In this paper, we propose a pricing model for stock option valuation. The model

is derived from the classical Black-Scholes option pricing equation via the application of the

constant elasticity of variance (CEV) model with dividend yield. This modifies the Black-

Scholes equation by incorporating a non-constant volatility power function of the underlying

stock price, and a dividend yield parameter

Edeki, S.O.

Owoloko, E. A.

Ugbebor, Olabisi O.

International Journal of Pure and Apllied Mathematics

S.O. Edeki

O.O. Ugbebor

E.A. Owoloko

Crossref

ON A DIVIDEND-PAYING STOCK OPTIONS PRICING MODEL (SOPM) USING CONSTANT ELASTICITY OF VARIANCE STOCHASTIC DYNAMICS

Covenant University Repository

International Journal of Pure and Applied MathematicsVolume 106 No. 4 2016, 1029-1036ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.eudoi: 10.12732/ijpam.v106i4.5PAijpam.euON A DIVIDEND-PAYING STOCK OPTIONS PRICINGMODEL (SOPM) USING CONSTANT ELASTICITYOF VARIANCE STOCHASTIC DYNAMICSS.O. Edeki1 §, O.O. Ugbebor2, E.A. Owoloko31,3Department of MathematicsCovenant UniversityCanaanland, Otta, NIGERIA2,3Department of MathematicsUniversity of IbadanIbadan, NIGERIAAbstract: In this paper, we propose a pricing model for stock option valuation. The modelis derived from the classical Black-Scholes option pricing equation via the application of theconstant elasticity of variance (CEV) model with dividend yield. This modifies the Black-Scholes equation by incorporating a non-constant volatility power function of the underlyingstock price, and a dividend yield parameter.AMS Subject Classification: 91B25, 37N40, 35R60Key Words: stock options, CEV, stochastic dynamics, dividend yield1. IntroductionIn financial mathematics, option pricing theory is a core aspect requiring fre-quent attention of financial analysts and financial engineers, actuaries and soon [1-3]. In recent years, ever since the 1987 market crash, several marketobservations have revealed the association between stock price and volatility;leverage effect is the phenomenon behind this [4]. The effect as explained in [4]states that “if a firm’s stock price falls, it will increase the debt-equity ratio ofReceived: January 2, 2016Published: April 1, 2016c© 2016 Academic Publications, Ltd.url: www.acadpubl.eu§Correspondence author1030 S.O. Edeki, O.O. Ugbebor, E.A. Owolokothe firm and therefore increases the variance of the stock’s return” It is worthmentioning that the Black-Scholes option pricing formula [5] does not reconcilethis leverage effect parameter.Black in [6], considered this financial leverage effect. This was later ex-tended and supported by other researchers in terms of empirical evidences [7].Consequently, a theoretical framework to capture this relationship (inverse) be-tween the stock price and its variance is built by Constant Elasticity of Variance(CEV) class of stock price distributions.In option pricing theory, Black and Scholes assumed that the underlyingstock price process, St, follows a lognormal distribution and solves the Stochas-tic Differential Equation (SDE):dSt = µStdt+ σStdWt, (1)where µ represents rate of return (drift coefficient), σ the stock price volatilityrate, and Wt is a standard Brownian motion.The basic assumptions of the Black-Scholes model include arbitrage-free op-portunities, no dividend paying equity, constant risk-free-rate, constant volatil-ity, and so on [8,9]. In this work, emphasis will be on how to address theassumptions of no dividend yield and that of constant volatility (stock pricestandard deviation). The assumption of the volatility being constant was tokeep the model linear and for analytical solvability. However, in practical set-tings, constant volatility is not realistic. We shall therefore resort to the fol-lowing form of SDE in the next subsection to modify the Black-Scholes modelfor stock option valuation.2. A Note on CEV Option Pricing Model (CEVOPM)The assumption of the constant volatility parameter in the Black-Scholes modelhas drawn the attention of many researchers even in terms of the analytical ornumerical solutions obtained via the different modifications of the Black-Scholesmodel [10-12] and the references therein. Delbean and Shirakawa proved thatarbitrage opportunities are allowed in the CEV model when the stock price ison strictly positive conditions [13].Cox and Ross [14] considered the CEV diffusion process governed by theSDE when q = 0:dSt = St (µ− q) dt+ σSξ2t d (Wt) , (2)whose solution at time t is St = S, ξ denotes an elasticity rate, q is a dividendyield parameter, while µ,σ, andWt remain as defined in the earlier section. TheON A DIVIDEND-PAYING STOCK OPTIONS PRICING... 1031elasticity rate parameter ξ, which is the central feature of the model, controlsthe relationship between the volatility and price of the underlying asset.The CEV diffusion model seems to be a natural extension of the GeometricBrownian Motion (GBM). In [15], Beckners considered the CEV and its impli-cations for option pricing on the basis of empirical studies and concluded thatthe CEV class could be a better descriptor of the actual stock price, in terms ofbehavior than the traditionally used lognormal model. Xiao et al. [16], appliedthe CEV model for the study of a defined-contribution pension plan where ben-efits are paid by annuity; with a focus on the model describing the dynamicmovements of the risky market price.In what follows, the Black-Scholes model will be modified using the SDEassociated with the CEV model in (2).2.1. The Modified Black-ScholesModel Derivation (CEV-B-S Model)Let St be the stock price at time t , satisfying the SDE in (4), such that thevalue of the contingent claim is V = V (S, t) with V ∈ C2,1 (R× [0, T ]), so byapplying Ito lemma, we have:dV =∂V∂tdt+∂V∂SdS +12∂2V∂S2〈dS〉=∂V∂tdt+∂V∂S(St (µ− q) dt+ σSξ2t d (Wt))+12∂2V∂S2〈St (µ− q) dt+ σSξ2t d (Wt)〉. (3)But〈St (µ− q) dt+ σSξ2t d (Wt)〉= σ2Sξt dt, (4)whereΘ • dWt ={dt, forΘ = dWt0, otherwise(5)Hence, (3) becomes:dV =∂V∂tdt+∂V∂S(St (µ− q) dt+ σSξ2t d (Wt))+12∂2V∂S2(σ2Sξt dt). (6)ThereforedV =(∂V∂t+ (µ− q)S∂V∂S+12σ2Sξ∂2V∂S2)dt+ σSξ2∂V∂SdW. (7)1032 S.O. Edeki, O.O. Ugbebor, E.A. OwolokoSuppose Ξ(t)is a delta-hedge-portfolio constructed by longing a contingentclaim, and shorting a delta unit of the concerned underlying asset such that:Ξ(t) =−∆S + V (S, t),dΞ(t) =dV (s, t)−∆dS,∆ =∂V∂S, anddΞ(t) =rΞ(t)dt.(8)In a bid to making the portfolio value riskless, (say bank account), where r isa riskless rate, we therefore have:dΞ(t) = dV (S, t)−∆dS,dΞ(t) =(∂V∂t+ (µ− q)S∂V∂S+12σ2Sξ∂2V∂S2)dt+ σSξ2∂V∂SdW−∆((µ− q)Sdt+ σSξ2dW), (9)⇒dΞdt=(∂V∂t+ (µ− q)S∂V∂S+12σ2Sξ∂2V∂S2)+ σSξ2∂V∂SdWdt−∂V∂S((µ− q)Sdt+ σSξ2ddWdt). (10)Thus:dΞdt=∂V∂t+12σ2Sξ∂2V∂S2≡ rΞ (11)So, combining (7), (8) and (11), we obtain∂V∂t+ rS∂V∂S+12σ2Sξ∂2V∂S2− rV = 0. (12)Equation (12) is referred to as the modified Black-Scholes model via CEV SDEwith dividend yield.3. Discussion of ResultWhen ξ = 2, (12) becomes the classical Black-Scholes model. It is also observedthat (12) coincides with the model obtained when the stock price St follows theSDE in (2) without a dividend yield [17], but the difference(s) between thestock prices SND andSD for the SDEs in (2) will be highlighted in the nextsubsection.ON A DIVIDEND-PAYING STOCK OPTIONS PRICING... 10333.1. Comparison of the SDE ModelsIn what follows, we shall compare the fundamental features of the associatedmodels presented above in this subsection.Let SND and SDbe the solutions of the SDE in (2), indicating no divi-dend and dividend yields respectively. Suppose further that V BSM0 denotesthe volatility of the Black-Scholes model, V CEVM0 is the volatility of the CEVmodel without dividend yield, V CEVM0D is the volatility of the CEV model withdividend yield, V arBSM the variance of the Black-Scholes model, VarCEVM the vari-ance of the CEV model without dividend yield, V arDCEVM the variance of the CEVmodel with dividend yield. Then, the following are thus deduced easily:For volatilities: V BSM0 = σ, VCEVM0 = σSξ2−1, V CEVM0D = σ∗Sξ2−1DFor variances: V arBSM = σ2, V arCEVM = σ2Sξ−2, V arDCEVM = σ2∗Sξ−2DFormean parameters: m̄NDCEVM = µ, m̄DCEVM = µ− q(13)Note : In both cases of the governing SDEs: with or without dividend pa-rameter, it is clear that the variances V CEVM0D = g (σ, SD)and VCEVM0D =h (σ, SND)are functions of the underlying asset prices SND and SD respectively.3.2. Interpretation of ResultsWe present a graph (see Figure 1) below to show the estimates of the distri-bution of (S (T )− 1, 0)+ using different values of the elasticity parameter forresults interpretation, and to show the economic implications of the elasticityrate. For the purpose of simulation, we consider the following values for theconcerned parameters: T = 2.0, σ = 0.3, µ = 0.05 and forξ = 0.1ξ = 2, andξ = 5 See [17] and the related references therein for details.3.3. Elasticity and Elasticity ParameterThe following relationship is used in the computation of elasticity [17]:e =(dV CEVMrdS)÷(V CEVMrdS), (14)whiledV CEVMrdS= (ξ − 2) σ2Sξ−3 and V CEVMr = σ2Sξ−2, (15)1034 S.O. Edeki, O.O. Ugbebor, E.A. OwolokoFigure 1: Estimates of the distributionsoe = ξ − 2. (16)Note. It is obvious from (13) that for any positive constant k with littlealgebra, we have:Set = kVCEVMr . (17)Remarks.1. Elasticity is said to be zero if ξ = 2; therefore, the stock price is lognor-mally distributed as indicated in the classical Black-Scholes model.2. Elasticity is −1 as proposed by Cox and Ross (see [14]), if ξ = 1.3. When ξ < 2, the so-called leverage effect observed mainly in equity mar-ket, shows that the stock volatility increases as the corresponding pricefalls.ON A DIVIDEND-PAYING STOCK OPTIONS PRICING... 10354. On the other hand, when ξ > 2, the so-called leverage effect observedmainly in commodity market, indicates that the volatility of the com-modity increases as the corresponding price increasesThe relationship between the stock price and the volatility of its return hasbeen shown empirically to be negative see [18, pg6]. This is guaranteed whenξ < 2 is considered.4. Concluding RemarksIn this paper, we proposed a pricing model for stock option valuation. Themodel is derived as a modification of the classical Black-Scholes pricing modelusing the constant elasticity of variance model with the incorporation of divi-dend yield parameter. This serves as an alternative to the traditional lognormalmodel for stock prices. In CEV model, the price variations of the underlyingasset are negatively correlated with variations in the level of volatility; thishelps in reducing the known volatility smile effects of the lognormal model.As a modification of the Black-Scholes pricing model, a non-constant volatilitypower function is introduced, and comparison between the CEV models withor without dividend yield is made.Acknowledgments and CommentsThe authors are indeed grateful to Covenant University for financial supportand provision of good working resources. We also thank the anonymous ref-eree(s) for their valuable time, constructive and helpful comments geared to-wards improving this paper.This work was carried out in collaboration by the concerned authors: SOE,OOU and EAO; with positive contribution. They all read and approved thefinal manuscript for publication.References[1] O.O. Ugbebor and S.O. Edeki, “On Duality Principle in Exponentially Levy Market”,Journal of Applied Mathematics & Bioinformatics, 3 (2), (2013), 159-170.[2] A. Habib, “The calculus of finance”, Universities Press (India) Private Ltd., (2011).[3] S.O. Edeki, I. Adinya, O.O. Ugbebor, “The Effect of Stochastic Capital Reserve on Ac-tuarial Risk Analysis via an Integro-differential equation”, IAENG International Journalof Applied Mathematics, 44 (2), (2014): 83-90.1036 S.O. Edeki, O.O. Ugbebor, E.A. Owoloko[4] N. Wang, “Price sensitivity to the exponent in the CEV model”, U.U.D.M. Project report,Uppsala Universitet, (2012).[5] F. Black, M. Scholes, “The pricing of options and corporate liabilities”, J. Pol. Econ. 81(1973): 637–659.[6] F. Black, “Studies of stock price volatility changes”, Proceedings of Meetings of the Amer-ican Statistical Association, Business and Economics Statistics Division, (1976).[7] R. Schmalensee, R. Trippi, “Common stock volatility expectations implied by optionpremia” Journal of finance, (1978).[8] E.A. Owoloko and M.C. Okeke, “Investigating the Imperfection of the B-S Model: A CaseStudy of an Emerging Stock Market”, British Journal of Applied Science & Technology4 (29) (2014): 4191-4200.[9] M. E. Adeosun, S. O. Edeki, O. O. Ugbebor, “Stochastic Analysis of Stock Market PriceModels: A Case Study of the Nigerian Stock Exchange (NSE)”, WSEAS Transactionson Mathematics, 14, (2015): 353-363.[10] J.D. Macbeth and L.J. Merville. “An empirical examination of the Black-Scholes calloption pricing model”, Journal of Finance, 34, (1979): 1173-1186.[11] S.O. Edeki, O.O. Ugbebor, E.A. Owoloko, Analytical Solutions of the Black–ScholesPricing Model for European Option Valuation via a Projected Differential TransformationMethod, Entropy, 17 (11), (2015): 7510-7521.[12] B. Lauterbach and P. Schultz, “Pricing warrants: an empirical study of the Black-Scholesmodel and its alternatives”, Journal of Finance, 45 (1990.), p. 1181.[13] F. Delbaen and H. Shirakawa, “A note of option pricing for constant elasticity of variancemodel”, Asia-Pacific Financial Markets, 9 (2), (2002): 85-99.[14] J. Cox and S. Ross, “The valuation of options for alternative stochastic processes”,Journal of Financial Economics, 3, (1976): 145-166.[15] S. Beckers, “The Constant Elasticity of Variance Model and Its Implications for OptionPricing”,The Journal of Finance, 35, (3) (1980): 661-673.[16] J. Xiao, Z. Hong, C. Qin, “The constant elasticity of variance (CEV) model and theLegendre transform-dual solution for annuity contracts” Insurance: Mathematics andEconomics, 40, (2007): 302-310.[17] S.O. Edeki, E.A. Owoloko, O.O. Ugbebor, “The Modified Black-Scholes Model via Con-stant Elasticity of Variance for Stock Options Valuation”, AIP Conference proceedings,1705, 020041 (2016); doi: 10.1063/1.4940289.[18] Y. L. Hsu, T. I. Lin, and C. F. Lee, “Constant elasticity of variance (CEV) option pricingmodel: Integration and detailed derivation”, Math. Comput. Simulation, 79 (2008),60–71.

Understanding How Dividends Affect Option Prices

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Understanding How Dividends Affect Option Prices

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