# 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?

## 1 Answer

A simple linear interpolation on implied variance along iso-moneyness lines is enough to garantee that there is no arbitrage between maturities as long as the inputted market data is arbitrage free.

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 $m=K/(Se^{rT})$ ? – JMC Dec 21 '20 at 23:41
• yes, it is the forward moneyness, adjusted for fixed dividends if there are any – Antoine Conze Dec 23 '20 at 8:19