Measuring Vega Risk Across a Volatility Smile
Summary
The document asks how to hedge the vega of an exotic option, such as a barrier, when its value depends on an implied-volatility smile rather than a single volatility input. It describes a workflow in which a pricing model is calibrated to a market smile and then used to value the exotic. The discussion considers whether to bump implied volatility strike by strike and hedge each resulting sensitivity with vanilla options, while noting that far out-of-the-money options may be illiquid.
The responses offer two risk representations. One is to group the volatility surface into strike and maturity buckets and measure sensitivity to bumps in each bucket. Another is to represent the smile with a model such as SABR or SVI and calculate sensitivity to its parameters. A further suggestion is to shift the full surface, recalibrate, and reprice the exotic to estimate parallel-shift vega. These are alternative sensitivity conventions; the brief exchange gives no numerical comparison or complete hedge construction.
Key ideas
- An exotic option’s vega can depend on the shape of the implied-volatility surface.
- Strike-by-strike volatility bumps can reveal local sensitivities but may imply hedges in illiquid options.
- Bucket vegas measure sensitivity to volatility changes within selected strike and maturity regions.
- A smile model allows risk to be expressed as sensitivity to its internal parameters.
- Parallel-shift vega can be estimated by bumping the full surface, recalibrating, and repricing.
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Full text
# Vega hedging with implied volatility smile # Vega hedging with implied volatility smile I have a problem with vega hedging. Consider the management of an exotic derivative, such as Barrier option. Typically we do the following tasks: - selecting a pricing model, say, a local volatility model such as CEV model. - choosing a relevant European option implied volatility smile/skew as the calibration instrument. - 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!!! ## Answer by AFK (score 3) https://quant.stackexchange.com/a/19460 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). ## Answer by mbison (score 1) https://quant.stackexchange.com/a/19458 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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