Lesson 18 of 20 · 8 min

How the Greeks interact in one position

Real outcomes are the sum of several Greeks acting at once, and they frequently cancel.

Every lesson so far has changed one input while holding the rest fixed. Reality changes several at once, and the result is a sum of effects that can point in different directions.

The decomposition is additive to a good approximation. The change in an option's price is delta times the price move, plus half of gamma times the price move squared, plus theta times the days elapsed, plus vega times the volatility change. Each term is a Greek you already know, multiplied by how much its input moved.

ΔPrice ≈ Δ·(ΔS) + ½·Γ·(ΔS)² + Θ·(days) + V·(Δσ)

Δ·(ΔS): The delta term: current sensitivity times the price move. Usually the largest single piece.
½·Γ·(ΔS)²: The gamma term: the curvature correction. Small for small moves, dominant for large ones.
Θ·(days): The theta term: decay times elapsed days. Always working against a holder.
V·(Δσ): The vega term: volatility sensitivity times the change in implied volatility. Frequently the largest term after an event.

Take a concrete case. A 105-strike call with the underlying at 100, 45 days out, implied volatility 28 percent: the model prices it at $2.16, with delta 0.346, gamma 0.0375, theta −$0.044 and vega $0.130. Ten days later the underlying is at 104 and implied volatility has fallen to 24 percent. The contract is worth $2.80.

Decompose the $0.64 gain. Delta contributed +$1.38. Gamma added +$0.30. Theta subtracted $0.44. Vega subtracted $0.52. Sum: +$0.72, against an actual +$0.64, with the eight-cent residual coming from higher-order and cross terms the approximation drops. The reader was right on direction by four percent and captured less than half of the delta gain, because the other three terms took most of it.

Key terms

Greek decomposition: Attributing a price change to its sources: delta, gamma, theta and vega terms, each an input move times a sensitivity.
Cross terms: Interaction effects the simple decomposition omits, such as delta itself changing as volatility changes. They explain the residual.
Offsetting exposures: The common situation where two Greek terms have opposite signs, so a correct directional view produces a small net result.

Worked example

Attributing one real move

Long a 105 call. Start: underlying 100, 45 days, volatility 28%, price $2.16, delta 0.346, gamma 0.0375, theta −$0.044, vega $0.130. End: underlying 104, 35 days, volatility 24%.

Delta term: +$1.38
Gamma term: +$0.30
Theta term: −$0.44
Vega term: −$0.52
Approximation total: +$0.72
Actual model price change: +$0.64

A four percent move in the right direction produced a 30 percent gain instead of a 64 percent one, because theta and vega took $0.96 of the $1.68 that delta and gamma produced.

Check yourself

In the worked example, delta and gamma contributed +$1.68 but the contract only gained $0.64. Where did the rest go?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.