Intuition would tell us that the spot price of the underlying, exercise price, risk-free interest rate, volatility of the underlying, time to expiration and dividends on the underlying (stock or index) should affect the option price.
Interestingly before Black and Scholes came up with their option pricing model, there was a widespread belief that the expected growth of the underlying ought to affect the option price. Black and Scholes demonstrate that this is not true.
The beauty of the Black and Scholes model is that like any good model, it tells us what is important and what is not. It doesn’t promise to produce the exact prices that show up in the market, but definitely does a remarkable job of pricing options within the framework of assumptions of the model. Virtually all option pricing models, even the most complex ones, have much in common with the Black–Scholes model.
Black and Scholes start by specifying a simple and well–known equation that models the way in which stock prices fluctuate. This equation called Geometric Brownian Motion, implies that stock returns will have a lognormal distribution, meaning that the logarithm of the stock’s re-turn will follow the normal (bell shaped) distribution.
Black and Scholes then propose that the option’s price is determined by only two variables that are allowed to change: time and the underlying stock price. The other factors - the volatility, the exercise price, and the risk–free rate do affect the option’s price but they are not allowed to change. By forming a portfolio consisting of a long position in stock and a short position in calls, the risk of the stock is eliminated.
This hedged portfolio is obtained by setting the number of shares of stock equal to the approximate change in the call price for a change in the stock price. This mix of stock and calls must be revised continuously, a process known as delta hedging.
Black and Scholes then turn to a little–known result in a specialized field of probability known as stochastic calculus. This result defines how the option price changes in terms of the change in the stock price and time to expiration. They then reason that this hedged combination of options and stock should grow in value at the risk–free rate.
The result then is a partial differential equation. The solution is found by forcing a condition called a boundary condition on the model that requires the option price to converge to the exercise value at expiration. The end result is the Black and Scholes model.
Interestingly before Black and Scholes came up with their option pricing model, there was a widespread belief that the expected growth of the underlying ought to affect the option price. Black and Scholes demonstrate that this is not true.
The beauty of the Black and Scholes model is that like any good model, it tells us what is important and what is not. It doesn’t promise to produce the exact prices that show up in the market, but definitely does a remarkable job of pricing options within the framework of assumptions of the model. Virtually all option pricing models, even the most complex ones, have much in common with the Black–Scholes model.
Black and Scholes start by specifying a simple and well–known equation that models the way in which stock prices fluctuate. This equation called Geometric Brownian Motion, implies that stock returns will have a lognormal distribution, meaning that the logarithm of the stock’s re-turn will follow the normal (bell shaped) distribution.
Black and Scholes then propose that the option’s price is determined by only two variables that are allowed to change: time and the underlying stock price. The other factors - the volatility, the exercise price, and the risk–free rate do affect the option’s price but they are not allowed to change. By forming a portfolio consisting of a long position in stock and a short position in calls, the risk of the stock is eliminated.
This hedged portfolio is obtained by setting the number of shares of stock equal to the approximate change in the call price for a change in the stock price. This mix of stock and calls must be revised continuously, a process known as delta hedging.
Black and Scholes then turn to a little–known result in a specialized field of probability known as stochastic calculus. This result defines how the option price changes in terms of the change in the stock price and time to expiration. They then reason that this hedged combination of options and stock should grow in value at the risk–free rate.
The result then is a partial differential equation. The solution is found by forcing a condition called a boundary condition on the model that requires the option price to converge to the exercise value at expiration. The end result is the Black and Scholes model.