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Options Pricing: The Foundation

Options are priced according to the Black-Scholes model, which uses five inputs to determine fair value. Understanding these is essential for algorithmic options trading.

Black-Scholes Formula

C = S×N(d₁) - K×e^(-rT)×N(d₂)

Where d₁ and d₂ are complex functions of S, K, r, T, and σ

The five inputs:

S = Current stock price
K = Strike price
T = Time to expiration (in years)
r = Risk-free rate
σ = Implied volatility (the market's guess of future volatility)

The Five Greeks

Greek	Measures	Range	Trading Use
Delta (Δ)	Price change per $1 move in stock	Call: 0-1, Put: -1-0	Directional exposure
Gamma (Γ)	Rate of delta change	0 to Max at ATM	Options: gamma risk
Theta (Θ)	Daily time decay	Negative for buyers	Expect daily loss for long options
Vega (ν)	Sensitivity to implied volatility	Positive for long	Bet on volatility changes
Rho (ρ)	Sensitivity to interest rates	Usually small	Less important for trading

python

from scipy.stats import norm
import numpy as np

def black_scholes(S, K, T, r, sigma):
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)

    call_price = S * norm.cdf(d1) - K * np.exp(-r*T) * norm.cdf(d2)
    delta = norm.cdf(d1)
    gamma = norm.pdf(d1) / (S * sigma * np.sqrt(T))
    vega = S * norm.pdf(d1) * np.sqrt(T) / 100  # Per 1% change in vol
    theta = (-S * norm.pdf(d1) * sigma / (2*np.sqrt(T)) - r*K*np.exp(-r*T)*norm.cdf(d2)) / 365

    return call_price, delta, gamma, vega, theta

# Example: AAPL $150 call, stock at $148
S, K, T, r, sigma = 148, 150, 30/365, 0.05, 0.25
price, delta, gamma, vega, theta = black_scholes(S, K, T, r, sigma)

print(f"Call price: ${price:.2f}")
print(f"Delta: {delta:.3f}  (price moves $0.72 if stock moves $1)")
print(f"Gamma: {gamma:.4f}  (delta changes by this much if stock moves $1)")
print(f"Vega: {vega:.2f}   (price changes $0.23 if IV changes 1%)")
print(f"Theta: ${theta:.3f}  (loses $0.02 per day from time decay)")

IV Rank & IV Percentile

Implied Volatility (IV) at this very moment—where does it sit in its 52-week range?

IV Rank > 50: Options are expensive relative to history
IV Rank < 50: Options are cheap relative to history
Buy options when IV Rank < 30 (cheap)
Sell options when IV Rank > 70 (expensive)

Portfolio Greeks

Your portfolio delta, gamma, theta, vega are the sums of your positions. Manage them as a portfolio, not individual options.

Practical Tip: Watch Theta Daily