The
Black-Scholes model is a mathematical form of the economic model of the market for equity. Here the price of the equity is a represented in a stochastic process. Robert C. Merton first developed the term "Black-Scholes" options pricing model, when he enhanced the theory published by Myron Scholes and Fischer Black. The model talks about the options pricing model. The foundation of the research was based on the work done by the scholars like A. James Boness, Louis Bachelier, Edward O. Thorp, Sheen T. Kassouf and Paul Samuelson. The prime focus of the Black-Scholes is that option is implicitly priced when the stock is traded.
In the year of 1997, Merton and Scholes received Nobel Prize in Economics for the development of this theory and other related works.
The major assumptions made in the development of Black-Scholes model are:
- Short selling of the underlying stock is possible.
- Stock trading is continuous.
- There are no taxes or transaction costs.
- Arbitrage opportunities are not available.
- There is the possibility of borrowing and lending cash at a fixed risk-free interest rate.
- The securities are divisible perfectly.
- The stock will not pay any dividend.
- The price of St, the underlying instrument, follows the geometric Brownian motion with constant volatility ? and drift ?:
dSt = μSt dt + σSt dWt
Black-Scholes model is extended further to have non-constant rates and volatilities. This model is also used to determine the value of European options on instruments paying dividends. In this case if the dividend is a known proportion of the stock price then closed-form solutions are available.
The practical implementations of Black-Scholes model are:
The volatility smile
All the parameters in the model other than the volatility are observed without ambiguity. Furthermore, it can also be said that under normal circumstances, the theoretical value of the option gives a monotonic increasing function of the volatility.
Valuing bond options
Because of the pull-to-par problem, Black-Scholes model can never be applied directly to bond securities. When the bond progresses for the maturity date, the volatility of the bond decreases, as the price involved with bond becomes known. But the simple Black-Scholes model does not reflect this phenomenon. Hence, a number of extensions to this model have been introduced to deal with such a phenomenon.
Interest rate curve
The interest rates are not constant. As they vary with tenor, an interest rate curve is obtained that helps to pick the appropriate rate to use in the Black-Scholes model.
Short stock rate
A short stock position is not free. Often a long stock position is lent at a low cost. In both the cases, this is considered as continuous dividend while evaluating Black-Scholes model.