Stochastic Portfolio Theory

Functionally Generated Portfolios Under Stochastic Transaction Costs: Theory and Empirical Evidence ArXiv ID: 2507.09196 “View on arXiv” Authors: Nader Karimi, Erfan Salavati Abstract Assuming frictionless trading, classical stochastic portfolio theory (SPT) provides relative arbitrage strategies. However, the costs associated with real-world execution are state-dependent, volatile, and under increasing stress during liquidity shocks. Using an Ito diffusion that may be connected with asset prices, we extend SPT to a continuous-time equity market with proportional, stochastic transaction costs. We derive closed-form lower bounds on cost-adjusted relative wealth for a large class of functionally generated portfolios; these bounds provide sufficient conditions for relative arbitrage to survive random costs. A limit-order-book cost proxy in conjunction with a Milstein scheme validates the theoretical order-of-magnitude estimates. Finally, we use intraday bid-ask spreads as a stand-in for cost volatility in a back-test of CRSP small-cap data (1994–2024). Despite experiencing larger declines during the 2008 and 2020 liquidity crises, diversity- and entropy-weighted portfolios continue to beat the value-weighted benchmark by 3.6 and 2.9 percentage points annually, respectively, after cost deduction. ...

Capital Asset Pricing Model with Size Factor and Normalizing by Volatility Index ArXiv ID: 2411.19444 “View on arXiv” Authors: Unknown Abstract The Capital Asset Pricing Model (CAPM) relates a well-diversified stock portfolio to a benchmark portfolio. We insert size effect in CAPM, capturing the observation that small stocks have higher risk and return than large stocks, on average. Dividing stock index returns by the Volatility Index makes them independent and normal. In this article, we combine these ideas to create a new discrete-time model, which includes volatility, relative size, and CAPM. We fit this model using real-world data, prove the long-term stability, and connect this research to Stochastic Portfolio Theory. We fill important gaps in our previous article on CAPM with the size factor. ...

Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized market ArXiv ID: 2411.13558 “View on arXiv” Authors: Unknown Abstract The strong relative arbitrage problem in Stochastic Portfolio Theory seeks an investment strategy that almost surely outperforms a benchmark portfolio at the end of a given time horizon. The highest relative return in relative arbitrage opportunities is characterized by the smallest nonnegative continuous solution of a Cauchy problem for a partial differential equation (PDE). However, solving this type of PDE poses analytical and numerical challenges, due to the high dimensionality and its non-unique solutions. In this paper, we discuss numerical methods to address the relative arbitrage problem and the associated PDE in a volatility-stabilized market, using time-changed Bessel bridges. We present a practical algorithm and demonstrate numerical results through an example in volatility-stabilized markets. ...

Macroscopic properties of equity markets: stylized facts and portfolio performance ArXiv ID: 2409.10859 “View on arXiv” Authors: Unknown Abstract Macroscopic properties of equity markets affect the performance of active equity strategies but many are not adequately captured by conventional models of financial mathematics and econometrics. Using the CRSP Database of the US equity market, we study empirically several macroscopic properties defined in terms of market capitalizations and returns, and highlight a list of stylized facts and open questions motivated in part by stochastic portfolio theory. Additionally, we present a systematic backtest of the diversity-weighted portfolio under various configurations and study its performance in relation to macroscopic quantities. All of our results can be replicated using codes made available on our online repository. ...

Calibrated rank volatility stabilized models for large equity markets ArXiv ID: 2403.04674 “View on arXiv” Authors: Unknown Abstract In the framework of stochastic portfolio theory we introduce rank volatility stabilized models for large equity markets over long time horizons. These models are rank-based extensions of the volatility stabilized models introduced by Fernholz & Karatzas in 2005. On the theoretical side we establish global existence of the model and ergodicity of the induced ranked market weights. We also derive explicit expressions for growth-optimal portfolios and show the existence of relative arbitrage with respect to the market portfolio. On the empirical side we calibrate the model to sixteen years of CRSP US equity data matching (i) rank-based volatilities, (ii) stock turnover as measured by market weight collisions, (iii) the average market rate of return and (iv) the capital distribution curve. Assessment of model fit and error analysis is conducted both in and out of sample. 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 empirical features (i)-(iv). We additionally simulate trajectories of the calibrated model and compare them to historical trajectories, both in and out of sample. ...

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. ...