A smooth fit for complex volatility surfaces

Quant shows a new way to capture implied vol with optimisers

Implied volatility surfaces are the elusive three-dimensional shapes that enable traders and quants to value exotic options and compute sensitivities.

These surfaces are not simple to build, though. Option prices give a discrete set of points, namely strikes and expiries, which form an initial skeleton for the surface. These bare bones must be fleshed out by interpolating between the points to build an unbroken skin.

Models in this space attempt to tackle the ever richer structures of some of the volatility surfaces. For example, slices of implied volatility may display W shapes, which are harder to capture than the characteristic smile shape. Solutions to this problem are normally proprietary and closely guarded by their intellectual owners – either vendors offering their services without revealing the underlying formulas, or banks, for which a good formula is the trick of the trade. 

Fabrice Deschâtres is founder of derivatives pricing firm Volptima, with previous quant roles at Goldman Sachs and several hedge funds. Not entirely satisfied with any of the existing volatility fitting methods, he set off to develop his own approach, convex volatility interpolation (CVI).

CVI is an algorithm for constructing implied volatility surfaces that is framed as a convex optimisation problem. As such, it is suitable to be processed by modern convex optimisation solvers: the CVXPY modelling language, introduced by Stanford University, and Clarabel solver, from Oxford University.

“Market participants want stability, accuracy and no arbitrage. They also want the fitting to be fast and ideally to have intuitive parameters that really relate to the risks the trader is managing,” says Deschâtres.

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Convex volatility interpolation 

The modelling of implied volatility surfaces is reframed as an optimisation problem

Traders often have to make trade-offs. Vanilla and exotic desks might adopt different volatility surfaces to respond to their specific needs. A vanilla desk would prefer accuracy, stability and intuitive parameters, whereas the exotic desk has the absence of arbitrage as its most important criterion. CVI is designed to “bridge the gap between the two”, says Deschâtres.

Central to the approach are B-splines, polynomials that interpolate a set of datapoints to create a smooth curve. Vladimir Lucic, head quant at Marex Solutions and a visiting professor at Imperial College London, thinks the original contribution of the work is to adapt the B-spline to include parameters that are normally used to handle the volatility surface, namely at-the-money volatility, smile and skew. 

“The spline is parameterised in a way that is more natural to the problem at hand. In addition, the author proposes a very efficient numerical method to solve the problem,” Lucic says. “This is a needed improvement to the literature already published on volatility parameterisation.”

A set-up that is flexible is often not free from arbitrage. However, Lucic says CVI finds a way to be flexible and ensure the arbitrage conditions are met.

Deschâtres explains the advantages of an algorithm that is specifically geared towards volatility rather than price. “Price-based volatility fitters have been used notably for exotics because they make the no-arbitrage condition easier to enforce. Whereas in the volatility space you tend to have more intuitive parameters and better stability on the wings.” His claim is that CVI can satisfy both.

Despite the applicability of CVI to both vanilla and exotics businesses, Deschâtres expects more interest for this method from the former, and he sees it as particularly suitable for equity and crypto.

Lucic endorses the new method and thinks it’s ready to be tested in production phase: “I definitely see it fit for the flow business and I think it’s got good potential for exotics too.”

Optimisation software tools such as CVXPY and Clarabel has helped unlock the method, too. Before the advent of CVXPY or other specialised modelling languages, Deschâtres says researchers needed high-level academic skills – “pretty much a PhD” – and a licence to a commercial solver to implement this type of approach.

“CVXPY made convex optimisation accessible to a lot of people and Clarabel made it fast,” he says. “The tools were ready. The contribution was finding the right formulation.”

“Still,” he admits, “if you want to make it as fast as possible you need a big engineering effort.” 

While seeing it as potentially fit for production, Lucic thinks the technique can be refined to make it more theoretically solid. “The method works on real examples, but there is no hard proof that the proposed linearisation is going to work under every condition, or, in other words, there is no clear mathematical proof for what are the conditions under which it actually works,” he says. 

Moreover, says Lucic, the no-arbitrage constraints in the strike space are soft, meaning that the penalties for being outside the bid/ask spread act as a correction mechanism rather than a guarantee of never crossing the lines. Incorporating hard constraints would strengthen the model. 

For Deschâtres, the first improvement to work on is making CVI a reliable tool under all scenarios: “At Volptima, one of the focuses is enhancing stability – in the far wings, within the listed strike range, everywhere.” 

Editing by Alex Krohn