When markets move, they do so with both sudden shocks and slow drifts. Yet for years, much of optimal trading theory has treated volatility as if it were static—a constant backdrop rather than a dynamic participant in the game. The recent paper by Chan, Sircar, and Zimbidis decisively challenges that assumption by embedding multiscale stochastic volatility into a classical dynamic trading model. The result? A more nuanced, volatility-aware framework that adapts trading speed and target positions based on the fast and slow undulations of risk.
From Smooth Sailing to Choppy Waters
The starting point is the elegant framework from Garleanu and Pedersen (2013), which showed how to optimally balance expected returns, risk exposure, and transaction costs in the presence of both instantaneous frictions (e.g., bid-ask spreads) and persistent price impacts (the lasting effect of trades on prices). But that model assumes constant volatility. In today’s markets, that’s akin to piloting a ship without watching the waves.
Chan et al. extend the model in two major ways:
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Volatility becomes stochastic, driven by two latent factors:
- A fast-scale factor (e.g., mean-reverting daily volatility noise).
- A slow-scale factor (e.g., macro regime shifts).
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They apply singular and regular perturbation techniques to solve the resulting nonlinear HJB equation, yielding second-order analytic corrections to both the optimal trading rate and the aim portfolio.
What Changes When Volatility Moves?
By treating volatility as a dynamic input rather than a static nuisance, the authors uncover intuitive yet powerful behavior shifts. Here’s how volatility modifies trading decisions:
| Volatility Regime | Key Correction | Trading Implication |
|---|---|---|
| Fast-scale (\u03b5) | $u^* = u^{(0)} - \frac{q}{K}\varphi(y)$ | De-leverage when $\sigma(y) > \bar{\sigma}$ |
| Slow-scale (\u03b4) | $u^* = u^{(0)} + \frac{1}{K} \sqrt{\delta} (B_q + \lambda B_l)$ | Trade slower if returns and volatility are positively correlated |
These corrections are interpretable. For instance, in fast-volatility regimes (like sudden news-driven turbulence), the correction dampens trading activity when current volatility is high. For slow regimes, when volatility and returns move together (e.g., bull runs accompanied by risk), the trader rationally pulls back.
Real Gains: Accuracy Without Complexity
Monte Carlo simulations in the paper show that these corrections are more than just theoretical. They lead to meaningful improvements:
- Fast volatility corrections improved PnL by 53bps with lower variance.
- Slow volatility corrections had similarly strong effects, even when $\delta$ was much smaller than $\varepsilon$.
Moreover, the authors introduce a first-order approximation for small price impact, making their solution computationally practical even in live systems. As they demonstrate, this approximation maintains accuracy while avoiding the need to solve a large nonlinear system at each time step.
Why It Matters for Algorithmic Traders
The trading strategies derived here don’t rely on black-box neural nets. Instead, they provide closed-form expressions for trading speed and target portfolios that adjust based on the observed or estimated volatility factors. This opens the door to hybrid architectures:
- Use machine learning to infer latent volatility states.
- Plug those into theory-driven controls that adapt execution in real-time.
Compared to pure deep reinforcement learning strategies, this approach preserves economic intuition while offering close-to-optimal performance—as confirmed by recent comparisons cited in the paper (Muhle-Karbe et al., 2024).
Future Horizons
Several questions remain ripe for exploration:
- What if price impact is nonlinear or stochastic? Extending the model to these richer frictions may close the realism gap further.
- How does partial information affect volatility inference? Integrating filtering theory or Bayesian learning (e.g., Cai & Yu, 2025) could make the strategy more robust to noise.
- Can we unify this with deep learning systems? A promising direction is to use LLMs or neural agents for return prediction while preserving this paper’s execution layer.
Final Thought
In algorithmic trading, speed is essential—but wisdom is better. This paper offers a framework where trading speed adapts to market uncertainty, and where both the strategy and the logic behind it are transparent. For portfolio managers who want to navigate both the sudden shocks and the slow tides, Chan et al. provide a second-order compass.
Cognaptus: Automate the Present, Incubate the Future.