# Vega with SVI Gatheral bumps

How would one go about computing a vega profile of an exotic derivative where the volatility surface is modeled using Gatheral's SVI parameterization? In particular, I am thinking about bumping each of the params up by epsilon and computing a Jacobian for each time slice. So the vega explain would be computed like this:

Consider a time series of SVI parameterizations: $w_t(k;a,b,\rho, m, \sigma)$. Let $\mathscr{V}(t)$ be the vega explain along each slice. Then the vega explain would something like:

$$\mathscr{V}(t) = \Delta a* \frac{\partial w_t}{\partial a} + \Delta b* \frac{\partial w_t}{\partial b} + \Delta \rho* \frac{\partial w_t}{\partial \rho}+ \Delta m* \frac{\partial w_t}{\partial m}+ \Delta \sigma* \frac{\partial w_t}{\partial \sigma} + \ldots$$