Machine Learning for Financial Engineering

The following sections are included:

• Preface

• Contents

The growth-optimal portfolio (GOP) is a portfolio which has a maximal expected growth rate over any time horizon. As a consequence, this portfolio is certain to outperform any other significantly different strategy as the time horizon increases. This property, in particular, has fascinated many researchers in finance and mathematics and has created a huge and exciting literature on growth-optimal investment. This chapter attempts to provide a comprehensive survey of the literature and applications of the GOP. In particular, the heated debate of whether the GOP has a special place among portfolios in the asset allocation decision is reviewed, as this still seems to be an area where some misconceptions exist. The survey also provides an extensive review of the recent use of the GOP as a pricing tool, in, for example, the so-called “benchmark approach”. This approach builds on the numéraire property of the GOP, that is, the fact that any other asset denominated in units of the GOP becomes a supermartingale.

This chapter provides a survey of discrete-time, multi-period, sequential investment strategies for financial markets. With memoryless assumption about the underlying process generating the asset prices, the best rebalancing is the log-optimal portfolio, which achieves the maximum asymptotic average growth rate. We show some examples (Kelly game, horse racing, St. Petersburg game) illustrating the surprising possibilities for rebalancing. Semi-log-optimal portfolio selection, as a small computational-complexity alternative of the log-optimal portfolio selection, is studied, both theoretically and empirically. For generalized dynamic portfolio selection, when asset prices are generated by a stationary and ergodic process, universally-consistent empirical methods are demonstrated. The empirical performance of the methods is illustrated for NYSE data.

Discrete-time, growth-optimal investment in stock markets with proportional transaction costs is considered. The market process is a sequence of daily relative prices (called returns) and is modeled by a first-order Markov process. Assuming that the distribution of the market process is known, we show sequential investment strategies such that, in the long run, the growth rate on trajectories achieves the maximum, with probability 1. Investments with consumption and with fixed transaction cost where the cost depends on the number of the shares involved in the transaction are also analyzed.

The growth-optimal strategy on non-leveraged, long-only memoryless markets is the best constantly-rebalanced portfolio (BCRP), also called the log-optimal strategy. Optimality conditions are derived for frameworks on leverage and short selling, and generalizing the BCRP by establishing no-ruin conditions. Moreover, the strategy and its asymptotic growth rate are investigated under a memoryless assumption, both from theoretical and empirical points of view. The empirical performance of the methods was tested for NYSE data, demonstrating spectacular gains for leveraged portfolios and showing the unimportance of short selling in the growth-rate sense, both in the case of BCRP and dynamic portfolios.

We present simple procedures for the prediction of a real-valued time series with side information. For squared loss (regression problem), we survey the basic principles of universally-consistent estimates. The prediction algorithms are based on a combination of several simple predictors. We show that, if the sequence is a realization of a stationary and ergodic random process, then the average of squared errors converges, almost surely, to that of the optimum, given by the Bayes predictor. We offer an analog result for the prediction of stationary Gaussian processes.

In this chapter, we study the empirical-pricing of American options. The pricing of American options is an optimal stopping problem, which can be derived from a backward recursion such that, in each step of the recursion, one needs conditional expectations. For empirical-pricing, [Longstaff and Schwartz (2001)] suggested replacing the conditional expectations by regression function estimates. We survey the current literature and the main techniques of nonparametric regression estimates, and derive new empirical-pricing algorithms.

The following sections are included:

• Index