Consistent calibration of HJM models to cap implied volatilities

Abstract

This paper proposes a calibration algorithm that fits multi-factor Gaussian models to the implied volatilities of caps using the respective minimal consistent family to infer the forward rate curve. The algorithm is applied to three forward rate volatility structures and their combination to form two-factor models. The efficiency of the consistent calibration is evaluated through comparisons with non-consistent methods. The selection of the number of factors and of the volatility functions is supported by a Principal Component Analysis. Models are evaluated in terms of in-sample and out-of-sample data fitting as well as stability of parameter estimates. The results are analyzed mainly focusing on the capability of fitting the market implied volatility curve and, in particular, of reproducing its characteristic humped shape.

International evidence on alternative models of the term structure of volatilities

“…This turns out to be similar to minimizing the relative errors in caplet volatilities (see Angelini & Herzel, 2005, for …”

“…In most studies, model parameters are estimated by minimizing the sum of squared pricing errors of all caps with different maturities. However, recent studies, e.g., Angelini and Herzel (2005) or Jarrow, Li, and Zhao (2007), use squared percentage pricing errors. According to these authors, this specification should help to mitigate differences in cap prices coming from a wide range of maturities.…”

“…The two new models provide more flexibility to adequately calibrate the observed cap prices, although this improved accuracy in replicating cap prices produces some instability in parameter estimates. 1 In the framework of the Heath, Jarrow, and Morton (1992) model, Angelini and Herzel (2005) evaluate the efficiency of the calibration method applied to three forward rate volatility structures and their combination to form two-factor models.…”

A Path-Independent Humped Volatility Model for Option Pricing

“…Indeed, contributions reporting that the volatility curve of forward rates presents a hump depending on their times to maturity are due to, among others, Kahn (1991), Litterman and Scheinkman (1991), , Amin and Morton (1994), Goncalves and Issler (1996) and, more recently, Angelini and Herzel (2005). when quadratic interpolation is used.…”

On Finite Dimensional Realizations of Two-Country Interest Rate Models

“…Also, seemingly reasonable estimates of discount function do not always lead to an acceptable yield curve shape. Angelini and Herzel (2005) and Angelini and Herzel (2002) thoroughly compared cubic splines and consistent interpolations (interpolation by mean of fitting to the consistent parameterized families) and concluded that consistent pricing gives the most stable estimates and the best performance in terms of pricing errors and forecasting capability. Besides that, the geometric approach connects the term structures to the joint set of factors which describes the joint evolution of the countries' economies.…”