Greeks, Hedging & Volatility Risk

An option responds to the underlying, volatility, time, rates, and sometimes correlation. A Greek estimates the change for a small bump in one factor, holding the others fixed under the chosen model. They are local. They are not a complete description of a jump, a gap, or a model change.

Delta is the first-order response to the underlying. Gamma is how delta itself changes. Vega is the response to implied volatility — confirm whether the desk reports it per 1.00 volatility unit or per one percentage point; that is a factor of 100. Theta is time. Rho is rates, and it can matter on long-dated options.

A Black–Scholes Greek is not automatically the same number under local volatility or Heston. The risk also depends on what is held fixed when an input is bumped.

Selling some underlying against a long call can offset delta at one measurement point. A delta-neutral book is not a risk-free book. Residual risks include gamma, vega, theta, rates, liquidity, costs, and jumps.

Frequent rehedging can reduce some risks and raise transaction costs. Continuous hedging is a mathematical ideal, not a practical guarantee.

Many long vanilla options have positive gamma and often negative theta. Many short-option books have the reverse. Those are common patterns, not rules.

Long gamma can benefit from a large enough realized move if the hedge is adjusted, and still lose time value if the market sits still. Short gamma can collect theta in a quiet tape and suffer in a sharp move. Positive gamma is not automatic profit. Assess the risks together.

A surface is indexed by strike and expiry. The book can be long one wing and short another. A single aggregate vega can hide that.

• Level: much of the surface moves up or down.
• Skew: downside and upside implied vols move differently.
• Term structure: short and long expiries move differently.
• Vol-of-vol, correlation, model risk, and gap risk sit beside that map.

Useful reports bucket by strike and expiry and add a few labelled surface shocks. The surface lesson is the quote map this risk sits on.

Ask the six questions before you call a hedge successful:

1. What is being hedged: delta, gamma, vega, rates, or a mix?
2. Is the hedge aligned to the liability’s drivers and horizon?
3. What residual risks remain?
4. How do costs, liquidity, and rebalancing change effectiveness?
5. Are valuation and hedge models consistent enough for the use?
6. Can the result be reproduced for audit?

Do not judge a hedge only by whether it reduced one Greek today. Look at P/L, stress, cash, slippage, and the horizon. Classroom only — not a production hedge policy.

On this site the path is: name the legs, read the calculator card, then shock the assumptions. Do not treat a scenario as a probability forecast unless a probability model is stated.

Build a strategy · What if · Monte Carlo. Label Greek and volatility units. Distinguish expiry payoff from a pre-expiry model value. Delayed prints are not fills.