Shape Invariant Modelling Pricing Kernels and Risk Aversion

Abstract

Pricing kernels play a major role in quantifying risk aversion and investors' preferences. Several empirical studies reported that pricing kernels exhibit a common pattern across different markets. Mostly visual inspection and occasionally numerically summarise are used to make comparison. With increasing amount of information updated every day, the empirical pricing kernels can be viewed as an object evolving over time. We propose a systematic modelling approach to describing the evolution of the empirical pricing kernels. The approach is based on shape invariant models. It captures the common features contained in the shape of the functions and at the same time characterises the variability between the pricing kernels based on a few interpretable parameters. The method is demonstrated with the European options and returns values of DAX index.

Uniform Confidence Bands for Pricing Kernels

“…A test on monotonicity of the PK has been proposed by Golubev, Härdle and Timofeev (2009), the extracted time varying parameter, realized either from a low dimensional model for PK or given the coverage probability, may thus be economically analyzed in connection with exogenous macroeconomic business cycle indicator, e.g. credit spread, yield curve, etc, see also Grith, Härdle and Park (2009).…”

Option pricing under time-varying risk-aversion with applications to risk forecasting

“…The resulting pricing kernel estimate is illustrated in Figure 2.3(b), which also exhibits the S-shape. Finally, the German market is inspected by Grith et al [89], who also recover S-shaped pricing kernels for the DAX. More precisely, Grith et al [89] estimate time series of pricing kernels by virtue of shape invariant modeling.…”

“…Finally, the German market is inspected by Grith et al [89], who also recover S-shaped pricing kernels for the DAX. More precisely, Grith et al [89] estimate time series of pricing kernels by virtue of shape invariant modeling. Thereby, the shape of some starting pricing kernel is only changed in scale and location.…”

“…In addition, the shape of pricing kernels is usually not constant but changes over time, which indicates that risk preferences are time-varying (see e.g. Rosenberg and Engle [138] and Grith et al [89]). …”

Nonparametric Estimation of Risk-Neutral Densities

“…In many applications, the empirical pricing kernel is the object of interest. In most of the studies Aït-Sahalia and Lo (2000), Brown and Jackwerth (2004), Grith et al (2009) it has been estimated as a ratio of two estimated densities:q computed as the second derivative of a smooth call function (as described in Section 2) andp based on historical returns. This approach leads to difficulties in deriving the statistical properties of the estimator.…”