Risk-Bounded Bertram Pairs Trading with Parameter Uncertainty

Summary

This paper studies Bertram’s optimal strategy for a pair of cointegrated assets whose price difference follows an Ornstein–Uhlenbeck process. The baseline objective is to maximize expected profit per unit time. The authors generalize the setup by constraining the volatility of profit per unit time, representing a limit on strategy risk such as one imposed by regulation.

The risk constraint can make the optimization problem nonconvex, but the paper shows that it remains efficiently solvable. It also addresses the practical problem that the price process parameters must be estimated from a finite sample. The analysis quantifies how estimation imprecision affects the optimal strategy and the resulting loss relative to a trader who knows the parameters exactly. The document focuses on optimization and statistical uncertainty; it provides no numerical performance results or empirical market validation in the supplied description.

Key ideas

• The strategy models the spread between cointegrated assets as an Ornstein–Uhlenbeck process.
• The basic objective maximizes expected profit per unit time.
• A risk-bounded version constrains profit volatility and can yield a nonconvex optimization problem.
• The constrained problem is shown to remain efficiently solvable.
• Finite-sample parameter uncertainty can change the optimal strategy and reduce profit relative to perfect parameter knowledge.

Tags

# Bertram's Pairs Trading Strategy with Bounded Risk


# Bertram's Pairs Trading Strategy with Bounded Risk









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.

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