Universal Portfolios

High order universal portfolios ArXiv ID: 2311.13564 “View on arXiv” Authors: Unknown Abstract The Cover universal portfolio (UP from now on) has many interesting theoretical and numerical properties and was investigated for a long time. Building on it, we explore what happens when we add this UP to the market as a new synthetic asset and construct by recurrence higher order UPs. We investigate some important theoretical properties of the high order UPs and show in particular that they are indeed different from the Cover UP and are capable to break the time permutation invariance. We show that under some perturbation regime the second high order UP has better Sharp ratio than the standard UP and briefly investigate arbitrage opportunities thus created. Numerical experiences on a benchmark from the literature confirm that high order UPs improve Cover’s UP performances. ...

Signature Methods in Stochastic Portfolio Theory ArXiv ID: 2310.02322 “View on arXiv” Authors: Unknown Abstract In the context of stochastic portfolio theory we introduce a novel class of portfolios which we call linear path-functional portfolios. These are portfolios which are determined by certain transformations of linear functions of a collections of feature maps that are non-anticipative path functionals of an underlying semimartingale. As main example for such feature maps we consider the signature of the (ranked) market weights. We prove that these portfolios are universal in the sense that every continuous, possibly path-dependent, portfolio function of the market weights can be uniformly approximated by signature portfolios. We also show that signature portfolios can approximate the growth-optimal portfolio in several classes of non-Markovian market models arbitrarily well and illustrate numerically that the trained signature portfolios are remarkably close to the theoretical growth-optimal portfolios. Besides these universality features, the main numerical advantage lies in the fact that several optimization tasks like maximizing (expected) logarithmic wealth or mean-variance optimization within the class of linear path-functional portfolios reduce to a convex quadratic optimization problem, thus making it computationally highly tractable. We apply our method also to real market data based on several indices. Our results point towards out-performance on the considered out-of-sample data, also in the presence of transaction costs. ...

Online Universal Dirichlet Factor Portfolios ArXiv ID: 2308.07763 “View on arXiv” Authors: Unknown Abstract We revisit the online portfolio allocation problem and propose universal portfolios that use factor weighing to produce portfolios that out-perform uniform dirichlet allocation schemes. We show a few analytical results on the lower bounds of portfolio growth when the returns are known to follow a factor model. We also show analytically that factor weighted dirichlet sampled portfolios dominate the wealth generated by uniformly sampled dirichlet portfolios. We corroborate our analytical results with empirical studies on equity markets that are known to be driven by factors. ...

Convex optimization over a probability simplex ArXiv ID: 2305.09046 “View on arXiv” Authors: Unknown Abstract We propose a new iteration scheme, the Cauchy-Simplex, to optimize convex problems over the probability simplex ${“w\in\mathbb{R”}^n\ |\ \sum_i w_i=1\ \textrm{“and”}\ w_i\geq0}$. Specifically, we map the simplex to the positive quadrant of a unit sphere, envisage gradient descent in latent variables, and map the result back in a way that only depends on the simplex variable. Moreover, proving rigorous convergence results in this formulation leads inherently to tools from information theory (e.g., cross-entropy and KL divergence). Each iteration of the Cauchy-Simplex consists of simple operations, making it well-suited for high-dimensional problems. In continuous time, we prove that $f(x_T)-f(x^) = {“O”}(1/T)$ for differentiable real-valued convex functions, where $T$ is the number of time steps and $w^$ is the optimal solution. Numerical experiments of projection onto convex hulls show faster convergence than similar algorithms. Finally, we apply our algorithm to online learning problems and prove the convergence of the average regret for (1) Prediction with expert advice and (2) Universal Portfolios. ...