# Black-Scholes calculator (option price and Greeks)

> Calculate European call and put prices and the Greeks (delta, gamma, theta, vega, rho) with the Black-Scholes-Merton model, including dividends.

Versión interactiva: https://www.calcopenly.com/es/finance/black-scholes-option-calculator
Tema: Calculadoras financieras

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.

## Datos

- **Stock price**
- **Strike price**
- **Time to expiry**: Days to expiry divided by 365 — you can type 30/365.
- **Volatility (per year)**
- **Risk-free rate (continuously compounded)**
- **Dividend yield (continuous)**

## Resultados

- Call price — resultado principal
- Put price
- Call delta
- Put delta
- Gamma
- Call theta (per day)
- Put theta (per day)
- Vega (per 1% volatility)
- Call rho (per 1% rate)
- Put rho (per 1% rate)
- d₁
- d₂

## Fórmula

$$
C = S e^{-qT} N(d_1) - K e^{-rT} N(d_2),\quad P = K e^{-rT} N(-d_2) - S e^{-qT} N(-d_1),\quad d_{1,2} = \frac{\ln(S/K) + (r - q \pm \sigma^2/2)T}{\sigma\sqrt{T}}
$$

## Ejemplos resueltos

### Hull example 15.6

- Stock price: 42
- Strike price: 40
- Time to expiry: 0.5 years
- Volatility (per year): 20%
- Risk-free rate (continuously compounded): 10%
- Dividend yield (continuous): 0%
- **Call price: 4.76**
- **Put price: 0.81**
- **d₁: 0.769263**
- **d₂: 0.627841**
- Fuente de comprobación: 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 chapter 19 Greeks

- Stock price: 49
- Strike price: 50
- Time to expiry: 0.3846 years
- Volatility (per year): 20%
- Risk-free rate (continuously compounded): 5%
- Dividend yield (continuous): 0%
- **Call price: 2.40**
- **Call delta: 0.5216**
- **Gamma: 0.06555**
- **Call theta (per day): -0.0118**
- **Vega (per 1% volatility): 0.1211**
- **Call rho (per 1% rate): 0.08907**
- Fuente de comprobación: 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

### Index option with dividend yield

- Stock price: 930
- Strike price: 900
- Time to expiry: 2/12 years
- Volatility (per year): 20%
- Risk-free rate (continuously compounded): 8%
- Dividend yield (continuous): 3%
- **Call price: 51.83**
- **Put price: 14.55**
- **d₁: 0.544479**
- Fuente de comprobación: Hull Example 17.1: c = 51.83, d1 = 0.5444; Python decimal (prec 50) Merton formula gives 51.8329568

### At the money with zero rate

- Stock price: 100
- Strike price: 100
- Time to expiry: 1 year
- Volatility (per year): 20%
- Risk-free rate (continuously compounded): 0%
- Dividend yield (continuous): 0%
- **Call price: 7.97**
- **Put price: 7.97**
- **Call delta: 0.5398**
- **Put delta: -0.4602**
- Fuente de comprobación: 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)

## Preguntas

### What is the Black-Scholes formula?

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.

### What does option delta mean?

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.

### What is put-call parity?

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.

### How does volatility affect an option's 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.

### Can Black-Scholes price American options?

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.

### ¿Qué precisión tiene «Black-Scholes calculator (option price and Greeks)»?

La precisión depende de tus datos y de los supuestos del método. El cálculo decimal usa 50 cifras significativas, pero las estimaciones, los métodos numéricos y los datos de origen pueden ser menos precisos; el redondeo mostrado no elimina esos límites. Ejemplos resueltos comprobados con fuentes independientes: 4. Por ejemplo, «Hull example 15.6» se comprueba con 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.

### ¿De dónde procede el método?

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).

## Fuentes

- 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)](https://doi.org/10.1086/260062)

_Solo para planificar. Prestamistas, autoridades fiscales y mercados aplican sus propios redondeos, comisiones y reglas; confirma las cifras con ellos antes de comprometerte._