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Black-Scholes-Optionpreisrechner

DatenEntwicklerMathe

[iotools_black_scholes_option_pricing_calculator]

Führung

Black-Scholes-Optionpreisrechner

Price European call and put options using the Black-Scholes-Merton model — entirely in your browser. Enter the spot price, strike, days to expiry, implied volatility, risk-free rate, and dividend yield, and you instantly get fair-value premiums plus the full set of Greeks: delta, gamma, theta, vega, and rho. Breakeven, intrinsic value, and time value are broken out for both calls and puts so you can size positions and read decay at a glance.

Nutzung

Enter the current spot price of the underlying.
Enter the option’s strike price.
Einführung days to expiry — the calculator converts to years using a 365-day basis.
Stellen Sie die implied volatility as an annualized percentage (e.g. 25 for 25%).
Stellen Sie die risk-free rate as an annualized percentage (Treasury-bill yield is a common proxy).
Optionally enter a continuous dividend yield for the underlying (use 0 for non-dividend assets).
Read the call and put prices, the Greeks, and breakeven points in the output panel.

Funktionen

European call and put pricing – Closed-form Black-Scholes-Merton premium for both option types.
Full Greeks – Delta, gamma, theta (per day), vega (per 1% vol), and rho (per 1% rate) for risk-managing positions.
Breakeven prices – Strike adjusted by the premium for both call and put, so you know where the position turns profitable at expiry.
Intrinsic vs time value split – Decompose each premium into its in-the-money payoff and remaining time premium.
Continuous dividend yield support – Handles dividend-paying stocks, ETFs, and FX-style assets via the Merton extension.
High-accuracy CDF – Cumulative normal distribution implemented with the Abramowitz & Stegun 26.2.17 approximation (absolute error under 1e-7).
Real-time, client-side – Calculations run locally on each input change. No data is sent to a server.

 Häufig gestellte Fragen

What assumptions does the Black-Scholes model make?

The model assumes the underlying follows geometric Brownian motion with constant volatility and drift, the risk-free rate is constant, there are no transaction costs or taxes, trading is continuous, no arbitrage opportunities exist, and the option is European-style (exercisable only at expiry). It also assumes log-normally distributed returns, which is a known simplification — real markets show fat tails and volatility smiles.
Why does the model only work for European options?

The closed-form Black-Scholes formula prices options that can be exercised only at expiry. American options can be exercised at any time before expiry, so they require numerical methods such as binomial trees, finite difference solvers, or Longstaff-Schwartz Monte Carlo to value the early-exercise premium.
What does each Greek tell me?

Delta measures how much the option price moves per $1 change in the underlying. Gamma is delta's rate of change — high gamma means delta is unstable. Theta is the daily time decay, almost always negative for long options. Vega is sensitivity to a 1-percentage-point change in implied volatility. Rho is sensitivity to a 1-percentage-point change in the risk-free rate. Together they describe the risk profile of an options position.
What is implied volatility and where do I get it?

Implied volatility is the volatility number that, when plugged into Black-Scholes, returns the option's current market price. It is the market's forward-looking estimate of how much the underlying will move. You can read it from your broker's option chain, public sources for index options (such as VIX-derived figures for SPX), or back it out by inverting Black-Scholes on a quoted premium.
How is the dividend yield used in the formula?

The Merton extension discounts the spot price by the continuous dividend yield over the option's life, since dividend payments reduce the value held by the option owner. For non-dividend assets, set the yield to 0 and the formula reduces to the original Black-Scholes. For currencies, the foreign risk-free rate plays the role of the dividend yield.
Why does the model misprice options in practice?

Realized return distributions have fat tails and skew, volatility itself is stochastic and clustered, jumps occur around earnings and macro events, and bid-ask spreads create execution slippage. Market practitioners adjust by quoting different implied volatilities at different strikes and maturities, producing the volatility surface — a smile or skew rather than the flat surface Black-Scholes assumes.

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