BlackScholesmodel

The Black-Scholes model is a mathematical framework for pricing European-style options and for constructing hedges. Developed by Fischer Black, Myron Scholes, and Robert Merton, it was introduced in the early 1970s and published in 1973, providing a closed-form solution for options prices under a specific set of assumptions.

In the model, the underlying asset price S_t follows geometric Brownian motion: dS_t = μ S_t dt + σ S_t dW_t, where μ is the drift and σ is the volatility. Under the risk-neutral measure, the drift μ is replaced by the risk-free rate r, giving dS_t = r S_t dt + σ S_t dW_t. The price V(S,t) of a derivative satisfies the Black-Scholes partial differential equation: ∂V/∂t + (1/2) σ^2 S^2 ∂^2V/∂S^2 + r S ∂V/∂S - r V = 0. For a European call option with maturity T and strike K, the solution is C = S N(d1) − K e^(−rT) N(d2), where d1 = [ln(S/K) + (r + σ^2/2) T] / (σ √T) and d2 = d1 − σ √T; the corresponding put price is obtained via put-call parity.

Key assumptions include no dividends during the option’s life, constant volatility and interest rates, no arbitrage, frictionless markets, and continuous trading with lognormal price dynamics. The model’s hedge ratio is Δ = N(d1).

Extensions address dividends, stochastic or local volatility, and jump processes. Limitations arise from its reliance on constant parameters, potential mispricing in volatile markets, and inapplicability to many American-style options without adjustments. The Black-Scholes model remains foundational in option pricing and risk management.