Module 6 — Professional Quant Topics

Options and the Greeks

Factor models, options pricing, risk, and microstructure — the curriculum interview desks expect.

Module lessons

Factor Models — CAPM to Fama-French to BarraOptions and the GreeksRisk Management — VaR, CVaR, and Position SizingMarket Microstructure — How Orders Become Prices

Learning objectives

  • ▸Read a Black-Scholes price.
  • ▸Interpret delta, gamma, vega, theta, rho.
  • ▸Understand why a delta-hedged long-vol position pays off when realised > implied.

FORMULA

Black-Scholes call

C = S·N(d1) - K·e^{-rT}·N(d2)
d1 = (ln(S/K) + (r + σ²/2)T) / (σ·sqrt(T))
d2 = d1 - σ·sqrt(T)

TEXT

The Greeks intuitively

Δ (delta): ∂C/∂S — how much option moves per $1 in spot. Γ (gamma): ∂²C/∂S² — how quickly delta changes. ν (vega): ∂C/∂σ — sensitivity to implied vol. Θ (theta): ∂C/∂t — daily P&L from time decay. ρ (rho): ∂C/∂r — interest-rate sensitivity (usually small).

TEXT

Long-vol P&L

A delta-hedged long option position earns: 0.5 · Γ · S² · (σ_realised² - σ_implied²) · dt. If realised vol exceeds implied, you make money. If it doesn't, theta eats you alive. This is the entire game of long-vol hedge funds.

CODE

BS in 10 lines

import numpy as np
from scipy.stats import norm

def bs_call(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)
    return S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)

print(bs_call(100, 100, 1, 0.04, 0.20))
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