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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?

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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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    $\begingroup$ Thanks a lot, this will be very helpful. I'll do research about "local stochastic volatility" models. $\endgroup$
    – EduardoBB
    Sep 4 at 16:11

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