Dynamical Optimization Theory of a Diversified Portfolio

We propose a frustrated and disordered many-body model of a stockmarket in which independent adaptive traders can trade a stock subject to the economic law of supply and demand. We show that the typical scaling properties and the correlated volatility arise as a consequence of the collective behavior of agents: With their interaction they determine a price which in turn affects their future way of investing. We introduce only one type of investors, since they all share the sa...

A new framework for portfolio diversification is introduced which goes beyond the classical mean-variance approach and portfolio allocation strategies such as risk parity. It is based on a novel concept called portfolio dimensionality that connects diversification to the non-Gaussianity of portfolio returns and can typically be defined in terms of the ratio of risk measures which are homogenous functions of equal degree. The latter arises naturally due to our requirement that...

The portfolio optimization problem in which the variances of the return rates of assets are not identical is analyzed in this paper using the methodology of statistical mechanical informatics, specifically, replica analysis. We define two characteristic quantities of an optimal portfolio, namely, minimal investment risk and concentrated investment level, in order to solve the portfolio optimization problem and analytically determine their asymptotical behaviors using replica ...

Multiplicative random processes in (not necessaryly equilibrium or steady state) stochastic systems with many degrees of freedom lead to Boltzmann distributions when the dynamics is expressed in terms of the logarithm of the normalized elementary variables. In terms of the original variables this gives a power-law distribution. This mechanism implies certain relations between the constraints of the system, the power of the distribution and the dispersion law of the fluctuatio...

By assuming the existence of the growth optimal portfolio (GOP), the stationarity of GOP-volatilities, and the maximization of relative entropy, the paper applies the benchmark approach to the modeling of the long-term dynamics of continuous markets. It reveals conservation laws, where the GOP is shown to follow a time-transformed squared Bessel process of dimension four. Moreover, it predicts the convergence of the averages of the GOP-volatilities with respect to the driving...

We present and discuss a stochastic model of financial assets dynamics based on the idea of an inverse renormalization group strategy. With this strategy we construct the multivariate distributions of elementary returns based on the scaling with time of the probability density of their aggregates. In its simplest version the model is the product of an endogenous auto-regressive component and a random rescaling factor designed to embody also exogenous influences. Mathematical ...

Several problems arising in Economics and Finance are analyzed using concepts and quantitative methods from Physics. Here is the abridged abstact: Chapter 1: By analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. A thermal machine which extracts a monetary profit can be constructed between two economic systems ...

I discuss some theoretical results with a view to motivate some practical choices in portfolio optimization. Even though the setting is not completely general (for example, the covariance matrix is assumed to be non-singular), I attempt to highlight the features that have practical relevance. The mathematical setting is Stochastic Portfolio Theory, which is flexible enough to describe most realistic assets, and it has been successfully employed for managing equity portfolios ...

In this paper, we develop the theory of functional generation of portfolios in an equity market with changing dimension. By introducing dimensional jumps in the market, as well as jumps in stock capitalization between the dimensional jumps, we construct different types of self-financing stock portfolios (additive, multiplicative, and rank-based) in a very general setting. Our study explains how a dimensional change caused by a listing or delisting event of a stock, and unexpe...

Traditional Markowitz portfolio optimization constrains daily portfolio variance to a target value, optimising returns, Sharpe or variance within this constraint. However, this approach overlooks the relationship between variance at different time scales, typically described by $\sigma(\Delta t) \propto (\Delta t)^{H}$ where $H$ is the Hurst exponent, most of the time assumed to be \(\frac{1}{2}\). This paper introduces a multifrequency optimization framework that allows inve...