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Black-Scholes

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Introduction

In the era of information technology, current business has been operated completed differently from previous. Of course, financial market also become more and more complicated in order to associate with change of business structure. Consequently, this had led to rising of financial risk to investors invisibly. Hence, investors would like to speculate and hedge the financial risk as much as possible when do financial decision. Derivative financial instruments such as options, futures and others have been introduced and more commonly used to manage financial risk for improving decision making in this dynamic competitive environment.

Options are defined as securities which one party has the right (no obligation) to buy or sell underlying assets with certain price within a certain/specific period of time (Hull, 2012). The option can be either call (right to buy) or put (right to sell) in the form of American options (exercised any time until expiry date) or European options (exercised on expiry date) as either traded options (standard option contracts) or overt-the-counter options (tailor made options). Due to various choices of options, different option pricing models such as Put-Call Parity, Black-Scholes, Cox-Rubenstein Binominal, Risk-Neutral valuation, the Greeks and others has been developed and applied in current financial market.

Black-Scholes Option Pricing Model (BS) BS is designed and introduced by Fisher Black and Myron Scholes in 1973 with the assumptions of the market is efficient, returns are lognormal distributed, no commission or transaction cost is charged, no dividend is paid, no penalties to short selling, terms of European option is used, interest rate is remained constant and known rate (Black & Scholes, 1973). Thereafter, the assumption of no dividends has been relaxed by Robert Merton in the same year. Besides that, assumption of no commission or transaction cost is charged and restriction of constant instant rate has been relaxed by Jonathan Ingerson and Robert Merton respectively in 1976 (Kwan, 2014). According to Bob Ryan (2009), BS not only can be used to value the real options embedded in capital investment projects but also provides a general framework for company valuation with same five key variables identified. The as shown in Table 1.

Advantages and Disadvantages of Black-Scholes Option Pricing Model The biggest strength of BS is the possibility of estimating market volatility of an underlying asset generally as a function of price and time instead of direct reference to specific investor characteristics. For instance, expected yield and utility function. This has led to widely usage of BS by options market participants for options valuation. Volatility of option prices is sensitive and subject to the underlying assets movement. Because the buyer has limited risk from adverse price movements, greater asset price volatility tends to raise the price of an option (Neely, 2005). Hence, the option price has to be measured carefully in order to have more accurate option values. This approach is popular and more often suggested as it is easy to be applied. The simple mathematic working has speed up the process resulted large volumes of option can calculated in a short time. This method is easy as it assumes neither transaction cost nor commission is charged. Consequently, this has become negative implication to BS. As the cost of options may be understated without consideration of any additional relevant cost incurred. It is quite impossible to pay nothing for an option other than exercise price in real world, there’s no such thing as a free lunch in real life. The other advantage of this model is straight forward with assumption of constant interest rate applied throughout the period of option. This is quite impossible that interest is remained unchanged. In reality, interest rate is rapidly changed although the fluctuation is insignificant normally. Whereas dramatic interest rate may be fluctuated due to uncertain economic in future. If the global economy accelerates, increase in long-term interest rates will be faster. Conversely, the interest rate will be pushed down of there is a global recession (Conerly, 2015). Hence, the accuracy of valuation option is doubt if constant interest rate especially longer time to maturity. For example, an explicit strategy in underlying assets and risk-less bonds whose terminal payoff is equally same as payment of derivative security at maturity date. The valuation of options not only affected by interest rate but also distorted by time to maturity. The greater flexibility of time constraint the greater value of options. Besides that, no arbitrage opportunity is allowed in this method due to assumption of markets are efficiency and self-replicating strategy or hedging such as an explicit strategy in underlying assets and risk-less bonds whose terminal payoff is equally same as payment of derivative security at maturity date (Teneng, 2011). In fact, it is quite impossible to beat the market in real life. The efficient market hypothesis maitains that all stocks are perfectly priced in associated with all relevant public and private information (Bergen, 2011). In addition, the model provides a reasonable value as guideline which allow investors to have the “objective price” for an option. So that, investor may consider the actual cost of the option based on the objective price calculated when do decision making or deal. On the other hand, the objective price of an option may mislead investors to perform wrong judgement if the accuracy of “objective price” is low as a result of wrong assumptions have been made. For instance, no dividends is paid during the option’s life in BS approach. Most companies would like to pay shareholders dividends not only as a return and a sign of company’s strength and profit to shareholders but also indication of positive expectation of future under their management and leadership and makes the shares become more attractive. Hence, BS model has been re-designed by Robert Merton later by discounting the value of future dividend and subtract it from the stock price (Bodie, Kane, Marcus and Jain, 2014). Last but not least, the key disadvantage of BS is designed in terms of European options only, i.e. exercised the option on the expiry date only. This has restricted the approached is allowed to use for evaluate European options only. The put-call parity is calculated with using BS working too. If the options are in American terms, then BS cannot be used. In contrast, Binominal can be used to evaluate American options with consideration of exercising at any time until maturity date but which is more complicated and expensive. American options are more popular and welcome due to greater flexibility but also more valuable. Case Study – Valuing Coca-Cola and Pepsico Options A real case study (Gardner & McGowan, Jr., 2012) for valuation of call options, i.e.: owners have the right to buy the stocks at pre-determined price by using the Black-Scholes option pricing is discussed here. Both selected companies, Coca-Cola (KO) and PepsiCo (PEP) are well-known established beverage companies with more than one hundred years history. Obviously, there is 2.70% dividend yield is shown in Figure 1 as per study case, however underlying of BS assumptions that the stock does not pay dividends has been applied in this case study. The approach without considering of dividends paid out will overstate the value of call options. This may be misleading the owner pays higher cost to buy the option which is not value for money. According to NASDAQ (2010), both companies paid dividends every year. Hence, the assumption of no dividends paid during option’s life should be excluded in this case study. The stock price should be adjusted accordingly by subtracting of discounted value of future dividends. According to Figure 2 KO options, there is more than 100 transaction of call options at strike price between USD 65 to USD 72.50. There is minimum transaction for strike piece lower than USD65 and higher than USD75. Hence, the option price set at USD50 which may not be so marketable in the options market. The history of stock price at last one year at range of approximately US$55 to USD70. The put options may not be accepted by any party as the stock price very unlikely to drop USD18 in less than one month time. No one will like to have options that there is a right but very unlikely to exercise the right unless the option price volatilities again stock price closely. Other than assumptions, wrong input of variables also will distort the valuation of call options. In the case study, the time to expiration of KO and PEP has been indicated as 24 days (maturity date 10 Dec 2010) and 19 days (expiry date, 17 Dec 2010) respectively in the Table 1 and Table 2 of case study. The variable has been wrongly recorded. The correct time to expiration for KO and PEP are 19 days and 24 days respectively. The longer duration to expiry date would be more costly as greater flexibility. The call valuation has been overstated of approximately 1% of 5 days variances due to wrong variables has been input. This would be consider significant impact to the valuations of options. Furthermore, the interest rate is fluctuated 0.02 for 4 weeks bank discount based on Figure 4 in the case study. Hence, the interest rate is expected to be fluctuated in the following seven months from May 2010 until maturity date, November 2010. In fact, the nearest date of maturity date should be used in order to calculate more accurate option price. Hence, assumption of interest rate remain constant and know rate should not applied in this case study. The valuation of option will be affected and this may lead to wrong judgement due to inaccuracy of objective price of option. Conclusion Although there are so many limitations of Black-Scholes model but which is still popular and widely used as valuation tools to improve financial decision making until today. The introduction of Black-Scholes has soared the option markets in 1970s. This is mainly because greater benefits has brought to users. The assumptions set are due to uncertainty which will be known only when it is happened. The simple and straight forward formula has saved a lot of times of options markets participants in the evaluation of option pricing. For instance, BS formula is strongly recommended by Accounting Board to calculate the stock options for compensation to staff after Enron case in Year 2001. Although it is not an actual price of options given, but objective price of options is calculated and allow investors to perform decision making based on the basis. In order to have more accuracy result, the variables can be set as few scenarios so that investors have more understanding and details when evaluate the options. There is no much time required in order to calculate option pricing by using Black-Scholes. The formula can be set up at excel worksheet easily and continuously used and cost effective. The options are more and more popular in the financial market due to highly competitive environment. Hence, it is essential to options market participants to fully understand several of pricing models in order to manage financial risk. In the wake of the high-tech collapse, it’s easy to see why people might be sceptical of a valuation tool that arguably exaggerated those companies’ growth potential (Copeland & Tufano, 2004). Therefore, it is very important to apply the appropriate option pricing models and also smart decision making. Again, the option pricing calculation is only a guideline to assist in decision making.

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