Relationship and Comparison Among Stock Greeks

The Greeks aren't independent; they dance together! Understanding their relationships is key to managing your option risks effectively. Think of them as interconnected parts of an engine, each influencing the others.

1. Delta (Δ) and Gamma (Γ): The Dynamic Duo

• Relationship: Gamma (Γ) is the rate of change of Delta (Δ). It's how quickly Delta is changing.
• In Simpler Terms: Imagine you're driving (Delta is your speed). Gamma is how quickly you're accelerating or decelerating. High Gamma means your speed changes rapidly with a tap on the gas pedal. Low Gamma means your speed is more stable.
• Comparison:
  • Delta tells you which direction your option price will move with a small change in the stock price.
  • Gamma tells you how much your Delta will change with that same small change in the stock price.
• Practical Implication: If you're trying to keep your portfolio Delta-neutral (unaffected by small stock price changes), you need to manage Gamma. High Gamma means you'll need to re-adjust your hedge more frequently.

2. Theta (Θ) and the Other Greeks: The Time Pressure

• Relationship: Theta (Θ) is affected by Delta (Δ) and Gamma (Γ), especially when hedging. A Delta-neutral and Gamma-neutral portfolio still loses value due to Theta (time decay).
• In Simpler Terms: You can perfectly balance your speed and acceleration (Delta and Gamma), but the clock is still ticking (Theta), and your option is losing value simply due to the passage of time.
• Comparison:
  • Delta, Gamma, Vega, and Rho tell you how your option is affected by changes in the market.
  • Theta tells you how your option is affected by time itself.
• Practical Implication: You need the stock to move in your favor quickly enough to offset the negative effect of Theta.
• Theta's Behavior
  • Theta is highest (most negative) for at-the-money options, because the stock is most likely to move into or out of the money, which is most impactful when the option is nearly at the strike price
  • Deep in-the-money and out-of-the-money options have lower Theta
  • The more volatile the underlying asset, the more the value is affected by time, so the higher the Theta
• Theta & Volatility
  • Theta's negative impact can be offset by an increase in volatility

3. Vega (ν) and Time to Expiration (T): The Uncertainty Window

• Relationship: Vega (ν) generally decreases as time to expiration (T) gets closer.
• In Simpler Terms: Volatility matters less as you get closer to the expiration date. There's simply less time for big price swings to happen.
• Comparison:
  • Vega is more important for longer-dated options.
  • Vega becomes less important as expiration approaches.
• Practical Implication: If you're trading volatility, focus on longer-dated options.

4. Delta (Δ) and the Probability of Finishing "In the Money" (ITM): A Rough Guide

• Relationship: Delta (Δ) can roughly estimate the probability of a call option expiring in the money.
• In Simpler Terms: An option with a Delta of 0.70 has roughly a 70% chance of being worth something at expiration.
• Important Caveat: This is just an approximation. It's not a precise probability calculation. The actual probability depends on many factors.

5. Rho (ρ) and Interest Rates

• Relationship Rho reflects the sensitivity of the option price to changes in interest rates. Calls generally have a positive rho and puts have a negative rho.
  • The longer the time to expiration, the greater the impact of interest rate changes on the option price.
• In Simpler Terms A call option benefits from higher interest rates, as it means the underlying asset can produce a higher return, while a put option is harmed as that return can be avoided by purchasing a call.
• Comparison:
  • Calls generally have a positive rho and puts have a negative rho
  • The longer the time to expiration, the higher the absolute value of Rho.

6. Put-Call Parity and Greeks

• Relationship: Put-call parity links the prices of a call option and a put option with the same strike price and expiration date. This relationship also extends to the Greeks.
• In Simpler Terms: If you know the price of a call option, you can calculate the price of a put option and vice-versa
• Comparison:
  • For example, the Delta of a call option and the Delta of a corresponding put option are related through the equation: Call Delta - Put Delta = e^(-qT), where q is the dividend yield.

Summary Table

The "d1" and "d2" Factors - The Hidden Engine Parts:

• Remember those "d1" and "d2" terms in the formulas? They're intermediate calculations based on the Black-Scholes model. They encapsulate the relationship between the stock price, strike price, time to expiration, volatility, and interest rates. You don't need to memorize their exact meaning, but understand that they're the ingredients that drive the Greek calculations. Important Note: These relationships are based on theoretical models and assumptions. Real-world market conditions can cause deviations.