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# Suitability of Black Scholes Model and Pricing Derivatives Finance Essay

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.

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“Suitability of Black Scholes Model and Pricing Derivatives Finance Essay”

## 1 Introduction of the Black-Scholes (BS) option pricing model

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.

## 2 The Suitability of the Black-Scholes (B-S) 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.

## 2.2 The accuracy of model input data

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.

## 2.3 The Suitability of the Black-Scholes (B-S) Model

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:

## 2.3.1 The Curve of the Exercise Price of Volatility

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.

## 2.3.2 The Structure of Volatility Period

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.

## 2.3.3 Volatility Matrix

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.

## 3 Conclusions

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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