Bertram's Pairs Trading Strategy with Bounded Risk

TL;DR

This paper develops a convex optimization framework for Bertram's pairs trading strategy with bounded risk, addressing parameter uncertainty and demonstrating efficient solvability despite non-convexity.

Contribution

It generalizes Bertram's model to include risk constraints, analyzes the impact of parameter estimation errors, and shows the problem remains efficiently solvable.

Findings

Risk-bounded trading strategies can be optimized via convex methods.

Parameter imprecision leads to quantifiable loss in strategy performance.

The optimization problem remains solvable despite non-convexity introduced by risk constraints.

Abstract

Finding Bertram's optimal trading strategy for a pair of cointegrated assets following the Ornstein--Uhlenbeck price difference process can be formulated as an unconstrained convex optimization problem for maximization of expected profit per unit of time. This model is generalized to the form where the riskiness of profit, measured by its per-time-unit volatility, is controlled (e.g. in case of existence of limits on riskiness of trading strategies imposed by regulatory bodies). The resulting optimization problem need not be convex. In spite of this undesirable fact, it is demonstrated that the problem is still efficiently solvable. In addition, the problem that parameters of the price difference process are never known exactly and are imprecisely estimated from an observed finite sample is investigated (recalling that this problem is critical for practice). It is shown how the imprecision affects the optimal trading strategy by quantification of the loss caused by the imprecise estimate compared to a theoretical trader knowing the parameters exactly. The main results focus on the geometric and optimization-theoretic viewpoint of the risk-bounded trading strategy and the imprecision resulting from the statistical estimates.