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A large literature finds evidence that pricing kernels estimated nonparametrically from option prices and historical returns are not monotonically decreasing in market index returns. We propose a new nonparametric estimator of the pricing kernel that reflects the information available to investors who set option prices. In simulations, the estimator outperforms current techniques. Our empirical estimates using S&P 500 index option data from 1996-2012 and FTSE 100 index option data from 2002-2013 suggest that the "pricing kernel puzzle'' is a byproduct of econometric technique rather than a behavioral or economic phenomenon.

The problem of nonlinear filtering of a random field observed in the presence of a noise, modeled by a persistent fractional Brownian sheet of Hurst index $(H_1,H_2)$ with $0.5, Comment: 24 pages

In (10) a "direct" stochastic transfer principle was introduced, which represented multiple integrals with respect to fractional Brownian mo- tion in terms of multiple integrals with respect to standard Brownian motion. The method employed in (10) involved an operator i (n) H , mapping a class of functions L 2 to L 2 . However, the operator does not map L 2 onto L 2 . Hence i (n) H is not invertible. The non-invertibility arises from the fact that i (n) H is defined in terms of n-fold Riemann-Liouville fractional integrals and it is well known that functions f 2 L p (a,b) cannot always be represented as fractional integrals of functions ' 2 L p (a,b). This paper establishes the inverse sto- chastic transfer principle. Our purpose is to explicitly define an operator, which, acting on a certain class of functions, gives the transfer principle in- verting that in (10). As a result, multiple stochastic integrals with respect to standard Brownian motion are represented in terms of multiple stochastic integrals with respect to persistent fractional Brownian motion. In order to do this, we first prove a characterization of a class of functions for which there exists ', defined on a compact set in R n , such that f is equal to the n-fold Riemann-Liouville fractional integral of '. We establish a method for computing ' and conclude the paper with an example of using the inverse transfer principle in filtering.

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.