Greeks: Option Risk Metrics

Greeks measure the sensitivity of an option's price to various market variables. They are essential tools for every options trader.

🔥 Vibe Prompt

"Calculate all five Greeks (Delta, Gamma, Theta, Vega, Rho) for Call and Put options in Python. Explain each metric's real trading meaning. Output formatted table and charts."

Five Greeks Overview

| Greek | Symbol | Measures | Unit | |-------|--------|----------|------| | Delta | $\Delta$ | Price sensitivity to $1 stock change | $/$ | | Gamma | $\Gamma$ | Delta's sensitivity to stock change | $/$/$ | | Theta | $\Theta$ | Time decay per day | $/day | | Vega | $\nu$ | Price sensitivity to 1% volatility change | $/% | | Rho | $\rho$ | Price sensitivity to 1% rate change | $/% |

Real Trading Applications

Delta

Delta = 0.6: Stock up $1, Call up $0.60
Delta Neutral Strategy: Buy Call + short Delta shares → overall Delta ≈ 0

Gamma

High Gamma: Delta changes rapidly, good for short-term trading
Low Gamma: Delta stable, good for long-term holding

Theta (Time Decay)

Theta is seller's friend: Option sellers earn time value
Last 30 days: Fastest time decay (Theta spikes)

Vega

High Vega: Option price very sensitive to volatility
Before earnings: Expected volatility rises → Vega matters

Practice Exercise

💡 Vibe Coding Practice: Ask AI to build a "Greeks Analysis Dashboard":

Input option parameters, display all Greeks

Plot Greeks vs stock price curves

3D surface of Greeks vs time

Find Delta-neutral combinations

Visualizing Option Prices

import matplotlib.pyplot as plt
import numpy as np

stock_prices = np.linspace(50, 150, 100)
prices = [black_scholes(S, 100, 0.5, 0.05, 0.2, 'call') for S in stock_prices]
plt.plot(stock_prices, prices)
plt.axvline(100, color='gray', linestyle='--', alpha=0.5)
plt.title('Call Option Price vs Stock Price')
plt.xlabel('Stock Price')
plt.ylabel('Option Price')
plt.grid(True, alpha=0.3)
plt.show()

Sensitivity Analysis

Impact of Volatility

vols = [0.1, 0.2, 0.3, 0.4, 0.5]
S, K, T, r = 100, 100, 0.5, 0.05
for sigma in vols:
    call = black_scholes(S, K, T, r, sigma, 'call')
    put = black_scholes(S, K, T, r, sigma, 'put')
    print(f'Vol {sigma:.0%}: Call ${call:.2f}, Put ${put:.2f}')

Time Decay

As expiration approaches, time value decreases
Theta accelerates in the final 30 days
ATM options have the highest time value

Practical Applications

Real-World Use Cases

Portfolio Insurance: Buy OTM puts to protect against market crashes
Income Generation: Sell covered calls on stocks you already own
Volatility Trading: Trade options based on volatility expectations
Earnings Play: Trade options around earnings announcements

Greeks in Practice

| Greek | Trader's Focus | |-------|---------------| | Delta | Directional risk, hedge ratio | | Gamma | Convexity, risk of large moves | | Theta | Time decay, option seller's friend | | Vega | Volatility risk, earnings plays | | Rho | Interest rate risk, long-dated options |

Delta (Δ) — Directional Risk

Delta measures how much an option's price changes when the underlying asset moves by $1.

| Option Type | Delta Range | Meaning | |-------------|------------|---------| | Call (ITM) | 0.50 to 1.00 | Moves with the stock | | Call (ATM) | ~0.50 | 50% chance of finishing ITM | | Call (OTM) | 0.00 to 0.50 | Low sensitivity | | Put (ITM) | -1.00 to -0.50 | Moves opposite the stock | | Put (ATM) | ~-0.50 | 50% chance of finishing ITM | | Put (OTM) | 0.00 to -0.50 | Low sensitivity |

Delta Examples

# Calculate position delta
call_delta = 0.65  # 65 delta
position_delta = call_delta * 10  # 10 contracts = 1,000 shares
print(f"Position delta: {position_delta}")  # Equivalent to owning 650 shares

# Delta-neutral hedge
stock_shares = 1000
call_delta = 0.65
contracts_to_hedge = stock_shares / (call_delta * 100)
print(f"Hedge with {contracts_to_hedge:.1f} contracts")

Delta Over Time

| Days to Expiry | ATM Delta | Deep ITM Delta | Deep OTM Delta | |---------------|-----------|----------------|----------------| | 60 | 0.50 | 0.95 | 0.05 | | 30 | 0.50 | 0.98 | 0.02 | | 7 | 0.50 | 0.99 | 0.01 | | 1 | 0.50 | 1.00 | 0.00 |

Gamma (Γ) — Delta Acceleration

Gamma measures how much delta changes when the underlying moves by $1.

| Gamma Value | Meaning | Best For | |-------------|---------|----------| | High (~0.10) | Delta changes rapidly | Short-dated ATM options | | Medium (~0.05) | Moderate sensitivity | Standard options | | Low (~0.01) | Delta is stable | Long-dated options |

Gamma Risk

# Gamma scalping — profiting from volatility
# Buy ATM option with high gamma
# Stock moves up → delta increases → sell more stock
# Stock moves down → delta decreases → buy back stock

entry_delta = 0.50
entry_gamma = 0.08

# Stock moves $1 up
new_delta = entry_delta + entry_gamma  # 0.58
# Sell 8 shares to remain delta-neutral

# Stock moves $1 down
new_delta = entry_delta - entry_gamma  # 0.42
# Buy 8 shares to remain delta-neutral

# Profit = number of scalps × spread captured

Theta (Θ) — Time Decay

Theta measures how much an option loses value each day.

| Option Type | Theta | Impact | |-------------|-------|--------| | Long Call | Negative (-$5/day) | Loses money daily | | Long Put | Negative (-$4/day) | Loses money daily | | Short Call | Positive (+$5/day) | Gains money daily | | Short Put | Positive (+$4/day) | Gains money daily |

Theta Decay Curve

| Days to Expiry | Daily Theta (ATM) | Decay Rate | |---------------|-------------------|------------| | 60 | -$0.80 | Slow | | 30 | -$1.50 | Moderate | | 14 | -$3.00 | Fast | | 7 | -$5.00 | Very fast | | 3 | -$10.00 | Exponential | | 1 | -$20.00 | Maximum |

Vega (ν) — Volatility Risk

Vega measures how much an option's price changes when implied volatility moves by 1%.

| Event | IV Change | Vega Impact | |-------|-----------|-------------| | Earnings announcement | +10-20% | Option prices spike | | Market crash | +20-50% | Put options surge | | Normal trading | ±2-5% | Gradual changes | | Post-earnings | -10-20% | IV crush, prices drop |

Vega Example

option_price = 5.00
vega = 0.15  # $0.15 per 1% IV change

# Earnings expected, IV rises 15%
price_change = vega * 15  # $2.25
new_price = option_price + price_change  # $7.25

# After earnings, IV drops 20% (IV crush)
price_change = vega * -20  # -$3.00
new_price = option_price + price_change  # $2.00

Rho (ρ) — Interest Rate Risk

Rho measures how much an option's price changes when interest rates change by 1%.

| Option | Rho Sensitivity | Current Impact | |--------|----------------|---------------| | Long-dated calls | Higher | Moderate ($0.10-0.50 per 1%) | | Short-dated calls | Very low | Negligible | | Puts | Negative | Puts lose value as rates rise |

Summary

The Greeks quantify option risk in different dimensions: delta (direction), gamma (acceleration), theta (time decay), vega (volatility), and rho (interest rates). Mastering these enables precise risk management.

Key takeaways:

Delta: directional exposure, 0-1 for calls, -1-0 for puts |
Gamma: delta's rate of change, highest for ATM options |
Theta: time decay, accelerates rapidly in the last 14 days |
Vega: volatility sensitivity, critical before earnings |
Rho: interest rate sensitivity, significant only for long-dated options |
Delta-neutral strategies isolate gamma and theta profits |
Short options profit from theta, long options profit from gamma |
Monitor vega around earnings to avoid IV crush |

What's Next: Combination Strategies

The next chapter covers combination strategies — spreads, straddles, and strangles.