Black-Scholes Calculator

For a European option on a non-dividend-paying underlying, the Black-Scholes price is

$C = S\,N(d_1) - K e^{-rT} N(d_2), \qquad P = K e^{-rT} N(-d_2) - S\,N(-d_1)$

$d_1 = \frac{\ln(S/K) + \left(r + \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}$

• $S$ — current price of the underlying
• $K$ — strike price
• $T$ — time to expiry in years (3 months is $T = 0.25$)
• $r$ — continuously compounded risk-free rate, as a decimal
• $\sigma$ — annualised volatility, as a decimal
• $N(\cdot)$ — standard normal cumulative distribution function

Read the two terms structurally: $K e^{-rT}$ is the strike discounted to today, $N(d_2)$ is the risk-neutral probability the option finishes in the money, and $N(d_1)$ is the call's delta. So the call price is the expected value of what you receive minus the discounted cost of what you pay, both weighted by those probabilities.

Put-call parity

$C - P = S - K e^{-rT}$

This holds by arbitrage regardless of the model, so it is the fastest check on any computed pair — and the easiest way to get the put once you have the call.

The main sensitivities

$\Delta_{\text{call}} = N(d_1), \qquad \Delta_{\text{put}} = N(d_1) - 1, \qquad \Gamma = \frac{N'(d_1)}{S\sigma\sqrt{T}}, \qquad \mathcal{V} = S N'(d_1)\sqrt{T}$

where $N'(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Vega $\mathcal{V}$ is per unit of volatility, so divide by 100 for a one-percentage-point move.

Implied volatility

Every input except $\sigma$ is observable, so given a market price you can invert the formula for $\sigma$. There is no closed form — the price is strictly increasing in $\sigma$, so bisection or Newton's method converges quickly.

The assumptions

The derivation assumes European exercise, lognormal returns, constant $r$ and $\sigma$, no dividends and frictionless trading. Real markets violate the last three routinely, which is why implied volatility varies by strike. This page computes the model exactly; it does not claim the model is the market, and nothing here is a recommendation to trade.

• $T$ in days or months: $T$ is in years. Ninety days is $T = 90/365 \approx 0.2466$, not $90$.
• Percentages left unconverted: $r = 0.05$ and $\sigma = 0.20$. Entering $5$ and $20$ produces nonsense of enormous magnitude.
• Using $N'$ instead of $N$: the price needs the cumulative normal, not the density. The density appears only in gamma and vega.
• Adding $\sigma^2/2$ to $d_2$: only $d_1$ carries the $+\sigma^2/2$; $d_2 = d_1 - \sigma\sqrt{T}$ always.
• Forgetting to discount the strike: the second term is $Ke^{-rT}N(d_2)$, not $KN(d_2)$.
• Pricing an American or dividend-paying option with this formula: American puts can be worth more than their European counterpart, and dividends require the Black-Scholes-Merton adjustment $S \to Se^{-qT}$.
• Skipping the parity check: if $C - P \ne S - Ke^{-rT}$, an arithmetic slip has crept in.