How this calculator works
The Black-Scholes-Merton model prices a European option, one that can only be exercised at expiry, on a stock that pays a continuous dividend yield. It assumes the stock price follows geometric Brownian motion with constant volatility and interest rates, no transaction costs, and continuous trading. The two intermediate terms d1 and d2 combine the distance between the stock price and the strike with the volatility and time remaining. N(d1) and N(d2) are standard normal probabilities used in the formula. Real markets depart from the assumptions: volatility changes over time, prices jump and early exercise can matter for American options. Treat the output as a model value rather than a market quote.
Formula
d1 = [ln(S/K) + (r - q + sigma^2/2) T] / (sigma sqrt(T)), d2 = d1 - sigma sqrt(T). Call = S e^(-qT) N(d1) - K e^(-rT) N(d2). Put = K e^(-rT) N(-d2) - S e^(-qT) N(-d1). S is the spot price, K the strike, r the risk-free rate, q the continuous dividend yield, sigma the volatility and T the time to expiry in years.
Worked example
A stock at $100 with a $100 strike, one year to expiry, 25% volatility, a 4.5% risk-free rate and a 1.5% dividend yield gives d1 = 0.245 and d2 = -0.005. The call is worth about $11.18 and the put about $8.27. Put-call parity holds: the call minus the put is $2.91, the same as the dividend-adjusted stock value of $98.51 minus the discounted strike of $95.60.
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Frequently asked questions
Does this work for American options?
No. The model values European options, which can only be exercised at expiry. American options can be exercised early, so they are worth at least as much, and a binomial model is usually needed.
What volatility should I use?
Use the market's implied volatility if you are comparing with a quote, or a historical estimate if you are valuing a scenario. Volatility is the input the model is most sensitive to.
How do dividends enter the model?
A continuous dividend yield q reduces the expected growth of the stock price, which lowers calls and raises puts. For discrete dividends, subtract their present value from the stock price or use a binomial model.
Why does the model use the risk-free rate?
The derivation is based on a riskless hedging portfolio, so the risk-free rate is the discount rate and the expected stock return drops out. The model does not depend on investors' risk preferences.