No forward-looking rates? No problem

A commonly used quant model could be the answer to the replacement of forward-looking Libor

As the industry prepares to take on the behemoth task of benchmark reform – labelled by some as a change as complex as Brexit – quants have started to wonder what the impact would be on interest rate models.

The new risk-free rates (RFRs) in various currencies – expected to replace Libor if or when the benchmark ceases after 2021 – are overnight rates, whereas Libor is a forward-looking term rate.

Industry consultations have shown a majority of derivatives market participants are leaning towards using a backward-looking term version of the RFR as a fallback to Libor.

This, no doubt, presents its own challenges for derivatives pricing and modelling, which, for decades, have relied on forward-looking rates.

Some market participants have raised the alarm that certain products may not be able to carry on at all without them.

Interest rate products such as forward rate agreements (FRAs), for instance, need forward-looking rates to work, or else outstanding contracts would have to be reworded. Other, more niche products like Libor-in-arrears swaps would have to be entirely discontinued, some have warned.

In Libor replacement: a modelling framework for in-arrears term rates, Andrei Lyashenko, head of market risk and pricing models at Quantitative Risk Management (QRM) in Chicago, and Fabio Mercurio, global head of quantitative analytics at Bloomberg in New York, argue this may not necessarily be the case if some tweaks are made to a popular pricing model.

The Libor Market Model (LMM) is an interest rate model that tries to price instruments by decomposing their payoffs into a set of forward rates. Each forward rate is simulated or evolved using knowledge of its volatility and correlations between them.

Forward rates are the current market expectations of the rates that are applicable at a time in the future, whereas forward-looking rates are rates that are known or realised in the beginning of the payment period, such as fixing a Libor rate at the beginning of the period. For backward-looking rates, this is done at the end.

In their new paper, the quants show the LMM can be extended to simulate forward values of both forward-looking and backward-looking rates using both historical and cross-sectional data.

The backward-looking rate is alive and continuously giving us information about what happens to the rate environment. The backward- and forward-looking rates only diverge when you enter the application period – before that they are the same

Andrei Lyashenko, Quantitative Risk Management

Typically, LMM models simulate forward rates up to the beginning of the accrual or application period and then stop. In the quants’ model, the model continues simulation up to the end of the accrual period to obtain the forward value of the associated in-arrears rate.

In the new RFR environment, markets from which forward-looking rates can be extracted may not be liquid enough to be used as inputs into this model, but that is not a problem, as backward-looking and forward-looking rates are the same until the corresponding application period is reached, so the model can simulate backward-looking rates to get the forward-looking ones at the same time.

“The backward-looking rate is alive and continuously giving us information about what happens to the rate environment. The backward- and forward-looking rates only diverge when you enter the application period – before that they are the same,” says Lyashenko.

One key advantage of this technique is that banks can expand on existing in-house implementations of the LMM to simulate the backward-looking in-arrears rates, using essentially the same inputs as before. The rate curves can be obtained from bonds or swaps, and the volatility of the rates can be obtained from options markets such as caps and swaptions – although it is not clear how liquidity in these markets would evolve post Libor reform. Correlations can be obtained from historical data on the rates.

“You don’t need to model a new rate or add extra risk factors into the model because the existing ones are already good enough,” says Mercurio. “You use the same stochastic processes and you make sure you extend the simulation of every single rate until the end of the corresponding application period, instead of stopping at the beginning of the application period. That’s basically the magic recipe that allows us to get the new in-arrears rates.”

Dynamics advantage

The quants also argue the resulting model has richer forward curve dynamics than the classical LMM model, so should be used even without Libor reform.

For firms that already have an LMM model in place, the extension might take a couple of months, the quants estimate. On the other hand, firms that might not typically use an LMM model, such as small end-users, may take a few years to implement the LMM from scratch.

The change from Libor to a fallback – whatever that rate may be – is going to be a major operational change for quants. So far, those interviewed by Risk.net have highlighted various modelling issues that might arise due to the reform, but many emphasised they haven’t yet been asked by senior management to find solutions to them.

Mercurio and Lyashenko’s paper takes a major step towards addressing one of the biggest challenges highlighted by quants in terms of adapting to the new regime – the move from a forward-looking rate environment to a backward-looking one. More such technical issues need to be ironed out before transition begins.