Local Volatility, Vomma, and Whether Stochastic Volatility Lowers Option Value

Summary

The document poses a model-risk question about valuing a short option under a Dupire local volatility model. The option is assumed to have negative vomma, meaning its value is locally concave with respect to volatility. The question asks whether this feature makes the local volatility price conservative, and whether introducing stochastic volatility would always reduce the price.

No answer, derivation, or empirical evidence is supplied, so the proposed relationship is unresolved. Negative curvature with respect to volatility describes sensitivity to changes in volatility under a specified pricing setup; by itself, it does not establish how prices compare across different volatility models. That comparison can depend on the volatility dynamics, calibration, payoff, and the way the models represent the volatility surface. The document is therefore a useful prompt about model assumptions and option convexity, but it does not provide a general pricing rule or certainty level.

Key ideas

• The question concerns pricing a short option with a local volatility model.
• Negative vomma means the option value is locally concave in volatility.
• The document asks whether stochastic volatility necessarily lowers value, but gives no answer or evidence.
• The sign of vomma alone does not settle price comparisons between differently specified volatility models.
• Model calibration, volatility dynamics, and payoff characteristics matter to such comparisons.

Tags

# Dupire Vomma and Stochastic volatility


# Dupire Vomma and Stochastic volatility












Suppose that you are short an option on asset $X_t$ following a pure diffusion. Suppose you are hedging your position using (Dupire) Local volatility model. Suppose that the option is concave with respect to the volatility, thus the second derivative with respect to the volatility is negative (ie having negative vomma).

Question : at what level of certainty we can say that the local volatility model will produce conservative price ? Is accounting for a stochastic behavior for the volatility will always result in smaller price in this case ?

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