Optimal mean-reversion strategy in the presence of bid-ask spread and delays in capital allocations

Abstract

A portfolio optimization problem for an investor who trades T-bills and a mean-reverting stock in the presence of proportional and convex transaction costs is considered. The proportional transaction cost represents a bid-ask spread, while the convex transaction cost is used to model delays in capital allocations. I utilize the historical bid-ask spread in US stock market and assume that the stock reverts on yearly basis, while an investor follows monthly changes in the stock price. It is found that proportional transaction cost has a relatively weak effect on the expected return and the Sharpe ratio of the investor's portfolio. Meantime, the presence of delays in capital allocations has a dramatic impact on the expected return and the Sharpe ratio of the investor's portfolio. I also find the robust optimal strategy in the presence of model uncertainty and show that the latter increases the effective risk aversion of the investor and makes her view the stock as more risky.

Keywords:

• Portfolio choice
• Mean-reverting security
• Transaction costs

JEL Classification:

• G11

Acknowledgments

The author is thankful to anonymous referees for the helpful remarks.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

† See, for example, Andrew (2008).

† For example, see Constantinides (1986), Davis and Norman (1990), Duffie and Sun (1990), Dumas and Luciano (1991), Shreve and Soner (1994), Vayanos (1998), Liu (2002), Liu (2004), and Jang et al. (2007), and many others. See Guasoni and Muhle-Karbe (2013), for a more detailed literature review.

‡ Transaction cost, either fixed, linear, or convex, has two effects on trading strategies: it causes an investor to skip on some investment opportunities or invest with delays and it makes an investor to underallocate if she chooses to undertake an investment opportunity. In this study, I emphasize the first effect even though the both are important for trading strategies. These effects are controlled by a trading volume. Moreover, two types of transaction cost are used to isolate contributions from the bid-ask spread and delays in capital allocations.

§ This bid-ask spread is close to its historical value in US stock market. See, for example, Jones (2002).

¶ See also Hansen and Sargent (1995).

† See, for example, Zhang and Zhang (2008), Kong and Zhang (2010), Zervos et al. (2013), Leung and Li (2015), Leung et al. (2015).

‡ Formally, , where is the number of shares held by an investor at time t and .

† For empirical detection and estimation of mean-reverting portfolios see, for example, Balvers et al. (2000), D'Aspremont (2011), and Akarim and Sevim (2013).

‡ See, for example, Bibbona et al. (2008).

§ I conjecture that parabolic partial differential equations for function g appearing in Propositions 1 to 4 have unique solutions. Proving the existence and uniqueness of these solutions lies beyond this paper. It is also assumed that the numerical solutions of these equations converge to the actual solutions in the limit

† A path of a process is simulated by using a finite-difference approximation , where , and is a set of independent standard normal random variables. is set to one day. The predictions of simulations are affected by the initial conditions only to a small extent due to a relatively long investment horizon.

† The chosen value for θ is justified by convenience. An alternative calibration for θ will not change the intuition behind the numerical results.

† A comparative statics with respect to parameter θ is also carried. The found results are fully consistent with the discussion of Example 6 and are not reported for the purpose of compactness.