# 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?

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

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 guarantee 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, 2020 at 23:41
• yes, it is the forward moneyness, adjusted for fixed dividends if there are any Dec 23, 2020 at 8:19
• @AntoineConze If I have IVs sampled at constant delta points over a set of maturities, is it valid (no arbitrage) to time interpolate between these (i.e. iso-delta lines)? Or do I strictly have to do this over iso-moneyness lines? In the BS space, delta = N(d1), where N( ) is the normal CDF. Sep 25, 2022 at 0:37

Moneyness is a good criteria for choosing the coordinates, and sure this obtains an arbitrage-free surface in the strike/moneyness dimension. It's questionable if a linear interpolation on the implied accumulated variance is necessarily globally arbitrage-free. Empirical evidence shows violation of arbitrage properties. I suppose your primary concern is term structure. For this, a better, more intuitive criteria is to choose two points with identical probability of the options---having different maturities---ending up in the money at maturities, as opposed to two points with identical moneyness. Moneyness has poor locality. The said probability is similar, though not identical, to Delta of the option. See the paper Arbitrage-free Asset Class Independent Volatility Surface Interpolation on Probability Space using Normed Call Prices by Pijush Gope and Christian Fries. Interestingly, as the paper shows, this results in a globally arbitrage free volatility surface.