Filling gaps in market data with optimal transport

Even the most highly traded derivatives can have illiquid points that create gaps in the implied volatility surfaces traders use for pricing. When those illiquid points are surrounded by liquid points, interpolation methods can be used to fill in the gaps. But for deep out-of-the-money strikes and long-dated maturities that are rarely traded, a reliable methodology for extrapolating prices is lacking, with traders often relying on crude approximations that are akin to rules of thumb.

“The most used approaches to fill volatility surface gaps are those that use interpolation techniques based on local volatility models or Bayesian methods,” explains Andrea Pallavicini, head of equity, FX and commodity models at Intesa Sanpaolo. “But those are less suitable for extrapolations of prices at illiquid or non-traded maturities.”

A conversation with traders at Julius Bear in Zurich prompted Valer Zetocha, a senior quantitative analyst at the firm, to search for a better solution.

A natural application I see for this method is the extrapolation of prices for dates that are not typically traded

Riccardo Longoni, Mediobanca

“I get a lot of inputs from traders,” he tells Risk.net. “Recently, I was asked about the stability of the implied volatility surface and how to modify it while it remains non-arbitrageable.”   

The traders wanted to ensure the prices they calculate for maturities that are not regularly quoted in the market are still accurate and not arbitrageable. But the crux of the problem is best illustrated with a narrower example.

Say a trader wants to increase the skew of a maturity that is not regularly quoted, either for generating scenarios or to test a sensitivity. One way to do this is by manipulating the implied volatilities. “The problem here is that you are never guaranteed that this will produce a valid distribution and it will be arbitrage-free,” says Zetocha.

In his latest paper, Zetocha explains how to generate a price for a new maturity that is not arbitrageable and also in line with what has been quoted before.

“My method consists of finding two distributions, similar to each other but with one displaying a higher skew,” he explains. “I then derive the map function that transforms the first into the second with higher skew. That function can then be applied to the original distribution, and you’re guaranteed that the resulting one will have a higher skew.”

The technique was inspired by the 18^th century French mathematician Gaspard Monge’s solution for moving a pile of sand from one place to another with the least possible effort. Both problems consist of three elements: an initial distribution, a final distribution, and a function that governs the transition between the two. For Monge, the initial and final distributions were known and the optimal function then needed to be discovered. In Zetocha’s case, the initial distribution is also known and the map function is derived as explained above. The final distribution is found using those elements.

The next step is to apply the distribution to the volatility surface. This comes with an additional complexity: the distribution marginals need to be in convex order to avoid time arbitrage. The paper explains how to solve this problem.

The paper is the latest example of optimal transport theory – introduced by Monge – being used in quant finance. Hadrien De March and Pierre Henry-Labordere used similar techniques to build arbitrage-free volatility surfaces, while Julien Guyon has cast the calibration of Vix options models as an optimal transport problem.

Riccardo Longoni, a senior quant in the model validation team at Mediobanca, says Zetocha’s work shows optimal transport techniques also work for extrapolation problems. “A natural application I see for this method is the extrapolation of prices for dates that are not typically traded, for example for products that are typically settled once a month,” he says.

Pallavicini at Intesa Sanpaolo agrees that it is “a useful technique for extrapolation where there is no strike priced at all”, adding that: “In this implementation, [this method] gives you the chance to calculate the closed-form transport function and relies on the assumption that marginal distributions change through time but the way they are connected can be considered invariant.”

Another application cited by Zetocha is changing an existing volatility surface – for instance, to increase the volatility of a process, add more skew or change the term structure.

The connection between the two applications is that they both involve modifying a set of distributions that are in a convex order to generate a new, arbitrage-free surface that has some desired features. The ability to modify the volatility surface could also give rise to other use cases, Zetocha says, such as the generation of new market data or the incorporation of external factors into the surface dynamics.