Even the most frequently traded derivatives can have points of illiquidity that create gaps in the implied volatility surface that traders use to set prices. If these non-liquid points are surrounded by liquid points, interpolation methods can be used to fill in the gaps. However, for deep out-of-the-money strikes and long maturities that are rarely traded, there is a lack of reliable methodologies for estimating prices, leaving traders with a rough rule-of-thumb-like approach. Often relies on approximations.
“The most used approaches to fill gaps in the volatility surface are those using interpolation techniques based on local volatility models or Bayesian methods,” says Andrea Parra, Head of Equities, Currencies and Commodities Models at Intesa Sanpaolo. Vicini explains. “However, these are not well suited for extrapolating prices for illiquid or untraded maturities.”
A conversation with traders at Julius Baer in Zurich led Valor Zetoka, a senior quantitative analyst at the firm, to look for a better solution.
I think a natural application of this method is extrapolation of prices for dates that are not normally traded. Riccardo Longoni, Mediobanca
“I get a lot of information from traders,” he tells Risk.net. “I was recently asked about the stability of the implied volatility surface and how to fix it while it remains non-arbitrable.”
Traders wanted to be sure that the prices they calculated for maturities that were not regularly quoted in the market were still accurate and not arbitrageable. But the core of the problem is best explained with a narrower example.
Suppose a trader wants to increase the skew of an unquoted maturity on a regular basis for scenario generation or sensitivity testing. One way to do this is to manipulate implied volatility. “The problem here is that there is no guarantee that this will produce a valid distribution and that no arbitrage will occur,” Zetocha says.
In his latest paper, Zetocha describes how to generate prices for new maturities that cannot be arbitraged and are consistent with previously quoted prices.
“My method consists of finding two distributions that are similar to each other, but one exhibiting a greater bias,” he explains. “We then derive a map function that transforms the first into a second with higher skew. Applying that function to the original distribution guarantees that the resulting distribution will be more skewed. It will be done.”
This technique was inspired by 18th century French mathematician Gaspard Monge’s solution for moving sand piles from one location to another with minimal effort. Both problems consist of three components: an initial distribution, a final distribution, and a function that controls the transition between the two. In Monge’s case, the initial and final distributions were known and the optimal function needed to be found. For Zetocha, the initial distribution is also known and the map function is derived as described above. The final distribution is determined using these elements.
The next step is to apply the distribution to the volatility surface. This involves additional complexity. To avoid time arbitrage, the distributive limits must be of convex order. This paper describes how to solve this problem.
This paper is the latest example of the optimal transport theory introduced by Monge being used in quantitative finance. Hadrien De March and Pierre Henry-Labordere used a similar technique to construct an arbitrage-free volatility surface. Meanwhile, Julien Guyon raised the adjustment of the Vix options model as an optimal transport problem.
Ricardo Longoni, a senior quant on Mediobanca’s model validation team, said Zetoka’s work shows that optimal transportation techniques also work for extrapolation problems. “A natural application of this method is estimating prices on dates that are not normally traded, such as for products that are typically settled once a month,” he says.
Intesa Sanpaolo’s Pallavicini agrees that this is a “useful technique for extrapolation when no strike price exists at all,” adding: It is based on the assumption that marginal distributions change over time, but the way they are connected can be considered unchanged. ”
Another application cited by Zetocha is to modify existing volatility surfaces. For example, you can increase the volatility of the process, add skew, or change the terminology structure.
The relationship between the two applications is that they both require modifying a set of convexly ordered distributions to generate a new arbitrage-free surface with some desirable characteristics. The ability to change the volatility surface could also create other use cases, Zetoka says, such as generating new market data or incorporating external factors into the surface dynamics.
