Instantaneous portfolio theory - Madan, Dilip B. (Quantitative finance, 2018)
Instantaneous risk is described by the arrival rate of jumps in log price relatives. As a consequence there is then no concept of a mean return compensating risk exposures, as zero is the only instantaneous risk-free return. From this perspective, all portfolios are subject to risk and there are only bad and better ways of holding risk. For the purpose of analysing portfolios, the univariate variance gamma model is extended to higher dimensions with an arrival rate function with full high-dimensional support and independent levels of marginal skewness and excess kurtosis. Investment objectives are given by concave lower price functionals 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. The time series estimation is conducted by digital moment matching applied to uncentred data and it is shown that data centring is a noisy activity to be generally avoided. The evaluation of the instantaneous investment objective requires the computation of measure distorted integrals. This is done using Monte Carlo applied to gamma distributed ellipitical radii with a low shape parameter. The resulting risk reward frontiers are between finite variation as the reward and measure distorted variations as risk. In the absence of an instantaneous risk-free return, portfolios on the efficient frontier are characterized by differences in asset variations being given by differences in asset covariations with the risk charge differential of the efficient portfolio. Portfolio variations seen as the equivalent of excess returns, may optimally be negative. Lower price maximizing portfolios are presented in two, six and twenty five dimensions.