Local Versus Stochastic Volatility Pricing of Barrier Options

Summary

The document compares prices for knock-out and knock-in barrier options under local volatility and stochastic volatility models. It reports an empirical pattern: knock-out barriers are usually more expensive under stochastic volatility and less expensive under local volatility. Given the parity relationship that a knock-out and its corresponding knock-in together equal a vanilla option, and the claim that vanilla prices match across the models, the implied ordering reverses for knock-in barriers: they are typically more expensive under local volatility.

The explanation offers an observed pricing tendency rather than a proven theorem. It acknowledges rare reported counterexamples and provides no data, derivation, model specifications, or conditions that establish when the pattern holds. The initial interview prompt concerns an in-the-money double knock-out, but the response states its conclusion broadly for knock-out barriers. The result should therefore be treated as a rule of thumb, not a universal ranking for every barrier structure or calibration.

Key ideas

• The response reports that knock-out barriers are generally priced higher under stochastic volatility than local volatility.
• Knock-in and knock-out prices sum to the corresponding vanilla option price under the stated parity relationship.
• If vanilla prices match across the models, knock-in barriers are generally more expensive under local volatility.
• The claimed price ordering is empirical and has no known mathematical proof in the document.
• Rare counterexamples are acknowledged, and no model setup or supporting dataset is given.

Tags

# local vol model and stochastic vol model for double knock out barrier option


# local vol model and stochastic vol model for double knock out barrier option












Here is an interview question:

> For a double knock out barrier option (ITM), which model gives higher price? local vol or stochastic vol model?

The answer is that local vol gives the higher price. However I only know that stochastic vol is the correct model to price barrier option to control the forward vol (vol of vol) as the product depends on the conditional distribution.

## Answer by Peter A (score 1)

https://quant.stackexchange.com/a/70120

Empirically, knock-out barrier options are more expensive in stochastic volatility models and less expensive in local volatility models. Since knock-out + knock-in = vanilla, and vanillas are priced the same in local vs stochastic volatility, knock-in barrier options are priced higher in local volatility.

There is no known mathematical proof for this, and I have seen rare claimed counterexamples. But it is certainly true almost always.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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