Getting a handle on model parameters

Mean reversion in rate parameters opens the door to dimensionality reduction

The parameters of popular factor models for interest rates, such as the Heath-Jarrow-Morton model, are known to be mean-reverting. The study of the speed at which the mean reversion happens has been largely neglected, but that information would be helpful for structuring more efficient hedging strategies.

More broadly, fresh research on interest rate modelling could be useful for banks and investors as they grapple with volatile rates during a period of macroeconomic uncertainty.

In derivatives desks’ practice, the speed of mean reversion is commonly considered constant to simplify models and their computation.

And it is often obtained via some rule of thumb, while ideally it should be provided by a rigorous and validated procedure and display good historical fit and stability.

A method to estimate the mean reversion speed of parameters is the subject of a paper by Vladimir Piterbarg, who heads the quantitative analytics and development team at NatWest Markets, and Igor Duchitskii, who was part of that team and has recently become a quantitative developer at a global hedge fund.

The method is based on a linear dependency between the variations of interest rates of different tenors that relates to a function of mean reversion speeds. Thanks to that linear dependency, the method can reduce the dimensionality of the problem, as it is possible to represent a series of rates as a function of fewer rates. 

Related

Estimating mean reversions in interest rate models 

Piterbarg and Duchitskii present a method for gauging the mean reversion speed of factors

To show how it works, let’s consider a basket of 10 rates and assume they depend on two factors – meaning a two-factor model would be used to capture their dynamics.

Two of those 10 rates are then arbitrarily selected and the factor model is applied to them. That gives a system of two equations and two unknown variables to describe the evolution of two rates.

The factors obtained by these two equations are then substituted into the equations for the other eight rates. This system provides the dynamics for those rates in terms of the initial two rates. The coefficients of the resulting relationships are not just 16 constants, but functions of the mean reversion speeds for the two parameters.

In essence, the authors reach a formulation for the 10 rates as a function of the mean reversion parameters that is fitted to historical data.

The errors of these relationships are weighted following a method given by the authors to detect the best suited covariance. That also makes the result agnostic to the choice of two rates used to solve for factors.

As the authors explain in the paper, a condition for this method’s applicability is that the variations of the interest rates depend linearly on the variations of the model factors, and that “the coefficients of this linear relationship are functions of tenors rather than absolute times”. Another necessary condition is that the model it’s applied to has separable volatility, meaning that within the model, volatility can be decomposed into both a term that depends on the current time and a term that depends on maturity.

The initial purpose of this method was to reduce the dimensionality and improve the accuracy of a framework for valuation adjustments of a portfolio of interest rate derivatives.

It can be plugged into many popular interest rate models, such as the Cheyette model, Hull-White, multi-factor Hull-White, Cox-Ingersoll-Ross and variations of them. They all have mean-reverting factors and there is little research into how to rigorously estimate mean reversion separately from volatility.

Those factors are generally unobservable, but in some cases they can be loosely interpreted. In a one-factor model, the factor is usually interpretable as the instantaneous short rate. In a two-factor model, the two factors might be interpreted as level and slope of the curve, and a three-factor model might have convexity as a third factor. But while this offers some intuition, it’s not a key feature of the method.