III AdvancedWeek 23 • Lesson 67Duration: 50 min

GRK The Greeks & Sensitivities

Delta, gamma, vega, theta — your portfolio's vital signs

Learning Objectives

•Understand each Greek and what it measures
•Learn how to use Greeks for risk management
•Apply sensitivity analysis concepts to non-options portfolios

Explain Like I'm 5

The Greeks tell you how your option reacts when things change. Delta = option moves when stock moves $1. Gamma = how delta itself changes (the second derivative). Vega = sensitivity to volatility. Theta = time decay eating your position alive. Knowing your Greeks is knowing your risks — they are the vital signs of any portfolio.

Think of It This Way

Greeks are like the dashboard gauges in a car. Delta = speedometer (direction and speed). Gamma = how fast you're accelerating. Theta = fuel gauge (time running out). Vega = weather sensitivity (how road conditions affect your speed). You wouldn't drive without a dashboard. Don't trade without knowing your Greeks.

1The Five Greeks

Delta ( $\Delta$ ): Change in option price per $1 change in underlying. - Call delta: 0 to +1 (ATM is approximately 0.5) - Put delta: 0 to -1 (ATM is approximately -0.5) - Portfolio delta = sum of individual deltas - Delta-neutral = portfolio delta near zero (market-neutral position) Gamma ( $\Gamma$ ): Change in delta per $1 change in underlying. - Highest for ATM options near expiration - Gamma risk = your delta changes rapidly, forcing constant rebalancing - Positive gamma is your friend (profits accelerate). Negative gamma is the killer. Vega ( $\nu$ ): Change in option price per 1% change in implied vol. - All options have positive vega (more vol = more valuable) - Long options = long vega (benefit from vol spikes) - Short options = short vega (get hurt by vol spikes) Theta ( $\Theta$ ): Change in option price per day. - Negative for long options (time decay) - Positive for short options (you collect the decay) - Accelerates near expiration — the last 30 days are particularly brutal Rho ( $\rho$ ): Change per 1% rate change. - Usually the least important Greek - Only matters for long-dated options or high-rate environments

2Delta Deep Dive

Delta is arguably the most important Greek because it has multiple interpretations: Hedge ratio: A call with delta 0.5 means you need to short 0.5 shares to be hedged. This is "delta hedging." Probability proxy: Delta approximately equals the probability of finishing in-the-money. A 0.3 delta call has roughly a 30% chance of being ITM at expiry. (This is approximate, not exact.) Position equivalent: 10 calls with delta 0.6 = equivalent exposure to 6 shares. This lets you convert complex options positions into "equivalent shares" for risk management. The tricky part: delta CHANGES as the stock moves (that's what gamma measures). So your hedge needs constant updating. This is the fundamental challenge of delta hedging — it's never a set-and-forget operation. Far OTM options have tiny delta. They're cheap but barely move with the stock. ATM options have ~0.5 delta — most responsive per dollar spent.

3The Gamma-Theta Tradeoff

This is one of the most important concepts in options trading that beginners tend to miss: Long gamma (long options): Your delta automatically adjusts in your favor. Stock goes up -> delta increases -> you make more per additional dollar. Stock goes down -> delta decreases -> you lose less. Sounds ideal. But you pay theta for that privilege. Every day, time decay eats your position. You need the stock to MOVE enough to overcome theta. Short gamma (short options): Opposite. You COLLECT theta daily, but your delta works against you on big moves. Small moves = profit (theta). Big moves = pain (gamma). This is the fundamental tradeoff: - Long gamma, short theta: Betting on movement. Pay daily, profit on big moves. - Short gamma, long theta: Betting on stability. Collect daily, bleed on big moves. This is why options market making is so difficult — you're constantly managing this tradeoff while trying to stay delta-neutral.

4Greeks Beyond Options

The concept of sensitivities applies everywhere, not just options: Sensitivity analysis for any portfolio: - "Delta" to market moves: How much does PnL change per 1% market move? This is your directional exposure. - "Vega" to volatility: How does performance change in high-vol vs low-vol regimes? Many systems adjust exits based on regime. - "Theta" equivalent: Is there time decay in your signal? Signal decay (information coefficient declining over time) operates the same way. - "Gamma" equivalent: Is your position sizing convex? DD-triggered scaling creates negative gamma (reduces exposure as losses mount). Sensitivity analysis is a UNIVERSAL framework: 1. Identify key inputs 2. Measure how output changes per unit change in each input 3. Manage the largest sensitivities This is what parameter sensitivity testing does in production — vary thresholds, risk levels, lookback periods, and see what breaks.

Call Option Delta vs Underlying Price (K=100)

Key Formulas

Black-Scholes Delta (Call)

Delta ranges from 0 (deep OTM) to 1 (deep ITM). ATM call delta is approximately 0.5. Delta also roughly represents the probability of finishing ITM.

Gamma

Gamma is highest for ATM options near expiration. High gamma means delta changes rapidly, requiring frequent rebalancing of hedges.

Hands-On Code

Computing Greeks

python

import numpy as np
from scipy.stats import norm

def compute_greeks(S, K, T, r, sigma, option_type='call'):
    """Compute all Greeks for a European option."""
    d1 = (np.log(S/K) + (r + sigma**2/2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    
    Nd1 = norm.cdf(d1)
    Nd2 = norm.cdf(d2)
    nd1 = norm.pdf(d1)
    
    if option_type == 'call':
        delta = Nd1
        theta = (-(S*nd1*sigma)/(2*np.sqrt(T)) - r*K*np.exp(-r*T)*Nd2) / 252
    else:
        delta = Nd1 - 1
        theta = (-(S*nd1*sigma)/(2*np.sqrt(T)) + r*K*np.exp(-r*T)*(1-Nd2)) / 252
    
    gamma = nd1 / (S * sigma * np.sqrt(T))
    vega = S * nd1 * np.sqrt(T) / 100  # per 1% vol
    rho = K * T * np.exp(-r*T) * (Nd2 if option_type=='call' else Nd2-1) / 100
    
    print(f"=== GREEKS ({option_type.upper()}) ===")
    print(f"  Delta: {delta:+.4f}")
    print(f"  Gamma: {gamma:+.4f}")
    print(f"  Vega:  {vega:+.4f}")
    print(f"  Theta: {theta:+.4f}")
    print(f"  Rho:   {rho:+.4f}")
    
    return {'delta': delta, 'gamma': gamma, 'vega': vega, 'theta': theta, 'rho': rho}

Computes all five Greeks for a European option using Black-Scholes closed-form expressions, providing a complete risk sensitivity profile.

Knowledge Check

Q1.You're short an ATM call with delta = -0.5. The stock moves up $2. Approximately how much do you lose?

Assignment

Compute all Greeks for a range of options. Plot delta vs stock price, gamma vs moneyness, and theta vs time to expiration. Which Greeks dominate near expiration vs far from expiration?

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