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Volatility Inputs and Smile Adjustments for Barrier Option Pricing

Article Quant Q&A · Author: EduardoBB
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Summary

This discussion asks how to use volatility from an implied-volatility surface in Reiner-Rubinstein formulas for an up-and-out call, expressed as a vanilla call minus an up-and-in call. The vanilla component may reference the strike while another component references the barrier, raising the question of whether each should use its own strike-dependent implied volatility and which volatility should enter the remaining formula terms.

The response says the Black-Scholes barrier formulas require a single volatility; inserting separate smile volatilities into different pieces is not a consistent use of that model. It then outlines approaches to account for smile effects: vanna-volga adjustments, local-volatility pricing, and local-stochastic-volatility models. Vanna-volga is described as historically useful but capable of producing arbitrage, while local volatility may not reproduce barrier prices perfectly. The discussion presents local stochastic volatility as a current practical approach, but offers no model specification, calibration procedure, or comparison data, so it is an overview rather than implementation guidance.

Key ideas

  • The standard Reiner-Rubinstein Black-Scholes barrier formulas use a single volatility input.
  • Plugging separate smile volatilities into distinct formula components is described as inconsistent.
  • Vanna-volga adjustments can account for smile effects but may create arbitrage in some regions.
  • Local-volatility methods avoid arbitrage under suitable construction but may miss barrier prices.
  • Local-stochastic-volatility models are cited as a later practical approach, without implementation details.

Tags

OptionsDerivatives pricingVolatilityRisk management
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# Different volatilities in Reiner Rubinstein barrier option model


# Different volatilities in Reiner Rubinstein barrier option model












Wanted to ask if a single or different volatilities should be used in Reiner-Rubinstein's option barrier pricing formulas given the following:

An Up-and-out call (Cuo) with strike K and barrier H is valued as Cuo = C - Cui where C is a plain vanilla call with strike K and Cui is an up-and-in barrier call.

The first part of the Cdi formula is a plain vanilla call with strike H. Since volatility values extracted from implied volatility surfaces are dependent on maturity and strike, K and H will yield different volatilities.

Should the two different volatility values be used, one for C and one for Cui? If so, which volatility is to be used for the rest of the terms of the formula for Cui? Which approach is possibly favored by market participants?

## Answer by Peter A (score 1)

https://quant.stackexchange.com/a/72123

If I understood right that you have in mind the formulas for Barrier options in the Black-Scholes model, you must use a single volatility. It is tempting, but wrong, to plug in Smile volatilities at strike and barrier levels.

We know that the true market, there is a smile, so the single volatility model is wrong. In the old days, FX quants adjusted the single vol price by adding on an estimate of the additional cost of Smile. This approach is called the "vanna-volga model". It was quite successful, but has problems of arbitrage in certain regions of parameter space.

After that, people began to price by solving Dupire's local volatility PDE. It's arbitrage free, but didn't hit barrier prices perfectly. Nowadays people use "local stochastic volatility" models.

Returning to your question, the vanna-volga approach was originally the closest thing to correcting the single vol formulas with smile, but the resulting prices can contain arbitrage. However, for interest (not for practical use) Yuan Li and I worked out a formula that doesn't contain any arbitrage, but is constructed from the standard single volatility formulas. It would not be used in practice, as local stochastic volatility is better. But I think it may be the true answer to your question:

Model-free valuation of barrier options

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