04 // Interactive Formula Compendium // FINANCEC = S₀ · N(d₁) − K · e^(−rT) · N(d₂)
Black-Scholes-Merton Option Pricing Formula — Call, Put & Greeks Sheet
The Black-Scholes formula prices a non-dividend-paying European call option as C = S₀ N(d₁) − K e^(−rT) N(d₂), where N(·) is the standard normal cumulative distribution function, S₀ is spot price, K is strike, r is risk-free rate, σ is volatility, and T is time to expiry.
Formula Reference
Main Formula
C = S₀ · N(d₁) − K · e^(−rT) · N(d₂)
Rearranged Forms
d₁ = [ ln(S₀ / K) + (r + ½σ²)T ] / (σ √T), d₂ = d₁ − σ √T
P = K · e^(−rT) · N(−d₂) − S₀ · N(−d₁) (European Put via Put-Call Parity)
Δ_call = N(d₁), Δ_put = N(d₁) − 1 (Option Delta Hedge Ratio)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| C | European Call Option Theoretical Price (C) | USD |
| S0 | Underlying Spot Price (S₀) | USD |
| K | Option Strike Price (K) | USD |
| r | Risk-Free Annual Rate (r) | 1/yr |
| sigma | Annualized Volatility (σ) | 1/√yr |
| T | Time to Expiration (T in years) | yr |
Inputs & Parameters
[USD]
[USD]
[1/yr]
[1/√yr]
[yr]
Calculated Result (European Call Option Theoretical Price (C))
C =10.45058USD
Formula Notes & Background
- Model the underlying asset via geometric Brownian motion dS = μS dt + σS dW and apply Itô's Lemma to option value V(S, t).
- Construct a dynamically delta-hedged portfolio Π = V − (∂V/∂S)S that eliminates the stochastic Wiener term dW, forcing dΠ = rΠ dt by no-arbitrage.
- Solve the resulting parabolic Black-Scholes PDE ∂V/∂t + ½σ²S²(∂²V/∂S²) + rS(∂V/∂S) − rV = 0 with terminal payoff max(S_T − K, 0).
Step-by-Step Examples
#1: Price an at-the-money 1-year European call option with S₀ = $100, K = $100, r = 5%, and σ = 20%.
S₀ = $100K = $100r = 0.05σ = 0.20T = 1.0 yr
d₁ = (0 + (0.05 + 0.02)×1) / 0.20 = 0.35 ⇒ N(d₁) = 0.6368; d₂ = 0.15 ⇒ N(d₂) = 0.5596
→ C = 100(0.6368) − 100 e^(−0.05)(0.5596) = $10.45 (Put P = $5.57)
#2: Verify Put-Call Parity C − P = S₀ − K e^(−rT) on the option above.
C = $10.4506P = $5.5735K e^(−0.05) = $95.1229
10.4506 − 5.5735 = 4.8771 = 100 − 95.1229
→ Parity holds to $4.88
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Frequently Asked Questions
What is the intuitive meaning of N(d₂) and N(d₁) in the Black-Scholes formula?+
N(d₂) is the risk-neutral probability that the call option finishes in-the-money (S_T > K), while N(d₁) is the option's Delta (∂C/∂S₀) and share-equivalent hedge ratio.
Why does the stock's expected return μ disappear from the Black-Scholes equation?+
Continuous delta hedging eliminates directional market risk, so under risk-neutral valuation all assets drift at the risk-free rate r.
What is Implied Volatility (IV)?+
Implied volatility is the unique value of σ that, when plugged into the Black-Scholes formula, matches the observed market price of the option.