Rough volatility moves to exotic frontiers

Since they were introduced in 2014, rough volatility models have attracted praise and scepticism in almost equal measure. But it is still too early to render a definitive verdict. Proponents of the models continue to build on the pillars of the theory and develop new applications. Another chapter in the story has just been written by one of the pioneers of rough volatility, Jim Gatheral, presidential professor of mathematics at Baruch College at the City University of New York, who won Risk.net’s quant of the year award in 2021 for his work in the field.

In 2018, Gatheral and Martin Keller-Ressel of TU Dresden introduced the class of so-called affine forward variance models, which include the Heston volatility model and its ‘rough’ cousin.

Their goal was to describe a class of models that allows for closed form computations, including of their characteristic function. However, their simulations proved unsatisfactory, particularly for rough Heston models.

“The question I wanted to answer was how to compute prices of options other than Europeans, how to compute barrier option prices, how to compute autocallable prices, and so on. Such prices are normally computed via simulation, and nobody knew how to simulate efficiently,” says Gatheral.

In his latest paper, Efficient simulation of affine forward variance models, Gatheral describes a fast simulation method that extends application of the rough Heston model to exotics pricing.

Simulation of affine forward variance models has long been a vexing problem. This was the case with the Heston model, which was published in 1994 and has been ubiquitous in banks since. Gatheral and Keller-Ressel demonstrated it belongs in the class of affine forward variance models. In fact, all affine forward variance models are essentially Heston models with a different choice of kernel – a term that quantifies how quickly the effects of moves in instantaneous variance die away. The kernel in the case of the classical Heston model is exponential; in the case of the rough Heston model, it is a power law.

“For 14 years basically nobody knew how to simulate the classical Heston model efficiently, until Leif Andersen came out with a very elegant solution in 2008,” says Gatheral. That solution was the so-called quadratic-expansion (QE) scheme, which ensures that simulated instantaneous volatility can never turn negative, as it would if a naive scheme is used.

The Monte Carlo simulation scheme Gatheral has developed for the rough Heston model combines the QE scheme with another simulation technique – the so-called Riemann-sum scheme. The hybrid approach could allow the industry to move away from existing solutions, such as those involving the simulation of hundreds of Heston models, which reach an output very slowly. The implementation is outlined in the paper, in what Gatheral promises is a very simple algorithm, for which he provides the code.

For 14 years basically nobody knew how to simulate the classical Heston model efficiently

Jim Gatheral, Baruch College

“For rough volatility models, it is typically very complex to get any analytical results,” says Pietro Rossi, a senior analyst in the data science unit at Italian consultancy firm Prometeia, and an adjunct professor of computational finance at the University of Bologna. “This method provides a way to perform the Monte Carlo simulation for a model that is not Markovian due to the rough nature of the volatility process. Prior to this contribution there was no viable method to do it.”

Rossi has himself tested the algorithm and agrees it is efficient and easy to implement. He is convinced of the importance of the paper within the realm of rough volatility. He acknowledges, though, that opinions on the matter may differ: one camp of quants still thinks rough vol is unnecessary; while another is certain that volatility displays a rough behaviour.

The primary motivation for Gatheral was academic rather than practical. “People keep asking me to do a second edition of my book, The volatility surface. For that I want a reasonable model, a rough volatility model, and for educational purposes the best modelling choice would be an affine forward variance model, because you can compute so many things in closed form. And in order to do anything interesting with this model you need to simulate, and this article shows how,” Gatheral says.

“For me, the primary application of this is to teaching – to give intuition for the behaviour of rough stochastic volatility models. Up to now, thanks to the analytical tractability of affine models, we could get easy intuition for the effect of rough volatility parameters on vanilla option prices; now, with the hybrid QE scheme, we can do the same for exotics.”

Outside the academic sphere, the implication for banks that use the classical Heston as their stochastic volatility backbone is that they can now easily replace it with the rough Heston model. Rough Heston fits the observed volatility surface and the smile dynamics of options much better than classical Heston, opening the door for comprehensive use in derivatives desks.