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The Greeks: The Language of Risk
The Greeks: What Moves the Price?
Once you have determined the price of an option using the Black-Scholes model, the next step is to answer questions like, “How much will the option price change if the stock price rises by $1?” In financial engineering, these answers are derived through mathematical derivatives, with each component represented by a Greek letter.
1. Delta (Δ): Sensitivity to Price Changes
Delta measures how much the option price changes for a one-unit change in the underlying asset’s price.
- Call Option Delta: Between 0 and 1 (option price rises as stock price rises).
- Put Option Delta: Between -1 and 0 (option price falls as stock price rises).
[!TIP] Delta is also used as an approximation of the probability that the option will expire in the money.
2. Summary of Key Greeks
Definition and Risk Management Significance of the Greeks
| Greek | Sensitivity Target | Core Meaning | Hedging Strategy |
|---|---|---|---|
| Delta (Δ) | Stock Price (S) | Directional Risk | Delta Neutral (Stock Trading) |
| Gamma (Γ) | Delta (Δ) | Curvature (Acceleration) Risk | Gamma Hedging (Option Trading) |
| Theta (Θ) | Time (T) | Time Value Erosion (Time Decay) | Disadvantageous for Buyers, Advantageous for Sellers |
| Vega (v) | Volatility (σ) | Volatility Risk | Price fluctuation due to IV changes |
| Rho (ρ) | Interest Rate (r) | Interest Rate Sensitivity | Opportunity cost due to interest rate changes |
3. Risk Profiling: Relationship Between Delta and Gamma
The rate at which Delta itself changes as the stock price changes is called Gamma. A large Gamma means that Delta changes rapidly even with minor stock price movements, which implies a very urgent situation for risk managers.
Buying or selling stocks to set the portfolio's total Delta to zero.
When the stock price moves, Delta is generated again due to Gamma.
Buying or selling stocks again to reset the Delta to zero.
💡 Professor’s Tip
Quants are not just people who guess “will it go up or down.” They are mathematical designers who combine these Greeks to selectively hold specific risks they desire, such as eliminating Delta while keeping Vega.