Statistics of Risk Aversion

Option prices are a valuable source of information concerning risk assessments from investors about future financial payoffs. The information is summarized in the state price densities (SPD), the continuous counterpart (normalized by a constant) from Arrow-Debreu security prices. Under no arbitrage assumptions the state price densities - corresponding to a risk neutral measure Q - are derived from option prices as in Breeden and Litzenberger (1978). In contrast to the state pricedensity, the historical density p(x) describes the random variations of the underlying price. In a utility based framework, investors facing financial risk have preference-indifference relations represented by a utility function. Standard economic theory enforces that the utility function is concave or equivalently, that investors are risk averse. Equilibrium and non-arbitrage arguments, asin Merton (1973), show that preferences from investors are related to the state price and historical densities, allowing to conclude the functional form of one given the functional form of the remaining two. Part of this relation is given by the pricing kernel (PK). In this paper we investigate pricing kernels from DAX and ODAX data in a time varying approach and consider their term structure. In order to approximate and analyse the complex dynamic structure from pricing kernels and risk preferences across different maturities, we use dynamic semiparametric factor models (DSFMs). The obtained PKs exhibit risk proclivity in certain regions ofreturns and maturities. The dynamics of risk aversion and proclivity from investors preferences is investigated through sensitivity analysis with respect to the basis functions. In the sequel pricing kernels are defined (section 2), their relation to utility functions and the DSFM estimation method are described (section 3). In the empirical part (section 4) pricing kernels are estimated from DAX and ODAX data sets and sensitivity analysis based on estimated loading coefficients and basis functions is perf