An Empirical Study of Stock Price Returns Using Black-Scholes Option Pricing Model

Abstract

This study examined empirically the predictive ability of Black-Scholes Option pricing model in Nigerian Stock Market by testing whether there is any significant difference between the market (underlying) price of the stock and the theoretical prices. The data used for this study was Coca-Cola Stock prices from 2018 to 2022 obtained from http://www.investing.com and were analyzed using Statistical packages such as Microsoft Excel and Minitab to obtain the result. The results showed that there is a significant difference between the underlying and the theoretical price (Black-Scholes Option Pricing Model). Based on the findings of this study, we conclude that the Black- Scholes model is not accurate in its price predictive ability on the Coca-Cola Stock prices over the years under study.

Introduction

I. INTRODUCTION

Stock is the proportion of a company's equity that is traded. Put differently, stock is one the instruments of a financial Market, otherwise, known as trading economy. A trading economy that concentrates on stock, only (a single portfolio) can be referred to as stock market. The price of stock is controlled by demand and supply vis-a-vis buyers and sellers, respectively. Particularly, the stock market prices fluctuate, due to some market forces. However, before investing into trading options, investors should have a good understanding of factors that determine the value of an option. These factors include the current stock price, the intrinsic value, time to expiration, volatility, cash dividends paid and macro-economic factors.

Speaking of the financial market, which is a trading environment that is composed of anything of value, that can be sold, bought or even exchanged, such as; stock bond, money, gold, structures, and many more. However, in this research our interest is in the first. In order to institutionalize and professionalize the trading in stock, the need to talk about institutes like the Nigerian stock exchange becomes necessary. The Nigerian stock exchange is a financial institution that trades exclusively on stock. Therein, we have the stock brokers, investors, a network of computer connected to a server, and many more. The Nigerian Stock Exchange started trading operation since 27th April, 1999, operates Automated Trading System (A. T. S). Unarguably, the stock exchange market is one of the important sectors of the Nigerian economy amongst which there is the Coca-Cola, a branch of the Nigerian Bottling Company which was introduced into the Nigerian market in 1951 and has since become a premium brand. With Nigeria as a developing country and the unstable nature of the economy, it is therefore important to test the market with a well-known model that is universally accepted for determining the price of derivative contracts.

The Black-Scholes Model for option pricing was developed in 1969 by Fisher Black and Myron Scholes, but was published in 1973 with the appearance in Chicago  Board as the first regulated market of negotiable options. This model was subsequently developed by Merton (1976).  It is noteworthy that for this scientific contribution, Myron Scholes and Robert C. Merton received the Nobel Prize for Economics in 1997 (the Swedish Academy of Sciences highlights the contribution of Fisher Black who was no longer alive at the time of the award).

The theory of Option pricing estimates a value of an option’s contract by assigning a price, known as a premium, based on the calculated probability that the contract will finish in the money (ITM) at expiration. Essentially, option pricing theory provides an evaluation of an option's fair value, which traders incorporate into their strategies.

While this model is useful, it is based on the following market assumptions that may hinder its accuracy, such as:

1. The short-rate interest rate and volatility are known to be constant through time.
2. No transaction costs or services associated with buying or selling of the option.
3. The options are European-style options which can only be exercised on the expiration date.
4. The returns on the underlying stock prices are normally distributed.
5. The Black-Scholes model assumes that markets are perfectly liquid and it is possible to purchase or sell any amount of options or their fractions at any given time.

Black and Scholes (1973) introduced a theoretical method to determine the options values, and they stated that the model follows a fixed systematic pattern based on relevant market indicators such as volatility, spot prices, time to expiration and expected risk-free rate of return. The first article that empirically examined the Black-Scholes model was written by MacBeth and Merville (1979). More recently, other empirical studies on the applications of Black-Scholes formula for option pricing were found in many well-structured  articles by Karoui, et al (1998); Kou (2002),  Haug and Taleb (2011) and, hence, a review of recent developments in the Black-Scholes models were synthesized in Saedi and Tularam (2018) and several other researchers like Frino and Khan (1991), Bakshi, et al (1997), Kim, et al (1997), Genkay and Salih (2003) found out that the BSOPM model was not the appropriate pricing tool in high volatility than in a low volatility.

 Angeli and Bonz (2010) tested the applicability and relevance of the Black–Scholes model for price stock index options and they determined the theoretical prices of options under the BSOPM model assumptions and then compared these prices with the real market values to find out the degree of variation in two different time zones and the result finally concluded that Black-Scholes model performed differently in the period before and after the financial crisis.

Sharma and Arora (2015) tested the relevance of Black Scholes Model in the Indian Stock market for the Option prices by using the model to calculate the theoretical option prices using the equation and then comparing it with the actual values. All the necessary assumptions were taken into consideration for option price calculation and the result concluded that the Black Scholes model values were not relevant to the market values of the stock options. Sethi and Nilakantan (2016) in their study explained that there was a critical contrast between the BSOPM call price and the market call price. As the quantity of perceptions expanded, the deviation of BSOPM price from the genuine market price expanded.  Several approaches have also been developed over the years to evaluate the real options value of an investment, see for example; Mckenzie, et al (2007); Grundy (1991);Kumar and Agrawal (2017); Cetin, et al (2006); Del Giudice et al., (2013), Sarkar, S (1995); Shinde and Takale (2012), Ugomma, et al (2023) and Ugomma and Benjamin (2023)  From the literature reviewed so far, we have seen the contributions of some authors and researchers that have made some remarkable contributions in the applications of Black-Scholes option pricing models in option or stock prices.

The pricing of Options in the market is dependent on certain factors such as spot price, volatility, and many more. Based on these factors, it is practically difficult to estimate the Option prices. Black-Scholes, in their contribution, developed the Black-Scholes Option pricing model, where they made some assumptions for the pricing of the option that makes it applicable to European Option pricing effectively. The question that begs for answer is whether Black-Scholes Option pricing model will still be effective for pricing real stock prices in the Nigeria Stock Market, since stock market experiences variations, with time in the spot price.

II. MATERIALS AND METHODS

A. The Black-Scholes Formula

The Black-Scholes formula can be derived for a call option on a non-dividend paying stock with strike price  and maturity . We assumed that the stock price follows a Geometric Brownian Motion. By the using the Ornstein-Uhlenbeck Processs as a solution to

Conclusion

The findings of this study reviewed that there is significant difference between the underlying price and the Black-Scholes Call price of Coca Cola\'s stock for each of the years, 2018 through 2022. Based on the findings, we conclude that the Black- Scholes model is not accurate in its price predictive ability on the Coca-Cola Stock prices over the years under study.

References

[1] Angeli and Bonz (2010). Changes in the Creditability of the Black-Scholes Option Pricing Model Due to Financial Turbulences, P. 57 [2] Bakshi, G.,Charles, C.,and Zhiwu, C.(1997). Empirical performance of Alternative Option Pricing Models, Journal of Finance, 3(5), 2–48 [3] Black, F. and Scholes, M.(1973).The Pricing of Options and Corporate Liabilities. The Journal of Political Economy, 81(3), 637-654. [4] Çetin, U., Jarrow, R., Protter, P. and Warachka, M.(2006).Pricing Options in an Extended Black Scholes Economy with Illiquidity: Theory and Empirical Evidence. The Review of Financial Studies, 19(2), 493-529 [5] Coca-Cola Sock data http://www.investing.com [Accessed on 3rd September 2022 [6] DelGiudice, M., Evangelista, F., and Palmaccio, M. (2016).Defining the Black and Scholes Approach: A first Systematic Literature Review. Journal of Innovative Entrepreneurship, 5(1), 1–13. [7] Frino, A., and Khan, E. (1991). The Black Scholes call option pricing model and the Australian Options Market: Where are we after 15 years, Accounting &Finance Working Papers No. 24, Faculty of Commerce 91/24, School of Accounting &Finance, University of Wollongong,1–21. [8] Genkay, R., and Salih, A. (2003). Degree of Mispricing with the Black–Scholes Model and Non-Parametric cures. Annals of Economics and Finance, 4(1), 73–101. [9] Grundy, B. D., (1991), ‘Option Prices and the Underlying Asset’s Returns Distribution. Journal of Finance, 46, 1045 – 1070. [10] Haug, E. and Taleb, N., (2011). Option Traders (very) Sophisticated Heuristics, never the Black-Scholes-Merton Formula. Journal of Behavior and Organization, 77, 97-106. [11] Karoui, N., Jeanblanc- Picqué, M. and Shreve, S.,(1998). Robustness of the Black and Scholes Formula. Mathematical Finance, 8(2), 3-126. [12] Kim, H., Jong, C. R., and Mohammed, F. K. (1997). An Empirical Investigation of Put Option Pricing: A specification Test of At-the-Money Option Implied Volatility. Journal of Financial And Strategic Decisions, 10(2), 75–83 [13] Kou, S.G., (2002). A Jump-Diffusion Model for Option Pricing. Management Science, 48(8), 1086-1101. [14] Kumar, R., and Agrawal, R. (2017). An Empirical Investigation of the Black–Scholes Call Option Pricing Model with reference to NSE. International Journal of BRIC Business Research (IJBBR), 6(2), 1–11. [15] MacBeth, J. and Merville, L. (1979). An Empirical Examination of the Black-Scholes Call Option Pricing Models. The Journal of Finance, 34(5), 1173-1186. [16] McKenzie, S., Gerace, D. and Subedar, Z., (2007). An Empirical Investigation of the Black-Scholes model: Evidence from the Australian Stock Exchange. Australasian Accounting, Business and Finance Journal, 1(4), pp. 71-82. [17] Merton, R. (1976). Option Pricing When Underlying Stock Returns Are Discontinuous. Journal of Financial Economics, 3,125-144. [18] Nilakantan, N. and Jain, A. (2014). Option Pricing with Skewness and Kurtosis. Second International Conference on Global Business, Economics, Finance and Social Sciences. Chennai: GB14. [19] Saedi, Y. and Tularam, G., (2018), A Review of the Recent Advances Made in the Black-Scholes Models and Respective Solutions Methods. Journal of Mathematics and Statistics, 14(1), 29-39. [20] Sarkar, S., (1995). Black-Scholes, As Compared to Observed Price: An Empirical Study. Managerial Finance, 21(10), 1-8 [21] Sharma, M. and Arora, D. (2015). Study of Relevance of Black-Scholes Model in Indian Stock Option Market. IJARIIE, 324-334. [22] Shinde, A.and Takale, K. (2012). Study of Black Scholes Model and its Application. Procedia Engineering, 270-279. [23] Ugomma, C.A and Benjamin, F.O (2023). An Empirical Verification on the Performance of Black-Scholes Option Pricing Model in Nigerian Stock Market. International Journal of Modern Science and Research technology. Vol. 1 (7). [24] Ugomma, C.A, Onuoha, N.O and Nkem, C.U (2023). Evaluation of the Credibility of Black-Scholes Option Pricing Model durin Turbulent and Normal days. International Journal of Novel Research and Development. Vol 8(6)

Copyright

Copyright © 2023 Chukwudi Anderson Ugomma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.