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Let's say i have an exotic structure that is to be vega hedged dynamically. I choose to price it with a local volatility (which means the model prices in your future vega hedges using all options for all strike and all maturities). In practice the hedging is done using a few options (equivalently a few implied volatilities). My questions are:

-What does vega for the exotic structure mean? Is it a bump of the whole volatility surface?

-Why do we need the model to be calibrated to the whole surface if in practice we will only buy/sell a few options?

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In his article, Dupire (1994) developed the local volatility approach under the assumption that options are traded for a continuum of maturities and strikes. In reality, only a finite number of options generating a grid of strikes and maturities is traded. Then the reconstruction of the local volatility function is obtained by interpolation methods. However, when replicating exotic structures, using Bredeen and Litzenberger one can prove that the (forward) price of the contract at time t for an initial (forward) price can be written in terms of a weighted combination of butterly spreads. In an options’s market with a continuum of quoted strikes, the risk neutral forward transition density equals the price of an ifinitesimal butterly portfolio strategy.

Thus, the asnwer to your second question comes from the dependency of Dupire approach from Breeden-Litzenberger.

Coming to your first questions, I do not fully understand it. Indeed, Dupire shows how the Local volatility function is a function of a calendar spread and a buttefly strategy. Moreover, Breeden-Litzenberger show that the static replication of an exotic structure is proportional to a couple of ATM put and call option. From this result, the whole VIX methodology stems.

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Although a local volatility model $$ dS_t = \sigma(S_t,t) S_t dW_t $$ is able to fit exactly quoted market prices of vanilla options, the concept of vega in a local volatility model is at best ill-defined, even for vanilla options.

However, if you insist on obtaining vega for an option in a pure local volatility model, then you could bump the functional form of the initially calibrated local volatility surface keeping the spot unchanged (a change in local volatility due to change in spot is basically part of local vol model delta): $\sigma(S_t,t) \rightarrow \sigma'(S_t,t)= \sigma(S_t,t) + \epsilon$. You do need to make sure that the new bumped local volatility surface does not lead to arbitrage, which as far as I know is not a trivial problem.

In a stochastic volatility model, $$ dS_t = \sigma_t S_t dW_t $$ $$ d\sigma_t = \eta \sigma_t dZ_t $$ vega can be defined as the change in value of the option, whether vanilla or exotic, by bumping the initial value of the instantaneous volatility $\sigma_0 \rightarrow \sigma_0' = \sigma_0 + \epsilon$. In stochastic volatility models vega is well-defined as the bump will not lead to arbitrage possibilities.

Both local volatility and stochastic volatility models can be translated into an implied volatility surface for vanilla options. But you can't use the vanilla options implied volatility surface to calculate vega for an exotic.

My two cents.

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