Whether Implied Volatility Convexity Ensures Convex Option Prices

Summary

The document poses a question about constructing a risk-neutral density from out-of-the-money options. Since the density is related to the second derivative of option price with respect to strike, it asks whether convexity of option prices is needed to obtain nonnegative probabilities.

It then asks whether imposing a convexity constraint directly on an implied-volatility surface necessarily produces convex option prices. The document offers no answer, derivation, model, or evidence, so it does not establish that the two forms of convexity are equivalent. It identifies a useful modeling question about the relationship between volatility-surface constraints, option-price shape, and density extraction, but leaves the result unresolved.

Key ideas

• A risk-neutral density can be related to the second strike derivative of option prices.
• The document asks whether convexity in implied volatility guarantees convexity in option prices.
• It provides no derivation or result resolving the relationship.
• The question concerns constraints used when constructing volatility surfaces and extracting densities.

Tags

# Does convexity in the IV space means convexity in the price space?


# Does convexity in the IV space means convexity in the price space?












Let's assume that we only look at OTM options to construct a Risk Neutral Density (RND).

As the RND is the second derivative of the price of the option with respect to the strike, we would expect convexity to get positive probabilities.

Now, let's assume a model to construct a volatility surface. If I am able to add a convexity constraint in the IV space, would it necessarily resul in convexity in the price space ?

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