Pairs Trading Under Drift Uncertainty and Risk Penalization

Abstract

Abstract. In this work, we study a dynamic portfolio optimization problem related to pairs trading, which is an investment strategy that matches a long position in one security with a short position in another security with similar characteristics. The relationship between pairs, called a spread, is modeled by a Gaussian meanreverting process whose drift rate is modulated by an unobservable continuous-time, finite-state Markov chain. Using the classical stochastic filtering theory, we reduce this problem with partial information to the one with full information and solve it for the logarithmic utility function, where the terminal wealth is penalized by the riskiness of the portfolio according to the realized volatility of the wealth process. We characterize optimal dollar-neutral strategies as well as optimal value functions under full and partial information and show that the certainty equivalence principle holds for the optimal portfolio strategy. Finally, we provide a numerical analysis for a toy example with a two-state Markov chain.

Optimal Converge Trading with Unobservable Pricing Errors

“…Let W h be the value of a portfolio h = (h (m) , h (1) , h (2) ), where quantities h (m) t , h (1) t and h (2) t denote fractions of the wealth invested at any time t ∈ [0, T ] in the market index S (m) and in the co-integrated assets with prices S (1) and S (2) , respectively. Consequently the percentage of wealth invested in the riskless asset is 1 − h (m) − h (1) − h (2) . We introduce now the suitable set of strategies.…”

“…, related to the co-integration between S (1) and S (2) , or equivalently, to pricing errors. The denominator, on the other hand, is akin to the idiosyncratic risk components, b 1 , b 2 and σ.…”

“…S (1) ). The role of ̺ 1 is actually to scale the contribution of pricing error and the independent idiosyncratic variance of S (2) in h (1) * . Naturally, ̺ 2 has the analogous interpretation.…”

“…Definition 4.1. An F-admissible portfolio strategy is a self-financing and F-predictable strategy h = (h (m) , h (1) , h (2) ) that satisfies integrability condition (3.1), and A F is the set of all F-admissible strategies.…”

“…Lee and Papanicolaou [19] solve the optimal pairs trading problem within a power utility setting, where the drift uncertainty is modeled by a continuous Gaussian mean-reverting process and necessitates Kalman filtering to extract the unobservable state process. Altay et al [2] extend the pairs trading model of Mudchanatongsuk et al [23] by incorporating regime switching under partial information and risk penalization. However, classical portfolio selection problems, which do not cover portfolios involving co-integrated assets, in a full or partial information and/or Markov regime-switching framework can be found, for example, in Zhou and Yin [28], Bäuerle and Rieder [6], and Sotomayor and Cadenillas [24] for the full information case with Markov regime switching or Bäuerle and Rieder [7], Björk et al [8] and Frey et al [15] for the partial information case.…”