BlackScholesPreisprozess

The Black-Scholes model, also known as the Black-Scholes-Merton model, is a mathematical framework used to price European-style options on stocks. Introduced in 1973 by Fischer Black, Myron Scholes, and Robert Merton, it provides a closed-form solution under a set of simplifying assumptions and has become foundational in modern financial theory and practice.

The model assumes the underlying asset price follows a geometric Brownian motion with constant volatility and drift. In pricing, the drift is replaced by the risk-free rate, leading to a no-arbitrage, risk-neutral valuation. Markets are frictionless, there are no transaction costs, and the option is European (exercisable only at expiration). If the underlying pays a continuous dividend yield, the framework can be adjusted to incorporate it.

The standard closed-form formulas give the prices of European call and put options. For a call, C = S0 N(d1) − K e^(−rT) N(d2), and for a put, P = K e^(−rT) N(−d2) − S0 N(−d1). Here S0 is the current stock price, K is the strike, T is time to maturity, r is the risk-free rate, σ is the volatility, and N(·) is the standard normal cumulative distribution function. The terms d1 and d2 are defined as d1 = [ln(S0/K) + (r + σ^2/2)T] / (σ√T) and d2 = d1 − σ√T. With a dividend yield q, the price can be adjusted by replacing S0 with S0 e^(−qT) and/or modifying the discounting.

Applications include option pricing, hedging (Greeks), and serving as a benchmark for more complex models. Limitations include the assumption of constant volatility, lognormal price dynamics, no jumps, and no early exercise for American options; extensions address stochastic volatility, jumps, and discrete dividends.