Into the quantiverse: real-world pricing goes arbitrage-free

In superhero movies, it is now almost de rigueur for the protagonist to battle an array of villains from parallel worlds, with the ultimate aim of restoring harmony across the ‘multiverse’.

A similar dynamic has been at work in quant finance, which – though it may not have the countless alternative realities of Spider-Man: Into the Spider-Verse or Everything Everywhere All At Once – has long been divided into two worlds.

In the first, quants use the so-called real-world or physical ‘P’ measure to describe markets, and manage the risks therein, based on observable data. These quants rely on factor models for applications such as market forecasts, portfolio management, value-at-risk or stress testing.

In the second, pricing models are developed using the risk-neutral ‘Q’ measure to satisfy the no-arbitrage condition.

These worlds have co-existed, with virtually no interaction, in every asset class.

Now, two quants at a Chicago-based consultancy have taken on the mantle of Miles Morales and Evelyn Wang, and claim to have bridged the two worlds – at least in the realm of fixed income.

Andrei Lyashenko, head of market pricing and risk models at Quantitative Risk Management, and his colleague, research analyst Yevgeny Goncharov, have developed a risk-neutral framework consisting of easily interpretable factors.

Their factor Heath-Jarrow-Morton (fHJM) term structure employs the same equations that would be used for the P measure, but with a twist that satisfies the no-arbitrage condition required for pricing models. That may sound like minor detail, but it is in fact a connection that has never been made before. Lyashenko and Goncharov believe it could quickly become a new market standard and replace the ubiquitous Libor Market Model (LMM) used for pricing interest rate derivatives.

Minding their Ps and Qs

The initial impetus came from QRM’s need to model prepayments, a significant risk factor in mortgages and mortgage-backed securities.

“The modelling of the whole curve consistently is of paramount importance for modelling mortgages, as well as their risk management or hedging,” says Lyashenko, a veteran rates quant and Risk.net’s Quant of the Year in 2020.

The two most important factors in modelling prepayments are short and long rates, which are typically modelled under the Q measure. Lyashenko tried to use two independent factor models to do this more flexibly, but soon realised this would not be consistent with the Q pricing, in which the fixing of the short rate would determine the long rate to ensure no arbitrage opportunities could arise.

“Prepayments are modelled using empirical data, which is part of the P world. But then, when you price mortgages, you stick to the Q measure and any inconsistency between the P world and Q world could show up,” says Lyashenko, referring to the risk of mispricing.

There are many situations in which firms need to model both risk-neutral and real-world rates, and there are obvious benefits to having these two models operate consistently. For example, risk-neutral models are used to compute valuation adjustments, but real-world models are preferred when it comes to setting counterparty credit limits. As these computations often need to be performed on the same portfolio, a unified approach would avoid pricing discrepancies.

Lyashenko struggled to find a satisfactory solution until he discovered a lost gem in the literature. In a 2011 paper, Jens Christensen, Francis Diebold and Glenn Rudebusch argued that the factor representation of a yield curve given by the Nelson-Siegel basis (a popular yield curve factor model) could be made arbitrage-free if the mean reversion matrix were set to a specific, and quite peculiar, form.

“I was mesmerised by this result and could not believe my eyes,” says Lyashenko. “I checked the computations several times and did not find any errors. I asked Diebold how in the world they came up with this mean reversion matrix. He replied that they reverse engineered it. This result stuck in my mind as an enigma.”

Lyashenko worked with Goncharov to find a way to impose the no-arbitrage condition on a factor model and eventually managed to generalise the solution proposed by Christensen, Diebold and Rudebusch. “Our research resulted in deriving the family of admissible bases that include [Christensen, Diebold and Rudebusch’s] solution as a specific case,” he says.

Every rate, everywhere, all at once

Goncharov stresses that the generalisation leads to versatility and intuitiveness: “Anything that can be done under the Q measure can also be done within our framework. And even if you take the P world out of the way, this framework is still valuable for Q quants for its simplicity and transparency. It is much lighter than LMM and more intuitive than Cheyette [a version of the HJM framework], which can be very valuable for development and more efficient implementation of the models.”

Alexander Sokol, founder of risk software provider CompatibL, hails Lyashenko and Goncharov’s work as a breakthrough. “For decades, P-measure quants have been working on dynamic Nelson-Siegel models, and Q-measure quants on Cheyette models,” he says. “Lyashenko and Goncharov discovered that they are all parts of a common framework. That’s the strength of this paper.”

The fHJM term structure has also contributed to Sokol’s own research: “With Oleksiy Kondratiev, we developed a term structure model based on machine learning and its starting point is the history of observed curve shapes. The initial objective was only to improve on the Nelson-Siegel model in the real-world measure. Thanks to [Lyashenko and Goncharov’s] paper, we realised that our model could be extended to the risk-neutral measure. As a result, the model we initially developed for calculating credit limits and yield curve forecasting can now also be used for pricing and XVA.”

If the solution proves robust, a divide in the price-modelling multiverse will have been successfully bridged.

Editing by Daniel Blackburn