Dynamic portfolio optimization with liquidity cost and market impact: a simulation-and-regression approach

We present a simulation-and-regression method for solving dynamic portfolio optimization problems in the presence of general transaction costs, liquidity costs and market impact. This method extends the classical least squares Monte Carlo algorithm to incorporate switching costs, corresponding to transaction costs and transient liquidity costs, as well as multiple endogenous state variables, namely the portfolio value and the asset prices subject to permanent market impact. To handle endogenous state variables, we adapt a control randomization approach to portfolio optimization problems and further improve the numerical accuracy of this technique for the case of discrete controls. We validate our modified numerical method by solving a realistic cash-and-stock portfolio with a power-law liquidity model. We identify the certainty equivalent losses associated with ignoring liquidity effects, and illustrate how our dynamic optimization method protects the investor's capital under illiquid market conditions. Lastly, we analyze, under different liquidity conditions, the sensitivities of certainty equivalent returns and optimal allocations with respect to trading volume, stock price volatility, initial investment amount, risk aversion level and investment horizon.     

Portfolio revision under mean-variance and mean-CVaR with transaction costs

The portfolio revision process usually begins with a portfolio of assets rather than cash. As a result, some assets must be liquidated to permit investment in other assets, incurring transaction costs that should be directly integrated into the portfolio optimization problem. This paper discusses and analyzes the impact of transaction costs on the optimal portfolio under mean-variance and mean-conditional value-at-risk strategies. In addition, we present some analytical solutions and empirical evidence for some special situations to understand the impact of transaction costs on the portfolio revision process.     

International equity portfolio allocations and transaction costs

In spite of the critical role of transaction cost, there are not many papers that explicitly examine its influence on international equity portfolio allocation decisions. Using bilateral cross-country equity portfolio investment data and three direct measures of transaction costs for 36 countries, we provide evidence that markets where transaction costs are lower attract greater equity portfolio investments. The results imply that future research on international equity portfolio diversification cannot afford to ignore the role of transaction costs, and policy makers, especially in emerging markets, will have to reduce transaction costs to attract higher levels of foreign equity portfolio investments.     

Deep hedging

We present a framework for hedging a portfolio of derivatives in the presence of market frictions such as transaction costs, liquidity constraints or risk limits using modern deep reinforcement machine learning methods. We discuss how standard reinforcement learning methods can be applied to non-linear reward structures, i.e. in our case convex risk measures. As a general contribution to the use of deep learning for stochastic processes, we also show in Section 4 that the set of constrained trading strategies used by our algorithm is large enough to ε-approximate any optimal solution. Our algorithm can be implemented efficiently even in high-dimensional situations using modern machine learning tools. Its structure does not depend on specific market dynamics, and generalizes across hedging instruments including the use of liquid derivatives. Its computational performance is largely invariant in the size of the portfolio as it depends mainly on the number of hedging instruments available. We illustrate our approach by an experiment on the S&P500 index and by showing the effect on hedging under transaction costs in a synthetic market driven by the Heston model, where we outperform the standard ‘complete-market’ solution.     

Multivariate utility maximization with proportional transaction costs

We present an optimal investment theorem for a currency exchange model with random and possibly discontinuous proportional transaction costs. The investor’s preferences are represented by a multivariate utility function, allowing for simultaneous consumption of any prescribed selection of the currencies at a given terminal date. We prove the existence of an optimal portfolio process under the assumption of asymptotic satiability of the value function. Sufficient conditions for this include reasonable asymptotic elasticity of the utility function, or a growth condition on its dual function. We show that the portfolio optimization problem can be reformulated in terms of maximization of a terminal liquidation utility function, and that both problems have a common optimizer.     

Portfolio optimization in discrete time with proportional transaction costs under stochastic volatility

This paper is devoted to evaluating the optimal self-financing strategy and the optimal trading frequency for a portfolio with a risky asset and a risk-free asset. The objective is to maximize the expected future utility of the terminal wealth in a stochastic volatility setting, when transaction costs are incurred at each discrete trading time. A HARA utility function is used, allowing a simple approximation of the optimization problem, which is implementable forward in time. For each of various transaction cost rates, we find the optimal trading frequency, i.e. the one that attains the maximum of the expected utility at time zero. We study the relation between transaction cost rate and optimal trading frequency. The numerical method used is based on a stochastic volatility particle filtering algorithm, combined with a Monte-Carlo method. The filtering algorithm updates the estimate of the volatility distribution forward in time, as new stock observations arrive; these updates are used at each of these discrete times to compute the new portfolio allocation.     

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

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.     

Investment strategies in the long run with proportional transaction costs and a HARA utility function

We consider an agent who invests in a stock and a money market in order to maximize the asymptotic behaviour of expected utility of the portfolio market price in the presence of proportional transaction costs. The assumption that the portfolio market price is a geometric Brownian motion and the restriction to a utility function with hyperbolic absolute risk aversion (HARA) enable us to evaluate interval investment strategies. It is shown that the optimal interval strategy is also optimal among a wide family of strategies and that it is optimal also in a time changed model in the case of logarithmic utility.     

A jackknife-type estimator for portfolio revision

This article proposes a novel approach to portfolio revision. The current literature on portfolio optimization uses a somewhat naïve approach, where portfolio weights are always completely revised after a predefined fixed period. However, one shortcoming of this procedure is that it ignores parameter uncertainty in the estimated portfolio weights, as well as the biasedness of the in-sample portfolio mean and variance as estimates of the expected portfolio return and out-of-sample variance. To rectify this problem, we propose a jackknife procedure to determine the optimal revision intensity, i.e. the percent of wealth that should be shifted to the new, in-sample optimal portfolio. We find that our approach leads to highly stable portfolio allocations over time, and can significantly reduce the turnover of several well established portfolio strategies. Moreover, the observed turnover reductions lead to statistically and economically significant performance gains in the presence of transaction costs.     

A comparison of static and dynamic portfolio policies

Garleanu and Pedersen (2013) show that the optimal static portfolio policy in light of quadratic transaction costs is a weighted average of the existing portfolio and the target portfolio. In this paper, we demonstrate the importance of the robust target portfolio in the static portfolio policy that considers quadratic transaction costs. By using both empirical and simulated data, we find no evidence that the optimal dynamic portfolio policy proposed by Garleanu and Pedersen (2013) is superior to the static portfolio policy that trades towards the robust target portfolio. The robust target portfolio is achieved by either introducing time-varying covariances or restricting portfolio weights. Furthermore, the static portfolio with time-varying covariances and the short sale-constrained static portfolio are both very efficient in reducing portfolio turnover. The good performance of the static portfolio policy is robust to parameter uncertainty and trading parameters.     

Optimal hedging in an extended binomial market under transaction costs

We develop an approach to optimal hedging of a contingent claim under proportional transaction costs in a discrete time financial market model which extends the binomial market model with transaction costs. Our model relaxes the binomial assumption on the stock price ratios to the case where the stock price ratio distribution has bounded support. Non-self-financing hedging strategies are studied to construct an optimal hedge for an investor who takes a short position in a European contingent claim settled by delivery. We develop the theoretical basis for our optimal hedging approach, extending results obtained in our previous work. Specifically, we derive a no-arbitrage option price interval and establish properties of the non-self-financing strategies and their residuals. Based on the theoretical foundation, we develop a computational algorithm for optimizing an investor relevant criterion over the set of admissible non-self-financing hedging strategies. We demonstrate the applicability of our approach using both simulated data and real market data.     

In search of a theory of debt management

The complete market approach to government debt management argues that a portfolio of non-contingent bonds at different maturities should be chosen so that fluctuations in market value offset changes in expected future deficits. However, this approach recommends huge fluctuations in positions, enormous changes in portfolios for minor changes in maturities and no presumption it is always optimal to issue long and invest short term in a wide array of model specifications. These extreme, volatile and unstable features are undesirable for two reasons. Firstly fragility of portfolios to small changes in assumptions means that it is often better to follow a balanced budget rather than issue the optimal debt portfolio under some possibly misspecified model. Secondly for even miniscule transaction costs, governments prefer a balanced budget rather than the large positions complete markets recommends. The complete market recommendations conflict with a number of features we believe are integral to bond market incompleteness, e.g. transaction costs, liquidity effects, robustness, etc. and which need to be explicitly incorporated into the portfolio problem.     

Constant Rebalanced Portfolio Optimization Under Nonlinear Transaction Costs

We study the constant rebalancing strategy for multi-period portfolio optimization via conditional value-at-risk (CVaR) when there are nonlinear transaction costs. This problem is difficult to solve because of its nonconvexity. The nonlinear transaction costs and CVaR constraints make things worse; state-of-the-art nonlinear programming (NLP) solvers have trouble in reaching even locally optimal solutions. As a practical solution, we develop a local search algorithm in which linear approximation problems and nonlinear equations are iteratively solved. Computational results are presented, showing that the algorithm attains a good solution in a practical time. It is better than the revised version of an existing global optimization. We also assess the performance of the constant rebalancing strategy in comparison with the buy-and-hold strategy.     

A new computational tool for analysing dynamic hedging under transaction costs

This paper highlights a framework for analysing dynamic hedging strategies under transaction costs. First, self-financing portfolio dynamics under transaction costs are modelled as being portfolio affine. An algorithm for computing the moments of the hedging error on a lattice under portfolio affine dynamics is then presented. In a number of circumstances, this provides an efficient approach to analysing the performance of hedging strategies under transaction costs through moments. As an example, this approach is applied to the hedging of a European call option with a Black–Scholes delta hedge and Leland's adjustment for transaction costs. Results are presented that demonstrate the range of analysis possible within the presented framework.     

A model for tax advantages of portfolios with many assets

Taxable portfolios present challenges for optimization models with even a limited number of assets. Holding many assets, however, has a distinct tax advantage over holding few assets. In this paper, we develop a model that takes an extreme view of a portfolio as a continuum of assets to gain the broadest possible advantage from holding many assets. We find the optimal strategy for trading in this portfolio in the absence of transaction costs and develop bounding approximations on the optimal value. We compare the results in a simulation study to a portfolio consisting only of a market index and show that the multi-asset portfolio’s tax advantage can lead either to significant consumption or bequest increases.     

Dynamic Investment Strategy with Factor Models Under Regime Switches

A model for dynamic investment strategy is developed where assets’ returns are represented by multiple factors. In a mean–variance framework with factor models under regime switches, we derive a semi-analytic solution for the optimal portfolio with transaction costs. Due to the existence of transaction costs, the optimal portfolio is characterized as a linear combination of current and target portfolios, the latter of which maximizes the value function in the current regime. For some special cases of interest, we also derive simplified analytical solutions. To see the effect of regime switches, the proposed model is applied to US equity market in which small minus big and high minus low are employed as factors. Investment strategy based on our model demonstrates empirically that the regime switching models exhibit superior performance over the single regime model for such performance measures as realized utility and Sharpe ratio which are of particular interest in practice. Taking a close look at the time series of portfolio returns, the result shows the usefulness of the regime switching model as investors flexibly optimize asset allocations depending on the state of the market.     

Fama–French factor timing: The long-only integrated approach

There is ample evidence that factor momentum exists in the standard long–short mixed approach to factor investing. However, the excess returns are put under scrutiny due to the high implementation costs. We present a novel real-life approach that relies on the long-only integrated approach to factor investing. Instead of exploiting the potential momentum in factor portfolios, our strategy builds on the momentum of the optimal factor score weights in the integrated approach, which allows us to additionally profit from the serial dependence in the factors' interaction effects. One limitation of short-term timing strategies is their high turnover. By including the information of the covariance matrix and minimising the strategy's risk to the market portfolio, we can substantially reduce turnover. The resulting timing alpha remains significant even after transaction costs in a robust statistical test framework across the major stock markets.     

Momentum profits in the Australian equity market: A matched firm approach

This paper examines momentum trading strategies within the Australian equity market over the period 1990 to 2007, inclusive. We analyse excess returns employing both Jegadeesh and Titman's (Jegadeesh, N., Titman, S., 1993. “Returns to buying winners and selling losers: implications for stock market efficiency”. The Journal of Finance, 48:65–91) zero cost investment portfolio approach and a matched control firm approach. We also allow for short sale restrictions, liquidity constraints and transaction costs in the form of bid-ask spreads. Testing reveals that both the Jegadeesh and Titman (Jegadeesh, N., and Titman, S. (1993). “Returns to buying winners and selling losers: implications for stock market efficiency”. The Journal of Finance, 48:65–91.) zero cost investment portfolio approach and the matched control firm approach yield excess profits. While the implementation of short sale restraints increases momentum profitability, the subsequent inclusion of bid-ask spreads results in a reduction in these gains. Further, we find that executing a momentum strategy in Australia results in statistically significant dollar profits.