# how to do interpolation in the term structure of volatility surface?

everyone~ I am a newbee in the quantitative finance and I meet a problem in working out an equity option volatility surface.

We use the reasonable market data to derive the implied volatility, then we use natural cubic spline to build the skew,but we do not know use which method to build the curve of term structure when we want to have a complete volatility surface?

Specifically just do a linear interpolation on $$T \mapsto \sigma(m F(T), T)^2 T$$ where $\sigma(K, T)$ is the implied volatility for strike $K$ and maturity $T$, $F(T)$ is the forward for maturity $T$, and $m$ is the option moneyness.
If there are discrete fixed dividends then start by working out the resulting affine relationship $$F(T) = a(T) S_0 + b(T)$$ then do the linear interpolation on $$T \mapsto \sigma(b(T) + m (F(T) - b(T)), T)^2 T .$$ This will garantee that there is no arbitrage between volatilities before and after ex-dividend dates.
• Is m the forward moneyness $m=K/(Se^{rT})$ ?