Calculadora de Opções (Black-Scholes)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
d1=(ln(S/K)+(r+σ²/2)T)/(σ√T); Call=S·N(d1)−K·e^(−rT)·N(d2)
About this calculator
The Black-Scholes Options Calculator prices call and put options using the famous mathematical model developed by Fischer Black, Myron Scholes, and Robert Merton. The model calculates the theoretical value of an option based on five parameters: underlying asset price, strike price, time to expiration, volatility, and risk-free interest rate. The formula incorporates the normal distribution to estimate the probability that the option will expire in the money.
The calculator follows the Black-Scholes formula: first it computes d1 and d2, which are normalized distance measures. d1 is given by (ln(S/K) + (r + σ²/2)T) / (σ√T), and d2 by d1 - σ√T. Then the call price is S·N(d1) - K·e^(-rT)·N(d2), and the put price is K·e^(-rT)·N(-d2) - S·N(-d1), where N() is the cumulative normal distribution function. The result is the theoretical fair value of the option.
This tool is useful for investors and traders who want to estimate option prices before trading, compare with market prices, or understand how changes in parameters (like volatility or time) affect value. It is widely used in finance for risk analysis and hedging strategies. However, remember that the model assumes efficient markets, constant volatility, and no dividends, which may not always reflect reality.
Important caveats: the Black-Scholes model does not account for dividends (unless adjusted), assumes constant volatility and fixed interest rates, and is not suitable for American-style options with early exercise. Additionally, implied volatility may differ from historical volatility. Use the calculator as a reference, but always consider actual market conditions and consult a professional for investment decisions.
Frequently asked questions
What does the calculator result mean?
The result is the theoretical fair value of the option according to the Black-Scholes model, i.e., the price the option should have in an efficient market.
Can I use this calculator for options that pay dividends?
Not directly. The basic model does not account for dividends. For options on dividend-paying stocks, you need an adjusted version (like Merton's model).
What is the difference between historical and implied volatility?
Historical volatility is calculated from past asset prices. Implied volatility is the volatility that makes the market price of the option equal to the Black-Scholes model. The calculator uses the volatility you input.
Does this calculator work for American options?
The Black-Scholes model is for European options (exercise only at expiration). American options may have additional value due to early exercise, so the result may underestimate the price.
What does 'in the money' and 'out of the money' mean?
A call option is in the money if the asset price is above the strike price. For a put option, it is the opposite. The model calculates the probability of the option expiring in the money.