### Performance Measure of Binomial Model for Pricing American and European Options

Received: 28 September 2014    Accepted: 6 October 2014    Published: 20 October 2014

Abstract

Binomial model is a powerful technique that can be used to solve many complex option-pricing problems. In contrast to the Black-Scholes model and other option pricing models that require solutions to stochastic differential equations, the binomial option pricing model is mathematically simple. It is based on the assumption of no arbitrage. The assumption of no arbitrage implies that all risk-free investments earn the risk-free rate of return and no investment opportunities exists that requires zero amount of investment but yield positive returns. It is the activity of many individuals operating within the context of financial market that, in fact, upholds these conditions. The activities of the arbitrageurs or speculators are often maligned in the media, but their activities insure that financial markets work. They insure that financial assets such as options are priced within a narrow tolerance of their theoretical values. In this paper we use binomial model to derive the Black-Scholes equation using the risk-neutral expectation formula. We also use binomial model for the valuation of European and American options. Lastly, the primary reason why the binomial model is used is its flexibility compared to the Black-Scholes model and it is also used to price a wide variety of options.

Keywords

American Option, Black-Scholes Model, Binomial Model, European Option

References
 [1] Björk, Tomas. Arbitrage theory in continuous Time, Oxford: OXFORD UP, 2004. Print. [2] Black F, and Scholes M. (1973): The pricing of options and corporate liabilities, Journal of political Economy, vol. 81(3).637-654. [3] Black F, and Scholes M.S (1973),”The pricing of options and corporate liabilities”, Journal of political Economy, vol.31, pp.637-659 [4] Black, F. (1976), ‘Studies of stock price volatility charges’, proceeding of the 1976 meeting of the American statistical Association, pp. 171-181, -79- [5] Blais, Marcel, Class lecture. Financial Mathematics 1, Worcester polytechnic institute, Worcester, MA. 9 Sept., 2009. [6] Boyle P,(1977): options: A Monte Carlo approach, Journal of Financial Economics, vol.4(3),323-338. [7] Brennan M. and Schwartz E.(1978): Finite Difference Methods and Jump processes arising in the pricing of contingent claims, Journal of Financial and Quantitative Analysis, vol.5(4),461-474. [8] Cox J., Ross S, and Rubinstein M.(1979): Option pricing: A simplified approach, Journal of financial Economics, vol.7,229-263. [9] Cox J.C and S.A Ross, “Option pricing: A simplified approach”, Journal of financial Economics, 1979. [10] Cox, J and Ross, S.(1976), ‘The valuation of options for Alternative stochastic process’, Journal of Financial economics, vol.3,pp. 145-166. [11] Fadugba S, Nwozo C. and Babalola T. (2012): The comparative study of Finite Difference method and Monte Carlo method for pricing European option, International Institute of Science, Technology and Education, Mathematical theory and Modeling, vol.2,No.4,60-66,USA. [12] Fadugba S.E et al (2012): “On the strength and weakness of Binomial model for pricing vanilla options”, International Journal of Advanced Research in Engineering and Applied Sciences, vol.1,No.1,13-22. [13] Fadugba S.E et al, (2013), On the Robustness of Binomial model and finite difference method for pricing European options, International Journal of IT, Engineering and Applied Sciences Research, Vol. 2, No.2, 2013. [14] Fadugba S.E, Okunola J.T and Adeyemi O.A, (2012),”On the strength and weakness of Binomial model for pricing Vanilla options”, International Journal of Advanced Research in Engineering and Applied Sciences, vol.1, No.1, 13-22. [15] Fadugba S.E. et al.(2012): On the stability and Accuracy of finite Difference Method for option pricing, International Institute of Science, Technology and Education, Mathematical theory and Modeling, Vol.2,No.6,101-108, USA. [16] Hull J.(2003): Options, Futures and other Derivatives, Pearson Education inc. Fifth Edition, Prentice Hall, New Jersey. [17] Merton, R.(1973), ‘Theory of Rotational option pricing’, The Bell Journal of Economics and management Science, vol. 4(1), pp.141-183. [18] Odegbile, O.O, “Binomial option pricing Model”. African institute of Mathematical Sciences [19] Review of option pricing literature and an online Real-time option pricing application development by shuLiu, 2007. [20] Robert A. Jarrow and James B. Wiggins (1997), “Option pricing and implied volatiles Journal of Economic survey”, Vol . 3, No 1, Pp. 125-144 [21] Ross A (2002), “The Britten-Jones and Neuberger smile-consistent with stochastic volatility option pricing model. A further analysis”, International Journal of Theoretical and Applied finance, Vol. 5, pp 1-31 [22] Ross S.M.(1999): An introduction to the mathematical Finance: Options and other Topics. Cambridge University Press, Cambridge. [23] Rubinstein M. (1983), “Displaced diffusion option pricing”, Journal of finance, Vol. 38, pp 213-217. [24] Rubinstein, M. (1994), “Implied Binomial Trees”, Journal of Finance, Vol. 49,pp. 717-818. [25] Ruppert, David. Statistics and Finance: An introduction, New York: Springer 2004, Print. [26] Sherwani, Y, “Binomial Approximation Methods for Option Pricing”, U.U.D.M Project Report 2007:22, Department of Mathematics, Uppsala University, 2007. [27] Trigeorgis, Lenos, “A log-transformed Binomial Numerical analysis method for valuing complex Multi-option investments”. The Journal of Financial and Quantitative analysis 26.3(1991):309-326,JSTOR.Web.19,April 2011 [28] Weston J., Copeland T., and Shastri K.(2005): Financial theory and corporate policy. Fourth Edition, New York, Pearson Addison Wesley.
• APA Style

Fadugba Sunday Emmanuel, Ajayi Olayinka Adedoyin, Okedele Olanrewaju Hammed. (2014). Performance Measure of Binomial Model for Pricing American and European Options. Applied and Computational Mathematics, 3(6-1), 18-30. https://doi.org/10.11648/j.acm.s.2014030601.14

ACS Style

Fadugba Sunday Emmanuel; Ajayi Olayinka Adedoyin; Okedele Olanrewaju Hammed. Performance Measure of Binomial Model for Pricing American and European Options. Appl. Comput. Math. 2014, 3(6-1), 18-30. doi: 10.11648/j.acm.s.2014030601.14

AMA Style

Fadugba Sunday Emmanuel, Ajayi Olayinka Adedoyin, Okedele Olanrewaju Hammed. Performance Measure of Binomial Model for Pricing American and European Options. Appl Comput Math. 2014;3(6-1):18-30. doi: 10.11648/j.acm.s.2014030601.14

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title = {Performance Measure of Binomial Model for Pricing American and European Options},
journal = {Applied and Computational Mathematics},
volume = {3},
number = {6-1},
pages = {18-30},
doi = {10.11648/j.acm.s.2014030601.14},
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abstract = {Binomial model is a powerful technique that can be used to solve many complex option-pricing problems. In contrast to the Black-Scholes model and other option pricing models that require solutions to stochastic differential equations, the binomial option pricing model is mathematically simple. It is based on the assumption of no arbitrage. The assumption of no arbitrage  implies that all risk-free investments earn the risk-free rate of return and no investment opportunities exists that requires zero amount of investment but yield positive returns. It is the activity of many individuals operating within the context of financial market that, in fact, upholds these conditions. The activities of the arbitrageurs or speculators are often maligned in the media, but their activities insure that financial markets work. They insure that financial assets such as options are priced within a narrow tolerance of their theoretical values. In this paper we use binomial model to derive the Black-Scholes equation using the risk-neutral expectation formula. We also use binomial model for the valuation of European and American options. Lastly, the primary reason why the binomial model is used is its flexibility compared to the Black-Scholes model and it is also used to price a wide variety of options.},
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AB  - Binomial model is a powerful technique that can be used to solve many complex option-pricing problems. In contrast to the Black-Scholes model and other option pricing models that require solutions to stochastic differential equations, the binomial option pricing model is mathematically simple. It is based on the assumption of no arbitrage. The assumption of no arbitrage  implies that all risk-free investments earn the risk-free rate of return and no investment opportunities exists that requires zero amount of investment but yield positive returns. It is the activity of many individuals operating within the context of financial market that, in fact, upholds these conditions. The activities of the arbitrageurs or speculators are often maligned in the media, but their activities insure that financial markets work. They insure that financial assets such as options are priced within a narrow tolerance of their theoretical values. In this paper we use binomial model to derive the Black-Scholes equation using the risk-neutral expectation formula. We also use binomial model for the valuation of European and American options. Lastly, the primary reason why the binomial model is used is its flexibility compared to the Black-Scholes model and it is also used to price a wide variety of options.
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Author Information
• Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria

• Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria

• Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria

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