How the calculator works.

Model

The calculator implements the Black-Scholes-Merton model for European options on a dividend-paying underlying. The closed-form solution gives an analytic price and analytic Greeks, so no numerical integration or simulation is required.

Inputs

S

Current stock price.

K

Strike price.

T

Time to expiration in years (computed as days / 365).

r

Risk-free interest rate, continuously compounded, as a decimal.

q

Continuous dividend yield, as a decimal.

σ

Implied volatility, annualized, as a decimal.

Intermediate terms

d1 = (ln(S / K) + (r - q + 0.5 * σ²) * T) / (σ * √T)
d2 = d1 - σ * √T

Where N(x) is the standard normal cumulative distribution function and n(x) is the standard normal probability density function. The site approximates N(x) using the Abramowitz & Stegun 7.1.26 polynomial, which is accurate to about 1.5×10^-7 — well below the precision needed for educational pricing.

Pricing

Call = S * e-qT * N(d1) - K * e-rT * N(d2)
Put  = K * e-rT * N(-d2) - S * e-qT * N(-d1)

Greeks

Vega

Vega = S * e-qT * n(d1) * √T / 100

Vega is the same for calls and puts. The division by 100 expresses vega as the dollar change in option price for a one percentage point change in implied volatility — the convention most traders use. To get vega per unit change in σ (a "100% volatility change"), multiply by 100.

Delta

Call delta = e-qT * N(d1)
Put delta  = e-qT * (N(d1) - 1)

Gamma

Gamma = e-qT * n(d1) / (S * σ * √T)

Gamma is the same for calls and puts.

Theta

Call theta = ( -(S * e-qT * n(d1) * σ) / (2√T)
             - r * K * e-rT * N(d2)
             + q * S * e-qT * N(d1) ) / 365

Put theta  = ( -(S * e-qT * n(d1) * σ) / (2√T)
             + r * K * e-rT * N(-d2)
             - q * S * e-qT * N(-d1) ) / 365

Theta is reported as the dollar change in option price per calendar day. Some platforms quote theta per trading day (252) instead; the calendar-day convention is more common for retail-facing tools.

Rho

Call rho =  K * T * e-rT * N(d2) / 100
Put rho  = -K * T * e-rT * N(-d2) / 100

Rho is reported per one percentage point change in the risk-free rate, matching the vega convention.

Assumptions

The Black-Scholes model carries well-known assumptions. Real markets violate most of them to some degree.

European exercise

The option can only be exercised at expiration. Most equity options listed in the U.S. are American style and may be worth more than this model suggests, particularly for puts on dividend-paying stocks.

Lognormal returns

The underlying follows geometric Brownian motion with constant drift and volatility. Real returns have fatter tails and clustered volatility.

Constant volatility

Implied volatility is treated as a single number across strike and time. In real markets, implied volatility varies by strike (the "smile" or "skew") and by tenor.

Continuous dividends

Dividends are modeled as a continuous yield q, not as discrete payments on ex-dividend dates.

Constant risk-free rate

A single rate r is used. Real curves are not flat.

No frictions

No transaction costs, bid/ask spreads, taxes, borrow costs, or short-sale restrictions.

What this means in practice

The numbers from this calculator are educational estimates. They are useful for understanding how vega and the other Greeks respond to changes in inputs — for example, how vega scales with time to expiration, or why gamma is highest near the money. They are not appropriate for live trading decisions, mark-to-market, or risk management.