In the usual adjusted Black Scholes world where BS vol varies with strike and with time to expiry as extracted from a market snapshot at moment 0, assuming a certain of interpolation method across strikes, that method being the same used for all expiries, is there a "logical" choice of method of how to extract the volatility applicable to a option with time to expiry between 2 listed expiries, and with a strike between 2 market forward strikes.
In other words,
spot: S0
Market times to expiry: T1 ... Tm
Market strikes:
X1_1 ... X1_n1
...
Xm_1 ... Xm_nm
Chosen interpolation method for strikes on T1, let's say spline or some polynomial or segments thereof, that matches market prices well. Same method is used for all Ti
Is there a "make sense" method to interpolate the vol for option with time to expriry
Ti <Tx < Ti+1
and with strike X
F(Xi_(j-1)) < X < F(Xi_j) F(X(i+1)_(k-1)) < X < F(X(i+1)_k)
A method that respects both across the strike dimension, and the time to expiry dimension?
