Interpolating Implied Volatility Across Maturities Without Arbitrage
Summary
The note addresses how to interpolate the maturity dimension when building an equity options volatility surface. One suggested method interpolates accumulated implied variance linearly with maturity along lines of constant forward moneyness. Under the stated condition that the input quotes are arbitrage-free, this is presented as preventing arbitrage between maturities. With discrete fixed dividends, the forward is represented as an affine function of spot, and the interpolation coordinate is adjusted to account for that relationship and ex-dividend dates.
A second response questions whether this variance interpolation guarantees global absence of arbitrage, noting that empirical violations have been observed. It proposes matching options with similar probabilities of finishing in the money across maturities, a coordinate related to but distinct from delta, and points to probability-space interpolation using normalized call prices. The discussion is conceptual: it provides no derivation, data, or implementation, and presents competing views on the guarantees of the simpler method.
Key ideas
- Interpolate accumulated implied variance across maturity along constant forward-moneyness lines as a basic surface construction method.
- The stated maturity-arbitrage condition depends on arbitrage-free input market data.
- Discrete fixed dividends require adjusting the interpolation coordinate to reflect the forward price relationship.
- Probability-based coordinates may offer more global arbitrage control than constant moneyness, though the note gives no implementation details.
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Full text
# how to do interpolation in the term structure of volatility surface? # 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? ## Answer by Antoine Conze (score 8, accepted) https://quant.stackexchange.com/a/22263 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. ## Answer by gobbledygook (score 1) https://quant.stackexchange.com/a/74622 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.
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