Instantaneous Portfolio Theory

Abstract

Instantaneous risk is described by the arrival rate of jumps in log price relatives. Aggregate arrivals are infinite. There is then no concept of a mean return compensating risk exposure. The only risk-free instantaneous return is zero. All portfolios are subject to risk and there are only bad and better ways of holding risk. The univariate variance gamma model is extended to higher dimensions with an arrival rate function with full support in high dimensions and independent levels of skewness and excess kurtosis across assets. Concave bid price functionals are formulated as measure distorted variations. Specific measure distortions are calibrated to data on S&P 500 index options and the time series of the index. It is shown that univariate estimation by digital moments applied to uncentered data dominates the use of centered data. Measure distorted integrals are computed using Monte Carlo applied to gamma distributed elliptical radii with a low shape parameter. Risk reward frontiers exist between the finite exponential variation as the reward and measure distorted exponential variations as risk charges. For market portfolios on the efficient frontier as an optimal portfolio, differences in asset exponential variations are given by differences in exponential asset covariations with the risk charge differential of the market portfolio. Optimal portfolio exponential variations may be negative. Bid price maximizing portfolios are presented in two, six and twenty five dimensions with an efficient frontier between exponential variations illustrated in dimension six.

Keywords: Lévy measure, Weak Subordination, Gamma distributed Elliptical Radius, Measure Distortion

JEL Classification: G10, G11, G13