📘 Black-Scholes Model: Overview & Formulas

Introduction

The Black-Scholes Model is a foundational framework in quantitative finance used to determine the fair value of European-style options. It assumes that the underlying asset follows a geometric Brownian motion with constant volatility and interest rates.

Core Variables

• S – Current stock price
• K – Strike price
• T – Time to expiration (in years)
• r – Risk-free interest rate
• q – Dividend yield
• σ – Volatility of the underlying asset
• N(x) – Cumulative distribution function of the standard normal distribution

Intermediate Terms

Define:
d₁ = [ln(S/K) + (r − q + σ²/2)·T] / (σ·√T)
d₂ = d₁ − σ·√T

Option Pricing Formulas

• Call Option Price: C = S·e−qT·N(d₁) − K·e−rT·N(d₂)
• Put Option Price: P = K·e−rT·N(−d₂) − S·e−qT·N(−d₁)

Greeks

• Delta (Δ):
  • Call: Δcall = e−qT·N(d₁)
  • Put: Δput = e−qT·(N(d₁) − 1)
• Gamma (Γ): Γ = [e−qT·N′(d₁)] / (S·σ·√T)
• Theta (Θ):
  • Call: Θcall = [−S·σ·e−qT·N′(d₁)] / [2·√T] − r·K·e−rT·N(d₂) + q·S·e−qT·N(d₁)
  • Put: Θput = [−S·σ·e−qT·N′(d₁)] / [2·√T] + r·K·e−rT·N(−d₂) − q·S·e−qT·N(−d₁)
• Vega (ν): ν = S·e−qT·√T·N′(d₁)
• Rho (ρ):
  • Call: ρcall = K·T·e−rT·N(d₂)
  • Put: ρput = −K·T·e−rT·N(−d₂)

Note: N′(d₁) is the standard normal probability density function: N′(x) = (1 / √(2π)) · e−x²/2