How Volatility Assumptions Affect Option Delta

Summary

The note asks whether implied volatility should be recalculated when estimating an option’s delta by changing the underlying price, and raises the same issue for theta. Its answer frames the choice as a modeling assumption about how the volatility surface moves as the underlying changes. Under a sticky-strike assumption, implied volatility stays tied to the strike, so the delta calculation excludes a volatility-driven vega contribution. Under sticky-delta, volatility moves with the option’s delta, so the total price response includes vega multiplied by the volatility change associated with the spot move.

The differential expression in the answer separates the direct spot effect from this volatility-mediated effect. Thus, a finite-difference delta that holds volatility fixed estimates a different sensitivity from one that updates volatility according to a surface rule. The note does not prescribe one convention or give market evidence for choosing between them; the appropriate calculation depends on the model and the question being asked. It also mentions theta without developing a corresponding treatment.

Key ideas

• Option delta can include both the direct spot effect and an effect transmitted through changing implied volatility.
• Sticky-strike holds implied volatility fixed by strike as spot moves.
• Sticky-delta allows implied volatility to change with delta, contributing a vega effect.
• The volatility-surface rule is a modeling choice that determines the sensitivity being measured.

Tags

# When calculating delta, should you recalculate vol?


# When calculating delta, should you recalculate vol?












Delta is the rate of change of price to the spot.

If the spot changes, so does the vol, so if you wanted to e.g. calculate delta as a finite difference

$$\frac{f(spot+h)-f(spot-h)}{2h}$$

would you recalculate a new volatility for the spots $spot - h$ and $spot + h$, or just ignore it?

Same question regarding calculating theta...

## Answer by Kermittfrog (score 3)

https://quant.stackexchange.com/a/59134

You are asking for the total effect of a (small) change in the underlying value on the option price, which is

$$ dO=\frac{\partial O}{\partial S}dS+\frac{\partial O}{\partial \sigma}\frac{\partial \sigma}{\partial S}dS $$

Effectively, this boils down to whether you want to apply, i.e. model a sticky-delta or sticky-strike rule. Other sources are Wiki and Derman

Sticky Strike implies that the implied volatility is a function of the strike level $K$, only and there should be no change due to a sensitivity w.r.t. implied volatility (vega). With a sticky delta assumption, you would see a vega effect.

I am afraid that this boils down to a modeling assumption on your part...

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.