Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Fractional Black Scholes Option Pricing with Stochastic Arbitrage Return

Bright O. Osu*, Chukwunezu A. Ifeoma

Department of Mathematics, Michael Okpara University of Agriculture, Umudike

Abstract: Option price and random arbitrage returns change on different time scales allow the development of an asymptotic pricing theory involving the options rather than exact prices. The role that random arbitrage opportunities play in pricing financial derivatives can be determined. In this paper, we construct Green’s functions for terminal boundary value problems of the fractional Black-Scholes equation. We follow further an approach suggested in literature and focus on the pricing bands for options that account for random arbitrage opportunities and got similar result for the fractional Black- Scholes option pricing.

Keywords: arbitrage returns, option pricing, green function, FBS equation

Citation: Bright O. Osu, Chukwunezu A. Ifeoma, Fractional Black Scholes Option Pricing with Stochastic Arbitrage Return. International Journal of Partial Differential Equations and Applications, 4 (2). 20-24. January 2016. doi:10.12691/ijpdea-4-2-1

1. Introduction

Fractional calculus has become of increasing use for analyzing not only stochastic processes driven by fractional Brownian processes [16], but also non -random fractional phenomena in physics [8], like the study of porous systems, for instance, and quantum mechanics [14]. Whichever the framework is, we believe that the very reason for introducing and using fractional derivative is to deal with non-differentiable functions. In financial literature for example, stochastic volatility models the Merton jump-diffusion model [9], non-Gaussian option pricing models [4,5], amongst others have been proposed. Each of these is based on the assumption of the absence of arbitrage. However, it is well-known that arbitrage opportunities always exist in the real world (see Refs. [6,15]). Of course, arbitragers ensure that the prices of securities do not get out of line with their equilibrium values, and therefore virtual arbitrage is always short-lived. An arbitrage possibility is essentially equivalent to the possibility of making a positive amount of money out of nothing without taking any risk. It is thus essentially a riskless money making machine. An arbitrage possibility is a serious case of mispricing in the market. It is well-known that arbitrage opportunities always exist in the real world [10]. Of course, arbitragers ensure that the prices of securities do not get out of line with their equilibrium values, and therefore virtual arbitrage is always short-lived. The first attempt to take into account virtual arbitrage in option pricing was made by Physicists Refs [1,7,13]. The authors assume that arbitrage returns exist, appearing and disappearing over a short time scale. Asma et al [2] applied the homotopy perturbation method for fractional Black-Scholes equation by using He’s polynomials and Sumudu transform to obtain the solution of fractional Black-Scholes equation. At this point, Belgacem et al. [3,9] had applied the Laplace transform and extended the theory and the applications of the Sumudu transform to the solution of fractional differential equations.

In this work, a technique is proposed for the construction of Green’s functions for terminal boundary value problems of the fractional Black-Scholes equation. The technique is based on the method of integral Laplace transform and the method of variation of parameters. It provides closed form analytic representations for the constructed Green’s functions [12]. We follow further an approach suggested in Ref. [15] and focus on the pricing bands for options that account for random arbitrage opportunities and got similar result for the fractional Black- Scholes option pricing.

2. Derivation of the Black-Scholes Equation

We base our derivation on replicating portfolio that ensures that no arbitrage opportunities are allowed. As in the discrete case, consider a portfolio , which is - measurable ( we can choose as we go, but at any point in time the choice is deterministic), denotes the proportion of shares invested at time , the rest of the money is invested in the money market account, giving risk-free rate of return, , say. In what follows, we state:

Theorem 2.1

Let a generic payoff function , the PDE associated with the price of derivative on the stock price is

[picture] | (2.1)

Proof:

The stock price follows the fractional Brownian motion process [9]

[picture] | (2.2)

and the wealth of an investor , follows a diffusion driven by (with time suppressed)

[picture] | (2.3)

Putting equation (2.2) into equation (2.3) yields;

[picture] | (2.4)

where is the risk premium. Suppose that the value of this claim at time is given by

[picture] | (2.5)

Applying the fractional Ito’s formula on equation 2.5, we have

[picture] | (2.6)

To track at all times, we have under the assumption of complete market that

[picture] | (2.7)

Thus

[picture] | (2.8)

and

[picture] | (2.9)

[picture] | (2.10)

While equation (2.6) with gives

[picture] | (2.11)

[picture] | (2.12)

Thus, the fractional Black-Scholes equation for valuing an option with value V is obtained.

3. Construction of Green’s Function for the Fractional Black-Scholes Equation

For a call option with maturity date , strike price , and payoff function , the value price satisfies the following fBm, [9],

[picture] | (3.1)

with homogeneous boundary value problem corresponding to

[picture] | (3.2)

and

[picture] | (3.3)

We set and , to get

or

[picture] | (3.4)

Let and , then we have

[picture] | (3.5)

Applying Laplace transform to (3.5) gives

or for

[picture] | (3.6)

To find a fundamental set of solutions to the homogeneous equation corresponding to (3.6), consider its characteristic equation:

Solving, we have

For simplicity, set , then .

Thus the roots are and where , while the parameters and . This yields two linearly independent solutions to the homogeneous equation corresponding to (3.6) as

and

with their linear combination given as

[picture] | (3.7)

Now representing according to the method of variation of parameters, the general solution to (3.6). Following the procedure of this method, one arrives at the well-posed system

of linear algebraic equations in the derivatives with respect to x of the coefficients and of the linear combination in (3.7). The solution of the above system is obtained as

and

Upon integration, the coefficients and are found in the form

and

Substitution of these in (3.7) yields the general solution of (3.5) in the form:

[picture] | (3.8)

The constants of integration M(s) and N(s) can be obtained upon satisfying the boundary conditions of (3’6) as

Substituting in (3.8), we obtain the solution to the boundary value problem in (3.5) and (3.6) in the form.

[picture] | (3.9)

Equation (3.9) can be re-written in a compact single-integral form as

[picture] | (3.10)

The solution to the initial boundary value problem stated by (3.1) can be obtained fromwith the aid of the inverse Laplace transform. In doing so, we keep in mind that the parameter w has earlier been introduced in terms of the parameter s of the Laplace transform as

This yields

[picture] | (3.11)

To obtain the solution we make the backward substitutions,

While the interval of integration in (3.11) transforms according to the relation to the interval with respect to . With all these in mind, one arrives at the solution to the terminal-boundary value problem as

Revealing the Green’s function to the problem as

[picture] | (3.12)

4. Fractional B-S Equation with Random Arbitrage Return

The associated option price obeys the following Stochastic P.D.E

[picture] | (4.1)

Subject to the initial condition

Here is the random arbitrage return that describes the fluctuations of return around , [7]. According to the law of large numbers converges in probability to the Black-Scholes price as One can split into the sum of the Black-Scholes price and the random field with the scaling factor giving

[picture] | (4.2)

Substituting (4.2) into (4.1), we have;

[picture] | (4.3)

Substituting (4.1) into (4.3), we have

Ergodic theory implies that in its integral form converges to zero as while converges weakly to a white Gaussian noise Furthermore, as the random field converges weakly to the field that obeys the asymptotic stochastic P.D.E given by

[picture] | (4.4)

With initial condition

Where .

Equation (4.4) can be solved in terms of the fractional Black-Scholes Green function; to give

[picture] | (4.5)

Where

It follows from (4.5) that since is the Gaussian noise with zero mean, is also the Gaussian field with zero mean.

The pricing bands for the options for the case of arbitrage opportunities can be given by

[picture] | (4.6)

Where

[picture] | (4.7)

and is the covariance

The variance quantifies the fluctuations around the fractional Black-Scholes price. It should be noted that it is independent of the detailed statistical characteristics of the arbitrage return.

It is note-worthy that an investor who employs the arbitrage opportunities band hedging sells the option for

[picture] | (4.8)

5. Conclusion

In this paper, we have introduced a technique for constructing the Green function for the fractional Black-Scholes equation. We also investigated the implications of random arbitrage return for option pricing. We derived the asymptotic equation for the random field that quantifies the fluctuations around the fractional Black-Scholes price and showed that it is independent of the detailed statistical characteristics of the arbitrage return.

References

[1]   | A.N. Adamchuk, S.E. Esipov; Collectively fluctuating assets in the presence of arbitrage opportunities and option pricing, phys. Usp. 40 (12) (1997) 1239-1248.
[2]   | Asma, A. E., Adem K. and Bachok, M. T; Homotopy Perturbation Method for Fractional Black-Scholes European Option Pricing Equations Using Sumudu Transform. Hindawi Publishing Corporation Mathematical Problems in EngineeringVolume 2013, Article ID 524852, 7 page.
[3]   | F. B. M. Belgacem, A. A. Karaballi, and S. L. Kalla, “Analytical investigations of the Sumudu transform and applications tointegral production equations,” Mathematical Problems in Engineering,no. 3-4, pp. 103-118, 2003.
[4]   | L. Borland, Theory of non-Gaussian option pricing, Quant. Finance 2 (2002) 415-431.
[5]   | L. Borland, J.-P. Bouchaud, Non-Gaussian option pricing model with skew, cond-mat/0403022, 2004.
[6]   | D. Galai; Tests of market efficiency and the Chicago board option exchange, J. Bus. 50 (1997) 167-197.
[7]   | K. Ilinski, How to account for the virtual arbitrage in the standard derivative pricing, preprint, condmat/990 2047.
[8]   | G. Jumarie. Schrodinger equation for quantum-fractal space-time of order n via the complex-valued fractional Brownian motion, Intern. J. of Modern Physics A 16(31), 5061-5084, 2001.
[9]   | Q. D. Katatbeh and F. B. M. Belgacem, “Applications of the Sumudu transform to fractional differential equations,” NonlinearStudies, vol. 18, no. 1, pp. 99-112, 201
[10]   | M. Melnikov, Y. A. Melnikov; Construction of Green’s function for the Black-Scholes equation, Electronic Journal of Differential Equations, Vol. 2007(2007), No. 153, Pp. 1-14.
[11]   | R.C. Merton, Continuous Time Finance, Blackwell, Oxford, 1992.
[12]   | B. O. Osu;An Alternative Approach for the Derivation of the Fractional Black-Scholes Equation for the Pricing of Options and its Solution via the Mellin Transform. International Journal of Applied Science and Mathematics, 2(6), 2394-2894, 2015.
[13]   | N.T. Shawagfeh. Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. And Comp. 131, 517-529, 2002.
[14]   | G. Sofianos, Index arbitrage profitability NYSE working paper 90-104, J. Derivatives 1 (N1) (1993).
[15]   | Sergei Fedotov, Stephanos Panayides; Stochastic arbitrage return and its implication for option pricing, Physica A 345 (2005) 207-217.
[16]   | W. Wyss. The fractional Black-Scholes equation, Fract. Calc. Appl. Anal. 3(1), 51-61, 2000.