This paper is concerned with the numerical calculation of European call stock option value using a nonlinear Schrödinger rather than Black-Scholes partial differential equation as the option pricing model. A nonlinear Schrödinger option pricing model is assumed to contain nonvanishing external potential energy terms.
Formulated in the 1970s, the Black-Scholes option pricing model is the foundation of the area of classical financial derivative pricing . This model is based on assumption that a stock price S, 0 ≤S < ∞, follows in time t, 0 ≤t < T, where T > 0 is a real number denoting the stock option maturity time, a geometric Brownian motion  with a drift µ, and a volatility σ given by the stochastic differential equation with the standard Wiener process W (1)
In mathematical finance the Black-Scholes partial differential equation is usually derived from the stochastic differential equation using the Itô lemma . This linear parabolic equation governing time-evolution of the market value u = u(t, S) of a stock option with the stock asset price S has the form (2)
where r is the short term risk free interest rate. The equation is completed by a suitable boundary conditions. For a European call option this boundary condition takes the form (3)
where K > 0 is a given strike price. In the framework of the economic approach the Black-Scholes option pricing theory is based on the efficient market hypothesis . Moreover it is assumed that the option price depends on the stock price which in turn is a random variable depending on time. For details concerning assumptions of Black-Scholes model see . Note that in the area of statistical physics equation (2) is derived from the stochastic differential equation (1) using the Stratonovich interpretation . Moreover, equation (1) resembles the backward Fokker-Planck equation describing the time evolution of the probability density function for a position occupied by a particle . This equation is obtained from the stochastic differential equation using the Kolmogorov probability approach .
Analytical investigations as well as market applications of the classical Black-Scholes model (1) indicate that it may be not capable of providing option price data of real markets, especially when market conditions are different than the model’s assumptions . Therefore different alternative option pricing models are being developed, such as stochastic volatility models , or models based on Lévy processes  aiming to follow large jumps of the stock price and leading computationally to solution of parabolic integro-differentional equations with singular kernels , as well as fractional or stochastic differential equations . Note that due to their complexity option pricing problems are also challenging problems in the field of computational mathematics or physics . The first computational option pricing methods included binomial tree methods or Monte Carlo methods . Although these methods are in many cases robust, it appears that they may not provide sufficiently accurate results and they may be inefficient in their basic forms. Since time evolution of option market values is governed by partial differential equations or variational inequalities, finite difference or finite element methods are used  to solve these equations. For numerical solutions of stochastic differential equations see .
A group of alternative option pricing models has been also developed in the framework of statistical physics. In the 1990s physicists turned their attention to economics and social sciences, and particularly financial economics . This interest has given rise to econophysics, i.e., a quantitative approach to social and economic phenomena using models and concepts coming mainly from statistical physics [18-20]. Option pricing problems are also considered in the framework of econophysics. Using the quantum probability formalism , the probability density function being a solution to a time dependent linear Schrödinger equation and corresponding to the market value of a stock option satisfying equation (1) rather than this market value itself may be considered. The linear Schrödinger equation governs the evolution of the complex-valued wave ψ function for which its square | ψ |2 defines the probability density function. The existence results for different types of linear Schrödinger equations can be found in book . Stock options pricing models based on linear Schrödinger equations and their relation to Black-Scholes models are reported in many papers [23-29]. Among others in the author’s previous paper , the European call option price based on the linear Schrödinger equation has been calculated. The solution to this equation has been found using a Fourier method, i.e., the option price is given by the following function (4) (5)
where k > 0 is a real constant reflecting particle expected energy and the parameter g is a constant obtained by data calibration. The obtained results indicated that the Schrödinger equation used for the option pricing needs to have a more complicated form than linear, especially the potential energy that describes the relation between particle and the entourage. Nevertheless, if the described Schrödinger equation varies from real form, it gives additional information about market forecasting, especially about trader leverage for the market. One of the reasons that is not possible to forecast stock or option prices on the basis of historical data is the influence of measurement impact (trader or market participant) on the measured object (stock, option). However, the general model can predict option prices more precisely.
Recently, based on an adaptive market hypothesis as well as Elliott wave market theory, a more general option pricing model based on a nonlinear Schrödinger equation has been proposed in [30, 31]. In  this model has been solved numerically using a domain decomposition method. Therefore, in this paper the nonlinear Schrödinger equation  is used to price European call options on index WIG20 listed on the Warsaw Stock Exchange (WSE). The model used in this paper contains an additional quantum potential as well as an internal potential rather than only an internal potential as in [30,31]. The nonlinear boundary value problem is solved numerically using a Runge-Kutta method. The obtained numerical results are discussed and compared to solutions found using a linear model. The obtained results indicate that the developed approach provides information about critical underlying asset price.
2 Financial optiom pricing with Schrödinger equation
As was shown in , the Black-Scholes equation can be replaced by a linear Schrödinger equation. However the main difference between the Black-Scholes and Schrödinger equations is that the Black-Scholes equation is aligned to the Euclidean space. In contrast, Schrödinger equation exists in Hilbert (complex) space. In general, the Black-Scholes equation can be written as a Wick rotated Schrödinger one. In this paper, an option pricing model based on the real part of the Schrödinger equation will be proposed. As a motivation of this approach, we will compare the Black-Scholes numerical solution with the real and imaginary parts of a linear and a nonlinear Schrödinger solution.
2.1 Black-Scholes equation numerical solution
Using Wolfram Mathematica, numerical solution (by method of lines) for the Black-Scholes (2) equation will be shown. We assumed that expiry time T = 1, volatility σ = 0.3, risk free interest rate r = 0.05, and strike price K = 10. The same values for the mentioned parameters will be used for the Schrödinger equation solutions. The boundary condition is described in (3) and u(1000, t) = 1000-K, u(0.001, t) = 0. Solution is shown in Figure 1.
2.2 Schrödinger equation numerical solution
In this section, numerical solutions for the Schrödinger equation will be showen. The boundary and initial conditions used for linear and nonlinear Schrödinger equation are: ψ(S, T) = max(S - K, 0), ψ(1000, t) = 1000 -K, ψ(0.001, t) = 0.
The linear Schrödinger equation for a free particle is given by the form below: (6)
Solutions for linear equation (6) are presented in Figure 2 (the real part) and in Figure 3 (the imaginary part) respectively. We see that only the real part of the solution of the linear Schrödinger equation Figure 2 has a similar form to the solution of the Black-Scholes equation presented in Figure 1. Similar computations have been made for a nonlinear Schrödinger equation (described in next section) and are given below:
only the real part of the solution of the Schrödinger equation (Figure 4) has a similar form as the solution of the Black-Scholes equation presented in Figure 1. The above numerical experiments proved that option pricing with Schrödinger equation can be obtained, but by only considering the real part of the solution.
3 Nonlinear Schrödinger equation with vanishing external potential as option pricing model
In quantum physics a nonlinear Schrödinger equation  describes a Bose-Einstein condensate , i.e., two identical quantum particles having the same state. Based on relations between the linear Schrödinger and the Black-Scholes partial differential equations, as well as to satisfy both efficient and behavioral markets and their complexity adaptive wave-form, a nonlinear and stochastic option pricing model based on a nonlinear Schrödinger equation has been proposed. This equation defining the option price wave function ψ = ψ(S, t), whose absolute square | ψ(S, t) |2 is the probability density function for the option price, in terms of the stock price and time has the form  (8)
where stands for the total potential energy with V representing the external potential. The Landau coefficient β = β(r, w) is interpreted as an adaptive market potential depending on interest rate r and/or control parameters w. First we shall consider the equation assuming no external potential, i.e., V(S) = 0.
For the existence results for nonlinear Schrödinger boundary value problems see .
The equation (8) will be solved exactly using the power series expansion method of Jacobi elliptic functions. Note that for low interest rate r, β(r) << 1, equation (8) can be approximated by the linear Schrödinger equation. Similarly we shall look for the solution to equation (8) in the form (9)
where ϕ(ξ) ∈ R is an unknown function depending on ξ = S - σkt, and k, ω ∈ R are constant parameters. These parameters in physics terms are interpreted as the wave number and circular frequency, respectively. Using (9) we can calculate the derivatives and the terms of equation (8) (10) (11) (12)
where the symbol Im defines the imaginary part of a complex number. k, S, ω, t are real, therefore we have (13)
As ϕ(S) > 0, this results in (14)
with boundary conditions: ϕ(ξ = 0) = ϕ0 and ϕ′(ξ = 0) = Recall that the equation (16) is and ordinary differential equation describing a nonlinear oscillator . The solution to this equation is supposed to have the form : (17)
where a0and a1 are unknown constants and sn(ξ) = sn(ξ, m) is defined  as the Jacobi elliptic function: (18)
with elliptic modulus m ∈[0,1]. Recall a few properties of Jacobi elliptic functions : (19) (20) (21) (22)
The function presented in equation (27) also contains an imaginary part. Using the splitting formula of any complex function into real and imaginary parts, i.e., exp ix = cos x+ i sin x and considering that β > 0, we see that the real part of equation (27) is given by: (29)
and denotes the real part of equation (28): (30)
4 Nonlinear Schrödinger model calibration
Let us calibrate equation (29) based on market data. These data are taken from the Warsaw Stock Exchange (WSE) listings of the option denoted as OW20F3280. This European call option has been listed in 2013. It is based on the index WIG20 assembling the stocks of the biggest companies listed on the WSE. The maturity date of this option was June 27,2013. Its strike price was equal to 2800 points.
Calibration was performed using the Levenberg-Marquardt  algorithm that was implemented using a Python application. It was included in the scipy.optimize package. In mathematics and computing, the Levenberg-Marquardt algorithm provides a numerical solution to the problem of minimizing a general non-linear function over the space of parameters of this function. These minimization problems arise especially in least squares curve fitting and non-linear programming. The Levenberg-Marquardt interpolates between the Gauss-Newton algorithm  and the method of gradient descent. Levenberg-Marquardt is more efficient than Gauss-Newton. This means that in many cases it finds a solution even if it starts very far from the minimum. On the other hand, for well-behaved functions and reasonable starting parameters, Levenberg-Marquardt is slower than Gauss-Newton. Levenberg-Marquardt can also be considered as Gauss-Newton using a trust region approach. The Pearson correlation coefficient was calculated using the scipy.stats package. The Correlation coefficient is calculated at the basis of estimated function and market data comparison. Charts were performed using the matplotlib library.
4.1 Stock WIG20 price calibration
A Python script has been created to estimate third-order polynomial parameters for approximating WIG20 trade using the Levenberg-Marquardt algorithm. Our calculations are presented in Figure 6. We achieved a correlation coefficient equal to 0.97 when we performed market approximation (WIG20 stock) using a third-order polynomial. A similar dark soliton (30) calibration will be performed. Next, we will calibrate the general solution given by equation (29). Computations will be performed for different m values.
4.2 Option OW20F3280 price calibration
and S(t) is stock price (WIG20), approximated using 3-order polynomial. The same simplification is proposed for function (30): (33)
Calibrating function (33), the dark soliton, with market data gives the results presented in Figure 7. with a correlation coefficient equal to 0.57. Now, we will check if calibration of the equation (31) can give different results than obtained for tha dark soliton. A set of values were created (sequence from 0.01 to 0.99 with 0.01 step for parameter m), and for each value, we have calculated the parameters z, σ and k. The stock price (WIG20) S(t) was approximated using a third-order polynomial. In Figure 8 we have presented computations for four cases. We see that highest correlation was achieved for point c, where the parameter m was equal to 0.69. For this case, the correlation coefficient is equal to 0.68 and is higher than it was for the dark soliton solution.
4.3 Nonlinear and linear model comparison
We will compare the linear and nonlinear Schrödinger models on the basis of market data calibration. The time range has been reduced for the option to check if we can get better correlation than was obtained when calibration was performed for the full time range. For the linear model, we have obtained a correlation coefficient equal to 0.91, presented in Figure 9. For the non-linear model, we have performed the calibration presented in Figure 10. An additional Python script was written to calculate the correlation coefficient on the basis of given parameter m. Coefficients were calculated for parameters m generated from 0.00001 to 0.99999 with a step equal to 0.00005. As the basis of these calculations we have chosen parameter m equal to 0.27455. The Correlation coefficient has increased for the nonlinear equation from 0.68 (full time range) to 0.87 (narrowed time range). As we see, the linear and nonlinear model can produce similar results when they are calculated for a narrower time range. It seems interesting to enhance the nonlinear model with additional external potentials.
5 Nonlinear Schrödinger model with external quantum potential
The lack of larger sets of the market data make it not possible to perform reliable computations. However, the model calibration was performed for a narrower time range and we have obtained satisfactory correlation with market data. This suggests that the considered model can be used for daily/weekly option pricing. If we had more market data to analysis, i.e., daily stock/option prices, then our model would probably forecast option prices more accurately.
Note that until now the simplest case of potential energy in equation (8) was considered, i.e., V(S) = 0 has been set. Additionally, improvement in the model based on the assumption that V(S) = 0 will be considered. In particular, the potential V(S) can be represented by different atomic potentials such as the Yukawa, Coulomb, Hulthen, and harmonic oscillator potentials , . Since the calculation of the analytical solution for nonlinear Schrödinger equation with V(S) = 0 is very complicated we will use numerical methods. Assume that the function ψ(S, t) can be split into (34)
which is called in literature the time independent Gross-Pitaevskii equation, and where the function ϕ(S) describes the ground state of a quantum system of identical bosons. This equation shows the relation between option price and asset price and is important to determine the maximum value of the asset price, i.e., critical underlying asset price, above which a crisis can occur. Recall that the square of the wave function is interpreted in quantum theory as the density probability of finding the particle at location S and time t. Let us write down the mentioned interpretation in the following mathematical way: (36)
which is only dependent on the ϕ(S) function. Equation (35) with function V(S) representing different atomic potentials will be solved numerically using a Runge-Kutta fourth-order method.
5.1 Quantum potentials
The first used potential, called the Yukawa potential, is given by following equation: (37)
where g is a magnitude scaling constant, mp is the mass of the affected particle, ξ is the radial distance to the particle, and k is a scaling constant. The variable ξ is identified with underlying asset price S. This potential reflects the fact that the nuclear strong force is carried by a particle with a mass approximately 200 times bigger than the electron mass .
The Coulomb potential describes the pairwise interaction between charged particles and has the form : (38)
Here e and k denote the elementary charge and a constant, respectively.
The Hulthen potential is a short-range potential which behaves like a Coulomb potential for small values of ξ and decreases exponentially for large values of distance. The Hulthen potential has been used in many branches of physics, such as nuclear, atomic, and solid state physics. It is given by: (39)
where α is the screening parameter and z is a constant which is identified with the atomic number when the potential is used for atomic phenomena.
The quantum oscillator potential is given by: (40)
where k is a constant. A diatomic molecule vibrates like two masses on a spring with a potential energy that depends upon the square of the displacement from equilibrium. The energy levels are quantized at equally spaced values.
5.2 Numerical solution of stationary nonlinear Schrödinger equation
Let us first solve numerically the nonlinear Schrödinger time independent equation. Next the solutions for different external potentials will be computed using a Runge-Kutta fourth-order method. The details of this method can be found in the Appendix. Let us write the second order equation (35) in the standard form (56) of the Runge-Kutta method: (41)
with the assumption that To solve the second order differential equation there is a need to introduce the new function: (42)
The price S, τ(S), and ϕ(S) are going to be discretized into set of n-points. Sn, τn, ϕn are the n-values for the nth point respectively.
The first step is to calculate the k1, k2, k3, k4 parameters to compute the τ(S) function. The right hand side functionfin (56) is defined for this step as equal to: (44)
The value for τn is calculated using: (45)
where τn is the value for function τ calculated at the previous point. The next step is to calculate the k1,k2,k3, k4 parameters to compute the ϕ(S) function. The right hand side function f (56) for this step is defined as: (46)
The value for ϕn+1 is calculated using: (47)
where ϕn is the value for function ϕ calculated at previous point.
5.3 Numerical results
Let us report on the obtained numerical results. First, a solution is provided for equation (35) with the assumption that V(S) = 0. This step is required to obtain the general shape of the solution and to compare it to solutions obtained for V(S) ≠ 0 potentials. In Figure 11 is presented the solution for the stationary Schrödinger equation with V(S) = 0. There are two parameters that have impact on the solution: the first one is the parameter β, and the second one is µ. It seems that increasing parameter µ increases the period of the ϕ(S) function, and increasing parameter β affects the value of the ϕ(S) function (ϕ(S) raises when β increases).
Another solution was obtained for a Coulomb potential Vc, defined in equation (38), with the assumption that the parameter e = 1. Using this additional potential provides another parameter k that has an effect on the solution ϕ(S). The results are displayed in Figure 12. The period of the ϕ(S) function is increasing when the parameter µ is raising. When the parameter k is raising, then the ϕ(S) function is decreasing.
Further computation is performed for the Yukawa potential Vy, given by (37) and assuming that m = g2 and a = kmp. In Figure 13, the numerical solution for equation (35) is provided. Note that the period of the solution is increasing when parameter µ is growing. The ϕ(S) function is increasing when the parameters g and a are decreasing.
Next, the equation (41) has been solved with a Hulthen potential Vh, given by (39) with the simplification that and The obtained solution is presented in Figure 14. It is observed that the period of the ϕ(S) function is increasing when the parameter µ is raising. Decreasing parameters g and p cause the value of the ϕ(S) function to grow.
When parameter µ is increasing, then the period of the ϕ(S) function is raising. When parameter k is raising then the value of the ϕ(S) function is growing. Note that the impact on the function ϕ(S) satisfying the equation (41) is different for the Coulomb, Yukawa and Hulthen potentials compared with the oscillator potential. The impact of the first three potentials on the Hamiltonian is negative while the impact of the oscillator potential is positive. However all of the potentials presented in Figures 11 – 15 have interesting properties in terms of option valuation. In each case the function ϕ(S) attains a maximum. It may be interpreted as critical value for the option price S. Exceeding this critical asset price, causes the ϕ(S) function to decrease rapidly. This means that probability of finding option for underlying asset price higher than critical one decreases to zero. THe presented results show that external potentials have changed the shape of the time independent, nonlinear Schrödinger equation.
In this paper an analytical option pricing model based on the nonlinear Schrödinger partial differential equation with vanishing external potential has been considered. Comparing the results generated by linear and nonlinear Schrödinger models, it seems that these models can provide similar results for a suitably selected time range. General and special solutions to nonlinear Schrödinger system have been computed and compared, and in addition the correlation between the model and the market data has been verified. Better results were obtained for the general than for the special, i.e., dark soliton, solution.
As a model enhancement, numerical solutions for stationary nonlinear Schrödinger equation with external atomic potentials have been proposed. The obtained results indicate that there exists a critical underlying asset price. Exceeding it causes the time independent nonlinear Schrödinger solution to decrease rapidly. This phenomenon can be associated with crises observed on the market. External potentials are changing the shape of the time independent nonlinear Schrödinger solution. Numerical solutions contribute to understanding this process and can be used in option price forecasting.
The introduced model requires further investigation, especially with respect to potential relation with Lévy processes. The selection of suitable elliptic moduli parameters or, in general, parameters influencing the potential term in the Schrödinger model may be formulated and investigated as optimal control problems for the nonlinear Schrödinger differential equation.
The author would like to thank the anonymous reviewers for constructive remarks which helped to improve the paper as well as Prof. Andrzej Myśliński for his support and inspiration.
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The Runge-Kutta algorithm  is the method that is behind the most of the physics simulations in this paper by letting us solve a differential equation numerically. It is known to be a very accurate and well behaved method for a wide range of problems. We wish to approximate the solution to a first order differential equation given by: (48)
with boundary condition: (49)
Function y(x) is an unknown function of x, which we would like to approximate. Another asumption is that y(x) is the a function of t and y(x) itself. At the boundary condition x0 the corresponding y value is y0. The function f (y(x), x) and the data x0, y0 are given. In general, the very basic numerical method to find solutions for y(x) as defined in equation (48) is called the Euler method and is defined as: (50)
where: yn+1 advances the solution from xn to xn+1 = xn + h, where h is the step size. The formula is unsymmetrical: it advances the solution through an interval h, but uses derivative information only at the beginning of that interval. That means (and this can be verified by expansion in a power series) that the step’s error is only one power of h smaller than the correction, i.e O(h2) added to equation (50). Unfortunately, the Euler method is not very accurate when compared to other methods running at the equivalent step size, and also this method is not very stable. That is why scientists are looking for more accurate and stable methods. One of the ideas to improve the Euler method is to use the midpoint, then use the value of both x and y at that midpoint to compute the real step across the whole interval. This method takes to extremes the idea of correcting the predicted value of the next solution point in the numerical solution. Finally, the solution for y function can be described as: (51)
where: (52) (53) (54) (55)
The parameter k1 is the same quantity as given by the Euler method (the vertical jump from the current point to the next Euler predicted point along with the numerical solution). Parameter k2 is evaluating the function f for with assumption that the y value will be changed by To summarize, the function f is being evaluated at a point that lies halfway between the current point and the Euler-predicted next point. Parameter k3 has a similar formula to k2. Essentially, the f value at this point is another estimation of the slope of the solution at the midpoint of the prediction interval. The y value of the midpoint is predicted already with the k2 value. Parameter k4 evaluates the f function at the xn + h point. The y value is equal to yn + k3 and is an estimation of the y value at the right of the interval. In general, this method is intended to find numerical solutions for first order differential equations. Solving higher order differential equations can be reduced to first order differential equations (assuming that the condition is known): (56)
then let us introduce a new function: (57)
The final step is to solve the equation below: (59)
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Published Online: 2017-05-04