The Black Scholes Model is one of the most important concepts in modern financial theory. It was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes and is still widely used today, and regarded as one of the best ways of determining fair prices of options.
The Black Scholes Model can be described as a model of price variation over time of financial instruments such as stocks that can, among other things, be used to determine the price of a European call option. The model assumes that the price of heavily traded assets follow a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option’s strike price and the time to the options expiry.
In other words, The Black–Scholes model is a mathematical description of financial markets and derivative investment instruments. The model develops partial differential equations whose solution, the Black–Scholes formula, is widely used in the pricing of European-style options.
ASSUMPTIONS
The Black–Scholes model of the market for a particular equity makes the following explicit assumptions:
- It is possible to borrow and lend cash at a known constant risk free interest rate. This restriction has been removed in later extensions of the model.
- The price follows a Geometric Brownian motion with constant drift and volatility. This often implies the validity of the efficient-market hypothesis.
- There are no transaction costs or taxes.
- Returns from the security follow a Log-normal distribution.
- The stock does not pay a dividend
- All securities are perfectly divisible (i.e. it is possible to buy any fraction of a share).
- There are no restrictions on short selling.
- There is no arbitrage opportunity
- Options use the European exercise terms, which dictate that options may only be exercised on the day of expiration.
LIMITATIONS
Among the most significant limitations are:
- the underestimation of extreme moves, yielding tail risk, which can be hedged with out-of-the-money options;
- the assumption of instant, cost-less trading, yielding liquidity risk, which is difficult to hedge;
- the assumption of a stationary process, yielding volatility risk, which can be hedged with volatility hedging;
- the assumption of continuous time and continuous trading, yielding gap risk, which can be hedged with Gamma hedging.