Black–Scholes model

The Black–Scholes model prices a European call option as C = S₀N(d₁) − Ke^(−rT)N(d₂), where d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T) and d₂ = d₁ − σ√T. The inputs are the current stock price S₀, strike price K, time to expiration T, risk-free interest rate r, and volatility σ. The model assumes constant volatility, a constant risk-free rate, no dividends, and that the underlying follows a geometric Brownian motion with continuous trading.

The model's key insight is that the option's price depends only on the current price, not on the expected return of the stock, because the option can be dynamically hedged to eliminate risk. This led to the concept of risk-neutral valuation, where the expected return is replaced by the risk-free rate.

The model was derived by solving a partial differential equation (the Black–Scholes equation) that arises from constructing a riskless portfolio of the option and the underlying stock. Robert C. Merton extended the model to allow for continuous dividends and stochastic interest rates, and his work was instrumental in the model's acceptance. The model also yields the Black–Scholes formula for put options via put-call parity.

Extensions include the Black-76 model for futures options, the Garman-Kohlhagen model for currency options, and the binomial options pricing model as a discrete-time alternative. The model's assumptions have been relaxed in various ways, such as incorporating jumps (Merton's jump-diffusion model) or stochastic volatility (Heston model).

The Black–Scholes model became the standard benchmark for option pricing and is widely used by traders and financial institutions. Its introduction spurred the growth of the options market and the development of financial engineering. The model's Greeks—delta, gamma, theta, vega, and rho—are used for hedging and risk management.

The model also influenced academic finance, leading to the concept of implied volatility, which is the volatility that makes the model price match the market price. The volatility smile and skew observed in markets highlight the model's limitations, but it remains a fundamental tool in quantitative finance.

Fischer Black died in 1995, two years before the Nobel Prize was awarded, so he was not honored. The model was initially rejected by several journals before being published in the Journal of Political Economy. The original paper was titled "The Pricing of Options and Corporate Liabilities" and also applied the model to corporate debt valuation.

The model's derivation assumes no transaction costs and continuous hedging, which are unrealistic but allow for a closed-form solution. The model has been criticized for its assumption of log-normal stock prices, which underestimates extreme moves. Despite this, it remains a cornerstone of financial theory and practice.

European option

An option that can only be exercised at expiration.

Volatility

A measure of the dispersion of returns for a given security.

Risk-free rate

The theoretical rate of return of an investment with zero risk.

Greeks

Sensitivities of the option price to various parameters.