], which primarily through information and uncertainty issues trying to obtain a Black–Scholes–Schrödinger model. It is important to mention also Ref. [8], that used Gauge theory to explain non-equilibrium pricing, deriving a

The Black–Scholes–Schrödinger semiclassical limit

Semiclassical methods have been used to find approximate solutions of the Schrödinger equation in different areas of theoretical physics, such as nuclear physics [22], quantum gravity [23], chemical reactions [24], quantum field theory [25] and path integrals [26]. When the system has interactions, the semiclassical approach gives an approximate solution for the wave function of the system, while for the free interaction case, the semiclassical approximation can give exact results [27]. In this

Some numerical illustrations

In order to explore the behavior of the semiclassical solution we make an analysis for three different bubble forms f(t): step, linear and parabolic functions as shown in Fig. 1.

We use as contract function Φ(S) those of a binary put Φ(S)={00 < S < K1K < S for which the pure BS solution πBS(S,t) is given by πBS(S,t)=er(Tt)[1N(d2(S,t))] where d2(S,t)=lnSK+(rσ22)(Tt)σ(Tt) and N(x) is the normal distribution function N(0,1).

Conclusions and further research

We have developed an interacting model for option pricing that generalizes the usual Black–Scholes formulation to include a more general case of quantum interactions, defined by arbitrage possibilities, and triggered by market imperfections. Specifically, we have proposed a new Black–Scholes–Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. [2].

Indeed, this new quantum model of option pricing incorporates arbitrage as an external time

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