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I think the question is about dynamic replication in a binomial context except it is worded a bit odd. Normally a tree would be given and one needs to find x(t) and y(t), where x(t) is amount of stock at time t and y(t) is amount of cash at time t. Then you do your replication argument and you have that the option price P(0) = X(0) + Y(0). The way this ...


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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 ...


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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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