BlackScholesMerton

The Black-Scholes-Merton model is a mathematical framework used to price European options on financial assets. Developed in the early 1970s by Fischer Black, Myron Scholes, and Robert Merton, it provides a closed-form formula for option prices under a set of simplifying assumptions. The model is foundational in modern financial theory and is widely used for pricing, hedging, and risk management.

The model rests on several key assumptions: frictionless markets with no arbitrage, continuous trading, constant volatility and interest rates, and an underlying asset whose price follows a geometric Brownian motion. In the risk-neutral view, the asset price S(t) evolves with drift r minus a dividend yield, and volatility σ, via the stochastic differential equation dS_t = (r - q)S_t dt + σ S_t dW_t, where r is the risk-free rate and q is the continuous dividend yield. The model leads to a partial differential equation that the option price must satisfy: ∂V/∂t + 1/2 σ^2 S^2 ∂^2V/∂S^2 + r S ∂V/∂S - r V = 0.

For a non-dividend-paying stock, the European call and put prices have closed-form solutions. The call price is C = S0 N(d1) - K e^{-rT} N(d2), and the put price is P = K e^{-rT} N(-d2) - S0 N(-d1), with d1 = [ln(S0/K) + (r + 1/2 σ^2) T] / (σ√T) and d2 = d1 - σ√T. When a continuous dividend yield q is present, S0 is effectively discounted by e^{-qT} and d1, d2 are adjusted accordingly: C = S0 e^{-qT} N(d1) - K e^{-rT} N(d2) with d1 = [ln(S0/K) + (r - q + 1/2 σ^2) T] / (σ√T).

Limitations include constant volatility and rates, no jumps, and European-exercise only; real markets exhibit volatility variation and early exercise for American options. Extensions incorporate stochastic volatility, jumps, and local volatility surfaces, while Merton also contributed models for dividends and corporate debt.