EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 4, 2017, 631-637
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Solving the Ivancevic Pricing Model Using the He’s
Frequency Amplitude Formulation

O. González-Gaxiola1,∗, S. O. Edeki2, O. O. Ugbebor2,3, J. Ruiz de Chávez4

1 Departamento de Matemáticas Aplicadas y Sistemas,
Universidad Autónoma Metropolitana-C, Mexico
2 Department of Mathematics, Covenant University, Nigeria
3 Department of Mathematics, University of Ibadan, Nigeria
4 Departamento de Matemáticas, Universidad Autónoma Metropolitana-I, Mexico

Abstract. In financial mathematics, option pricing theory remains a core area of interest that
requires effective models. Thus, the Ivancevic option pricing model (IOPM) is a nonlinear adaptive-
wave alternative for the classical Black-Scholes option pricing model; it represents a controlled
Brownian motion (BM) in an adaptive setting with relation to nonlinear Schrödinger equation.
The importance of the IOPM cannot be overemphasized; though, it seems difficult and complex to
obtain the associated exact solutions if they exist. Therefore, this paper provides exact solutions
of the IOPM by means of a proposed analytical method referred to as He’s frequency amplitude
formulation. Cases of nonzero adaptive market potential are considered. The method is shown to
effective, efficient, simple and direct in application, even without loss of generality.

2010 Mathematics Subject Classifications: 91G80, 35L05, 97M30

Key Words and Phrases: Ivancevic pricing model, nonlinear Black-Scholes model, option pric-
ing, Amplitude-frequency formulation

1. Introduction

The classical Black-Scholes model (BSM) serves as a remarkable financial model for
option pricing and valuation. The BSM describes the time-evolution of the market value
of financial equity such as European or stock option [4, 17, 19]. The main assumptions
associated with this classical arbitrage pricing theory (BSM) include the following: the
asset price S (or the underlying asset) following a geometric Brownian motion (GBM),
the drift parameter, µ and the volatility rate, σ are assumed constants, lack of arbitrage
opportunities (no risk-free profit), frictionless and competitive markets [12, 8, 2]. Thus,

∗Corresponding author.

Email addresses: ogonzalez@correo.cua.uam.mx (O. González-Gaxiola) soedeki@yahoo.com (S. O.
Edeki), ugbebor1@yahoo.com (O. O. Ugbebor), jrch@xanum.uam.mx (J. Ruiz de Chávez)

http://www.ejpam.com 631 c© 2017 EJPAM All rights reserved.



O. González-Gaxiola, S. O. Edeki et al / Eur. J. Pure Appl. Math, 10 (4) (2017), 631-637 632

the stock price S = S(t), at time t, (0 ≤ t ≤ T ) follows the stochastic differential equation
(SDE):

dS = S(µdt+ σdWt), S ∈ [0,∞) (1)

where µ, σ > 0, and Wt are mean rate of return of S, the volatility, and a standard
Brownian motion respectively.
So, for an option value u = u(S, t), the Black-Scholes partial differential equation (PDE)
associated to (1) can be expressed as:

∂u

∂t
+ rS

∂u

∂S
+

1

2
S2σ2

∂2u

∂S2
− ru = 0 (2)

with u(0, t) = 0, u(S, t)→ 0 as S →∞, u(S, T ) = max(S − E, 0), E is a constant and

S(t) = S0e
(µ−σ

2

2
)t+σWt , S0 = S(0). (3)

In literature, detailed and extensive work on the importance of (2) with respect to exact,
analytical, approximate or numerical methods of solutions have been referred [21, 9, 5, 10].
Vukovic [24] in a recent study, established the interconnectedness of the Schrödinger and
the Black-Scholes equations via the tools of quantum physics in the sense of Hamilto-
nian operator. It was noted that while the Black-Scholes Hamiltonian was anti-Hermitian
causing the eigenvalues to be complex, the Schrödinger Hamiltonian was Hermitian. It
was further showed that the Black-Scholes equation can be derived from the Schrödinger
equation via the application of quantum mechanics tools [1, 7]. In [3], [25] and [27] the
solution of linear and nonlinear Schrödinger equations was obtained by Homotopy pertur-
bation method, variational iteration method and He’s frequency formulation respectively.
Recently, in [22] an analytical option pricing model based on the nonlinear Schrödinger
partial differential equation with vanishing external potential has been considered. The
facts incorporated include the points that: the Schrödinger equation requires a complex
state function while the Black-Scholes equation is a real PDE that yields a real valued
expression for the option price at all time.
The Black-Scholes model (2) can be applied to a reasonable number of one dimensional op-
tion models ascribed to u and S, say for puts/calls and stocks/dividends respectively [17].
As noted in [11, 23], one could consider the associated probability density function (PDF)
resulting from the backward Fokker-Planck equation using the classical Kolmogorov prob-
ability method instead of the market value of an option obtained via the Black-Scholes
equation.

2. The Ivancevic Option Pricing Model (IOPM) [13]

As an alternative method for obtaining the same PDF for the market value of a stock
option, Ivancevic [18] applied the quantum-probability formation as a solution to a time-
dependent Schrödinger equation (linear or nonlinear) for the evolution of the complex-
valued wave function, and proposed an adaptive, wave-form nonlinear model [6, 20].



O. González-Gaxiola, S. O. Edeki et al / Eur. J. Pure Appl. Math, 10 (4) (2017), 631-637 633

Henceforth, such nonlinear adaptive model is referred to as Ivancevic option pricing model
as follows:

i
∂w

∂t
+

1

2
σ2
∂2w

∂S2
+ β|w|2w = 0, i2 = −1 (4)

where w = w(S, t) denotes the option pricing wave-function at time t , |w|2 = |w(S, t)|2
represents the PDF for the option price with regard to stock price and time, σ represents
a constant or stochastic process as the dispersion frequency volatility coefficient while β is
referred to as the Landau coefficient representing adaptive market potential. The model
(4) becomes linear if β = 0 . In this work, a case of non-zero adaptive market potential
(β 6= 0) will be considered in terms of analytical solutions using a proposed semi-analytical
method referred to as He’s frequency amplitude formulation.

3. He’s amplitude frequency formulation

He’s frequency amplitude formulation is the development of an ancient Chinese algo-
rithm [14, 15, 16]; it is very effective to nonlinear oscillators as shown by the authors in
[26].
We consider a generalized nonlinear oscillator in the form

u′′ + f(u) = 0, u(0) = A, u′(0) = 0. (5)

We use two trial functions u1(t) = Acost and u2(t) = Acosωt, which are, respectively, the
solutions of the following linear oscillator equations:

u′′ + ω2
1u = 0, ω2

1 = 1 (6)

and
u′′ + ω2

2u = 0, ω2
2 = ω2. (7)

where ω is assumed to be the frequency of the nonlinear oscillator, equation (5).
For the case of equation (5), the residuals are

R1(t) = −cost+ f(Acost) (8)

and
R2(ωt) = −ω2cosωt+ f(Acosωt). (9)

We will use the following frequency-amplitude formulation:

ω2 =
ω2
1R2(ωt = 0)− ω2

2R1(t = 0)

R2 −R1
. (10)

In order to use He’s amplitude frequency formulation, we choose two trial functions u1(S, t)
and u2(S, t) with which we calculate the residuals R1(S, t) and R2(S, t) respectively. The
trial functions u1(S, t) and u2(S, t) for the application of the present study are solutions
of the following linear Schrödinger equation

i
∂w

∂t
+ ω

∂2w

∂S2
= 0. (11)



O. González-Gaxiola, S. O. Edeki et al / Eur. J. Pure Appl. Math, 10 (4) (2017), 631-637 634

He’s amplitude frequency formulation reads [16]

ω2 =
R2(S, 0)− ω2R1(S, 0)

R2(S, 0)−R1(S, 0)
. (12)

From Eq (12) we can obtain approximate solutions for a certain type of differential equa-
tions whose solutions are a priori periodic.

3.1. Numerical Illustrative Examples

In this subsection, we consider the following cases for numerical examples.

Example 1. Suppose β = 2 and σ =
√

2. Then the corresponding Ivancevic option pricing
model given by Eq (4) is: {

∂w
∂t = i

(
∂2w
∂S2 + 2|w|2w

)
,

w(S, 0) = e2iS .
(13)

We use the trial functions

w1(S, t) = ei(2S+t), w2(S, t) = ei(2S+ωt). (14)

By calculation, we obtain

w1,t = iei(2S+t), w1,S = 2iei(2S+t), w1,SS = −4ei(2S+t) (15)

w2,t = iωei(2S+ωt), w2,S = 2iei(2S+ωt), w2,SS = −4ei(2S+ωt). (16)

Substituting the two trial functions and their derivatives into Eq. (13) gives the residuals:

R1(S, t) = −3ei(2S+t), R2(S, t) = −(2 + ω)ei(2S+ωt) (17)

According to He’s frequency formulation, we have

ω2 =
R2(S, 0)− ω2R1(S, 0)

R2(S, 0)−R1(S, 0)
=

3ω2e2iS − ωe2iS − 2e2iS

e2iS − ωe2iS
. (18)

Simplifying, we obtain ω = −2. The solution is obtained as follows

w(S, t) = e2i(S−t). (19)

Showing that (19) satisfies (13) is obvious and straightforward.

Example 2. Suppose β = 3 and σ = 1. Then the corresponding Ivancevic option pricing
model given by Eq (4) is: {

∂w
∂t = i

(
∂2w
∂S2 + 6|w|2w

)
,

w(S, 0) = e2iS .
(20)



REFERENCES 635

We use the trial functions

w1(S, t) = ei(2S+t), w2(S, t) = ei(2S+ωt). (21)

Now, by calculating, we obtain

w1,t = iei(2S+t), w1,S = 2iei(2S+t), w1,SS = −4ei(2S+t) (22)

w2,t = iωei(2S+ωt), w2,S = 2iei(2S+ωt), w2,SS = −4ei(2S+ωt). (23)

Substituting the two trial functions and their derivatives into Eq. (20) gives the residuals:

R1(S, t) = −3ei(2S+t), R2(S, t) = −(2 + ω)ei(2S+ωt) (24)

According to He’s frequency formulation, we have

ω2 =
R2(S, 0)− ω2R1(S, 0)

R2(S, 0)−R1(S, 0)
=

(2− ω)e2iS − ω2e2iS

(2− ω)e2iS − e2iS
. (25)

Simplifying, we obtain ω = 2. The solution is obtained as follows

w(S, t) = e2i(S+t). (26)

It is straightforward to verify that w(S, t) = e2i(S+t) satisfies the problem given by Eq.
(20).

4. Concluding Remarks

In this work, the Ivancevic option pricing model (IOPM) is considered. This nonlinear
adaptive-wave model serves as alternative for the classical Black-Scholes option pricing
model based on a controlled Brownian motion in an adaptive setting relating to nonlinear
Schrödinger equation. By considering cases of nonzero adaptive market potential, exact
solutions of the IOPM by means of a proposed He’s frequency amplitude formulation
method (HFAFM) were obtained. The results revealed that the proposed HFAFM is
simple, direct, and effective as the obtained solutions coincide exactly with the exact
solutions without any form of linearization, perturbation, or discretization.

References

[1] B E Baaquie. Quantum Finance: Path Integrals and Hamiltonians for Options and
Interest Rates. Cambridge University Press, Cambridge, 2004.

[2] G Barles and H M Soner. Option pricing with transaction costs and a nonlinear
Black-Scholes equation. Financ. Stoch, 2: 369-397, 1998.

[3] J Biazar and H Ghazvin. Exact solutions for non-linear Schrödinger equations by He’s
homotopy perturbation method. Phys. Letters A, 366: 79-84, 2007.



REFERENCES 636

[4] F Black and M Scholes. The pricing options and corporate liabilities. J. Political
Econ., 81: 637-654, 1973.

[5] R Company, E Navarro, J R Pintos and E. Ponsoda. Numerical solution of linear and
nonlinear Black-Scholes option pricing equations. Comput. Math. Appl., 56: 813-821,
2008.

[6] R Cont. Empirical properties of asset returns: stylized facts and statistical issues.
Quant. Finance, 1: 223-236, 2001.

[7] M Contreras, R Pellicer, M Villena and A Ruiz. A Quantum Model of Option Pricing:
When Black-Scholes meets Schrödinger and Its Semi-Classical Limit. Physica A, 389:
5447-5459, 2010.

[8] S O Edeki, E A Owoloko and O O Ugbebor. The Modified Black-Scholes Model
via Constant Elasticity of Variance for Stock Options Valuation. AIP Conference
proceedings, 1705: 020041, 2016.

[9] S O Edeki, O O Ugbebor and E A Owoloko. Analytical Solutions of the Black-
Scholes Pricing Model for European Option Valuation via a Projected Differential
Transformation Method. Entropy, 17: 7510-7521, 2015.

[10] S O Edeki, O O Ugbebor and E A Owoloko. He’s Polynomials for Analytical Solutions
of the Black-Scholes Pricing Model for Stock Option Valuation. Lecture Notes in En-
gineering and Computer Science: Proceedings of the World Congress on Engineering,
WCE 2016, London, U.K., 632-634, 2016.

[11] C W Gardiner. Handbook of Stochastic Methods. Springer, Berlin, 1983.

[12] O González-Gaxiola, J Ruiz de Chávez and J A Santiago. A Nonlinear Option Pricing
Model Through the Adomian Decomposition Method. Int. J. Appl. Comput. Math.,
2: 453-467, 2016.

[13] O González-Gaxiola and J Ruiz de Chávez. Solving the Ivancevic option pricing model
using the Elsaki-Adomian decomposition method. Int. J. of Applied Math., 28: 515-
525, 2015.

[14] J H He. Some asymptotic methods for strongly nonlinear equations. Int. J. Mod.
Phys. B, 20(10): 1141-1199, 2006.

[15] J H He. Comment on He’s frequency formulation for nonlinear oscillators. Eur. J.
Phys. 29: L19-L22, 2008.

[16] J H He. An improved amplitude-frequency formulation for nonlinear oscillators. Int.
J. Nonlinear Sci. Numer. Simul., 9(2): 211-212, 2008.

[17] V G Ivancevic. Adaptive Wave Models for Sophisticated Option Pricing. Journal of
Mathematical Finance, 1: 41-49, 2011.



REFERENCES 637

[18] V G Ivancevic. Adaptive-Wave Alternative for the Black-Scholes Option Pricing
Model. Cognitive Computation, 2: 17-30, 2010.

[19] R C Merton. Theory of Rational Option Pricing. The Bell Journal of Economics and
Management Science, 4: 141-183, 1973.

[20] J Perello, R Sircar and J Masoliver. Option pricing under stochastic volatility: the
exponential Ornstein-Uhlenbeck model. J. Stat. Mech. P06010, 2008.

[21] M R Rodrigo and R S Mamon. An alternative approach to solving the Black-Scholes
equation with time-varying parameters. Appl. Math. Lett., 19: 398-402, 2006.

[22] M Wróblewski. Nonlinear Schrödinger approach to European option pricing. Open
Phys., 15 : 280-291, 2017.

[23] J Voit. The Statistical Mechanics of Financial Markets. Springer, Berlin, 2005.

[24] O Vukovic. On the Interconnectedness of Schrödinger and Black-Scholes Equation.
Journal of Applied Mathematics and Physics, 3: 1108-1113, 2015.

[25] A M Wazwaz. A study on linear and nonlinear Schrödinger equations by the varia-
tional iteration method. Chaos, Solitons and Fractals, 37: 1136-1142, 2008.

[26] H L Zhang. Application of He’s frequency-amplitude formulation to an x1/3 force
nonlinear oscillator. Int. Journal of Nonlinear Sciences and Numerical Simulation, 9:
297-300, 2008.

[27] Y-N Zhanga, Fei Xu and Ling-ling Deng. Exact solution for nonlinear Schrödinger
equation by He’s frequency formulation. Comput. Math. Appl., 58: 2449-2451, 2009.