Authors: Chukwudi Anderson Ugomma
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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.
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:
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
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.
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