American Call (Bjerksund-Stensland)
Computes the price of an American call option with the Bjerksund-Stensland (1993) approximation, valid when the asset pays dividends (cost of carry below the interest rate). It defines an exercise trigger price and, below it, combines exponential terms to approximate the value with early exercise. It's faster than a binomial tree and widely used in practice for American calls on dividend-paying stocks. Enter the spot price, the strike, the rate, the cost of carry, the volatility and the term.
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American Call (Bjerksund-Stensland)
Computes the price of an American call option with the Bjerksund-Stensland (1993) approximation, valid when the asset pays dividends (cost of carry below the interest rate). It defines an exercise trigger price and, below it, combines exponential terms to approximate the value with early exercise. It's faster than a binomial tree and widely used in practice for American calls on dividend-paying stocks. Enter the spot price, the strike, the rate, the cost of carry, the volatility and the term.
The American call when there are dividends
For an American call on a non-dividend stock, the rule is simple: never exercise before expiry. But when there are dividends, exercising early can be worthwhile, and the American call becomes worth more than the European. Bjerksund and Stensland gave, in 1993, an elegant and fast approximation for this case.
The method defines a trigger price: if the asset rises to it, exercising immediately is optimal. Below the trigger, the value is approximated by a combination of exponential terms that capture the probability of exercising over time. It's far faster than a binomial tree and has excellent accuracy for most practical cases.
Enter the spot price, the strike, the interest rate, the cost of carry (below the rate when there are dividends), the volatility and the term. The tool returns the American call value. The larger the dividends, the lower the trigger drops and the bigger the early-exercise premium embedded in the price.
Related Tools
American Call with Dividend (Roll-Geske-Whaley)
Computes the price of an American call option on a stock paying a discrete dividend, with the Roll-Geske-Whaley model. Unlike the European call, the American one can be exercised early, and that's only optimal precisely an instant before the dividend, when the price will drop. The model finds the critical exercise price and combines probabilities via the bivariate normal. Enter the spot price, the strike, the rate, the volatility, the term, the dividend and the date it's paid.
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.
American Put (Barone-Adesi-Whaley)
Computes the price of an American put option with the Barone-Adesi-Whaley quadratic approximation. Unlike the European put, the American one can be exercised at any time, and that right has value — the so-called early-exercise premium. The method iteratively finds the critical price below which exercising already pays off, and adds that premium to the European put value. It's fast and accurate, with no need for a binomial tree. Enter the spot price, the strike, the rate, the cost of carry, the volatility and the term.
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.