Black-Scholes Model
The Black-Scholes model is a mathematical formula that estimates the fair price of a European-style option. It uses five inputs: the current price of the underlying asset, the option's strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying. By combining these, the model produces a theoretical option value, and it remains the foundational framework for modern option pricing.
Worked example
For an at-the-money option where the stock price equals the strike, a useful approximation of the call value is 0.4 × S × σ × √T. Take a stock at S = $100, volatility σ = 20% (0.20), and time T = 1 year (√1 = 1). The approximate call price = 0.4 × $100 × 0.20 × 1 = $8. So the option is worth roughly $8 per share, or $800 for one 100-share contract.
Why it matters
The Black-Scholes model matters because it gave markets a consistent, repeatable way to price options and to derive implied volatility, underpinning the entire derivatives industry. Its main pitfall is its assumptions: constant volatility, a lognormal price distribution, no dividends in the basic form, and European-style exercise. Real markets violate these, so traders adjust the model rather than trust it blindly.
Frequently asked questions
What is the difference between European and American options in this model?
Black-Scholes prices European options, which can only be exercised at expiration. American options, exercisable any time before expiry, need adjusted or numerical methods such as binomial trees, though for non-dividend-paying calls the values often coincide.
Who created the Black-Scholes model?
It was published in 1973 by Fischer Black and Myron Scholes, with key contributions from Robert Merton. Scholes and Merton received the 1997 Nobel Memorial Prize in Economic Sciences for the work; Black had died in 1995 and was ineligible.
Related terms: Option Greeks, Implied Volatility