BlackScholes

The Black-Scholes model is a mathematical framework used to price European-style options on non-dividend-paying stocks. Developed in 1973 by Fischer Black and Myron Scholes, with further contributions by Robert Merton, it became a cornerstone of modern financial engineering. The model derives an explicit formula for option prices based on no-arbitrage arguments and hedging arguments, using a risk-neutral valuation approach.

The key result is the Black-Scholes pricing formula. For a European call option on a non-dividend-paying stock, the price is C = S0 N(d1) − K e^(−rT) N(d2), where d1 = [ln(S0/K) + (r + 0.5 σ^2) T] / (σ√T) and d2 = d1 − σ√T. For a put option, the price is P = K e^(−rT) N(−d2) − S0 N(−d1). Here S0 is the current stock price, K the strike, T the time to expiration, r the risk-free interest rate, σ the volatility, and N(·) the standard normal cumulative distribution function.

Assumptions include: no dividends during the option’s life, no transaction costs, continuous trading, constant volatility and interest rates, and that the stock price follows a geometric Brownian motion. Options priced are European, meaning exercise only at expiration.

Limitations involve its applicability to American options, dividend-paying stocks, and markets with frictions. It also assumes constant volatility and cannot capture features like the volatility smile or jumps. Extensions include the Black-Scholes-Merton framework, local and stochastic volatility models, and other generalizations.

The model profoundly influenced finance, providing a tractable pricing mechanism and forming the basis for numerous derivatives and risk-management practices. In 1997, Robert Merton and Myron Scholes received the Nobel Prize for the theory.