BlackScholesPreis

BlackScholesPreis refers to the theoretical value of a European option as determined by the Black-Scholes pricing model. The model provides closed-form formulas for call and put prices when the underlying asset follows a lognormal process under a risk-neutral measure. For a non-dividend-paying stock, the European 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), where N(·) is the standard normal cumulative distribution function. The terms d1 and d2 are defined as d1 = [ln(S0/K) + (r + 0.5 σ^2) T] / (σ sqrt(T)) and d2 = d1 − σ sqrt(T). For stocks that pay a continuous dividend yield q, the formulas adjust to C = S0 e^(−qT) N(d1) − K e^(−rT) N(d2) and P = K e^(−rT) N(−d2) − S0 e^(−qT) N(−d1), with d1 and d2 defined using (r − q + 0.5 σ^2) in place of r.

Parameters include S0 (current spot price), K (strike), T (time to maturity), r (risk-free interest rate), σ (volatility), and optionally q (dividend yield). The model rests on assumptions of frictionless markets, no arbitrage, constant volatility and rates, and trading of European options only. It uses risk-neutral valuation and a partial differential equation derivation.

History and use: the model was introduced in 1973 by Fischer Black, Myron Scholes, and Robert Merton, transforming option pricing and risk management. While widely used, it has limitations in real markets, such as volatility dynamics, dividends, and the existence of early exercise for American options. Extensions address stochastic or local volatility and jumps to cope with observed market features.