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I have a problem with vega hedging.

Consider the management of an exotic derivative, such as Barrier option. Typically we do the following tasks:

  1. selecting a pricing model, say, a local volatility model such as CEV model.
  2. choosing a relevant European option implied volatility smile/skew as the calibration instrument.
  3. Calibrating the pricing model to the calibration instrument and use the calibrated model to find the value of exotic.

My question is how to find the vega greek? As it's the whole imp vol smile/skew that affects the value of the exotic, should I calculate vega(K) by bumping each imp vol with strike K one at a time, and construct the hedging portfolio by using vanilla options across all strikes? (I don't think it's going to happen in practice considering the illiquidity of OTM option.)

It would be best if anyone could provide some reference dealing with this issue. Thanks!!!

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Instead of just considering a parallel shift of the whole volatility surface, you can decompose the surface into maturities/strikes domains, so called buckets and consider Vega buckets which are sensitivities wrt to bumps of each of these domains.

The vol smile is often inter/extra-polated using a model calibrated to market prices, e.g. the SABR model or SVI. So you can measure risk by considering sensitivities to bumps of the internal parameters of the model instead ($\alpha,\beta$ and $\rho$ risks for SABR).

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since vega is the sensitivity to a parallel shift of the entire vol surface, why do you not simply bump the entire input surface all at the same time? You can bump all your vanillas simultanous then recalibrate your model to the bumped surface. The use your model to reprice, which gives you your vega.

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