A market-making model for an options portfolio

Options market-making models are among the financial industry’s most closely guarded trade secrets. So much so that researchers looking for an example in the financial literature often end up recreating the ‘confused John Travolta’ meme from Pulp Fiction.

“I can’t recall a portfolio-level market-making model [being] published,” says Vladimir Lucic, a visiting professor at Imperial College London and head of quants at Marex Solutions. “There are some looking at single options, but not more realistic ones at portfolio level, where you look at the risk of the whole book and try to manage it.”

Lucic teamed up with Alex Tse, a former equity derivatives trader and lecturer at University College London, to fill the gap.

Options market-making is a complex problem, especially at the portfolio level, which explains both the lack of publicly available models and practitioners’ reticence around sharing their solutions.

It naturally integrates the view of a vanilla options trader into a market-making model

Alex Tse, University College London

“It’s an interesting problem, because it is a high-dimensional one,” says Tse. “Each option has risk exposure to many risk factors, like spot prices, realised and implied vols, rates. Then, there are multiple strikes and maturities for the same underlying. If you add up all the different assets, it becomes a very high-dimensional problem.”   

The model they developed and published in Risk.net earlier this month follows the principles established by Avellaneda and Stoikov in 2008 – which have proven popular in equity market-making due to their elegance and intuitive appeal – and applies them to options.   

The key inputs are the market implied volatility surface, which reflects current mark-to-market valuations, and the user’s subjective view of realised volatility. The latter is crucial and means the model could have practical applications beyond research projects.

Tse explains that vanilla options traders usually take a calibrated implied volatility surface and form their own view on realised volatility, often incorporating macro factors and company news. They then decide whether they have an edge in a particular instrument by comparing their expectations for realised vol with market-implied levels.

“This is what makes our model more interesting or appealing to practitioners,” says Tse. “It naturally integrates the view of a vanilla options trader into a market-making model.”

The second set of inputs in the model are numerical quantities that represent market order flow, which consists of two parts. The first is the expected activity for a particular option, which may indicate, for example, the number of orders that might arrive at a given mid-price. The second is an elasticity measure, which represents the sensitivity of the order flow to wider or tighter spreads.

Once these are combined, the next step is to determine if new orders present an opportunity for alpha or volatility arbitrage. The model generates signals that indicate whether a trade is expected to be profitable given, for example, the realised versus implied volatility differential. The model also includes risk control measures, which allow users to derive optimal bid-ask quotes to execute the strategy.

Options specialists may notice that the model features some unintuitive assumptions, such as constant volatility. The authors, though, insist this has little impact on its outputs.

“It can be justified by the fact that recalculation of the market-making quotes usually takes place much more frequently than that of the implied volatility surface,” says Tse, adding that users can always recalibrate the implied vol surface any time and compute the optimal quotes against the most recent implied volatility values. “This practice is also rather common in derivatives trading where a pricing model is constantly recalibrated and hedges are re-computed.”

The natural application of the model is in European-style options, though the authors believe it can be extended to the American version.

The model can also incorporate data-driven techniques.

“We will continue to develop this model,” says Lucic. “As we have a Hamilton-Jacobi-Bellman [HJB] equation at hand here, a natural extension would be to look at reinforcement learning, as it is well-suited for solving problems that involve optimal control policies.”

HJB equations are particularly well-suited to problems involving multi-period decision-making.

Lucic and Tse’s collaboration began when they met to discuss research interests in 2023, and they plan to continue with new projects in other asset classes.

“The model received some attention from hedge funds that specialise in the market-making of crypto-currencies,” says Lucic. “I believe our framework is already well-versed for that purpose and allows them to build their own adapted version of the model.”