In quantum physics a nonlinear Schrödinger equation [33] describes a Bose-Einstein condensate [34], 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 [30]
$$\begin{array}{rl}i\frac{\mathrm{\partial}\psi (S,\phantom{\rule{thinmathspace}{0ex}}t)}{\mathrm{\partial}t}=& -\frac{1}{2}\sigma \frac{{\mathrm{\partial}}^{2}\psi (S,\phantom{\rule{thinmathspace}{0ex}}t)}{\mathrm{\partial}{S}^{2}}\\ & +V(x)\psi (S,\phantom{\rule{thinmathspace}{0ex}}t)+\beta |\psi (S,\phantom{\rule{thinmathspace}{0ex}}t){|}^{2}\psi (S,\phantom{\rule{thinmathspace}{0ex}}t),\end{array}$$(8)

where $i=\sqrt{-1}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}V(S)+\beta |\psi (S,t){|}^{2}$ 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 [35].

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
$$\psi (S,t)=\varphi (\xi )\mathrm{exp}i(kS-\omega t),$$(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)
$$i\frac{\mathrm{\partial}\psi (S,\phantom{\rule{thinmathspace}{0ex}}t)}{\mathrm{\partial}t}={e}^{i(kS-t\omega )}\left(\omega \varphi (\xi )-ik\sigma \frac{\mathrm{\partial}\varphi (\xi )}{\mathrm{\partial}\xi}\right),$$(10)
$$\begin{array}{l}-\frac{1}{2}\sigma \frac{{\mathrm{\partial}}^{2}\psi (S,t)}{\mathrm{\partial}{S}^{2}}=\\ \phantom{\rule{2em}{0ex}}\frac{1}{2}{e}^{i(kS-t\omega )}\sigma \left({k}^{2}\varphi (\xi )-2ik\frac{\mathrm{\partial}\varphi (\xi )}{\mathrm{\partial}\xi}-\frac{{\mathrm{\partial}}^{2}\varphi (\xi )}{\mathrm{\partial}{\xi}^{2}}\right),\end{array}$$(11)
$$\begin{array}{ll}\beta |\psi (S,t){|}^{2}\psi (S,t)& =\\ & {e}^{i(kS-t\omega )-2Im(kS-t\omega )}\beta |\varphi (\xi ){|}^{2}\varphi (\xi ),\end{array}$$(12)

where the symbol *Im* defines the imaginary part of a complex number. *k*, *S*, *ω*, *t* are real, therefore we have
$$Im(kS-t\omega )=0.$$(13)

As *ϕ*(*S*) > 0, this results in
$$|\varphi (\xi ){|}^{2}\varphi (\xi )=\varphi (\xi {)}^{3}.$$(14)

Applying (13) and (14) into (12), we find that:
$$\beta |\psi (S,t){|}^{2}\psi (S,t)={e}^{i(kS-t\omega )}\beta \varphi (\xi {)}^{3}.$$(15)

After inserting (10), (11), and (15) into equation (8), we get the final equation to solve:
$$\frac{{\mathrm{\partial}}^{2}\varphi (\xi )}{\mathrm{\partial}{\xi}^{2}}+(\omega -\frac{1}{2}\sigma {k}^{2})\varphi (\xi )-\beta \varphi (\xi {)}^{3}=0,$$(16)

with boundary conditions: *ϕ*(*ξ* = 0) = *ϕ*_{0} and *ϕ*′(*ξ* = 0) = ${\varphi}_{0}^{1}.$ Recall that the equation (16) is and ordinary differential equation describing a nonlinear oscillator [33]. The solution to this equation is supposed to have the form [30]:
$$\varphi (\xi )={a}_{0}+{a}_{1}sn(\xi ),$$(17)

where *a*_{0}and *a*_{1} are unknown constants and *sn*(*ξ*) = *sn*(*ξ*, *m*) is defined [36] as the Jacobi elliptic function:
$$sn(\xi )=sn(\xi ,m),$$(18)

with elliptic modulus *m* ∈[0,1]. Recall a few properties of Jacobi elliptic functions [36]:
$$\begin{array}{r}sn(\xi ,0)=sin(\xi ),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}sn(\xi ,1)=tanh(\xi ),\end{array}$$(19)
$$\begin{array}{r}cn(\xi ,0)=cos(\xi ),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}dn(\xi ,0)=1,\end{array}$$(20)
$$\begin{array}{r}\frac{d}{d\xi}(sn(\xi ))=cn(\xi )dn(\xi ),\end{array}$$(21)
$$\begin{array}{r}\frac{d}{d\xi}(cn(\xi ))=-sn(\xi )dn(\xi ).\end{array}$$(22)

Using (17), the first and second order derivatives of function (17) are calculated as equal to:
$$\frac{\mathrm{\partial}\varphi (\xi )}{\mathrm{\partial}\xi}={a}_{1}cn(\xi )dn(\xi ),$$(23)

and
$$\begin{array}{rl}\frac{{\mathrm{\partial}}^{2}\varphi (\xi )}{\mathrm{\partial}{\xi}^{2}}=& -{a}_{1}\left(sn(\xi )[1-{m}^{2}s{n}^{2}(\xi )]\right)\\ & +{m}^{2}sn(\xi )[1-s{n}^{2}(\xi )].\end{array}$$(24)

After substituting function (17) into the nonlinear oscillator equation (16) and using formulas (23)-(24) we get:
$$\begin{array}{c}{a}_{0}=0,\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{a}_{1}=\pm \sqrt{\frac{\sigma}{\beta}},\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\omega =\frac{1}{2}(1+{m}^{2}+{k}^{2}),\\ \varphi (\xi )=\pm m\sqrt{\frac{\sigma}{\beta}}sn(\xi ),\phantom{\rule{1em}{0ex}}m\in [0,1),\end{array}$$(25)
$$\begin{array}{r}\varphi (\xi )=\pm \sqrt{\frac{\sigma}{\beta}}tanh(\xi ),\phantom{\rule{1em}{0ex}}m=1.\end{array}$$(26)

Using the functions (25) - (26) we obtain the periodic solution (9) to the equation (8) in the form:
$$\begin{array}{l}\psi (S,t)=\pm m\sqrt{\left(\frac{\sigma}{\beta}\right)}sn(S-\sigma kt)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{exp}i(kS-\frac{1}{2}\sigma t(1+{m}^{2}+{k}^{2})),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}m\in [0,1),\end{array}$$(27)

and
$$\begin{array}{l}{\psi}^{d}(S,t)=\pm \sqrt{\left(\frac{\sigma}{\beta}\right)}tanh(S-\sigma kt)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{exp}i(kS-\frac{1}{2}\sigma (2+{k}^{2})),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}m=1.\end{array}$$(28)

Functions (27) and (28) denote a general solution for *m* ∈ [0,1) and the so-called dark soliton solution for *m =* 1, respectively. We expect the function *ψ*(*S*, *t*) to be real [30].

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:
$$\begin{array}{ll}{\psi}_{r}(S,t)& =\pm m\sqrt{\left(\frac{\sigma}{\beta}\right)}sn(S-\sigma kt)\\ & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{cos}(kS-\frac{1}{2}\sigma t(1+{m}^{2}+{k}^{2})),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}m\in [0,1)\end{array}$$(29)

and ${\psi}_{r}^{d}(S,t)$ denotes the real part of equation (28):
$$\begin{array}{ll}{\psi}_{r}^{d}(S,t)& =\pm \sqrt{\left(\frac{\sigma}{\beta}\right)}tanh(S-\sigma kt)\\ & \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{cos}(kS-\frac{1}{2}\sigma t(2+{k}^{2})),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}m=1.\end{array}$$(30)

Functions (29) and (30) will be calibrated with market data. We will check the correlation between the model that uses the above equations and compare it to the market data.

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