Stock Option Pricing Using Binomial Trees with Implied Volatility

Aimmatul Ummah  Alfajriyah; Endah R.M  Putri; Daryono Budi  Utomo; Moch. Taufik  Hakiki

doi:10.20956/j.v20i3.34476

Authors

Aimmatul Ummah Alfajriyah Departemen Aktuaria, Institut Teknologi Sepuluh Nopember
Endah R.M Putri Departemen Matematika Institut Teknologi Sepuluh Nopember
Daryono Budi Utomo Departemen Matematika Institut Teknologi Sepuluh Nopember
Moch. Taufik Hakiki Departemen Aktuaria Institut Teknologi Sepuluh Nopember

DOI:

https://doi.org/10.20956/j.v20i3.34476

Keywords:

Binomial Tree, Black-Scholes, Implied Volatility, Option

Abstract

The Black-Scholes model provides an analytical solution in option pricing and has been widely used in finance. This model assumes constant volatility. Pricing option incorporating implied volatility is conducted using implied binomial tree. This study aims to simulate the prices of put options and call options using implied binomial trees, binomial trees and the Black-Scholes model and determine the factors that influence option prices. The simulation was conducted using Matlab. The option price resulted from implied binomial tree and binomial tree are compared with the option prices of the Black-Scholes model to determine the difference of option prices with constant volatility and option prices incorporating implied volatility. The implied binomial tree method provides better option prices than the binomial tree based on small relative error value to the Black-Scholes model. This is caused by the transition probability value of stock price movements in the implied binomial tree at each point is different, whereas in the binomial tree the value of transition probability is same. Furthermore, increasing the time step causes the option prices obtained from the implied binomial tree converge to the Black-Scholes. It is concluded that these three methods can be used in option pricing. Factors that influence the option price are stock price, strike price, interest rate and maturity date, are also obtained

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References

. Barle, S. & Cakici, N., 1998. How To Grow a Smiling Tree. The Journal of Financial Engineering, Vol. 7, No. 2, 127-146.

. Black, F. & Scholes, M., 1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, Vol. 81, No. 3, 637-654.

. Brigham, E. F. & Daves, P. R., 2019. Intermediate Financial Management, Thirteenth Edition. Cengage Learning., United States of America.

. Derman, E. & Kani, I., 1994. Riding on A Smile. Risk, Vol. 7, No. 2, 32-39.

. Derman, E. & Miller, M. B., 2016. The Volatility Smile. John Wiley & Sons, Inc., Canada.

. Emmanuel, F., Adedoyin, A. O. & Hammed, O. O., 2014. Performance Measure of Binomial Model for Pricing American and European Options. Applied and Computational Mathematics, Vol. 3, No. 6, 18-30.

. Gatheral, J., Jaisson, T. & Rosenbaum, M., 2022. Volatility Is Rough. In Commodities, Second Edition. Chapman and Hall., London.

. Härdle, W. K., Hautsch, N. & Overbeck, L., 2009. Applied Quantitative Finance, Second Edition. Springer., Berlin.

. Huang, D., Schlag, C., Shaliastovich, I. & Thimme, J., 2019. Volatility-of-Volatility Risk. Journal of Financial and Quantitative Analysis, Vol. 54, No. 6, 2423-2452.

. Hull, J., 2022. Options, Futures, and Other Derivatives, Eleventh Edition. Pearson Education Inc., New York.

. Horvath, B., Muguruza, A. & Tomas, M., 2021. Deep Learning Volatility: A Deep Neural Network Perspective on Pricing and Calibration in (Rough) Volatility Models. Quantitative Finance, Vol. 21, No.1, 11-27.

. Jiang, L. & Li, C., 2005. Mathematical Modeling and Methods of Option Pricing. World Scientific Publishing Co. Pte. Ltd., Singapore.

. Jiang, Q., 2019. Comparison of Black–Scholes Model and Monte-Carlo Simulation on Stock Price Modeling. Advances in Economics, Business and Management Research, Vol. 109, 135-137.

. Kim, Y. S., Stoyanov, S., Rachev, S. & Fabozzi, F. J., 2019. Enhancing Binomial and Trinomial Equity Option Pricing Models. Finance Research Letters, Vol. 28, 185-190.

. Liu, Y. F., Zhang, W. & Xu, H. C. 2014. Collective Behavior and Options Volatility Smile: An Agent-Based Explanation. Economic modelling, Vol. 39, 232-239.

. Maulida, V., Siswanah, E. & Nisa, E. K., 2019. Penentuan Harga Opsi Tipe Eropa dengan Model Binomial. Journal of Mathematics and Mathematics Education, Vol. 1, No. 1, 65-72.

. Muroi, Y., 2022. Discrete Malliavin Greeks. In Computation of Greeks Using the Discrete Malliavin Calculus and Binomial Tree. Springer., Singapore.

. Muroi, Y. & Suda, S., 2022. Binomial Tree Method for Option Pricing: Discrete Cosine Transform Approach. Mathematics and Computers in Simulation, Vol. 198, 312-331.

. Ni, S. X., Pearson, N. D., Poteshman, A. M. & White, J., 2021. Does Option Trading Have A Pervasive Impact on Underlying Stock Prices?. The Review of Financial Studies, Vol. 34, No. 4, 1952-1986.

. Nian, K., Coleman, T. F. & Li, Y., 2021. Learning Sequential Option Hedging Models from Market Data. Journal of Banking & Finance, Vol. 133, 106277.

. Roul, P., 2022. Design and Analysis of A High Order Computational Technique for Time‐Fractional Black–Scholes Model Describing Option Pricing. Mathematical Methods in the Applied Sciences, Vol. 45, No. 9, 5592-5611.

. Rubinstein, M., 1994. Implied Binomial Trees. The Journal of Finance, Vol. 49, No. 3, 771-818.

. Simarmata, J. E. & Ahzan, Z. N., 2021. The Use of The Black Scholes Model in Determining the Price of the European Type Option. Journal of Research in Mathematics Trends and Technology, Vol. 3, No. 2, 2021, 15-25.

. Soini, V. & Lorentzen, S., 2019. Option Prices and Implied Volatility in The Crude Oil Market. Energy Economics, Vol 83, 515-539.

. Srivastava, A. & Shastri, M., 2020. A Study of Black–Scholes Model’s Applicability in Indian Capital Markets. Paradigm, Vol. 24, No. 1, 73-92.

. Subartini, B., Riaman, R., Nabiilah, N. & Sukono, S., 2021. Analisis Penerapan Metode Pohon Binomial dan Metode Black-Scholes dalam Penentuan Harga Opsi Beli. Teorema: Teori dan Riset Matematika, Vol. 6, No. 2, 260-266.

. Vagnani, G., 2009. The Black–Scholes Model as A Determinant of The Implied Volatility Smile: A Simulation Study. Journal of Economic Behavior & Organization, Vol. 72, No.1, 103-118.

. Vulandari, R. T. & Sutrima., 2020. Black-Scholes Model of European Call Option Pricing in Constant Market Condition. International Journal of Computing Science and Applied Mathematics, Vol. 6, No. 2, 46-49.