Black-Scholes-Merton Call (Dividends)
Computes the price of a European call option with the Black-Scholes-Merton model, the extension that incorporates a continuous dividend yield q. The premium is S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2), and the dividend lowers the call's value, because part of the asset's return leaks to whoever holds the stock. It's the standard model for options on dividend-paying stocks and on indices. Enter the spot price, the strike, the interest rate, the dividend yield, the term and the volatility.
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Black-Scholes-Merton Call (Dividends)
Computes the price of a European call option with the Black-Scholes-Merton model, the extension that incorporates a continuous dividend yield q. The premium is S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2), and the dividend lowers the call's value, because part of the asset's return leaks to whoever holds the stock. It's the standard model for options on dividend-paying stocks and on indices. Enter the spot price, the strike, the interest rate, the dividend yield, the term and the volatility.
Black-Scholes when the stock pays dividends
The original Black-Scholes assumes the stock pays nothing during the option's life. In the real world, stocks pay dividends and indices have a continuous yield, and ignoring this overstates the value of calls. Robert Merton extended the model to include a continuous dividend yield, and that version is the one the market actually uses for options on dividend-paying stocks and indices.
The adjustment is elegant: the asset term is now discounted by the dividend yield, S·e^(−qT), because the option holder doesn't receive the dividends the stockholder does. The larger the expected dividend, the lower the call premium and the higher the put. It's the same mechanism behind Garman-Kohlhagen, where the foreign rate plays the role of the dividend.
Enter the spot price, the strike, the risk-free rate, the dividend yield, the term in years and the volatility. The tool returns the fair call premium. Like every Black-Scholes model, it assumes constant volatility and European exercise, so treat the number as a theoretical reference, not an exact quote.
Related Tools
Black-76 Call Price (Options on Futures)
Works out the premium of a European call option on futures with the Black-76 model, the variant of Black-Scholes used when the underlying is a futures or forward contract. The price is e^(−rT)·[F·N(d1) − K·N(d2)], with d1 and d2 built from the futures price, the strike, the volatility and the term. Because the future already embeds the cost of carry, the discount hits both terms and the interest rate never shows up inside d1. It is the standard for options on commodities, indices and rates. Enter the futures price, the strike, the risk-free rate, the term in years and the annual volatility.
Implied Volatility (Black-Scholes)
Computes the implied volatility of a European call option by inverting the Black-Scholes formula via bisection: given the market price, it finds the volatility the model would need to reach it. It's the volatility the market is actually pricing in, the number behind the volatility smile and the VIX. Unlike the other Greeks, it has no closed form and requires a numerical solution. Enter the spot price, the strike, the interest rate, the term and the market price of the call.
Option Price (Heston Model)
Prices a European call under the Heston (1993) model, where the asset volatility is itself stochastic and mean-reverting — reproducing the volatility smile and skew that constant-vol Black-Scholes misses. The price comes out in semi-closed form by numerically integrating the characteristic function (Little Heston Trap formulation, Gauss-Legendre quadrature). Enter spot, strike, maturity, rate, dividend and the five model parameters: initial variance, kappa, theta, vol of vol and correlation.
Contingent-Premium Call Option
Computes the fair premium of a contingent-premium call option, also called pay-later. The buyer pays nothing upfront: the premium is only due at expiry, and even then only if the option finishes in the money. For the deal to be fair, that deferred premium must be larger than a plain call's, compensating for the risk the seller receives nothing. The formula divides the Black-Scholes value by the exercise probability. Enter price, strike, rate, dividend, volatility and term.
Option Veta (Vega Decay)
Computes the veta of an option, the Greek that measures how much the vega changes with each passing day. Since vega captures the premium's sensitivity to volatility, veta shows whether that sensitivity is shrinking over time, and it does: near expiry vega tends to zero, so a call's veta is usually negative. It's a second-order Greek that helps anticipate how the volatility exposure will behave. The result comes per year and per day, for a call with no dividends. Enter the spot price, the strike, the interest rate, the term in years and the volatility.
Call Dual Delta
Computes the dual delta of a call option: the premium's sensitivity to the strike price, that is, dC/dK. While ordinary delta measures the reaction to the underlying's price, dual delta measures how much the option would change if the strike were slightly different. For a call it equals −e^(−rT)·N(d2) and has a practical reading: in absolute terms it approximates the risk-neutral probability of the option finishing in the money. Enter the spot price, the strike, the interest rate, the term in years and the volatility.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.