The widening of the spread between the Libor and overnight indexed swap (OIS) rates post-crisis spurred the development of the so-called multicurve modelling framework.
The pricing of derivatives with the Libor rate as the underlying, such as swaps, requires the modelling of the Libor forward rate curve on which the cashflows of the derivative are based. In order to price the derivative, the cashflows are then discounted using an appropriate discounting rate, which used to be Libor prior to the crisis.
However, as the Libor and OIS rates began to widen and collateralisation of derivatives increased, awareness grew that derivatives need to be discounted based on the rate of the collateral posted. For cash-only credit support annexes, this is the OIS rate, which could be earned by investing the cash in the money markets.
Multicurve pricing is now common for standard linear interest rate derivatives, but is not so advanced for products such as futures, which rely on an important convexity adjustment to forward rates to account for their non-linear relationship with interest rates. The pricing of products such as Libor futures, for instance, requires a multicurve modelling framework that can factor in this convexity adjustment, which uses the volatility of both the Libor and OIS rates and the correlation between them as inputs.
One particular issue is that banks typically assume the basis between Libor and OIS to be deterministic – that is, the spread is fixed – which, in turn, results in the assumption of a perfect correlation of 1 between Libor and OIS rates. This can result in overestimated convexity adjustments. More advanced banks use stochastic basis spreads that sometimes result in unrealistically high basis between the two rates.
In this month’s first technical, The present of futures, Fabio Mercurio, the head of quantitative analytics at Bloomberg in New York, shows that introducing something called a minimal basis volatility could help solve the problem.
Minimal basis volatility
The minimal basis volatility model creates a case in which the Libor-OIS correlation is smaller than 1, and for any level of correlation guarantees the basis has the smallest possible volatility.
“If we assume an imperfect correlation between Libor and OIS, the Libor-OIS basis is necessarily stochastic, and that can create unrealistic basis scenarios. So, to reduce the occurrence of that, the best we can do is to minimise the basis volatility,” says Mercurio.
The Libor curve in the framework is modelled using a one-factor shifted-lognormal Libor market model, whereas the OIS rates are modelled using a one-factor Cheyette model.
The author then uses stochastic differential equations and approximation techniques to model the adjustment using a closed form formula, and finds that the resulting prices fit market data very well.
The convexity adjustment modelled this way also allows for the calibration of the OIS curve in the multicurve framework based on just the Libor parameters and the Libor-OIS correlation. This is an added advantage, given the absence of appropriate calibration instruments for the OIS curve.
Even a decade after the emergence of the multicurve modelling framework, no industry standard currently exists, mainly because of the limitations surrounding the modelling of the Libor-OIS basis. As a result, simplifying assumptions such as a deterministic basis spread have been made, which result in unrealistic prices.
While appropriately modelling the convexity adjustment for the pricing of futures is a key aspect of capturing multicurve dynamics, regulatory developments such as benchmark reform and non-cleared margin are likely to make modelling these aspects more complex than it already is. For instance, how do you price the death of an old benchmark and the start of a new one for a long-dated swap? What if the new market is not liquid enough? How do you then calibrate the rate curve?
“With the increase of regulation on variation margin, potential changes in benchmarks – in particular Libor, the underlying of the futures – and the launch of new related products such as Sonia three-month futures, it is more important than ever to understand the minutiae of the contracts and their valuation impacts,” says Marc Henrard, the head of quantitative research at vendor OpenGamma in London.
For these reasons, research in this area still has a long way to go.
You can listen to Fabio Mercurio talking about this feature here.
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