**Research**

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Today, the Black-Scholes model has been widely accepted as the most commonly used option pricing framework. In addition, there have been quite a number of empirical tests for the BS models. In most cases the studies have been positive in the sense that Black-Scholes model provides option values which are fairly close to the actual prices. However, in recent years the performance of BS model has been significantly deteriorating in the sense that option of the same stock with the same strike prices which should have a similar implied volatility actually exhibit progressively different implied volatilities. This brings a debate on whether the model should be applicable or reliable in the US business environment. The reason is because the US government decided to launch stock options very soon, hence it is necessary to assess whether BS model is appropriate to price the US stock option.

Short empirical project testing Black-Scholes option

The BS formula is derived from the no-arbitrage principle with the idea that to construct a riskless portfolio which is supposed to represent a self-financing replicating hedging strategy for the writer of the option. This shows that self-financing means that the writer of the option is not compelled to finance the hedging position, but instead apply the premium of the option to enter into the position. By replicating it means that risky position in the option is covered in all cases, no matter the direction taken by the price of the underlying asset. The Black-Scholes formula needs to have five input parameters. These include:

- The price of the underlying stock (
*A*or the stock index taken at time, t=0_{t}) - The free interest rate (
*r*) - The strike price of the option (
*K*) - Time taken to mature (T)
- The volatility of the underlying stock or the stock index (σ).

The BS model assumes that the asset price follows geometric Brownian motion with constant volatility. Thus, the expected future volatility plays a crucial role in finance theory. Finance researchers depend on the historical behavior of asset prices in order to develop expectations focusing on the volatility noting down the movements in volatility as they relate to prior volatility and the investors information set. However, in some cases BS implied volatilities tend to differ across exercise prices and times to expiration. For instance, options which are deep in the money or out of the money are said to have higher implied volatilities than at-the-money options. However, there is an assumption that failure of BS model to describe structure of the reported option price arises from its constant volatility.

The Geometric Brownian Motion Assumption

This assumption is very restrictive since reality provides many examples which contradict the idea that stock price or stock index movements have the same properties similar to Geometric Brownian Motion. The first assumption is that stock returns are normally distributed. However, some studies support the idea that large movements in stock returns are more likely than a normally distributed stock price model would predict. Practically, this means that likelihood of a large downward movement of equity prices is strongly underestimated. The second assumption is that the underlying implied volatility is constant. This assumption is inherent in many calculations, although working with volatility varies day by day.

Black-Scholes model theory review

The Black-Scholes model is one of the most significant concepts in today’s financial theory both in terms of approach and applicability. More focus in the theory of option pricing came to be understood in 1973, with publication of a paper titled *The Pricing of Options and Corporate Liabilities* by Black and Scholes who coined the closed form formula to ascertain prices of the European calls and puts, which were based on the assumptions which showed how to edge in a continuing manner the exposure of the short position of an option. This paper reviews the theory applied in the Black-Scholes model of option pricing and provides a more detailed analysis of assumptions of the model, as well as mathematical derivation processes of the model and evaluation of the inherent setbacks of the theory. However, the point to note is that the concepts behind the Black and Scholes provides a framework to think about option pricing.

According to Hull (2006), the BS model is a crucial breakthrough in the pricing of stock options. The model has a huge influence on the manner in which traders price and hedge options. However, there are some important assumptions to note which underlie the BS model. First and foremost, it is vital to understand that BS model assumes that stock prices move lognomarlly at a constant volatility rate σ. Thus, in the BS model, the underlying prices depend on the geometric Brownian motion, as well as the logarithmic growth rate of the prices which are distributed normally. Hull (2006) posits that the percentage changes in asset prices are lognomarlly distributed. However, according to Bodie et al. (2006), sudden extreme fluctuations, such as those experienced after an occurrence of a takeover attempt, are ruled out. Consequently, in the BS model, standard deviation of the percentage change in stock prices over a given time period* dt *is considered to be the same. Hull (2006) argues that this continues to be the same, regardless of stock price. Thus, the stock price movement over the infinite time *dt *is described using the following stochastic differential equation:

dS/S=μdt + σdZ

In the above differential equation, S is the stock price; *μ *is the expected rate of return on the stock. On the other hand, *dZ *is regarded as aWiener process with a zero mean and a constant standard deviation *σ *being the same as √*dt. *Bodie et al. (2006) also explained that another key component of the BS model is the risk-neutral valuation. This means that in a risk-neutral environment different people are indifferent to the risk; hence, they need a compensation for risk. In other words, BS model does not involve any variable which are affected by investor’s risk preferences. Therefore, investors can value options as though the expected rate of return on all the securities is to be a risk-free investment rate. In addition, another assumption of BS model is that the market is efficient; hence, there is no arbitrage activity in the markets. Thus, the option value should indicate all available data regarding the option and the underlying assets. Thus, the participants in an efficient market are unable to trade options in order to earn riskless profits. Because of the mentioned underlying assumptions, the BS model has various advantages. For instance, since it asserts that price of an option is function of the current stock prices *S, *the exercise price *K *and the risk free interest rate *r, *time to expiration t, stock price volatility *σ. *All these are considered independent variables of a trader’s risk preferences. Thus, BS model needs little information to price options, whereas the computational demands of the model require little information to price options. Generally, the main appeals of BS model are its ease of implementation and its simplicity.

Implied volatility

The volatility of the stock price is the only parameter of the Black-Scholes model which cannot be observed from the market data and terms which are specified by the market contracts. Therefore, the availability of a particular option of the market price can result into estimation of the stock market price, which can be obtained by numerically inverting the BS formula, hence yielding the implied volatility which makes the option value derived BS model equivalent to the market value. The accuracy of BS model should provide the measure of the expected volatility of the underlying asset over the remaining life of the option contract. Currently, many investors quote option market prices focusing on the implied volatility.

The manner in which the implied volatility varies with strike price for options of a fixed expiration is referred to as the volatility smile. Currently, the volatility smile is being used as a way of testing the validity of BS model by seeing if the implied volatility is independent of strike price as assumed by BS model. On the other hand, the fundamental assumption of the BS model is that the stock prices move with constant volatility. Thus, the implied volatility derived by inverting the BS formula should be similar to the options on the same underlying with the same maturities but with different strike prices. In addition, if the BS model is correct, it means that the implied volatility should be flat.

Black and Scholes option real data from NYSE pricing option

In the last two decades the stock market crash in the U.S. led the New York Stock Exchange to the launch of circuit-breaker mechanism which was applied to manually stock trading if stock prices fluctuated in an abnormal manner, in order to stabilize the market to protect the investors. Since then, the mechanism had been applied to minimize the stock fluctuations and reassure investors. Black and Sholes designed the first trading simulation test of the option valuation model by applying a sample of 2,039 six-month call option transactions on 545 NYSE securities for 766 trading days, which were derived from an option broker starting from May 1966 through July 1969. The values derived from BS model are computed daily. In addition, the volatility parameters are based on the daily returns of the underlying stock over the past year. The six month duration on the paper rate acts as a proxy for a risk-free interest. Therefore, to evaluate whether the BS model values are too high or too low, a strategy is conducted to ascertain whether the valuation model values are on average high or low. In addition, to ascertain whether the option writers’ premiums are too high or too low on average, all calls are purchased at market prices.

The volatility *σ *of the financial asset refers to a statistical quantity which needs to be determined beginning from the market information. It refers to the standard deviation of asset return or equivalent of algorithm price changes of the asset. There are different methodologies which are applied to refer to volatility estimation from information derived from the market ranging from direct calculation from the past historical volatility to the computation of the volatility implied in evaluation of an option price computed from using the Black and Scholes formula or some variants of it.

Some of the results are obtained in investigation of the statistical properties of volatility for the top 100 capitalized stocks in the US equity over a four year period. The empirical data were derived from the Trade and Quote (TAQ) database. This data base is maintained by the NYSE. Particularly, the data from the TAQ data base cover a period of 1,011 trading days ranging from January 1995 to December 1998. This is an important data base which contains all the information of all transactions which occurred for each stock which were traded in the U.S. equity markets. In this case study the capitalization considered is derived from the records of August 1998. Therefore, for each stock and for each transaction day, the time series of stock recorded transaction by transaction is considered.

The Trade and Quote database of the NYSE contains time stamped data on all trades and quotes on the NYSE. The TAQ database contains time calculated to the nearest second, price and volume of each transaction, quotations, as well as the time and the bid, and asks quotes. For instance, considering data obtained for the first quota of 1995, which is equivalent to 56 trading days, the analysis begins with the sixty most actively traded stocks on the NYSE, based on the average daily trading volume, which does not split the sample period as stocks split may detrimentally influence the trading activities of the stocks. Therefore, the preponderance of the price discovery for the NYSE stocks occurs on the NYSE rather than on other stock exchanges. When the most active contracts have less than five days to mature, the next most active contract is selected to eliminate the option expiration effects which are documented elsewhere. This means that each stock will have 58 option days, where a matching sample of stock option data is available. The reason is that the volume of measurements is required over a short period of time. Therefore, the option days need to be deleted with less than 20 trades for stock, the call or the put. On the other hand, since approach of an ex-dividend can result into an unusual activity in the option, all the option days are eliminated just before the dividend. This leaves 14 stocks, as well as a total of 231 option days.

The BS model formula is the most commonly used option pricing model today. The model has a massive influence on the manner in which investors price and hedge options. Therefore, this model relates to the option values relating to the five parameters, which are: risk-free interest, strike price, time to maturity, underlying asset price and volatility of the underlying asset return. The Black-Scholes model is based on the following ideal assumptions as shown in the reviewed literature. First, the underlying asset prices move with constant volatility. Second, the underlying asset returns are normally distributed. Third, all the traders are indifferent to the risks associates, meaning that they are risk-neutral. Finally, there are no riskless arbitrage opportunities. Because one of the fundamental assumptions underlying BS is that the asset prices move with constant volatility, implied volatilities which are derived by numerically inverting the BS formula. However, there are many reasons to guess why BS still plays a crucial role in actual option. This is an analytical approach meaning that it is easily comprehensible from the practitioner’s standpoint.