Calculate European call and put prices and the Greeks (delta, gamma, theta, vega, rho) with the Black-Scholes-Merton model, including dividends.
Atualizado Exemplos verificados: 4
The call is worth $8.26 and the put $5.79. For each 1.00 rise in the stock price the call gains about 0.5909, and with nothing else changing one day of time takes about 0.0258 off its value.
| Medidas | Call | Put |
|---|---|---|
| Price | 8.26 | 5.791 |
| Delta | 0.5909 | −0.4091 |
| Gamma | 0.022 | 0.022 |
| Theta per day | −0.0258 | −0.0124 |
| Vega per 1% volatility | 0.2747 | 0.2747 |
| Rho per 1% rate | 0.2541 | −0.2335 |
Theta is per calendar day (annual theta ÷ 365); vega and rho are per one percentage point. European exercise; the model assumes constant volatility and rates.
The Black-Scholes-Merton model prices a European option from the stock price S, the strike K, the time to expiry T in years, the volatility σ, the risk-free rate r and a continuous dividend yield q. A call is worth S·e^(−qT)·N(d₁) − K·e^(−rT)·N(d₂), where N is the standard normal distribution, and the put follows from the same terms. The Greeks measure sensitivity: delta to the stock price, gamma to delta itself, theta to the passing of time, vega to volatility and rho to the interest rate.
With the defaults, a stock at 100, a strike of 100, six months to expiry, 25% volatility and a 5% rate, the call is worth 8.26 and the put 5.79. The call's delta of 0.5909 means it gains about 0.59 for each 1 rise in the stock, and its theta shows it losing about 0.026 a day.
The model assumes constant volatility and rates, log-normal prices and exercise only at expiry. US stock and ETF options are American-style and can be exercised early; index options such as SPX are European and cash-settled.
Fonte de verificação: Hull, Options, Futures, and Other Derivatives, Example 15.6: c = 4.76, p = 0.81, d1 = 0.7693, d2 = 0.6278; Python decimal (prec 50, erf series) gives 4.7594224, 0.8085994
Fonte de verificação: Hull ch. 19 (S=49, K=50, 20 weeks): price 2.40, delta 0.522, gamma 0.066, theta −4.31/yr = −0.0118/day, vega 12.1 (0.121 per 1%), rho 8.91 (0.0891 per 1%); Python decimal values to 6 dp
Fonte de verificação: Hull Example 17.1: c = 51.83, d1 = 0.5444; Python decimal (prec 50) Merton formula gives 51.8329568
Fonte de verificação: With r = q = 0 and S = K, d1 = σ√T/2 = 0.1 and C = P = S(2N(0.1) − 1) = 7.9655675 (Python decimal and math.erf agree)
C = S·e^(−qT)·N(d₁) − K·e^(−rT)·N(d₂), with d₁ = (ln(S/K) + (r − q + σ²/2)T) ÷ (σ√T) and d₂ = d₁ − σ√T. For the defaults, d₁ = 0.2298 and d₂ = 0.0530. Fischer Black and Myron Scholes published it in the Journal of Political Economy in 1973 (vol. 81, no. 3, pp. 637–654); Robert Merton and Scholes received the 1997 economics prize for the work, Black having died in 1995.
Delta is how much the option price changes for a 1 change in the stock price. The default call has a delta of 0.5909 and the put −0.4091; without dividends they always differ by exactly 1. One US equity option contract usually covers 100 shares, so a call with a delta of 0.59 moves roughly like 59 shares of the stock.
For European options with the same strike and expiry, call − put = S·e^(−qT) − K·e^(−rT). With the defaults, 8.26 − 5.79 = 2.469, which equals 100 − 100·e^(−0.05 × 0.5). If market prices break this relation, a riskless profit exists before trading costs, so the put price follows directly from the call price.
Higher volatility raises both call and put prices, because it widens the range of prices the stock can reach before expiry. The default call's vega is 0.2747, so each extra 1% of volatility adds about 0.27; raising volatility from 25% to 35% lifts the call from 8.26 to 11.01 and the put from 5.79 to 8.54.
Only some. US stock and ETF options are American-style, so they can be exercised before expiry. For a call on a stock that pays no dividends, early exercise is never worth it, so the Black-Scholes price is also the American price (Hull, Options, Futures, and Other Derivatives). American puts, and calls just before a dividend, are worth more than the model shows and need a binomial tree or similar method.
A precisão depende dos dados inseridos e das hipóteses do método. O cálculo decimal usa 50 algarismos significativos, mas estimativas, métodos numéricos e dados de origem podem ter menor precisão; o arredondamento exibido não elimina essas limitações. Exemplos resolvidos verificados com fontes independentes: 4. Por exemplo, “Hull example 15.6” é verificado com Hull, Options, Futures, and Other Derivatives, Example 15.6: c = 4.76, p = 0.81, d1 = 0.7693, d2 = 0.6278; Python decimal (prec 50, erf series) gives 4.7594224, 0.8085994.
Hull — Options, Futures, and Other Derivatives, ch. 15 (Black-Scholes-Merton), 17 (dividend yield) and 19 (Greeks); Black & Scholes (1973), The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81(3).
Apenas para planejamento. Credores, autoridades fiscais e mercados aplicam seus próprios arredondamentos, tarifas e regras; confirme os valores com eles antes de assumir um compromisso.
Esta calculadora inclui 4 exemplos resolvidos com respostas de fontes independentes. Eles fazem parte do conjunto de testes e você também pode executá-los aqui.
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