Mario Ayala, Benjamin Vallejo Jiménez
5 min
Abstract
We revisit the classical Merton consumption--investment problem when risky-asset returns are modeled by stochastic differential equations interpreted through a general $α$-integral, interpolating between Itô, Stratonovich, and related conventions. Holding preferences and the investment opportunity set fixed, changing the noise interpretation modifies the effective drift of asset returns in a systematic way. For logarithmic utility and constant volatilities, we derive closed-form optimal policies in a market with $n$ risky assets: optimal consumption remains a fixed fraction of wealth, while optimal portfolio weights are shifted according to $θ_α^\ast = V^{-1}(μ-r\mathbf{1})+α\,V^{-1}\operatorname{diag}(V)\mathbf{1}$, where $V$ is the return covariance matrix and $\operatorname{diag}(V)$ denotes the diagonal matrix with the same diagonal as $V$. In the single-asset case this reduces to $θ_α^\ast=(μ-r)/σ^{2}+α$. We then show that genuinely state-dependent effects arise when asset volatility is driven by a stochastic factor correlated with returns. In this setting, the $α$-interpretation generates an additional drift correction proportional to the instantaneous covariation between factor and return noise. As a canonical example, we analyze a Heston stochastic volatility model, where the resulting optimal risky exposure depends inversely on the current variance level.
Alex: All from how you read the noise. That might explain why standard models seem too careful in bumpy markets.
Sam: It does. High-frequency data has extra bumps from trading frictions. The left-edge rule understates growth. α corrects it without changing the prices.
Alex: In choppy markets, the right α means more risky assets. How does it work for one risky asset?
Sam: Wealth grows from safe bonds and one wiggly stock. The goal is steady long-term growth of your total pot. For log utility, put a fixed share in the stock: its edge over the bond, divided by its bumpiness squared.
Alex: Noise tweaks that edge?
Sam: Yes. Standard rule: share is (μ - r)/σ², with μ as stock growth, r bond rate, σ² bumpiness. Midpoint—α half—adds half to the share directly. The noise looks like hidden growth.
Alex: Does that work for multiple stocks?
Sam: Yes. Each stock's growth shifts by α times its bumpiness. Optimal shares tilt toward riskier ones—higher α means more overall risk.
Alex: The math sorts how much each gets. But what if bumpiness changes over time?
Sam: Bumpiness can rise with market fear, from a factor like X that wiggles partly in sync with the asset, via correlation ρ. The averaging adds a term: α times ρ times how bumpiness changes with X, times X's bumpiness. Optimal share gains from that, divided by current bumpiness squared.
Alex: If ρ is positive, fear boosts perceived growth under higher α?
Sam: Yes—portfolios get bolder right when volatility spikes. It adjusts with the market mood.
Alex: In models where volatility wiggles with returns, α corrects based on that link.
Sam: Right. No link means no effect. The paper derives this cleanly.
Alex: The shift is always from averaging style—fixed for steady bumps, changing with correlated factors. Traders might pick α for high-frequency noise.
Sam: Yes. Math choices reshape strategies via perceived growth. The paper notes limits: log utility only, needs admissible strategies, α calibration from data.
Alex: It shows how noise handling shapes portfolios—a meaningful way to rethink noisy data.
Sam: That's the point. It refines risk reads without breaking core ideas.
Alex: Well put, Sam. Thanks for joining me on ResearchPod.