[Paper Review] Stock Market Trading Via Stochastic Network Optimization

Michael J. Neely|ArXiv.org|Sep 22, 2009

Stochastic processes and financial applications15 references4 citations

TL;DR

This paper proposes a stochastic network optimization-based trading policy for dynamic stock market trading under constraints on maximum shares held and per-period trades. Using Lyapunov optimization, it achieves time-average profit arbitrarily close to the optimal offline policy, with tradeoffs in convergence speed and maximum inventory levels, even under non-ergodic price processes.

ABSTRACT

We consider the problem of dynamic buying and selling of shares from a collection of $N$ stocks with random price fluctuations. To limit investment risk, we place an upper bound on the total number of shares kept at any time. Assuming that prices evolve according to an ergodic process with a mild decaying memory property, and assuming constraints on the total number of shares that can be bought and sold at any time, we develop a trading policy that comes arbitrarily close to achieving the profit of an ideal policy that has perfect knowledge of future events. Proximity to the optimal profit comes with a corresponding tradeoff in the maximum required stock level and in the timescales associated with convergence. We then consider arbitrary (possibly non-ergodic) price processes, and show that the same algorithm comes close to the profit of a frame based policy that can look a fixed number of slots into the future. Our analysis uses techniques of Lyapunov Optimization that we originally developed for stochastic network optimization problems.

Motivation & Objective

• To design a universal, adaptive trading policy that maximizes long-term time-average profit in a stochastic stock market with random price fluctuations.
• To limit investment risk by constraining the maximum number of shares held and the number of shares traded per period.
• To achieve performance arbitrarily close to an ideal policy with perfect future price knowledge, despite lacking such knowledge.
• To ensure robustness under non-ergodic price processes by using frame-based performance bounds.
• To provide a provable tradeoff between convergence speed, maximum inventory levels, and profit optimality.

Proposed method

• Applies Lyapunov optimization techniques originally developed for stochastic network control to stock trading problems.
• Uses a max-weight functional that incorporates current prices, transaction costs, and stock queue levels to determine optimal buy/sell actions.
• Imposes constraints: $\mu_n^{\text{max}}$ shares per stock per period and $Q_n^{\text{max}}$ maximum shares held to limit risk.
• Derives a dynamic trading algorithm that adapts to real-time price changes without requiring knowledge of future prices or underlying distributions.
• Establishes performance bounds via convergence analysis of time-averaged profit relative to an ideal offline policy.
• Uses a frame-based analysis to bound performance under arbitrary (possibly non-ergodic) price processes.

Experimental results

Research questions

• RQ1 Can a universal trading policy achieve near-optimal profit without prior knowledge of future price distributions?
• RQ2 What is the tradeoff between maximum inventory levels ($Q_n^{\text{max}}$) and convergence speed to optimal profit?
• RQ3 How does the policy perform under non-ergodic price processes where statistical stationarity does not hold?
• RQ4 Can the algorithm maintain performance close to an ideal offline policy that knows future prices?
• RQ5 What is the impact of per-period trade limits ($\mu_n^{\text{max}}$) on long-term profit and risk exposure?

Key findings

• The proposed algorithm achieves a time-average profit arbitrarily close to the optimal offline policy, with the closeness determined by the choice of $Q_n^{\text{max}}$ and $\mu_n^{\text{max}}$.
• The convergence time to near-optimality scales inversely with the magnitude of $Q_n^{\text{max}}$, meaning smaller inventory limits lead to faster convergence.
• Under ergodic price processes with mild memory decay, the algorithm ensures that the time-average profit converges to within $\epsilon$ of the optimal value for any $\epsilon > 0$.
• For arbitrary (possibly non-ergodic) price processes, the algorithm achieves performance within a constant gap of a frame-based policy that can look ahead a fixed number of time slots.
• The algorithm is robust to non-stationary and non-ergodic price dynamics, maintaining performance guarantees without requiring knowledge of underlying probability distributions.
• The long-term wealth growth is linear due to the constraints, but the algorithm enables consistent short-term gains through bounded inventory and trade levels.