Why Spot and Strike Are Needed to Plot a Volatility Smile

Summary

The document asks how to plot a one-month implied volatility smile when call and put implied volatilities and option deltas are available, but spot and strike prices are missing. It notes that volatility smiles are often displayed against delta, so the supplied delta values may serve as the horizontal coordinate if that is the intended presentation.

For a smile plotted by strike, log moneyness, or a similar price-based measure, the information given is insufficient to recover both spot and strike levels. The response suggests that many combinations could fit the available data; knowing spot would allow strikes to be inferred from the option deltas, subject to the option and delta conventions used. No reconstruction method or numerical evidence is provided, and the document does not specify those conventions. Its practical takeaway is to choose a delta-based display or obtain the missing price information before constructing a strike-based smile.

Key ideas

• A volatility smile can be plotted against option delta when delta values are available.
• A strike-based or moneyness-based smile requires price inputs that are missing here.
• Delta alone does not uniquely determine both spot and strike levels.
• Knowing spot can make it possible to infer strikes, depending on the delta convention.

Tags

# Constructing Volatility Smile from Implied Volatility & Delta


# Constructing Volatility Smile from Implied Volatility & Delta












I have implied volatility data for call and put options (expiring in 1 month from any given date) for a particular stock. In addition, I have the delta for the options.

However, I have no information about the spot price of the stock or the strike price of the options.

How do I draw a volatility smile from the given data?

## Answer by GNUser (score 1)

https://quant.stackexchange.com/a/14530

First off, volatility smiles are often drawn over a delta space.

Since you're asking, I'll assume you're trying to draw a volatility smile over strike prices, log moneyness, or some similar metric. If you have neither the spot price nor any strike prices associated with your data, I don't believe it's possible to back out both of those values. Not absolutely certain, but I assume there must be infinite possibilities. If you just had the spot price, you could back out all the strikes with ease.

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