The Greek Letters

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📈 As we saw, Delta Hedging is a very effective tool for hedging the risk of a sold option.
⚖️ When a portfolio becomes Delta Neutral, changes in the underlying asset’s price no longer affect profitability.
🔄 We also saw that the option’s delta changes with the underlying asset price, so hedging requires periodic portfolio rebalancing (Dynamic Hedging).

❗ However, Delta Hedging alone is not sufficient for complete risk protection, and in practice, other “Greek letters” are also monitored, which play their roles in the relationship between the stock and the option.

📌 It is important to understand that if delta hedging is effective only for a single type of option, when we consider other Greeks here, we are talking about an options portfolio that depends on a single underlying asset.

💼 Options dependent on a single asset can vary by strike price, maturity, type (Put/Call), and quantity. The portfolio may also include futures…

📑 Below, the Greeks are discussed in general, and at the end, formulas and an Excel file are provided as an example from the source*.

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Gamma (Γ) – measures the curvature of the relationship.
📉 It is a second-order derivative that affects the sensitivity of the option price when the underlying price moves in a wide range.
🔹 (For small changes, delta is usually enough, as the curvature has little impact).
🎯 For ideal hedging, the portfolio needs to be both delta and gamma neutral.

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THETA (θ) – is the rate at which the value of an option portfolio changes over time, assuming all other parameters remain constant.

🕒 θ does not serve the same hedging purpose as gamma and delta, since in their case there is uncertainty,
⏳ theta is simply related to the passage of time.
✅ Nevertheless, it is still considered an important parameter, as it is symmetrically related to gamma.

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Vega (V) – is the rate at which the value of an option portfolio changes due to volatility in the underlying asset.
💹 In other words, the larger the Vega (positive or negative), the more sensitive the portfolio value is to changes in the underlying asset’s price volatility (Sensitivity on Volatility).

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Rho (ρ) – is the rate at which the value of an option portfolio changes due to interest rate changes.
💰 In other words, it measures how sensitive the portfolio value is to changes in the market interest rate.

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How are the parameters managed in practice?

🔹 In practice, it is not possible for a portfolio to constantly balance all parameters.
📊 Typically, individual traders manage an options portfolio dependent on a single asset (e.g., a derivatives portfolio linked to the Euro).
💻 Software automatically calculates the Greek letters and sets limits, which the trader cannot exceed, and/or requires additional authorization.

📅 Delta is balanced on a daily basis.
👀 Other Greeks are mainly monitored, and reactions to them occur less frequently.

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Formulas:

📂 Excel – Greeks – this file contains examples with formulas for a dividend-free stock and a portfolio of short European calls.

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📚 Source: Options, Futures & Other Derivatives, John C. Hull

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Greek	Symbol	Unit	What it Measures	Practical Meaning
Delta	Δ	shares	∂V / ∂S	Value change for $1 move in underlying
Gamma	Γ	shares per $	∂Δ / ∂S	Change in delta for $1 move
Vega	ν	currency per 1.00 vol	∂V / ∂σ	Value change for 100% vol change
Theta	Θ	currency per year*	∂V / ∂t	Value lost (or gained) as time passes
Rho	ρ	currency per 1.00 rate	∂V / ∂r	Value change for 100% rate change