Black-Scholes Call Price

Options

Updated Apr 2026 • Has calculator

The theoretical fair value of a European call option derived from the Black-Scholes model.

What is BS Call?

The Black-Scholes call price gives the theoretical fair value of a European call option on a non-dividend-paying stock. It is derived from a no-arbitrage argument that constructs a risk-free portfolio by continuously delta-hedging. The formula takes the current stock price, strike price, time to expiration, risk-free rate, and implied volatility as inputs and returns the call premium an option buyer should pay.

Formula

C = S·N(d₁) − K·e^(−rT)·N(d₂)

Worked Example

Representative Q1 2024 market conditions

Step 1 Stock price (S): $185.00

Step 2 Strike price (K): $185.00 (at-the-money)

Step 3 Risk-free rate (r): 5.25% (3-month T-Bill)

Step 4 Time to expiry (T): 30 days = 0.082 years

Step 5 Implied volatility (σ): 28%

Step 6 d₁ = [ln(1) + (0.0525 + 0.0392)×0.082] / (0.28×0.286) = 0.114

Step 7 d₂ = 0.114 − 0.080 = 0.034

Step 8 C = 185×N(0.114) − 185×e^(−0.0043)×N(0.034) ≈ $6.11

Step 9 → The 30-day ATM call costs approximately $6.11 per share ($611 per contract)

Source: Black & Scholes (1973) — Journal of Political Economy (2024-01-15)

Calculate BS Call

Stock Price

Current market price of the underlying stock

Strike Price

Price at which the option can be exercised

Risk-Free Rate

Annual risk-free rate (e.g. 3-month T-Bill yield)

Time to Expiry

Time to expiration in years (e.g. 0.25 = 3 months)

Implied Volatility

Annualised implied volatility of the underlying stock

Call Premium

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Not investment advice.

How to Interpret BS Call

< 0

Pricing error — call price cannot be negative

0 – 1

Low premium — far OTM or short-dated option

1 – 10

Typical ATM premium — near-term option

> 10

Deep ITM or long-dated — high intrinsic/time value

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Sources