FixedIncomeRisk Forum

Though the HJM and LIBOR models at first appear more general than the traditional short rate models, these models suffer from the criticism of “smoothing.” In the discussion to follow, I define the term “smoothing” to imply fitting financial models to a set of observed prices without an underlying economic rationale. The concept of smoothing is different from overfitting in that the former implies fitting without an economic rationale, while the latter implies fitting based upon some economic rationale, but using more parameters than needed to obtain a good fit. Smoothing may overlook some important relationships that could potentially be modeled endogenously, while overfitting fits to the noise present in the data. In other words, smoothing allows the modeler to ignore some important economic relationships by making entirely ad-hoc adjustments to fit the model to observed prices (thus, fail to deal with the misspecification error caused by some hidden variables), while overfitting allows the modeler to invent economic relationships that don’t exist but are artifacts of the noise present in the observed prices.

A simple example of smoothing is using the Black and Scholes model for pricing equity call options of different strikes, and using different volatilities corresponding to different strikes to fit the “smile” with a third-order polynomial function. If the dynamics of the smile are not modeled based on some economic fundamentals, then a trader may not know why and how the option smile changes over time. The option smile obviously represents some systematic economic factor(s), but the incorporating these factor(s) into the option prices is beyond the scope of the Black and Scholes model. Perhaps, a stochastic volatility/jump model is needed to fit the smile. Yet, if traders continue to use the Black and Scholes model to price options by adjusting the implied volatilities across different strikes to fit the smile using a third-order polynomial, then they are “smoothing.” Smoothing basically allows the option trader to price an option of a given strike, given the observed prices of options with strikes surrounding the given strike. However, traders can achieve such smoothed prices even by performing a giant Taylor series expansion, without any knowledge of stochastic processes that drive the stock price movements.

Similarly, it would be wise to be aware of the dangers of smoothing while considering HJM and LIBOR models, especially those with a high degree of time-inhomogeneity in the volatility process, used in calibrating these models to observed prices of caps and swaptions. The origins of time-inhomogeneous volatilities as smoothing variables can be traced to the extended versions of the models of Black, Derman, and Toy [1990], Black and Karasinski [1991], and Hull and White [1990]. Though practitioners have mostly discarded these earlier generation models, LIBOR market models with time-inhomogeneous volatilities remain quite popular. It is unclear how well the LIBOR model performs over time, especially since little research exists on the hedging effectiveness of this model using the approach of Fan, Gupta, and Ritchken [2003].

In this forum, please share your thoughts on the smoothing problem for the HJM and LIBOR models, and recommend any alternatives to these models.