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On the Effect of Alpha Decay and Transaction Costs on the Multi-period Optimal Trading Strategy

Chutian Ma, Paul Smith

Abstract

We consider the multi-period portfolio optimization problem with a single asset that can be held long or short. Due to the presence of transaction costs, maximizing the immediate reward at each period may prove detrimental, as frequent trading results in consistent negative cash outflows. To simulate alpha decay, we consider a case where not only the present value of a signal, but also past values, have predictive power. We formulate the problem as an infinite horizon Markov Decision Process and seek to characterize the optimal policy that realizes the maximum average expected reward. We propose a variant of the standard value iteration algorithm for computing the optimal policy. Establishing convergence in our setting is nontrivial, and we provide a rigorous proof. Addtionally, we compute a first-order approximation and asymptotics of the optimal policy with small transaction costs.

On the Effect of Alpha Decay and Transaction Costs on the Multi-period Optimal Trading Strategy

Abstract

We consider the multi-period portfolio optimization problem with a single asset that can be held long or short. Due to the presence of transaction costs, maximizing the immediate reward at each period may prove detrimental, as frequent trading results in consistent negative cash outflows. To simulate alpha decay, we consider a case where not only the present value of a signal, but also past values, have predictive power. We formulate the problem as an infinite horizon Markov Decision Process and seek to characterize the optimal policy that realizes the maximum average expected reward. We propose a variant of the standard value iteration algorithm for computing the optimal policy. Establishing convergence in our setting is nontrivial, and we provide a rigorous proof. Addtionally, we compute a first-order approximation and asymptotics of the optimal policy with small transaction costs.

Paper Structure

This paper contains 13 sections, 5 theorems, 74 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Multi-period portfolio selection
  3. Our model
  4. Goal
  5. Outline
  6. Preliminaries
  7. Markov Decision Process
  8. Dynamic programming solution
  9. Approximate optimal policy with small transaction costs
  10. First-order expansion
  11. Numerical simulation
  12. Conclusion
  13. Convergence analysis

Key Result

Theorem 2.1

Let $s$ denote the current state, and $g$, $T$, and $P$ be the reward function, system transition function, and probability distribution of the random disturbance, respectively. If there exists a constant $\lambda$ and a measurable function $h : S \rightarrow \use@mathgroup \M@U \symAMSb{R}$ that sa then the $\pi$ that obtains the maximum is average-cost optimal and $\lambda$ equals the maximum av

Figures (5)

  • Figure 1: Optimal policy: no-trade zone
  • Figure 2: $\rho_0 = 0.3, \rho_1 = 0.8$
  • Figure 3: $\rho_0 = 0.8, \rho_1 = 0.3$
  • Figure 4: probability of switching position with G and $G_{naive}$ respectively
  • Figure 5: difference in the probability

Theorems & Definitions (14)

  • Definition 2.1: State space
  • Definition 2.2: Control space
  • Definition 2.3: System equation
  • Definition 2.4: Policy
  • Definition 2.5: Reward function
  • Theorem 2.1: schal1993average, feinberg2012average
  • Theorem 2.2
  • Remark
  • Lemma A.1
  • Proposition A.1
  • ...and 4 more