# Instability in risks using local volatility

I am valuing vanilla call options using FDM with Crank Nicolson discretization and Rannacher smoothing (mind you, I am having the same issue on MC) and I am getting unstable delta, gamma and theta. I am computing risks by simple bump and revalue using 1 day theta and 1% central gamma and delta.

By unstable I mean that I see oscillation in my risks when:

• I use a different bump value (e.g. 2 day theta or 2% delta/gamma)
• Making my grid finer by scaling down $$dt$$ and $$dS$$

My option values are always within ~ 40bps of the B-S value at my coarsest grid values ($$dS = 0.005S_{0}$$, $$dt = 0.025$$) but I keep getting my risks, especially gamma and theta, off by up to 40%. They do converge when I use an extremely fine grid, but by that point the computation is prohibitively slow.

I am using natural cubic spline for interpolation in strike and linear interpolation in time in total variance on my surface $$w(t,K) := t\space\sigma_{Imp}^{2}(t, K)$$. I have tried taking $$\frac{\partial w}{\partial K}$$, $$\frac{\partial^{2} w}{\partial K^{2}}$$ and $$\frac{\partial w}{\partial t}$$ both analytically from the interpolation method or by using finite differences and linearly interpolating between nodes on the LV surface. I am of course using the version of Dupire's formula that uses these derivatives.

My interpolation on $$w(t, K)$$ does admit arbitrage. In fact, I have noticed that surfaces that have more arbitrage give me worse values.

Why am I getting unstable risks? Would a parametrized no-arbitrage IV surface help here? Do I have to do some smoothing on my LV surface? Should I instead impose a shape on my LV surface (e.g. bicubic spline) and solve for the node points as an optimization problem? What is standard practice?