# Suitability Of Black Scholes Model And Pricing Derivatives Finance Essay

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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.

## 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, ÃŽÂ¼ 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.