Veranstaltungen

Building on the previously proposed volatility stabilized models (Fernholz & Karatzas, 2005) in the framework of stochastic portfolio theory, we propose a rank-based extension to model an equity market over long time horizons. In this extended model, a collection of d stock capitalizations is driven by rank-dependent drift and diffusions coefficients normalized by their inverse market weight. Under an explicit condition on the parameters, we establish global weak existence to the market weight system as well as ergodicity of the induced ranked market weights. We also establish uniqueness of the reflected stochastic differential equation that the ranked system satisfies. Moreover, we show that the model admits relative arbitrage; that is the market portfolio can be outperformed with probability one in finite time. We then calibrate the model to three stable features of US equity data: (i) volatility of the ranked market capitalizations, (ii) frequency and intensity of "collisions" (i.e. changes in rank) and (iii) the capital distribution curve. Despite having only two parameters per asset to fit three distinct criteria, we are able to show that the model parsimoniously fits these stylized features of long-term US equity modelling. To the best of our knowledge this is the first model exhibiting relative arbitrage that has statistically been shown to have a good quantitative fit with the empirically estimable features (i)-(iii) above. We also generate simulated sample paths of the calibrated model and compare them to historical trajectories, both in and out of sample. Based on joint work in progress with Martin Larsson.