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Adaptive-Wave Alternative for the Black-Scholes Option Pricing Model

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Abstract

A nonlinear wave alternative for the standard Black-Scholes option-pricing model is presented. The adaptive-wave model, representing controlled Brownian behavior of financial markets, is formally defined by adaptive nonlinear Schrödinger (NLS) equations, defining the option-pricing wave function in terms of the stock price and time. The model includes two parameters: volatility (playing the role of dispersion frequency coefficient), which can be either fixed or stochastic, and adaptive market potential that depends on the interest rate. The wave function represents quantum probability amplitude, whose absolute square is probability density function. Four types of analytical solutions of the NLS equation are provided in terms of Jacobi elliptic functions, all starting from de Broglie’s plane-wave packet associated with the free quantum-mechanical particle. The best agreement with the Black-Scholes model shows the adaptive shock-wave NLS-solution, which can be efficiently combined with adaptive solitary-wave NLS-solution. Adjustable ‘weights’ of the adaptive market-heat potential are estimated using either unsupervised Hebbian learning or supervised Levenberg–Marquardt algorithm. In the case of stochastic volatility, it is itself represented by the wave function, so we come to the so-called Manakov system of two coupled NLS equations (that admits closed-form solutions), with the common adaptive market potential, which defines a bidirectional spatio-temporal associative memory.

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Notes

  1. Note that the domain of validity of the ‘quantum probability’ is not restricted to the microscopic world [27]. There are macroscopic features of classically behaving systems, which cannot be explained without recourse to the quantum dynamics (see [28] and references therein).

  2. For example, the general pendulum equation:

    $$ \alpha^{\prime \prime }(t,\phi )+\sin [\alpha (t,\phi )]=0 $$

    has the elliptic solution:

    $$ \alpha (t,\phi )=2\sin ^{-1}\left[ \sin \left( {\frac{\phi}{2}}\right) \right] \hbox{sn}\left[ t,\sin ^{2}\left( {\frac{\phi} {2}}\right) \right] . $$

  3. A shock wave is a type of fast-propagating nonlinear disturbance that carries energy and can propagate through a medium (or, field). It is characterized by an abrupt, nearly discontinuous change in the characteristics of the medium. The energy of a shock wave dissipates relatively quickly with distance and its entropy increases. On the other hand, a soliton is a self-reinforcing nonlinear solitary wave packet that maintains its shape while it travels at constant speed. It is caused by a cancelation of nonlinear and dispersive effects in the medium (or, field).

  4. A closely related solution of an anharmonic oscillator ODE:

    $$ \phi^{\prime \prime }(s)+\phi (s)+\phi^{3}(s)=0 $$

    is given by

    $$ \phi (s)=\sqrt{{\frac{2m}{1-2m}}} \hbox{cn}\left( \sqrt{1+{\frac{2m}{1-2m}}} s, m\right) . $$

  5. Controlled stochastic volatility here represents volatility evolving in a stochastic manner but within the controlled boundaries.

  6. The hypothesis that financial leverage can explain the leverage effect was first discussed by F. Black [61].

  7. The Manakov system (19)–(20) has the following Lax pair [86] representation:

    $$ \begin{aligned} \partial_{x}\phi &=M\phi \hbox{ and }\partial_{t}\phi =B\phi , \hbox{ or }\partial_{x}B -\partial_{t}M=[M,B],\qquad \hbox{with}\\ M(\lambda )&=\left( \begin{array}{ccc} \hbox{i}\lambda & \psi_{1} & \psi_{2}\\ -\psi_{1} & \hbox{i}\lambda & 0 \\ -\psi_{2} & 0 & \hbox{i}\lambda \end{array} \right) \qquad \hbox{and}\\ B(\lambda )&= -\hbox{i}\left( \begin{array}{ccc} 2\lambda ^{2}-|\psi_{1}|^{2}-|\psi_{2}|^{2} & 2i\psi_{1}\lambda -\partial_{x}\psi_{1} & 2i\psi_{2}\lambda -\partial_{x}\psi_{2}\\ -2i\psi_{1}^{\ast }\lambda -\partial_{x}\psi_{1}^{\ast } & -2\lambda ^{2}+|\psi_{1}|^{2} & \psi_{1}^{\ast }\psi_{2} \\ -2i\psi_{2}^{\ast }\lambda -\partial_{x}\psi_{2}^{\ast } & \psi_{1}\psi_{2}^{\ast } & -2\lambda ^{2}+|\psi_{2}|^{2} \end{array} \right). \end{aligned} $$
    (25)

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  1. Defence Science & Technology Organisation, Adelaide, Australia

    Vladimir G. Ivancevic

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Appendix: Manakov System

Appendix: Manakov System

Manakov’s own method was based on the Lax pair representation.Footnote 7

Alternatively, for normalized value of the market-heat potential, β = r = 1, Manakov system allows solutions of the form:

$$ \sigma (s,t)=\varphi (s) \hbox{e}^{{\rm i}w_\sigma^{2}t},\qquad \psi (s,t) =\phi (s) \hbox{e}^{{\rm i}w_\psi^{2}t}, $$
(26)

where φ, ϕ are real-valued functions and w σ, w ψ are positive wave parameters for volatility and option-price. Substituting (26) into the Manakov equations we get the ODE-system [68]

$$ \varphi^{\prime \prime }(s) =w_\sigma^{2}\varphi (s)-[\varphi^{2}(s)+\phi ^{2}(s)] \varphi (s), $$
(27)
$$ \phi^{\prime \prime }(s) =w_\psi^{2}\phi (s)-[\phi^{2}(s)+\varphi^{2}(s)] \phi (s). $$
(28)

For w σ = w ψ = w, Eqs. 27 and 28 have a one-parameter family of symmetric single-humped soliton solutions (see the left part of Fig. 14) given by

$$ \varphi (s)=\pm \phi (s)=w \hbox{sech}(w s), $$
(29)

as well as periodic solutions:

$$ \varphi (s)=A\cos (Bs)\qquad \hbox{and}\qquad \phi (s)=A\sin (Bs), $$
(30)

where \(A=\sqrt{w^{2}+B^{2}}\) (with B an arbitrary parameter). For 0 < w < 1 there is also another, in general asymmetric, one-parameter family of solutions for each fixed w [68]

$$ \begin{aligned} \varphi (s) &= \sqrt{2(1-w^{2})}\cosh (ws)/\kappa,\\ \phi (s) &= -w\sqrt{2(1-w^{2})}\sinh (s-s_0)/\kappa,\qquad \hbox{where}\\ \kappa &=\cosh (s)\cosh (w s)-w\sinh (s)\sinh (w s), \end{aligned} $$
(31)

in which φ is symmetric and ϕ antisymmetric.

Fig. 14
figure 14

Initial envelopes for volatility |σ0| and option-price |ψ0| within the Manakov solitons: (left) bright compound soliton (29), (middle) dark compound soliton (32), and (right) compound kink-shaped soliton (36). These initial envelopes can be used for numerical studies of the Manakov system and its various generalizations (modified and adapted from [87])

Full size image

On the other hand, for negative values of the potential β, the Manakov equations accept dark soliton solutions of the form [87]

$$ \sigma (s,t)=\psi (s,t)=k\left[ \tanh (ks)-\hbox{i}\right] \hbox{e}^{\hbox{i}(ks-5k^{2}t)}, $$
(32)

which are localized dips on a finite-amplitude background wave (see the middle part of Fig. 14). In this very interesting case, volatility and option-price fields are coupled together, forming a dark compound soliton. Note that their respective relative amplitudes are controlled by the corresponding nonlinearities and frequency. For β =  −1 the Manakov equations alow also solutions of the form:

$$ \sigma (s,t)=\varphi (s) \hbox{e}^{-iw_\sigma^{2}t},\qquad \psi (s,t)=\phi (s) \hbox{e}^{iw_\psi^{2}t}. $$
(33)

Introducing (33) into the Manakov equations, we get the ODE-system:

$$ \varphi^{\prime \prime }(s)= [\varphi^{2}(s)+\phi^{2}(s)] \varphi (s)-w_\sigma^{2}\varphi (s), $$
(34)
$$ \phi^{\prime \prime}(s) =[\phi ^{2}(s)+\varphi ^{2}(s)] \phi (s)-w_\psi^{2}\phi (s), $$
(35)

which, for w σ = w ψ = w, allow for kink-shaped localized soliton solutions (see the right part of Fig. 14) given by [87]

$$ \varphi (s)=\pm \phi (s)=({w}/\sqrt{2})\tanh ({w} s/\sqrt{2}), $$
(36)

as well as periodic solutions (30). Inserting (14) back into (33) gives the double-kink solution for the Manakov system:

$$ \sigma (s,t)=\pm({w}/\sqrt{2})\tanh ({w} s/\sqrt{2}) \hbox{e}^{-iw^{2}t},\qquad \psi (s,t)=\pm({w}/\sqrt{2})\tanh ({w} s/\sqrt{2}) \hbox{e}^{-iw^{2}t}. $$
(37)

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Ivancevic, V.G. Adaptive-Wave Alternative for the Black-Scholes Option Pricing Model. Cogn Comput 2, 17–30 (2010). https://doi.org/10.1007/s12559-009-9031-x

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