Stochastic factors can matter: improving robust growth under ergodicity

TL;DR

This work develops a robust, ergodic-growth framework for incomplete markets where asset returns drift is uncertain but constrained by a joint diffusion with a nontraded stochastic factor $Y$. The authors derive a variational problem whose solution $\phi^*$ satisfies an Euler–Lagrange PDE in $x$ (indexed by $y$), yielding a robust growth rate $\lambda_{\mathcal P}=\tfrac{1}{2}\int_F (\nabla_x\phi^*)^\top c_X(\nabla_x\phi^*) p$ and a robust optimal policy $\theta^*_t=\nabla_x\phi^*(Z_t)$. They construct a worst-case measure ${\mathbb P}^*$ under which $\theta^*$ is growth-optimal, and compare with the prior Itkin et al. results where the optimal strategy depends only on $X$; knowledge of $Y$ can strictly improve robust growth except in edge cases. The theoretical development is illustrated through Gaussian (OU) environments and a comprehensive set of pair-trading examples, including Central Tendency OU, fat tails, and stochastic volatility, showing both gains from leveraging stochastic factors and risks if those factors are mis-specified. Overall, the paper provides a tractable, explicitly solvable route to incorporate stochastic factors into robust growth strategies with clear financial interpretation and practical implications for statistical arbitrage and paired-trading contexts.

Abstract

Drifts of asset returns are notoriously difficult to model accurately and, yet, trading strategies obtained from portfolio optimization are very sensitive to them. To mitigate this well-known phenomenon we study robust growth-optimization in a high-dimensional incomplete market under drift uncertainty of the asset price process $X$, under an additional ergodicity assumption, which constrains but does not fully specify the drift in general. The class of admissible models allows $X$ to depend on a multivariate stochastic factor $Y$ and fixes (a) their joint volatility structure, (b) their long-term joint ergodic density and (c) the dynamics of the stochastic factor process $Y$. A principal motivation of this framework comes from pairs trading, where $X$ is the spread process and models with the above characteristics are commonplace. Our main results determine the robust optimal growth rate, construct a worst-case admissible model and characterize the robust growth-optimal strategy via a solution to a certain partial differential equation (PDE). We demonstrate that utilizing the stochastic factor leads to improvement in robust growth complementing the conclusions of the previous study by Itkin et. al. (arXiv:2211.15628 [q-fin.MF], forthcoming in $\textit{Finance and Stochastics}$), which additionally robustified the dynamics of the stochastic factor leading to $Y$-independent optimal strategies. Our analysis leads to new financial insights, quantifying the improvement in growth the investor can achieve by optimally incorporating stochastic factors into their trading decisions. We illustrate our theoretical results on several numerical examples including an application to pairs trading.

Figures (4)

• Figure 1: Boxplots of growth rates for $\theta^*$ and $\widehat{\theta}$ under $\mathbb{P}^*$ and $\widehat{\mathbb{P}}$ obtained from 10,000 simulations with time horizon $T \in \{10,20,30\}$, increasing from light to dark green, with outliers omitted. The triangle in each box represents the mean and the dashed lines are the theoretical growth rates $\lambda_{\mathcal{P}}, \lambda_{\Pi}$ and $g(\theta^*;\widehat{\mathbb{P}})$ appearing in descending order.
• Figure 2: Slices of $\partial_x \phi^*(\cdot,y)$ for $y$ between $-2$ and $2$ for 11 equally spaced points (solid lines with y increasing from dark to light) plotted along side $\widehat{\phi}'$ (dashed line).
• Figure 3: Slices of $\partial_x \phi^*(\cdot,y)$ for $y$ between $-3$ and $3$ for 11 equally spaced points (solid lines with y increasing from dark to light) plotted along side $\widehat{\phi}'$ (dashed line).
• Figure 4: The left panel plots slices of $\partial_x \phi^*(\cdot,y)$ for $y$ between $0.0225$ and $0.0575$ for 11 equally spaced points (solid lines with y increasing from dark to light) plotted along side $\widehat{\phi}'$ (dashed line). The right panel depicts only $\widehat{\phi}'$ (solid line) together with the theoretical limiting holdings of the strategy $\lim_{x \to \pm \infty} \widehat{\phi}'(x) = \mp \frac{\alpha^2}{\sqrt{2}\beta^{3/2}}$ (dashed lines).

Theorems & Definitions (14)

• Definition 2.2: Admissible class of measures
• Remark 2.3
• Remark 3.1
• Remark 4.2
• Definition 4.3: Finite growth class
• Lemma 5.1: Characterization of the optimizer
• Theorem 5.2: Main result
• Proposition 5.3: Worst-case measure
• Remark 7.1
• Lemma A.1