Greeks and Risk Management

The Greeks are the sensitivities of an option’s value to changes in underlying variables.
They quantify how price, volatility, time, and interest rates affect an option’s value — the foundation of risk management and hedging.

1. Overview of Greeks

Greek	Symbol	Definition	Interpretation
Delta	$\Delta = \frac{\partial V}{\partial S}$	Change in value for small move in $S$	Hedge ratio (shares per option)
Gamma	$\Gamma = \frac{\partial^2 V}{\partial S^2}$	Curvature of value wrt $S$	Convexity of option value
Vega	$\nu = \frac{\partial V}{\partial \sigma}$	Sensitivity to volatility	Exposure to implied vol changes
Theta	$\Theta = \frac{\partial V}{\partial t}$	Sensitivity to time decay	Daily erosion of time value
Rho	$\rho = \frac{\partial V}{\partial r}$	Sensitivity to interest rate	Rate exposure
Phi	$\phi = \frac{\partial V}{\partial q}$	Sensitivity to dividend yield	Dividend exposure

2. Delta — Directional Sensitivity

For a European call under Black–Scholes:

\[\Delta_{call} = e^{-qT} N(d_1), \qquad \Delta_{put} = e^{-qT} (N(d_1) - 1).\]

• $\Delta_{call}$ ranges from 0 → 1.
• $\Delta_{put}$ ranges from -1 → 0.

Interpretation

• Delta ≈ number of shares needed to hedge one option.
• Example: $\Delta = 0.6$ means you buy 0.6 shares per short call to be delta-neutral.

3. Gamma — Convexity and Delta Stability

\[\Gamma = e^{-qT}\frac{N'(d_1)}{S_0\sigma\sqrt{T}}.\]

• Measures how fast delta changes with $S$.
• High near-the-money, low deep ITM/OTM.
• Gamma determines the hedging frequency required to remain delta-neutral.

Gamma is positive for both long calls and long puts — long options benefit from volatility because they are convex.

4. Vega — Volatility Sensitivity

\[\nu = S_0 e^{-qT} N'(d_1)\sqrt{T}.\]

• Measures change in option value per 1% change in volatility.
• Largest for ATM, long-dated options.
• Traders use Vega-neutral portfolios to isolate volatility trading from directional moves.

5. Theta — Time Decay

\[\Theta = -\frac{S_0 e^{-qT} N'(d_1)\sigma}{2\sqrt{T}} - r K e^{-rT} N(d_2) + q S_0 e^{-qT} N(d_1).\]

• Typically negative for long calls and puts — time decay erodes value.
• Positive theta occurs in short option positions (income strategies).
• Time decay accelerates as expiration approaches.

6. Rho and Phi — Rate and Dividend Exposure

\[\rho_{call} = K T e^{-rT} N(d_2), \qquad \rho_{put} = -K T e^{-rT} N(-d_2).\] \[\phi_{call} = -T S_0 e^{-qT} N(d_1), \qquad \phi_{put} = T S_0 e^{-qT} N(-d_1).\]

Interest rates and dividends matter primarily for long-dated or high-yield assets.

7. Higher-Order Greeks (optional)

Symbol	Name	Description
$\text{Vanna}$	$\frac{\partial^2 V}{\partial S \partial \sigma}$	How delta changes with volatility
$\text{Vomma}$	$\frac{\partial^2 V}{\partial \sigma^2}$	Curvature wrt volatility (volga)
$\text{Charm}$	$\frac{\partial \Delta}{\partial t}$	Time decay of delta
$\text{Color}$	$\frac{\partial \Gamma}{\partial t}$	Time decay of gamma

These are used in advanced risk management systems.

8. Risk Management and Hedging

8.1 Delta Hedging

A delta-hedged portfolio offsets price risk:

\[\Pi = V - \Delta S.\]

Regularly rebalance $\Delta$ to remain neutral.
However, imperfect hedging introduces gamma and theta risk — known as Gamma–Theta tradeoff.

8.2 Gamma–Theta Tradeoff

• Long Gamma → benefits from volatility but loses to time decay.
• Short Gamma → earns Theta but loses from large price swings.

Market makers dynamically manage this tradeoff through delta–gamma hedging.

8.3 Vega Hedging

To hedge volatility exposure, hold an opposite Vega position — for example, long ATM calls and short OTM calls.
Complex portfolios are constructed to remain delta–vega–theta neutral at the book level.

8.4 Portfolio Greeks

In a portfolio of $n$ options:

\[\Delta_P = \sum_i \Delta_i Q_i, \quad \Gamma_P = \sum_i \Gamma_i Q_i, \quad \nu_P = \sum_i \nu_i Q_i, \dots\]

These aggregate Greeks form the basis of risk limits and scenario stress testing.

9. Visualization: Greek Sensitivities

10. Summary

• The Greeks quantify directional, curvature, volatility, time, and rate sensitivities.
• They enable hedging, risk control, and capital allocation.
• Real-world traders manage entire portfolios through Greek neutrality.
• The balance between Gamma and Theta defines most market-making behavior.

Next up: Volatility Surface and Smile