What is the Black-Scholes formula?

The Black-Scholes model is a closed-form solution for European option prices derived from no-arbitrage assumptions. It computes theoretical fair value from spot price, strike, time to expiration, risk-free rate, and volatility. See the methodology page for the exact formulas we use.

What is delta in options?

Delta measures how much the option price changes when the underlying stock moves $1. Call deltas range from 0 to +1; put deltas range from 0 to −1. Delta also serves as a rough proxy for the probability the option finishes in-the-money.

What is theta and how does time decay work?

Theta measures how much the option loses per day from time decay. Theta accelerates as expiration approaches, especially for at-the-money options. Premium sellers benefit from theta; premium buyers lose to it. The Black-Scholes calculator shows theta per day so you can quantify the daily decay on any contract.

What is implied volatility (IV)?

Implied volatility is the volatility figure that, when plugged into the Black-Scholes formula, returns the market price of the option. It represents the market's forecast of future volatility. IV expands and contracts independently of the underlying price.

Black-Scholes calculator and Greeks — price, delta, gamma, theta, vega, IV solver

Theoretical price for European calls and puts, full Greek table, implied volatility solver, and interactive sensitivity curves.

The Black-Scholes model gives a closed-form theoretical price for a European call or put option using six inputs: spot price, strike, time to expiration, risk-free rate, dividend yield, and implied volatility. This calculator returns that theoretical price plus all five first-order Greeks — delta, gamma, theta, vega, and rho — and can reverse-solve for the implied volatility that reconciles a market price. Assumptions: no early exercise (European style), constant volatility, and continuous trading; real US equity options are American-style and can deviate from Black-Scholes near ex-dividend dates.

Inputs

Option type

Spot price (S)

Strike (K)

Days to expiry

Risk-free rate (r)

decimal, e.g. 0.045 = 4.5%

Implied volatility (σ)

decimal, e.g. 0.25 = 25%

Dividend yield (q)

Solve implied volatility

Market price

Implied volatility: —

Results & Greeks

Theoretical price

—

Delta

—

Δ stock = $1

Gamma

—

Δ of Δ

Theta

—

per day

Vega

—

per 1 vol point

Rho

—

per 1% rate

Probability ITM

—

Risk-neutral

Sensitivity

Black-Scholes pricing methodology, formulas, and worked examples

Formula

Call price C = S₀N(d₁) − K·e^−rTN(d₂). Put price P = K·e^−rTN(−d₂) − S₀N(−d₁). Where d₁ = [ln(S₀/K) + (r + σ²/2)T] ÷ (σ√T) and d₂ = d₁ − σ√T.

Inputs

Spot price (S₀)
Strike (K)
Time to expiration (T, in years)
Risk-free rate (r, default 4.5%)
Implied volatility (σ, annualized)
Option type (call / put)

Conservative worked example

NVDA spot $400, $410 call, 30 days to expiration (T = 0.0822 years), r = 4.5%, σ = 35%. d₁ = 0.082, d₂ = -0.018. Call price ≈ $13.50. Delta ≈ 0.53.

Aggressive worked example

NVDA spot $400, $440 call (deep OTM), 7 days to expiration (T = 0.0192), σ = 50% (high IV pre-earnings). Call price ≈ $2.10. Delta ≈ 0.21. Time-decay dominant.

Losing outcome example

You sold the $440 call from the aggressive example for $2.10. NVDA gaps up to $470 on earnings. Your short call is now intrinsically worth $30 (intrinsic = $470 - $440) plus residual extrinsic. Realized loss = ($30 + ext) - $2.10 ≈ -$28+ per share = -$2,800+ per contract. IV crush mitigates but does not eliminate the loss.

Commissions and slippage

BSM is a pricing model — no commissions or fees baked in. Add your broker fees separately.
BSM gives theoretical price. Real fills depend on bid-ask spread; treat the model price as a midpoint estimate.

What this calculator does NOT model

American-style early-exercise premium (BSM is European).
Dividends (use a dividend-adjusted variant if needed).
Volatility smile / skew across strikes.
Stochastic volatility regimes (jumps, GARCH dynamics).

Related metric definitions

For the full mathematical methodology, see methodology. Educational only — not investment advice. See the disclaimer.