The Black-Scholes model is a mathematical model used to calculate the theoretical value of European-style call and put options. It was developed by Fischer Black and Myron Scholes in 1973 and later extended by Robert Merton. The model assumes that the underlying asset follows a geometric Brownian motion and that the risk-free rate, volatility, and time to expiration are all known and constant.
The model helps to calculate the fair value of options by taking into account five input variables:
1. Underlying asset price: This is the current market price of the underlying asset, such as a stock or a commodity.
2. Strike price: This is the predetermined price at which the option holder has the right to buy or sell the underlying asset.
3. Time to expiration: This is the time remaining until the option expires, and the holder must exercise their right to buy or sell the underlying asset.
4. Risk-free rate: This is the rate of return on a risk-free investment, such as U.S. Treasury bonds, that the option holder can earn over the life of the option.
5. Volatility: This is a measure of how much the underlying asset’s price fluctuates over time. Options on assets that are more volatile are generally more expensive.
Using these input variables, the Black-Scholes model calculates the theoretical price of a call or put option. The formula for calculating the fair value of a European call option using the Black-Scholes model is:
C = S0N(d1) − Ke − rTN(d2)
where:
- C is the fair value of the call option
- S0 is the current market price of the underlying asset
- K is the strike price
- r is the risk-free rate
- T is the time to expiration
- N() is the cumulative standard normal distribution function
- d1 and d2 are calculated as follows:
$$d_1 = \frac{ln\left(\frac{S_0}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)T}{\sigma \sqrt{T}}$$
$$d_2 = d_1 - \sigma \sqrt{T}$$
The formula for calculating the fair value of a put option is similar but involves different inputs:
P = Ke − rTN( − d2) − S0N( − d1)
where P is the fair value of the put option.
In summary, the Black-Scholes model provides a mathematical framework for determining the fair value of options based on various input variables. It is widely used by option traders and investors to estimate the value of options and to make informed trading decisions.