Convex Volatility Interpolation: A New Approach to Implied Volatility Surfaces

The world of options trading relies on accurately pricing instruments beyond simple calls and puts – the so-called “exotic” options. Central to this is the construction of implied volatility surfaces, complex three-dimensional models that represent market expectations of future volatility at different strike prices and expiration dates. Building these surfaces isn’t straightforward; it requires filling in gaps between observed option prices with sophisticated interpolation techniques. Now, a fresh approach, dubbed convex volatility interpolation (CVI), is gaining traction for its potential to bridge the gap between the needs of traders dealing with standard options and those focused on more complex derivatives.

The challenge lies in the shape of these volatility surfaces. While the classic “smile” shape – where out-of-the-money options are more expensive than at-the-money options – is common, real-world surfaces can be far more complex, exhibiting “W” shapes or other distortions. Capturing these nuances accurately is crucial for correct pricing and risk management. Traditionally, solutions have been proprietary, developed by vendors or banks and closely guarded as competitive advantages.

A Convex Approach to Volatility

Fabrice Deschâtres, founder of derivatives pricing firm Volptima, and previously a quant at Goldman Sachs and several hedge funds, sought a more robust and accessible method. Dissatisfied with existing techniques, he developed CVI, an algorithm framed as a convex optimization problem. This formulation allows it to be processed using modern optimization solvers like CVXPY, developed at Stanford University, and Clarabel, from Oxford University.

“Market participants seek stability, accuracy and no arbitrage. They too want the fitting to be fast and ideally to have intuitive parameters that really relate to the risks the trader is managing,” Deschâtres explained. The need for speed and intuitive parameters is particularly acute given the dynamic nature of financial markets. Volatility surfaces need to be updated frequently to reflect changing market conditions.

Historically, vanilla options desks (those trading standard options) and exotic options desks have often used different volatility surfaces, reflecting their differing priorities. Vanilla desks prioritize accuracy, stability, and parameters that are easy to understand, while exotic desks place the highest importance on avoiding arbitrage opportunities – ensuring that no risk-free profit can be made. CVI aims to reconcile these competing demands, offering a single surface suitable for both.

B-Splines and Parameterization

At the heart of CVI are B-splines, mathematical functions used to create smooth curves that interpolate a set of data points. Vladimir Lucic, head quant at Marex Solutions and a visiting professor at Imperial College London, highlights the innovation in how Deschâtres adapted B-splines for volatility surface construction. “The spline is parameterised in a way that is more natural to the problem at hand. The author proposes a very efficient numerical method to solve the problem,” Lucic said. “This is a needed improvement to the literature already published on volatility parameterisation.”

Traditional volatility parameterization methods can be cumbersome. CVI’s approach allows for more intuitive control over key parameters like at-the-money volatility, the degree of the smile, and the skew (the asymmetry of the volatility surface). This makes it easier for traders to understand and manage the risks associated with their positions.

A flexible approach to volatility surface construction doesn’t automatically guarantee the absence of arbitrage. However, Lucic notes that CVI successfully navigates this challenge. Deschâtres further explains the advantage of working directly in the volatility space rather than focusing on price-based fitting. “Price-based volatility fitters have been used notably for exotics due to the fact that they build 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.” He argues that CVI achieves both.

Accessibility and the Rise of Optimization Tools

The development of CVI was also facilitated by advancements in optimization software. Deschâtres points out that before tools like CVXPY and Clarabel became available, implementing this type of approach required significant academic expertise – “pretty much a PhD” – and a commercial solver license. “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.”

However, even with these tools, achieving optimal performance requires substantial engineering effort. While CVI shows promise for production use, Lucic believes further refinement is needed to solidify its theoretical foundations. “The method works on real examples, but there is no hard proof that the proposed linearisation is going to work under every condition,” he cautions. He also suggests strengthening the no-arbitrage constraints, moving from “soft” penalties for crossing bid/ask spreads to “hard” constraints that guarantee avoidance of arbitrage.

Next Steps: Stability and Refinement

Deschâtres acknowledges the need for continued development, with a primary focus on enhancing stability. “At Volptima, one of the focuses is enhancing stability – in the far wings, within the listed strike range, everywhere.” He anticipates greater initial interest in CVI from vanilla options desks, particularly those trading equity and cryptocurrency options.

Lucic echoes this sentiment, endorsing the method as production-ready for flow business and seeing potential for exotics as well. The ability to create a more stable and intuitive volatility surface could lead to more accurate pricing, reduced risk, and improved trading strategies. The ongoing refinement of CVI, addressing the theoretical concerns raised by Lucic, will be crucial to unlocking its full potential and establishing it as a standard tool in the world of derivatives pricing.