The Black-Scholes model is a mathematical framework for pricing European options, developed in 1973 by Fischer Black, Myron Scholes, and Robert Merton. This groundbreaking model revolutionized options trading by providing a theoretical estimate of the price of European-style options. The model requires five key inputs: the current stock price, the option's strike price, time remaining until expiration, the volatility of the underlying asset, and the risk-free interest rate. On the right, you can see the payoff diagrams for both call and put options at expiration, with the strike price set at 50.
The Black-Scholes formula for a European call option is given by: C equals S times N of d1, minus K times e to the power of negative r times T minus t, times N of d2. Where d1 equals the natural logarithm of S over K, plus r plus sigma squared over 2, times T minus t, all divided by sigma times the square root of T minus t. And d2 equals d1 minus sigma times the square root of T minus t. For put options, the formula is: P equals K times e to the power of negative r times T minus t, times N of negative d2, minus S times N of negative d1. The function N represents the cumulative normal distribution function, which gives the probability that a standard normal random variable is less than or equal to a specific value. On the right, you can see the graph of this function in blue, along with the normal probability density function in red.
The Black-Scholes model is built on several key assumptions. First, it only applies to European-style options, which can only be exercised at expiration. Second, it assumes no dividends are paid during the option's life. Third, markets are efficient with no arbitrage opportunities. Fourth, there are no transaction costs or taxes. Fifth, the risk-free interest rate remains constant. Sixth, returns on the underlying asset follow a log-normal distribution. And seventh, volatility stays constant throughout the option's life. These assumptions create limitations for the model. In reality, volatility changes over time, many stocks pay dividends, and not all options are European-style. On the right, you can see how the Black-Scholes price of a call option varies with the stock price. The blue curve shows the option price according to Black-Scholes, while the red line shows the intrinsic value. Notice how the option price is always higher than the intrinsic value due to time value. Let's see what happens when we change the time to expiration and volatility.