Since the Black-Scholes (B-S) Model was proposed, it became a widely used pricing model in the options market. This paper critically discusses the suitability of using the Black-Scholes model for pricing derivatives from two points: its own accuracy and the accuracy of input data. Finally, it is safety to conclude that the B-S pricing model is only the best tool currently.
Since the option firstly came into the market in1973, it became one of the best choices among derivatives for investors to invest, speculate and hedge. Then with the option being extensively and fruitfully applied, a lot of models for pricing are proposed by many researchers after in-depth study and exploration, such as the Black-Scholes (B-S) Model (Black and Scholes, 1973), the Binomial Pricing Model (Chalasani,1999 and Lee, S. Park, H., and Jeon, 2007), Monte Carlo Simulation (Rubinstein, 1981), Finite Difference Method and so on. And the most influential one is Black-Scholes (B-S) model created by Fisher Black and Myron Scholes (1973). It has already been considered as one of the most successful models in applied economics. Based on the assumptions that stock prices follows a geometric Brownian motion and the logarithm of stock prices obeys normal distribution, a portfolio including a stock and its derivative is constructed. The proceeds of two positions in the portfolio are highly negative correlated and the stock earnings (loss) are always offset by derivative securities losses (gains). As the portfolio is risk-free, the yield is equal to the risk-free interest rate in the case of Risk-free in a small time interval. Therefore, the present value of the portfolio is determined by the risk-free rate and the duration. The BS model is as follows: In this expression, AŽA¼ is the instantaneous expected return on the stockError: Reference source not found is the instantaneous volatility of the return, and z (t) is a standard Brownian motion or a Wiener process and S is the underlying asset. According to the model, the Black-Scholes equation, was derived by setting an instantaneous riskless portfolio composed by appropriately weighted stocks, options and bonds. The B-S model’s specific pricing formulas are as follows:
for the call option, and
for the put option, where Where R is the constant risk-free interest rate and N(x) is the normal cumulative density function, K is exercise price, Error: Reference source not found is the standard deviation of stock returns, T is the time to maturity options. According to Bruno Dupire, “implied Black-Scholes volatilities strongly depend on the maturity and the strike of the European option under scrutiny”. Then, this problem was solved easily by Merton in 1973. That makes the BS model more applicable. Just like what Black had pointed out during his lifetime, the B-S model for option pricing should really be called Black-Merton-Scholes model.
The output accuracy of any theoretical pricing model depends on the exactness of input and the model itself. Therefore, perfect combination of accurate models and accurate input data creates perfect result. Both are indispensable.
The accuracy of Model assumptions essentially determined whether the model is a perfect description of the real world. In deriving the B-S model, many important assumes are used: 2.1.1 Market is frictionless market with transaction efficiency (Black and Scholes, 1973) Under this assumption, in essence, it implies that the underlying assets can be freely traded without any restrictions. Everybody is free to borrow funds at the same rate. Undoubtedly, it does not take the impact of taxes into account. However, the market’s operation is not without friction in the real world. In some markets, such as China’s stock market, the underlying asset cannot be traded freely, because there are restrictions on daily change limits. What’s more, there is also a complete ban or limits on short selling in some of the stock market. And sometimes the proceeds from short selling cannot be fully used. Wherefore, if the freedom of short selling is not given, the demand of securing the short equity using the put option will be more intense. This may result in a higher price of the put option than the call option. While options trading are not taxed, almost no one option investment will be better than another under the influence of tax policy issues. Thus, most traders do not usually consider the tax implications. In the real world, traders cannot borrow unlimited funds. If the funds can be borrowed for free, traders can always borrow money, and then put them into the Options Clearing House. Since the lending rates are equal, the Options Clearing House should also pay the interests because of the initial performance bond (margin). Therefore, obtaining the initial performance bond is not a problem. However, traders may be forced to close the contract before expiration of the option because of not making up the maintenance margin. Even if borrowing unlimited funds, traders have to face different interest rates on loans, always higher than lending rates. Finally, if the difference between the rates of borrowing and loading is more obvious, pricing models offer less reliable data. The transaction cost in the options investment is another factor which has to be considered as well. Broker commissions, clearing fees and membership fees and market maker’s bid-ask spread are all transaction costs. Once considering transaction costs, many investment portfolios theoretically acceptable are often not feasible. 2.1.2 “The Stock Pays no Dividends” (Black and Scholes, 1973) The B-S model assumes that the stock pays no divides or other distributions. This is clearly an unrealistic assumption. Payment of dividends will increase the intrinsic value of the put option and reduce the value of the call option. As there are protective measures for options paying dividends in the early OTC (Over-the-Counter) market. For example, the adjustment of exercise price is used to eliminate the impact of dividends paid on the option value. But the current contractual terms of stock options will not be adjusted with cash bonus payment correspondingly in floor trading and the OTC market. Finally, the dividends become an important factor pricing the options. 2.1.3 “The Short-Term Interest Rate is Constant through Time” (Black and Scholes, 1973) The interest rate is assumed to be risk-free and fixed in the B-S pricing model. In most markets, risk-free interest rate is the interest rates on government securities. However, the actual market risk-free interest rate is usually not fixed but variable. Because the impact of changes in interest rates is a function of the options duration, and most of all listed options contract period of less than 9 months, it is not sufficient to constitute a significant impact on value, unless the interest rate changes to a large extent and the actual value is high. Therefore, the interest rate is not an important factor compared to the underlying asset price or price volatility. But this does not mean that traders can completely ignore the possibility of fluctuation in interest rates. Particularly, after the long-term stock options were introduced into the market, the interest rate impact is more and more important. 2.1.4 “The variance rate of the return on the stock is constant” (Black and Scholes, 1973) The B-S option pricing model assumes that when option expires, the standard deviation of expected returns of stock remain unchanged. That means that the future stock price volatility is constant. But actually, stock price volatility is influenced not only by stock prices, but also by the time the option expires, and other factors. It cannot remain constant in the life of the option. If the stock price volatility associated with stock prices, the stock price may deviate from the lognormal distribution in the B-S pricing model (Corrado, and Miller, 1993). Then pricing error may exist according to the B-S pricing model. If the stock price and volatility are positive correlated, the B-S pricing model will tend to underestimate the value of the call option in the virtual the state and overestimate the value of the state of the put option in the virtual the state. When the stock price goes up, the volatility is also rising. That means higher stock price appears in a higher probability in the geometric Brownian motion. Correspondingly, when stock prices fall, the volatility falls down. That means lower stock prices arise in a less probability in the geometric Brownian movement. In contrast, if the stock price and volatility are negative correlated, the B-S pricing model will tend to overestimate the value of call option price in the virtual the state and underestimate the value of the price of a put option in a virtual state for the following reasons. When volatility declined with stock prices rising, a high stock price is difficult to achieve. When volatility increased with stock prices falling, a very low stock price is easy to achieve. 2.1.5 Underlying Asset Price Changes in a Continuous Manner (Black and Scholes, 1973) The underlying asset price changes have the following three forms (Thomas, Copeland, Fred Weston Kuldeep Shastr. 2010): diffusion form, beating form and diffusion in the form beating. In the diffusion form, the price changes in a continuous smooth manner, such as the changes in temperature, a typical spread in the form. In pure beating form, the prices remained unchanged during a period and then instantly jump to another price, such as changes in interest rates set by Central Bank in China. The same situation continues to happen. The form of beating diffusion is the beat of the combination of the form of diffusion and the forms of beating. In addition to the occasional beating in the price, it changes in a continuous smooth way in general. The BS pricing model assumes that stock price movements are spread in the confusion form, and the transaction will be continuous forever. There is no the price gap existing. Obviously, it is a kind of convenient but not precise form of the hypothesis. The option price of the underlying asset does not render the proliferation of forms in the real world as the Exchange will not open 24 hours a day. There is a closing price at the end of each trading day. The next day’s opening price is not necessarily equal to the closing price the day before. This is clearly will cause the price spread of the gap and is not allowed by the confusion form. Even in normal trading hours, the proliferation of forms of assumption may not be set up. Once the big news is released in the market, it may result in the price gapped up or down suddenly. 2.1.6 Logarithm of the Stock Prices Obey Normal Distribution at Maturity (Black and Scholes, 1973) The B-S pricing model assumes that stock prices have a lognormal distribution. It means that the logarithm of stock prices conform to normal distribution when the option expires. Actually, the stock price is not strictly logarithmic normal distribution. The left tail of its distribution curve is a longer length or the right tail rather long and the broad peak is flat or tall. The kurtosis and skewness of its distribution curve is not necessarily zero. If the logarithmic of stock price is not accurate normal distribution at the end of option, it may generate pricing bias. Therefore, it is the BS model that does not depict the world perfectly in some degree. In response to these shortcomings, the options researchers conducted study and exploration of the B-S model in-depth and put forward many amendments to the pricing model. There are a series of amendments (Bakshi. Cao and Chen, 1997), such as the one proposed by Merton in considering of the effects from bonus in 1973, the American call option pricing model made by Roll in 1977, the futures option pricing model established by Black in 1976, the random interest rate pricing model set up by Moton, the stochastic volatility model created by Hull and White in 1987 and the pure hopping model built by Cox and Ross and so on. Although the extended model is closer to the actual situation in the real world, the increased variables make the complex mathematical structure is more difficult to understand. The accuracy of the input data is also harder to ensure. However, since each of those above models is just an extension of a part of many defects in the model, no model has yet been sent out to overcome all the shortcomings. Even if someone tries to develop such a model, there is nothing but only a bunch of complicated mathematical formula with no practical value will be got.
Based on the B-S option pricing model, the stock price, exercise price, option period, risk-free interest rate and stock price volatility all are the model input variables. Apart from the volatility, the remaining four variables can be observed directly in the market. Thus, the accuracy of the input data is mainly determined by the volatility (James. Doran. Ehud. Ronn. 2005). Usually, there are two ways for estimating volatility: calculating the standard deviation of returns based on historical stock price data and weighting average on the implicit volatility implied in the market price. In spite of the volatility compared to the data which the model requires may have some bias when the option expires, it is a good approximation. After all it is impossible to get an accurate data on the future.
Although the B-S pricing model assumptions cannot perfectly describe the real world with many drawbacks, it is still widely used in practice. The reason is that the model is not only easy to understand, but also the model input variables are relatively simple. To some certain extent, this ensures the accuracy of input data. In the practical application process of the B-S pricing model, the actual employees can adopt some simple extension models to overcome its shortcomings. For the random price changes, more trading techniques are utilized to overcome the problem of pricing bias rather than the use of the more complex extended model. Usually, the trading techniques are that taking different pricing volatilities for the different price and different maturity options. There are three specific operating methods as follows:
According to the different option exercise prices, actual practitioners calculate the corresponding implicit price volatility and drawn the volatility curves with the changes of the option exercise price. If the curve is concave, it were called the volatility “smile” curve; If it is convex, it is known as the “frown” curve. According to the implementation curve of price volatility, it is impossible to estimate the different option exercise prices with different volatilities for the same stock.
Actual practitioners can also draw the curve of the structure of volatility period based on the implied volatility. The curve reflects the relationship between the volatility and options expiration time. In the light of the curve, it can price the different option exercise prices with different volatilities for the same stock to show the volatility changes under it duration.
A coordinate of the volatility matrix is the strike price and the other coordinate is the time to maturity. The data in matrix are the implied price volatilities calculated from the BS pricing model. If a specific execution price and the option price in expiration date cannot be directly observed from the market, the option’s implied price volatility can be determined by linear interpolation. When there is a need for a new valuation of options, the corresponding strike price and the implicit pricing volatility in expiration date can be found from the matrix. Actually the relationship between the volatility structure and its changes with the exercise price are taken into account in the volatility matrix.
Although the B-S pricing model is not very accurate, it is better than other option valuation methods and is still an indispensable trading analysis tool. Most of options traders think that the deficiencies of the BS pricing model should be offset by trading experience rather than more complex models. Owning the B-S pricing model in options market just likes holding a candle into the dark room. Sometimes the flickering candlelight may lead us to judge wrongly. With more and more study and research, there will be more appropriate option pricing model in the future than the B-S pricing model undoubtedly.
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