# How can I calculate bucket vega using dupire local volatility surface?

I am trying to calculate the bucket vega of the portfolio which includes mainly vanilla options and some exotic options. I am pricing the value of portfolio with fdm by using dupire local volatility surface calculated from fitted implied volatility surface. I have tried the following 2 methods to get the bucket vega, but there was something wrong.

1. Bump specific bucket of local volatility surface
2. Bump specific bucket of implied volatility surface and calculate dupire local volatility

In Black-Scholes, the same price of vanilla option can be seen when the implied volatility of the 1-year to 2-year maturity on the implied volatility surface only increased by 1%(of course, the surface are assumed to be differentiable for all K and T) and the volatility of all maturities increased by 1% on the implied volatility surface(parallel shift) because implied volatility of BS is constant. However, when using dupire local volatility, the price is different between above two situation, and it involves the problem of bucket vega. Because there are exotic options combined in portfolio, local volatility surface is being used for pricing, and is there any way to calculate bucket vega?