Chapter 3 of 9

Delta, gamma, theta and vega

Use option Greeks and local sensitivities to explain option behavior, without treating Greeks as fixed forecasts or exact probabilities.

3 min read · Original educational chapter

In this chapter

Delta and gamma describe local price sensitivity
Time and volatility affect the same premium
Greeks interact and change
Decision lab: Greeks describe a local model, not a complete path
Worked example
Common mistakes
Check your understanding

Delta and gamma describe local price sensitivity #

Delta approximates the premium change for a small change in the underlying, holding other inputs fixed. A delta of 0.40 suggests roughly 0.40 premium units for a one-unit underlying move near that state. It is not a guarantee or a universal probability of profit.

Gamma measures the rate of change of delta with respect to the underlying. Standard long vanilla options generally have positive gamma; their delta changes as the underlying moves. A large move can therefore make a delta-only estimate inaccurate.

Time and volatility affect the same premium #

Theta describes sensitivity to the passage of time under a stated convention and other inputs held fixed. It is commonly quoted per day, but units and sign conventions must be checked. Time decay is not a fixed daily rupee payment.

Vega describes sensitivity to a change in implied volatility. It is often quoted per one percentage-point change, not per 100% change. A volatility decline can offset a favorable underlying move. Longer-dated options can have substantial volatility exposure even when their day-to-day underlying delta looks modest.

Greeks interact and change #

A local approximation can combine delta × price change, half gamma × price change squared, vega × volatility-point change and theta × elapsed days. It remains an approximation, omits cross-effects and is less reliable for large changes or near discontinuities.

For a multi-leg portfolio, sensitivities can be added only with consistent units, direction and quantities. Near-zero net delta does not imply no risk: gamma, volatility, gap and funding exposures can remain. Greeks from different models or timestamps should not be combined casually.

Decision lab: Greeks describe a local model, not a complete path #

Delta can change as price and time move. Gamma, vega and theta interact with the scenario and with the pricing assumptions. A single small-change estimate can become unreliable across an event jump or a large movement.

Construct separate price, volatility and time scenarios for a fictional option and record where a linear delta estimate fails. Keep units explicit: volatility points versus decimals, one day versus a year, and per-unit values versus total contracts. A neat Greek number does not guarantee an executable hedge.

Hypothetical worked example

Direction is only one driver of a premium change

A hypothetical long call has delta 0.40, gamma 0.02, vega 0.50 per volatility point and theta −0.10 per day. The underlying rises 2, implied volatility falls 3 percentage points and one day passes.

The local approximation is 0.40 × 2 + 0.5 × 0.02 × 2² + 0.50 × (−3) − 0.10 = −0.76 premium units. A favorable underlying move can coexist with a premium loss.

For 100 units, this approximation implies −₹76 before costs. Actual repricing may differ because the sensitivities themselves move. The inputs are fictional and do not provide an executable option quote.

Common mistakes

Reading delta as a guaranteed probability of a profitable trade.
Mixing vega quoted per volatility point with a decimal-volatility change.

Check your understanding

What does positive gamma imply for a standard long option, locally?

Make this chapter useful

Can you defend the reasoning?

Before moving on, answer the self-check in your own words and write why each alternative fails. Use the worked example to test that explanation. Then apply it in the module research task; keep the assumptions and the evidence that could change your answer.

Sources and further reading

Further-reading references from the curriculum source list. Contract sizes, expiry calendars, settlement references, margin parameters and tax treatment can change; verify the current operative documents for the actual product. Worked figures are illustrative, not current quotes.

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General financial education. No securities recommendation, assured outcome or personal advice. Read the risk disclosures.