Portfolio Optimization with Mean-reverting Assets ... · Jing Ye A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy

Portfolio Optimization with

Mean-reverting Assets: Combining

Theory with Deep Learning.

Jing Ye

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Operations Research and Financial Engineering

Adviser: Professor John. M. Mulvey

November 2018

c© Copyright by Jing Ye, 2018.

All rights reserved.

Abstract

We study the finite-horizon dynamic portfolio model involving mean reverting

(Ornstein-Uhlenbeck) assets in the presence of transaction costs. The goal is to

maximize the expected constant relative risk aversion (CRRA) utility of terminal

wealth. We investigate the problem in two parts. In the first part, we analyze the

problem without transaction costs for three types of portflios: we analytically solve

the problem for a single risky asset and a risk free asset, and then derive the solution

to a portfolio of OU assets and a portfolio of OU and geometric Brownian motion

(GBM) assets via a system of ordinary differential equations (ODEs). In the second

part, we extend the problem by adding proportioanl transaction cost. The portfolio

optimization problem becomes intractable both analytically and computationally

when there exist transaction costs and multiple assets. To address transaction costs,

we propose a novel numerical approach employing deep neural networks, building on

the previous ODE solution and given the standard no-trade zone policy rules. Our

method readily extends to high-dimensional portfolio problems wherein traditional

methods fail. Experiments with synthetic and market data show the numerical

benefits of the developed algorithms.

iii

Acknowledgements

I would like to express my dearly gratitude to my advisor, Professor John Mulvey.

His remarkable insights in finance, patient guidance and dedication to teaching have

shaped my understanding of research and influenced my personal values. This thesis

will not come to fruition without his support and guidance. I was very fortunate

to have an advisor who allows me the greatest degree of freedom to explore the

research topics that interested me most, while providing guidance and help when I

face challenges. My highest appreciation to Professor Mulvey’s help, support and

enlightenment is not to utter words, but to live by them.

I am also very grateful to the members of my dissertation committee: Professor

Amir Ali Ahmadi and Professor Mengdi Wang, for kindly being on my committee

and providing constructive comments. And I would also like to thank Professor

Woo Chang Kim for serving as a reader for this thesis, as well as providing valuable

feedbacks.

I would like to express my sincere gratitude to Ms. Margaret Holen. During

the time when I was a member of the summer research project led by Ms. Holen, I

was so influenced by her enthusiasm, optimism and dedication to work. Ever since

our first collaboration, she has generously shared her personal experiences with me,

constantly provided me invaluable advices both on academics and on career, and she

always encourages me to step out and strive for more as a woman scholar.

Graduate school would not have been the same without the great colleagues I

worked with and friends I made here. I would like to thank my colleagues Yifan

Sun, Han Hao and Kun Lu, with whom I have collaborated on many projects. And I

would like to thank all my fellow PhD students: Huanran Lu, Ziwei Zhu, Peiqi Wang,

Jiequn Han, Yuan Liu for their sincere friendship.

Last but not least, I would like to thank my parents and my families. I will never

be able to receive the world-class education from Princeton University without theiriv

support of my decision to come to the U.S. to pursue study. They also played critical

roles in supporting and inspiring my academic journey: they helped me go through

the most stressful times with encouragement and comfort; they shared with me the

joy and happiness when I succeeded. All of my achievements have been made possible

only due to the unconditional love and support of my families. They inspired me to

pursue my dreams, and to never give up. This thesis is dedicated to them.

v

To my parents Jianhe Ye and Ying Ye.

vi

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction 1

1.1 Dynamic Asset Allocation . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Deep Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Mean Reverting Assets . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Contribution and Outline . . . . . . . . . . . . . . . . . . . . . . . . 13

2 General Framework for Finite-horizon Portfolio Optimization 15

2.1 The Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Zero Transaction Cost . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Proportional Transaction Cost . . . . . . . . . . . . . . . . . . . . . . 19

3 Optimal Strategy Under Zero Transaction Cost 23

3.1 Optimal Strategy With One OU Asset . . . . . . . . . . . . . . . . . 23

3.2 Optimal Strategy With Multiple Correlated OU Assets . . . . . . . . 30

3.3 Optimal Strategy With Mixture Correlated Risky Assets . . . . . . . 36

3.4 Properties of Optimal Trading Strategy . . . . . . . . . . . . . . . . 47

3.5 Simulations of Optimal Trading Strategies . . . . . . . . . . . . . . . 52

vii

4 Optimal Strategies with Transaction Cost - Deep Neural Network

Approach 61

4.1 Learning The No-Trade Zone Using Deep Neural Network . . . . . . 61

4.2 Single Asset No-Trade Zone Parameterization . . . . . . . . . . . . . 63

4.3 One-asset Trading Strategy Parameterization . . . . . . . . . . . . . . 66

4.4 Training DNN for One Asset . . . . . . . . . . . . . . . . . . . . . . . 69

4.5 Parameterizing Multi-assets NT Zone and Trading Strategy . . . . . 72

4.6 Training DNN for Multiple Assets . . . . . . . . . . . . . . . . . . . . 73

4.7 Neural Network Structure and Training Procedure . . . . . . . . . . . 76

4.8 Relations Between No-Trade Zone and Transaction Cost . . . . . . . 78

4.9 Performance Based on Simulation . . . . . . . . . . . . . . . . . . . . 81

5 Numerical Experiments With Market Data 82

5.1 Creating Mean Reverting Asset . . . . . . . . . . . . . . . . . . . . . 83

5.2 Experiment with One Asset . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 Experiment with Multi-Asset Portfolios . . . . . . . . . . . . . . . . . 92

5.4 Computation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 Conclusion 95

7 Future Work 97

7.1 Refining and Extending the Current Model . . . . . . . . . . . . . . . 97

7.2 Applying the Core Concept to Reinforcement Learning. . . . . . . . . 100

Bibliography 102

A Technical Details: Proofs and Computations 111

A.1 Solution to single-asset HJB equation . . . . . . . . . . . . . . . . . . 111

A.2 Solution to multi-asset HJB equation . . . . . . . . . . . . . . . . . . 117

A.3 Proof of Optimal Strategy Properties . . . . . . . . . . . . . . . . . . 120

viii

A.4 Proof of Multi-Asset Environment Dynamics . . . . . . . . . . . . . . 125

B Parameter Estimation 127

B.1 Parameter Estimation of univariate mean-reverting process . . . . . . 127

B.2 Parameter Estimation of multivariate mean-reverting process . . . . . 129

C Summary of Data 134

C.1 Information of ETF and Stocks . . . . . . . . . . . . . . . . . . . . . 134

C.2 Estimated Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 137

ix

List of Tables

5.1 Number of parameters and computation time used for different number

of assets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

C.1 Information of ETF universe as of July 31st, 2017 . . . . . . . . . . . 135

C.2 Information of Stocks as of July 31st, 2017 . . . . . . . . . . . . . . . 136

C.3 Λ for Year 2014-2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

C.4 Σ for Year 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C.5 Σ for Year 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

C.6 Σ for Year 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.7 Σ for Year 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

x

List of Figures

1.1 VIX from year 2004 to 2016. . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 Sign and Horizon Dynamics of Inter-temporal Hedging Demand . . . 49

3.2 Signs of Myopic Demand and Inter-temporal Hedging Demand . . . . 51

3.3 Optimal Strategy Corresponding to One Sample Price Path. . . . . . 53

3.4 Sign and Horizon Dynamics of Inter-temporal Hedging Demand.

a) upper: fixed price level at 15, higher than −B(τ)/C(τ) and

−B′(τ)/C ′(τ). b) lower left, fixed price level at 15.3, lower than

−B(τ)/C(τ), cross −B′(τ)/C ′(τ). c) lower right, fixed price level at

15.45, cross −B(τ)/C(τ) and −B′(τ)/C ′(τ). . . . . . . . . . . . . . . 54

3.5 Average Wealth Growth and Average Utility Growth over 10,000 Monte

Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6 Terminal Wealth Distribution over 10,000 Monte Carlo Simulations . 56

3.7 Optimal Strategy Corresponding to one Sample Price Path. . . . . . 59

3.8 Average Wealth Path and Average Utility Path . . . . . . . . . . . . . 60

3.9 Average Wealth Path and Average Utility Path . . . . . . . . . . . . . 60

xi

4.1 A single layer feedforward neural network fθ(x). The input is a m-

dimensional vector (x1, ...xm). The parameters θ to be estimated are

w1...wm and b. The activation function is f which is usually taken as

a Sigmoid function or Rectified Linear Unit(ReLU). The output of the

network is f(b+∑wixi). . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 A typical NT zone and corresponding rebalance operations. The mean

reverting level is 15.4463 and transaction cost is 2.0%. i) From point

a to a’: point a is outside the NT zone, we rebalance it to point a’.

We convert the position from investing 50% of wealth in shorting asset

to investing 32% of wealth in longing the asset because the current

observed asset price is 14.73 which is way below the mean reverting

level 15.4463. ii) From b to b’: b is outside the NT zone, we rebalance

it to b’. We convert the position from investing 75% of wealth in

longing asset to investing 71% of wealth in shorting the asset because

the current observed asset price is 16.13 which is way above the mean

reverting level 15.4463. iii) At c: because point c is within NT zone,

no rebalance operation is needed. . . . . . . . . . . . . . . . . . . . . 67

4.3 Computational graph of the deep neural network. . . . . . . . . . . . 71

4.4 No trade zone at t=0.4 under different transaction cost. Larger trans-

action costs leads to wider NT zone. . . . . . . . . . . . . . . . . . . 78

4.5 Price level at which boundary hits the leverage ratio limit. a) Upper

left: Price level at which lower boundary hits +1 limit. b) Upper right:

Price level at which upper boundary hits -1. c) Lower left: Price level

at which lower boundary hits -1. d) Lower right: Price level at which

upper boundary hits +1. . . . . . . . . . . . . . . . . . . . . . . . . 80

xii

4.6 Mean utility over 10000 sample paths. The blue curve is mean utility

over time following the optimal strategy we derived under 0 transaction

cost. The orange curve is the mean utility overtime following the NT

zone policy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.1 Back test result for year 2014. The mean reverting pair is created using

listed stock JP Morgan Chase(JPM) and XFL ETF. a) Cumulative

wealth. b) Mean reverting level estimated using historical data and

asset price. c) Percentage invested in risky asset. . . . . . . . . . . . 86




asset price. c) Percentage invested in risky asset. . . . . . . . . . . . . 88









5.5 Cumulative wealth when backtest using 48 mean reverting assets from

2014 to 2017. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

xiii

Chapter 1

Introduction

1.1 Dynamic Asset Allocation

The portfolio optimization problem is an essential problem in mathematical finance

that has been studied ever since 1950s. Markowitz (1952, 1956, 1959) establishes

the theory of portfolio selection based on optimization of expected investment return

subject to expected portfolio return variance constraints. Following Markowitz’s

work, Sharpe (1964) and Lintner (1965) develop a single period equilibrium model,

Sharpe-Lintner capital asset pricing model (CAPM), which has a great impact on the

modern capital market theory, but was criticized for homogeneous expectations and

the single-period nature. Merton (1969, 1971) extends CAPM from a static (single-

period) model to a continuous time model with stochastic variation in investment

opportunities. He explicitly solves continuous-time portfolio optimization problem

where the investor can invest between stocks modeled as geometric Brownian Motions

(GBMs) and a money market account with a fixed risk-free rate to maximize the

expected utility of consumption and terminal wealth. Merton also shows that the

optimal strategy is a fix-mix strategy for the constant relative risk aversion (CRRA)

utility when there is no transaction cost. In addition, Mossin (1968), Samuelson

1

(1969) and Hakanson (1970) solve analytically a similar problem in discrete settings.

There has been a significant literature extending Merton’s problem from differing

perspectives. One main stream is to incorporate stock return predictability into

portfolio optimization. Brennan et al. (1997) analyze the continuous time portfolio

problem of a long-lived investor who invest in bonds, stock and cash driven by three

predictor variables. Their approach is to numerically solve the partial differential

equation derived originally by Merton (1973). Campbell and Viceira (1999) are able

to find an analytical approximation to the more general problem of deriving both

optimal consumption and portfolio trading strategy for an infinite-horizon investor

with Epstein-Zin utility. Many researchers, in particular, model the risk premiums

as mean reverting (Ornstein-Uhlenbeck) processes. Kim and Omberg (1996) derive

an exact solution for finite-horizon investor under Hyperbolic absolute risk aversion

(HARA) utility by limiting the investor to maximizes utility over terminal wealth

only. Wachter (2002) solves, in closed form, the optimal portfolio choice problem for

an investor with utility over both consumption and terminal wealth under complete

market assumption. Liu (2004) extends the work to multiple risky assets and explic-

itly solve up to the solution of ODE systems when investor has a constant relative

risk aversion (CRRA) utility and the asset returns follow a quadratic process which

include both the affine process and the Ornstein-Uhlenbeck process as special cases.

Munk et al. (2004) consider not only stocks with mean-reverting excess return, but

also bond trading and inflation when solving the optimal portfolio choice problem.

Lv and Meister (2009) solve a problem to maximize growth rate, also known as the

Kelly Criterion, with multiple OU processes in a complete market.

Other researches incorporate regimes into return predictability. Bae et.al (2014)

employ the hidden Markov Model (HMM) to identify regimes in stock, bond, and

2

commodity markets, then construct a stochastic program to optimize portfolios un-

der the regime switching framework and use scenario generation to mathematically

formulate the optimization problem. Reusa and Mulvey (2016) improve currency

carry trade strategy by using trade return regimes identified by HMM to determine

when to open and close positions, and employing a mean-semivariance allocation

model to improve allocations when positions are opened. Mulvey and Liu (2016)

utilize a nonparametric machine learning algorithm, called trend filtering, to identify

regimes of the financial markets in a data-driven and model-free context. Mulvey et

al. (2016) also employ trend filtering, a machine learning technique, to distinguish

regimes possessing relatively homogeneous patterns, propose a scenario-based port-

folio model to address multiple regimes.

In the above mentioned studies, authors either solve or approximate the optimal

trading strategy by assuming no transaction cost. However, dynamic portfolio opti-

mization often requires frequent rebalancing and hence transaction costs are usually

not negligible. Such trading costs are caused by several factors such as the bid-offer

spread, execution commissions, market impact or tax, etc. Magill and Constantinides

(1976) initiated the research by incorporating transaction costs into Merton’s prob-

lem. They propose that the investors only trades in securities when the variation

in the underlying security prices forces his portfolio proportions outside a no-trade

zone. Davis and Norman (1990) were the first to provide a detailed formulation and

analysis, along with an algorithm and numerical computations of the optimal policy

for an infinite-horizon investment and consumption decision problem. Shreve and

Soner (1994) relax some assumptions in Davis and Norman’s work and conduct an

exhaustive and rigorous analysis of the optimal trading and strategies in an infinite

horizon. They prove existence, uniqueness and regularity of the value function. Liu

and Loewenstein (2002) studied for the first time the finite-horizon optimal trading

3

problem with single risky asset. The multi-asset portfolio optimization problem,

on the other hand, is much more difficult to solve. Akian et al. (1996) consider

the multiple-stock version of the Davis and Norman model when stock returns are

uncorrelated. They prove existence and uniqueness of a viscosity solution to the

Hamilton-Jacobi-Bellman (HJB) equation and present some numerical results for the

1- and 2-risky-asset cases. Liu (2004) obtains an almost closed form solution for fixed

and proportional costs in continuous time for infinite lived constant absolute risk

aversion (CARA) investors under the assumption that asset returns are uncorrelated.

Muthuraman and Kumar (2006) have developed numerical methods to solve the free

boundary HJB equation for the case of two correlated risky assets. They adapt the

finite element method and use an iterative procedure to convert the free boundary

problem into a sequence of fixed boundary problems with runtimes grow super

exponentially with dimension. Muthuraman and Zha (2008) have further developed

numerical methods based on combining simulation with the boundary update proce-

dure and provide a computational scheme whose runtime scales polynomially in the

number of assets. Lynch and Tan (2010) numerically solved in finite discrete horizon

a similar problem with two risky assets as in Muthuraman and Kumar (2006) but

they incorporate return predictability for the first time. Their methods are based on

a grid approximation of the state space for the associated dynamic program. Yet,

there is no guarantee that the aforementioned numerical methods are optimal or any

indication of how much better one might do with an optimal strategy. Brown et

al. (2009, 2011) developed a dual bounding technique that can be used to evaluate

the quality of the trading strategies. Broadie and Shen (2016, 2017) provide three

lower bounds for the optimal solution of the problem with 20 risky assets and 40

investment periods: the value function optimization (VF), the hyper-sphere and the

hyper-cube policy parameterizations (HS and HC). They also achieved tighter upper

4

bounds by improving the duality method in Brown et al. (2009, 2011).

In addition to proportional transaction cost, we mention researches on non-linear

transaction cost. Grinold (2006) derives the optimal steady state position with

quadratic trading costs and a single predictor of returns per security. Garleanu and

Pedersen (2013) derive a closed-form solution for a model with linear dynamics for

return predictors, quadratic functions for transaction costs, and quadratic penalty

terms for risk. Chan and Sircar (2016) consider a class of dynamic portfolio opti-

mization problems incorporating return predictability, and stochastic volatility with

quadratic transaction cost. Their approach is to view this as a perturbation problem

and construct an asymptotic approximation of the solution.

Besides solving for no-trade zone when transaction costs exist, rebalancing a multi-

asset portfolio back to the no-trade zone is a practically important problem. Mulvey

et al. (2001, 2002) set up a multi-stage optimization model for investing in assets over

an extended time horizon which is able to address transaction costs in a comprehensive

manner. In particular, they show that the rebalancing problem can be posed as a

generalized network with side constraints and propose a search algorithm for finding

a feasible solution with the lowest transaction cost.

5

1.2 Deep Learning

Deep learning is part of a broader family of machine learning methods aiming to

model complex nonlinear structures in high-dimensional data sets. It is an artificial

neural network (ANN) with multiple hidden layers of units between the input and

output layers. By using the back-propagation algorithm, one can compute how

a machine should change its internal parameters that are used to compute the

representation in each layer from the representation in the previous layer. It has had

a long and rich history, which dates back to the 1940s. McCulloch and Pitts (1943)

first introduce the idea of artificial neural networks, to simulate neurons to model

how learning happens or could happen in the biological brain. In the 1980s, with the

introduction of the concept of distributed representation (Hinton et al., 1986) - each

input to a system should be represented by many features, and each feature should

be involved in the representation of many possible inputs - and the successful use of

back-propagation algorithm (Rumelhart et al., 1986a; LeCun, 1987) to train deep

neural network models, deep learning has undergone great development. Finally

during recent years, with the development of computing resources and availability of

big data as well as an increasing number of different neural network models, deep

learning has made a lot breakthroughs in many areas such as image recognition (see

[53, 28, 97]), speech recognition (see [43, 88, 100]), natural language understanding

(see [20]), language translation (see [49, 107]), reconstructing brain circuits (see [46])

and gene expression and disease (see [63, 116]) etc. For the a comprehensive review

of deep learning models and training methods, see [58, 30].

Although deep learning has achieved immense success in science and technology,

there are not many published research of deep learning applications in finance.

Problems in the financial markets are very different from those typical deep learning

applications from various aspects: first and foremost, there are a lot noises but little6

signals in financial markets. Unlike recognizing an image or responding appropriately

to verbal requests, where deep learning models can extract key features from the

big data, it’s difficult, for instance, to obtain signals of a stock that is likely to

perform well in the future. Also, financial markets are highly unstable. A well

trained model might fits well during certain periods, but quickly fall out of use

due to frequent and unpredictable market changes. Nevertheless, there are still a

broad range of opportunities for developing new deep learning models and methods

for financial applications. Because in theory at least, given enough units, the deep

learning architectures are flexible enough to approximate arbitrarily well continuous

functions on compact sets, no matter how complex or nonlinear (Hornik et al, 1990;

Hornik, 1991). This is very different from simple linear factor models, traditional

financial econometrics models or ad hoc methods of statistical arbitrage and other

quantitative asset management techniques.

We list some representative deep learning applications in financial markets from

different subareas. Hutchinson et al. (1994) were the first to use radial basis

function (RBF) networks to approximate the Black-Scholes option pricing formula.

Sirignano (2016) design a new deep neural network architecture for modeling spatial

distributions and uses this “spatial neural network” to model the joint distribution of

best ask and best bid prices at a future time conditional on the current state of the

limit order book. Heaton et al. (2016) provide a four-step deep learning algorithm

to construct portfolios. Sirignano et al. (2017) analyze multi-period mortgage risk

- mortgage prepayment, delinquency, and foreclosure risk - at loan and pool levels

using a multi-layer feedforward neural network model trained and tested over 120

million mortgages originated across the US between 1995 and 2014. Dixon et al.

(2017) apply deep neural networks to financial time series data to classify market

movement directions and develop trading strategies for 43 different Commodities and

7

FX futures. Culkin and Das (2017) survey how and why deep learning can influence

the field of finance in a very general way with a specific revisit to the original work

to reproducing the Black and Scholes option pricing formula to a high degree of

accuracy by training a fully-connected feed-forward deep neural network.

Aside from directly using deep neural works to predict return, risk or other fi-

nancial market quantities, another strain of deep learning applications is to combine

with financial math. Despite prior work on transaction-cost models, there are few

scalable solutions on finding the optimal trading strategy with multiple risky assets

due to the curse of dimensionality - the computational cost scales exponentially with

the number of assets. In their recent working papers, E et al. (2017) propose a new

method for solving high-dimensional fully nonlinear second-order partial differential

equations (PDEs) herein. The PDEs are reformulated as a control theory problem

with the gradient of the unknown solution approximated by neural networks, like deep

reinforcement learning with the gradient acting as the policy function. Sirignano and

Spiliopoulos (2017) take a different approach to solve high-dimensional PDEs. They

approximate the solution with a deep neural network which is trained to satisfy the

differential operator, initial condition, and boundary conditions. Their techniques

has inspired us to use deep neural network to solve the free boundary HJB equation

associated with the transaction cost portfolio optimization problem.

8

1.3 Mean Reverting Assets

Finally in this section, we discuss the application of our results to the financial

markets. There has been numerous empirical and theoretical studies showing that

many asset prices exhibit mean reversion. One example would be the Chicago

Board Options Exchange (CBOE) Volatility Index, commonly known as the VIX

index. The mean reversion dynamics is well studied in many literatures (see, e.g.

[24, 34, 106, 117]). To visualize the mean-reverting pattern, the following graph

shows the time series of VIX from year 2004 to 2016.

2004 2006 2008 2010 2012 2014 2016

1020

3040

5060

time

Fut

ure

Pric

e

Figure 1.1: VIX from year 2004 to 2016.

While VIX itself cannot not be traded, investors can gain exposure to the index

by trading VIX futures, options, exchange-traded notes (ETNs) or exchange-traded

funds (ETFs) designed to track the index. Besides VIX, Bessembinder et al. (1995)

9

use the term structure of futures prices to test whether investors anticipate mean

reversion in spot asset prices and provide evidence of mean reversion in the prices

of several real and financial assets including financials, metals, and agriculturals.

Schwartz (1997) shows there is strong mean reversion in the commercial commodity

prices by comparing three different factor models of the stochastic behavior of com-

modity prices. Larsen and Sørensen (2007) study the mean-reverting properties of

exchange rates with two diffusion models in a target zone. Balvers et al. (2000) find

strong evidence of mean reversion in relative stock index prices across 18 countries.

Their findings imply a significantly positive speed of reversion with a half-life of three

to three and one-half years. In addition, bond pricing also heavily rely on mean

reversion models as interest rate and credit default risk often have mean reversion

properties. Except for various asset classes with mean reverting properties, investors

can also create artificial mean reverting tradables such as pairs trading. The original

idea of pairs trading, first initiated in Morgan Stanley in the 1980s, is strait-forward:

to find a pair of stocks whose spread exhibit potentially mean-reverting behavior.

Since then research of pairs trading or statistical arbitrage has flourished over the

next three decades, see Pole (2007) for a comprehensive review on statistical arbitrage.

Following the mean reverting properties of the asset prices, the problem of how

to construct an optimal trading strategies has been extensively studied. Gatev

et al. (2006) test the pairs trading strategy with daily stock data over 1962-2002

by a simple buy-low-sell-high rule. Avellandeda and Lee (2010) apply Component

Analysis (PCA)- and ETF- based methods to the broad universe of U.S. stocks

to construct the mean-reverting tradable and select the entry and exit point when

residuals deviate by certain standard deviations from equilibrium. Other than ad

hoc thresholds to enter/exit a trade, Elliott et al. (2005) model the entry time as

the first passage time of a OU process, followed by an exit at a fixed finite horizon.

10

Leung and Li (2014), Leung et al. (2014) and Leung et al. (2015) formulate the

trade as an optimal double stopping problem that gives the optimal entry and exit

decision rules. Specifically, they model the spot dynamics by the Ornstein Uhlenbeck

(OU), Cox-Ingersoll-Ross (CIR), or exponential Ornstein-Uhlenbeck (XOU) model.

Similarly, for OU, CIR or XOU underlying dynamics, Zhang and Zhang (2008), Song

et al. (2009), Kong and Zhang(2010), Song and Zhang (2013), Leung et al. (2015)

consider an optimal switching problem when investor can enter and exit the position

infinitely many times with either fixed or proportional transaction cost. By similar

methods, Leung and Li (2016) also study the optimal entry and exit boundaries

when trading financial derivatives such as futures, options and credit derivatives with

transaction costs (see [60] ).

On the other hand, we would like to mention some representative literatures on

pairs-trading strategies taking stochastic control approach as they are mostly closely

related to the methodologies we adopted in this dissertation. Benth and Karlsen

(2005) analyze the classical Merton’s portfolio optimization problem when portfolio

contains one risky asset following an exponential Ornstein-Uhlenbeck (XOU) process.

Jurek and Yang (2007) consider a similar problem as we did in this dissertation,

a finite portfolio optimization consisting OU asset subject to CRRA utility and

Epstein-Zin recursive utility, but they only provide explicit solutions to optimal

trading strategy with single mean-reverting asset and uncorrelated multi-assets.

Tourin and Yan (2013) solve explicitly, under exponential utility and zero risk free

rate, the finite horizon portfolio choice problem comprising of a risk-free asset and

two co-integrated and correlated stocks. Liu and Timmermann (2013) consider a

similar problem as Tourin and Yan (2013), but they consider two cases - recurring

arbitrage opportunities and nonrecurring arbitrage opportunities, and provide a

closed solution and numerical solution under first and second case respectively. Yet

11

none of the aforementioned researches assumed transaction cost in the portfolio op-

timization problem, which is particularly unrealistic in pairs trading as the creating

mean-reverting tradable incur more transaction cost than normal trade.

For constructing the mean-reverting tradable, we slightly modify the stock-ETF

pairs trading methods in Avellaneda and Lee (2010). They use Principal Compo-

nent Analysis (PCA) and regressing stock returns on sector Exchange Traded Funds

(ETFs) to construct trading signal and model the idiosyncratic returns as mean-

reverting processes. The exact construction process and related carry cost will be

discussed in the corresponding section. In addition, there are other applications of

the methodology to asset allocation as well as asset and liability management.

12

1.4 Contribution and Outline

In this dissertation, we consider the problem of finite-horizon dynamic portfolio opti-

mization problem with mean-reverting asset prices. Our general model considers risk

aversion and proportional transaction costs. we highlight our contributions as follows:

First, we choose to model the asset prices, rather than asset returns or risk

premiums as in many other literature, as Ornstein-Uhlenbeck process and extend the

model to correlated multi-assets portfolio when there is no transaction cost. This is

meaningful in reality when portfolios contains mean reverting assets such as CBOE

Volatility Index (VIX) future or assets created by pairs trading.

Second, we provide a novel numerical method to solve the same optimal portfolio

choice problem when proportional transaction costs exist. We use Deep Neural

Network (DNN) to parameterize the trading boundaries of the no trade zone (NT

zone). The training of the DNN heavily relies on the optimal policy derived un-

der no transaction cost assumption in both single risky asset and multiple risky

assets cases. Our study shows when there is transaction cost, the DNN method

significantly outperforms the policy we derived under no transaction cost assumption.

Moreover, our method is scalable to high dimensional case. We have conducted

the experiment with 48 correlated mean-reverting assets, which we can solve in less

than 7 hours using 1 GPU. To our best knowledge, previous work has not been

done on studying the transaction boundary in high dimensional case with correlated

Brownian Motion driving the asset prices.

Finally, our combination of analytical solutions with deep learning approach is

not specific to this particular portfolio choice problem, but rather, generic enough13

to be adapted into other models with prices driving by different dynamics such as

geometric Brownian motion or exponential Ornstein-Uhlenbeck (XOU) processes

or even a mix of different stochastic dynamics, provided that the optimal trading

strategy can be derived under no transaction cost assumption. There is also a great

flexibility of the choice of the DNN architecture. In our method, we employ the most

straightforward feedforward neural network, but more advanced and complicated

architecture can be explored.

The organization of this paper is as follows: In chapter 2, we propose the general

framework. In chapter 3, we derive the theoretical results of finite horizon optimal

allocation problem under no-transaction-cost assumption for three types of portfolios

consisting mean-reverting assets and analyze some properties of the trading strategies.

Simulations based on synthetic data is provided. In chapter 4, we describe our deep

neural network (DNN) method to parametrize the no-trade zone and trading strategy

based on no-trade zone, provides training procedure of the DNN and numerical results

from synthetic data. Results in this section shows the no-trade zone characterized

by DNN meets our intuition. chapter 5 shows backtest results of our DNN algorithm

applied to real data of stock-ETF pairs trading with high transaction cost. chapter

6 summarizes current and future work.

14

Chapter 2

General Framework for

Finite-horizon Portfolio

Optimization

In this chapter, we establish a general framework for solving the finite horizon portfolio

optimization problem without and with proportional transaction cost. That being

said, we don’t specify price dynamics of the underlying assets or utility function for

now. The framework we propose in this chapter can be applied to many different

situations. In the following chapters, we then provide specific examples.

2.1 The Model Setup

Throughout this dissertation, we assume a probability space (Ω,F , P ) and a filtration

Ft. We consider a financial market consisting of n + 1 assets: one risk-less asset,

X0 and n risky assets X = (X1, ..., Xn)T driven by a n-dimensional Brownian motion

B(t) = (B1(t), ..., Bn(t))T on the filtered probability space with with Bi(0) = 0 almost

15

surely and constant coefficients of correlation E(dBidBj) = ρijdt:

dX0 = rX0dt

dX = µ(X)dt+ Σ(X)dB(t)(2.1)

where r is the fixed risk-free rate, µ(X) and Σ(X) are functions of X:

µ(X) =[µ1(X1), ..., µn(Xn)

]T

Σ(X) =

σ1(X1)

. . .

σn(Xn)

(2.2)

Notice if:µi(Xi) = µiXi

σi(Xi) = σiXi, i = 1, ..., n(2.3)

Then this problem is reduced to the classic Merton’s problem. We leave µ(X) and

Σ(X) in a generic form in this chapter and specify in chapter 3 the functional forms

that allow for explicit solutions and meet our investment criteria.

Assume π = (π1, ..., πn) is the proportion of total wealth W invested in X, the rest

of the wealth (1−πT1)W is invested in X0. The wealth dynamics of the self-financing

portfolio satisfies:

dW

W=

n∑i=1

πidXi

Xi

+ (1− πT1)rdt

= πT diag(X)−1dX + (1− πT1)rdt(2.4)

The investor’s goal is to maximize the following expected utility of terminal wealth:

maxπ

E(U(WT )) (2.5)

16

We also leave the utility function in general format and specify it later in chapter 3.

17

2.2 Zero Transaction Cost

First, we assume a frictionless market where assets can be continuously transacted

without any cost or market impact. With the stochastic control approach, we let

J(W,X, τ) denote indirect utility function, where τ = T − t is the horizon of the

investment, the The Hamilton-Jacobi-Bellman equation can be derived as:

maxπ

− ∂J

∂τ+ L J

= 0 (2.6)

where the operator L is defined as:

L =[πT diag(X)−1 µ(X) + (1− πT1)r

]W

∂

∂W+ µ(X) ∂

∂X

+ 12π

T diag(X)−1Σ(X)Σ(X)T diag(X)−1πW 2 ∂

∂W 2

+ πT diag(X)−1Σ(X)Σ(X)TW ∂2

∂W∂X+ 1

2tr(Σ(X)H(X)Σ(X)T )

(2.7)

H(X) is the hessian of J with respect to X. diag(X) is the matrix with X on the

diagonal, and zeros elsewhere. tr(.) is the trace function. J(W,X, τ) also satisfies the

boundary condition:

J(W,X, 0) = U(W ) (2.8)

By solving the PDE (2.6) - (2.8), which we will explain in detail in chapter 3, we can

obtain the optimal allocation π∗(W,X, τ).

18

2.3 Proportional Transaction Cost

Now we assume a proportional transaction cost occurs at each transaction (buy or

sell of the risky assets). Assuming an investor has a portfolio X = (X1, X2, ..., Xn),

where Xi is the dollar values in i-th risky asset and X0 is the dollar values in the

bank account. In the presence of transaction costs, the dynamics follow:

dX0 = rX0dt−n∑i=1

(1 + li)dLi +n∑i=1

(1−mi)dMi

dXi = µi(Xi)dt+ σi(Xi)dBi + dLi − dMi

(2.9)

Li(t) and Mi(t) are nondecreasing, right continuous adapted processes with Li(0) =

Mi(0) = 0, representing cumulative dollar values of buying and selling the i-th risky

asset. li ∈ [0,∞) and mi ∈ [0, 1), i = 1, 2, ..., n, accounts for the proportional trans-

action costs incurred in buying and selling the i-th risky asset. The solvency region

S is defined as the set of dollar amounts in each assets such that the net wealth is

always positive :

S =x = (x0, x1, ..., xn) ∈ Rn+1 : x0 +

n∑i=1

[(1−mi)x+

i − (1 + li)x−i]> 0

(2.10)

Given initial positions x0 = (x0, x1, ..., xn) ∈ S in the risk-free and risky assets

respectively, the trading strategy (Li, Mi) is admissible for a position x from

times ∈ [0, T ), if Xi, i = 0, ..., n follows (2.9) with Xs = x is in S. We denote by

As(x) the set of all admissible investment strategies for x from time s. Finally, the

investor’s goal is to maximize the utility of terminal wealth among all admissible

strategies:

sup(Li,Mi)∈A0(x0)

E[U(WT )] (2.11)

Following Davis et al (1993), Shreve and Soner (1994), Muthuraman and Zha (2007)

or Dai and Zhong (2010), the solvency region can be divided into three type of regions:

19

In the no-trade-zone (NT zone, defined as none of the assets trade), the indirect utility

function J(W,X, τ) satisfies the following HJB equation:

− ∂J

∂τ+ L J = 0 (2.12)

In the i-th asset’s buy-zone (BZi), the marginal cost of decreasing the amount in the

bank account must be equal to the marginal benefit of buying the risky asset:

BiJ = 0 (2.13)

In the i-th asset’s sell-zone (SZi), the marginal benefit of increasing the amount in

the bank account must be equal to the marginal cost of selling the risky asset:

SiJ = 0 (2.14)

with terminal condition:

J(x, T ) = Ux0 +

n∑i=1

[(1−mi)x+

i − (1 + li)x−i]

(2.15)

where:

L = 12

n∑i,j=1

ρijσi(xi)σj(xj)∂2J

∂xi∂xj+

n∑i=1

µi(xi)∂J

∂Xi

+ rx0∂J

∂X0

Bi = −(1 + li)∂J

∂x0+ ∂J

∂xi

Si = (1−mi)∂J

∂x0− ∂J

∂xi

(2.16)

The problem becomes a free-boundary problem, all that needs to be found are the

boundaries of the regions such that the respective equations hold within the regions

20

and the following equation holds in the entire state space:

max− ∂J

∂τ+ L J,BiJ,SiJ

= 0, x ∈ S , t ∈ [0, T ) (2.17)

Previous researches focus on numerical implementation of PDE (2.13) -(2.17) as

analytical solutions are difficult to obtain. In this dissertation, we propose a different

approach of learning the two boundaries riu and ril for each asset i by neural network.

Instead of directly parameterize riu and ril without prior knowledge, we specifically

parameterize the two boundaries as follows:

riu(t,X,W ) ≈ π∗(t,X,W ) + f θu(t,X,W )

ril(t,X,W ) ≈ π∗(t,X,W ) + f θl (t,X,W )(2.18)

where π∗(t, x,W ) is the optimal solution of the same portfolio choice problem

under zero transaction cost assumption as shown in section 2.2, and fu(t,X,W )

and fl(t,X,W ) are small positive values that will be approximated by the neural

network. The intuition for this parameterization is that when there is no transaction

cost, the optimal strategy is to always rebalance the position to back to π∗(t,X,W ).

meaning riu(t,X,W ) = ril(t,X,W )) = π∗(t,X,W ). When transaction cost gradually

increases, the NT zone will widen and π∗(t,X,W ) should fall into the NT zone. By

utilizing π∗(t,X,W ), we can significantly improve the training results and avoid the

neural network falling into local optimum.

The training of the neural network is as follows: we simulate n sample price paths;

at each time step, rebalance the portfolio according to riu(t,X,W ) and ril(t,X,W ))

until time T . Finally, the loss function is defined as:

loss = 1n

n∑j=i

(J(W,Xj, T )− U(W )(j)

)(2.19)

21

where U(W )(j) is the j-th path’s terminal utility following neural network trading

strategy (2.18) and J(W,Xj, T ) is the optimal terminal utility with zero transaction

cost strategy. It is clear that the quantity J(W,Xj, T ) is an proper upper bound

for U(W )(j), the portfolio choice problem with transaction cost. By minimizing the

loss, we can find the NTZ boundaries approximated by the neural networks. The

parameterization and training details are elaborated in chapter 4.

22

Chapter 3

Optimal Strategy Under Zero

Transaction Cost

3.1 Optimal Strategy With One OU Asset

Following a similar setting in Jurek and Yang (2007), we assume the investor can

invest in two assets in the market. The risky asset has price Xt following an Ornstein-

Uhlenbeck (OU) process, the risk-less asset is denoted by Yt. Their dynamics are as

follows:dXt = λ(µ−Xt)dt+ σdZt

dYtYt

= rdt(3.1)

where λ ∈ R+, σ ∈ R+ and µ ∈ R are speed of reversion, instantaneous volatility

and long term mean level of the OU process respectively. Zt denotes the standard

Brownian Motion. r ∈ R+ is the deterministic risk free rate. In reality, Xt could

be the price of an ETF that tracks VIX index or the value of a long-short portfolio

constructed from pairs-trading. In latter case, longing (shorting) 1 unit of the spread

is equivalent to longing (shorting) 1 unit of the overvalued security and shorting

(longing) β units of the undervalued security, where β is to be determined from

23

history.

Furthermore, we also adopt the following assumptions on capital market in Merton

(1973):

Assumption 1. All assets have limited liability.

Assumption 2. There are no transactions costs, taxes, or problems with indivisibil-

ities of assets.

Assumption 3. There are a sufficient number of investors with comparable wealth

levels so that each investor believes that he can buy and sell as much of an asset as

he wants at the marketprice.

Assumption 4. The capital market is always in equilibrium(i.e., there is no trading

at non-equilibrium prices).

Assumption 5. There exists an exchange market for borrowing and lending at the

same rate of interest.

Assumption 6. Short-sales of all assets, with full use of the proceeds, is allowed.

Assumption 7. Trading in assets takes place continually in time.

As noted in Merton (1973), Assumptions 1-6 are the standard assumptions of

a perfect market, and Assumption 7 follows directly from Assumption 2 since in

frictionless market, investor would like to be able to revise their portfolios at any

time. Although in reality, the frictionless continuous-time setting is unattainable, the

analytical solution derived under these assumptions are indispensable for relaxing

the frictionless assumption in the next chapter.

24

Let π(t) ∈ R (i.e. shorting and leverage allowed) be the current proportion of

wealth invested in the risky asset at time t, the total wealth of the portfolio Wt

follows:dWt

Wt

= πtdXt

Xt

+ (1− πt)dYtYt

(3.2)

Our goal is to maximize the expected utility of the terminal wealth WT at a given

finite horizon T :

maxπ

E[U(WT )] (3.3)

where we choose U(W ) to be the constant relative risk aversion (CRRA) utility:

U(W ) =

W 1−γ − 11− γ if γ > 0 and γ 6= 1

ln(W ) if γ = 1

(3.4)

The CRRA utility function (3.4) has constant relative risk aversion level γ with γ = 0

corresponding to risk neutrality as utility is linear in W , and γ > 0 corresponding

to risk averse agents. The larger γ is, the more risk averse the agent is. The first

derivative U ′(W ) > 0 indicates investors prefer more wealth than less and the second

derivative U ′′(W ) < 0 exhibits diminishing marginal utility. There has been many

empirical studies and surveys showing that most individuals have risk aversions

between 1 and 10 (see, e.g., [54, 78]). Hence in this section, we focus on solving the

portfolio problem analytically with γ > 1.

Let τ = T−t be the horizon of the investment period. Assume J(W,X, τ) to be the

indirect utility function which satisfies the boundary condition J(W,X, 0) = U(W ).

With the standard stochastic control methods, we can derive the Hamilton-Jacobi-

25

Bellman equation:

maxπ

− Jτ +

[πλ

X(µ−X) + (1− π)r

]WJW + 1

2(πσX

)2W 2JWW

+ λ(µ−X)JX + 12σ

2JXX + σπσ

XWJWX

= 0

(3.5)

with boundary condition

J(W,X, 0) = U(W ) (3.6)

where Jτ ,JW , and JX denote the derivatives of J with respect to t, W , and X respec-

tively. Similarly, JWW , JXX and JWX denote the higher derivatives. From Eq.(3.5),

we can compute the optimal asset allocation:

π∗(τ,X) = − JWWJWW

[λ(µ−X)/X − r

(σ/X)2

]− XJWX

WJWW

(3.7)

The first term in Eq.(3.7) is the scaled mean-variance efficient portfolio weight, also

called the myopic demand because this is the vector of portfolio weights for an agent

who only optimizes over one single period. Within the first term, the first part

−JW

WJWW

is the reciprocal of the absolute risk aversion factor of the indirect utility

function, and the second partλ(µ−X)/X − r

(σ/X)2 is the expected excess returns on

risky asset X scaled by the inverse of the variance of risky asset return. The second

term in Eq.(3.7) represents the inter-temporal hedging demand, which arises in a

multi-period portfolio choice problem when an investor accounts for changes in the

investment opportunity set and tries to hedge against adverse future shocks. In our

model, the investment opportunity set is time varying since expected returns on

risky asset is time varying.

26

The HJB equation (3.5) is solved by first “guessing” a general form for the solution

and then verified later. Following past literatures on portfolio optimization of assets

with mean-reverting properties (see [6, 50, 52, 113]), we conjecture the indirect utility

takes the following form:

J(W,X, τ) =(Wφ(τ))1−γ − 1

1− γ (3.8)

where

φ(τ) = exp(A(τ) +B(τ)X + C(τ)X2/2) (3.9)

A(0) = B(0) = C(0) = 0 (3.10)

Substituting π∗ (3.7) and the ansatz (3.8) - (3.10) into Eq.(3.5) generates the following

PDE:

−[A′(τ) +B′(τ)X + 1

2C′(τ)X2

]+ σ2

2

C(τ) + (1− γ)

[B(τ) + C(τ)X

]2−σ

2γ

2[(−λ+ r

γσ2 X + λµ

γσ2 ) + 1− γγ

(B(τ) + C(τ)X)]2

+ λ(µ−X)[B(τ) + C(τ)X

]+[λ(µ−X)− rX

][(−λ+ r

γσ2 X + λµ

γσ2 ) + 1− γγ

(B(τ) + C(τ)X)]

+ r

+σ2(1− γ)[(−λ+ r

γσ2 X + λµ

γσ2 ) + 1− γγ

(B(τ) + C(τ)X)][B(τ) + C(τ)X

]= 0

(3.11)

The left-hand side of the above Eq.(3.11) is the instantaneous expected change in the

indirect utility function J(W,X, τ), which must equal to zero to satisfy the optimality

condition. Notice, Eq.(3.11) is a quadratic equation of X. Making coefficients of X,

X2 and constant term zeros, we can obtain the following ODE system of A(τ), B(τ)

and C(τ):

C ′(τ) = aC2(τ) + bC(τ) + c (3.12)

B′(τ) = aB(τ)C(τ) + b

2B(τ) + dC(τ) + g (3.13)

27

A′(τ) = a

2B(τ)2 + dB(τ) + σ2

2 C(τ) + (λµ)2

2γσ2 + r (3.14)

with boundary condition:

A(0) = B(0) = C(0) = 0 (3.15)

and parameters:

a =1− γγ

σ2, b =2(γr − r − λ)

γ

c =(λ+ r)2

γσ2 , d =λµ

γ, g = −λµ(λ+ r)

γσ2

(3.16)

The first equation (3.12) in the ODE system only contains C(τ), the second equation

(3.13) contains B(τ) and C(τ) and the third equation (3.14) contains all of the A(τ),

B(τ) and C(τ). Hence we can solve Eq.(3.12) - (3.14) sequentially. We state the

results in the following theorem.

Theorem 3.1.1. Given risky asset price X follows (3.1), fixed risk free rate r, and

investment horizon τ , the optimal asset allocation of total wealth to risky asset that

maximizes the expected CRRA utility of terminal wealth E[U(WT )] is:

π∗(τ,X) = 1γ

[λ(µ−X)/X − r

(σ/X)2

]+ 1− γ

γ

[C(τ)X +B(τ)

]X (3.17)

Where

C(τ) = 2c(1− e−ητ )2η − (b+ η)(1− e−ητ ) (3.18)

B(τ) = −4gr(1− e−ητ/2)2 + 2gη(1− e−ητ )η[2η − (b+ η)(1− e−ητ )] (3.19)

A(τ) =∫ a

2B(τ)2 + dB(τ) + σ2

2 C(τ) + (λµ)2

2γσ2 + rdτ (3.20)

with

η =√b2 − 4ac (3.21)28

and parameters a, b, c, d, g defined in (3.16).

We leave A(τ) in the integral form since this term does not appear in our optimal

investing proportion π∗, nevertheless, the integral can be computed by checking the

integral table. We include the detailed proof and computation in Appendix A. While

the closed solution derived in Theorem 3.1.1 assumes γ > 1, which ensures the solution

is well behaved and will be used in the following chapters, the ODE system (3.12) -

(3.16) can be solved for all γ > 0. See Appendix A for a complete set of solutions.

29

3.2 Optimal Strategy With Multiple Correlated

OU Assets

In this section, we further extend our analysis to multiple correlated risky assets.

Suppose there are n risky assets, which jointly follow the multi-dimensional OU pro-

cess:

dX = µX(X)dt+ σXdZ (3.22)

where drift coefficient µX(X) being:

µX(X) = Λ(M −X) (3.23)

with mean reverting speeds characterized by n-by-n diagonal matrix Λ:

Λ =

λ1

λ2

. . .

λn

(3.24)

long term mean levels by n-by-1 vector M :

M = (µ1, µ2, ..., µn)T (3.25)

and diffusion coefficient σX is the n-by-n diagonal matrix:

σX =

σ1

σ2

. . .

σn

. (3.26)

30

Z is a vector of n correlated Brownian Motions. The correlation matrix of the risk

factors is characterized by:

dZdZT = σρdt =

1 ρ12 ... ρ1n

ρ21 1 ρ2n

.... . .

ρn1 ρn2 1

dt (3.27)

Y is the value of risk-free money market account with a deterministic growth rate r:

dY = rY dt (3.28)

Let π = (π1, π2, ..., πn)T ∈ Rn be the portfolio weights invested in X = (X1, X2, ..., Xn)T

and the rest of the wealth is invested in the risk free account. Then the total wealth

dynamics satisfies :

dW

W=[πTµW + r(1− πT1)

]dt+ πTσWdZ (3.29)

with

µW =[ λ1

X1(µ1 −X1),

λ2

X2(µ2 −X2), ...,

λn

Xn

(µn −Xn)]T

(3.30)

σW =

σ1/X1

σ2/X2

. . .

σn/Xn

(3.31)

where µW is a n-by-1 vector, σW is a n-by-n diagonal matrix and 1 represents the

n-by-1 all-ones vector.

31

The investor aims to maximize the expected CRRA utility of terminal wealth:

maxπ

E[U(WT )] (3.32)

Following the aforementioned method, we assume J(W,X, τ) to be the indirect

utility function where τ is the horizon of the investment period. We can derive the

following HJB equation:

maxπ

− Jτ + 1

2W2πTΣWπJWW +W

[πT (µW − r1) + r

]JW

+WπTΣWXJWX + µXTJX + 1

2Tr(ΣXJXXT )

= 0(3.33)


J(W,X, 0) = U(W ) (3.34)

whereΣW = σWσρσ

TW

ΣWX = σWσρσTX

ΣX = σXσρσTX

(3.35)

similar to single asset case, Jτ , JW , and JX denote the derivatives of J with respect to

τ , W , and X respectively; JWW , JWX and JXXT denote the higher derivatives; tr(.)

denotes the matrix trace function. The optimal trading strategy is:


ΣW−1(µW − r1)− 1

WJWW

ΣW−1ΣWXJWX (3.36)

With a similar interpretation in the previous single risky asset model, the first term

in Eq.(3.36) is the mean-variance efficient portfolio weights or the myopic allocation,

which is the vector of expected excess returns on risky assets, µW − r1, scaled by

the inverse of the variance-covariance matrix of risky asset returns, ΣW−1 and the

32

reciprocal of the coefficient of relative risk aversion, − JWWJWW

. The second term in

Eq.(3.36) is the inter-temporal hedging demand.

Similar to the ansazt we used in single risky asset model, we assume the solution

of J(W,X, τ) takes the form:

J(W,X, τ) =(Wφ(τ,X))1−γ − 1

1− γ (3.37)

with

φ(τ,X) = expA(τ) +B(τ)TX + 1

2XTC(τ)X

(3.38)

and boundary condition:

A(0) = 0, B(0) = 0, C(0) = 0 (3.39)

The format of φ(τ,X) is a natural extension of Eq.(3.9) in the single asset case to

high dimensions. A(τ) is a scalar, B(τ) is a n-by-1 vector and C(τ) is a n-by-n

matrix. Substituting the ansatz of J(W,X, τ) and optimal allocation π∗ into the

HJB equation (3.33), we can obtain the following PDE:

−[A′(τ) +B′(τ)TX + 1

2XTC ′(τ)X]

+ 12γ (µW − r1)TΣ−1

W (µW − r1) + r

+1− γγ

(µW − r1)TΣTWXΣ−1

W

[B(τ) + C(τ)X

]+ µTX

[B(τ) + C(τ)X

]+1− γ

2γ[B(τ) + C(τ)X

]TΣX

[B(τ) + C(τ)X

]+ 1

2Tr

ΣXC(τ)

= 0

(3.40)

The left-hand side of the above PDE is, up to a multiplicative factor J , the instan-

taneous expected change in the indirect utility function J(W,X, τ). Observe that

Eq.(3.40) is quadratic in X. Because the equation holds for all X, the coefficients of

second order, first order and constant term have to be zero. By matching the orders,

33

we can obtain the ODE systems for A(τ), B(τ) and C(τ). We state the results in the

following theorem.

Theorem 3.2.1. Given risky asset prices X follow (3.22)-(3.27), fixed risk free rate r,

and investment horizon τ , the optimal asset allocation to risky assets that maximizes

the expected CRRA utility of terminal wealth E[U(WT )] is:

π∗(τ,X) = −1γ

ΣW−1

(r1− µW )− (1− γ)ΣWX

[B(τ) + C(τ)X

](3.41)

whereC ′(τ) =1− γ

γC(τ)TΣXC(τ)− 1

γ

[ΛC(τ) + C(τ)TΛ

]−2r(1− γ)

γC(τ) + 1

γ(Λ + rI)TΣ−1

X (Λ + rI)(3.42)

B′(τ) =[1− γ

γΣXC(τ)− 1

γΛ− 1− γ

γrI]TB(τ)

−1γ

(Λ + rI)TΣ−1X ΛM + 1

γC(τ)TΛM

(3.43)

A′(τ) = 12γ (ΛM)TΣ−1

X (ΛM) + 1γMTΛB(τ)

+1− γ2γ B(τ)TΣXB(τ) + r + 1

2Tr[ΣXC(τ)

] (3.44)

with boundary conditions:

A(0) = 0, B(0) = 0, C(0) = 0 (3.45)

where Λ,M, µW ,ΣW ,ΣX ,ΣWX are defined in (3.24), (3.25) and (3.35) respectively.

1 is the n-by-1 all-ones vector and I is the n-by-n identity matrix.

The detailed proof and computation is included in the Appendix A.2. To observe

the ODE system more carefully, the differential equation for C(τ) in (3.42) is the

famous matrix Riccati equation, which can be solved analytically (see [92]). But the

differential equation for B(τ) in (3.43) contains time varying coefficients, hence cannot

34

be solved analytically. Although the ODE system (3.42) - (3.45) does not admit

a closed-form solution, it can be solved numerically by discretizing the ODEs and

simulating them using the initial conditions. To be specific, at tn+1, we approximate

Ctn+1 using Ctn and (3.42), we approximate Btn+1 using Btn and Ctn and (3.43), we

approximate Atn+1 using Btn and Ctn and (3.44).

35

3.3 Optimal Strategy With Mixture Correlated

Risky Assets

In the previous sections, we have established the model for optimal portfolio strategy

with assets following mean-reverting processes. Yet in reality, a portfolio manager has

not only mean-reverting assets in his/her portfolio, but also more traditional assets

such as stocks. Hence, it is natural to consider the problem of optimal portfolio

strategy when the investor has a wider investment choice. In this section, we extend

the investor’s choice by allowing him/her to invest in two types of risky assets: one

following OU process, the other following geometric Brownian Motion(GBM) process.

The formulation of the portfolio choice problem is similar to a multi-OU portfolio.

We first derive the closed form solution of the portfolio that only contains one asset

from each category, then we extend the model to a portfolio with multiple OU and

GBM assets, in which case a solution up to an ODE system can be attained.

3.3.1 Mixture Portfolio: Single OU Asset and GBM Asset

We denote the risky asset following OU process by X1 and the risky asset following

GBM process by X2, the risk-less asset by Y . Their dynamics are as follows:

dX1 = λ1(µ1 −X1)dt+ σ1dZ1

dX2 = µ2X2dt+ σ2X2dZ2

dYt = Ytrdt

(3.46)

where λ1 ∈ R+, σ1 ∈ R+ and µ1 ∈ R are speed of reversion, instantaneous volatility

and long term mean level of the OU process respectively. And µ2 ∈ R, σ2 ∈ R+ are

the drift and volatility of the GBM process. Z1 and Z2 denote two standard Brownian

36

Motions with correlation ρ:

dZ1dZ2 = ρdt (3.47)

r ∈ R+ is the deterministic risk free rate.

Let π1, π2 ∈ R be the current proportions of wealth invested in X1 and X2 at time

t respectively and the rest invested in risk-free asset, the total wealth of the portfolio

Wt follows:

dWt

Wt

= π1dX1

X1+ π2

dX2

X2+ (1− π1 − π2)dYt

Yt

=[π1λ1(X1 − µ1)

X1+ π2µ2 + (1− π1 − π2r

]dt+ σ1

X1dZ1 + σ2dZ2

(3.48)

With the goal of maximizing the expected CRRA utility of the terminal wealth

E[U(WT )], we can derive the Hamilton-Jacobi-Bellman equation:

maxπ1,π2

− Jτ +

[π1λ1

X1(µ1 −X1) + π2µ2 + (1− π1 − π2)r

]WJW

+ 12[(π1σ1

X1)2 + 2ρπ1π2

σ1σ2

X1+ (π2σ2)2

]W 2JWW

+ (π1σ2

1X1

+ π2ρσ1σ2)WJWX1 + (π1ρσ1

X1σ2X2 + π2σ

22X2)WJWX2

+ λ1(µ1 −X1)JX1 + µ2X2JX2

+ 12(σ2

1JX21

+ 2ρσ1σ2X2JX1X2 + (σ2X2)2JX22)

= 0

(3.49)

with boundary condition

J(W,X1, X2, 0) = U(W ) (3.50)

37

where τ = T − t is the horizon of the investment period and J(W,X1, X2, τ) is the

indirect utility function. From Eq.(3.49), we can compute the optimal asset allocation:

π∗1(τ,X1, X2) = − JWWJWW

[λ1(µ1 −X1)/X1 − r

(σ1/X1)2 − ρX1(µ2 − r)σ1σ2

]+ X1JWX1

WJWW

π∗2(τ,X1, X2) = − JWWJWW

[µ2 − rσ2

2− ρλ1(µ1 −X1)/X1 − ρr

σ1σ2/X1

]+ X2JWX2

WJWW

(3.51)

Since in the Merton’s problem, the optimal strategy is to invest a fixed fraction

of wealth into risky asset, in other words, the value function is the constant Sharpe

ratio value function, we conjecture the indirect utility takes the following form:

J(W,X1, X2, τ) =(Wφ(τ))1−γ − 1

1− γ (3.52)

where φ(τ) only contains linear and quadratic terms in X1:

φ(τ) = exp(A(τ) +B(τ)X1 + C(τ)X21/2) (3.53)

A(0) = B(0) = C(0) = 0 (3.54)

Substituting π∗1, π∗2 (3.51) and the ansatz (3.52) - (3.54) into Eq.(3.49) generates the

following PDE:

−[A′(τ) +B′(τ)X1 + 1

2C′(τ)X2

1

]+ 1− γ

γ

[λ1(µ1 −X1)X1

− r](B(τ) + C(τ)X1)X1

+r + 11− ρ2

[λ1µ1 − (λ1 + r)X1

σ1− ρµ2 − r

σ2

]λ1µ1 − (λ1 + r)X1

σ1

−[ρλ1µ1 − (λ1 + r)X1

σ2− µ2 − r

σ2

]µ2 − rσ2

+ σ2

1(1− γ)2

2γ (B(τ) + C(τ)X1)2

+λ1(µ1 −X1)[B(τ) + C(τ)X1

]+ σ2

12[C(τ) + (1− γ)(B(τ) + C(τ)X1)2

]= 0

(3.55)

The left-hand side of the above Eq.(3.55) is the instantaneous expected change in

the indirect utility function J(W,X1, X2, τ), which must equal to zero to satisfy the

38

optimality condition. Notice, Eq.(3.55) is a quadratic equation of only X1. Then by

making the coefficients of each term zero, we can obtain the following ODE system

of A(τ), B(τ) and C(τ):

C ′(τ) = aC2(τ) + bC(τ) + c (3.56)

B′(τ) = aB(τ)C(τ) + b

2B(τ) + dC(τ) + g (3.57)

A′(τ) = a

2B(τ)2 + dB(τ) + σ21

2 C(τ) + r

+ 12γ(1− ρ2)

[(λ1µ1

σ1)2 − 2ρλ1µ1(µ2 − r)

σ1σ2+ (µ2 − r)

σ2)2] (3.58)


A(0) = B(0) = C(0) = 0 (3.59)

and parameters:

a =1− γγ

σ21, b =

2(γr − r − λ1)γ

c =(λ1 + r)2

γσ21(1− ρ2), d =

λ1µ1

γ

g = − 12γ(1− ρ2)

λ1µ1(λ1 + r)γ(1− ρ2)σ2

1+ ρ(λ1 + r)(µ2 − r)

γ(1− ρ2)σ1σ2

(3.60)

Notice, the ODE system (3.56) - (3.58) has the same format as (3.12) - (3.14) in

section 3.1, but with different parameters a, b, c, d, g. Therefore, we can solve it in a

similar way and the results are stated in the following theorem.

Theorem 3.3.1. Given risky asset prices X1 and X2 follow (3.46), fixed risk free

rate r, and investment horizon τ , the optimal asset allocation of total wealth to risky

39

asset that maximizes the expected CRRA utility of terminal wealth E[U(WT )] is:

π∗1(τ,X1, X2) = −1γ

[λ1(µ1 −X1)/X1 − r

(σ1/X1)2 − ρX1(µ2 − r)σ1σ2

]+ 1− γ

γ

[B(τ)X1 + C(τ)X2

1

]π∗2(τ,X1, X2) = −1

γ

[µ2 − rσ2

2− ρλ1(µ1 −X1)/X1 − ρr

σ1σ2/X1

](3.61)

Where

C(τ) = 2c(1− e−ητ )2η − (b+ η)(1− e−ητ ) (3.62)

B(τ) = −4gr(1− e−ητ/2)2 + 2gη(1− e−ητ )η[2η − (b+ η)(1− e−ητ )] (3.63)

A(τ) =∫ a

2B(τ)2 + dB(τ) + σ21

2 C(τ) + r

+ 12γ(1− ρ2)

[(λ1µ1

σ1)2 − 2ρλ1µ1(µ2 − r)

σ1σ2+ (µ2 − r)

σ2)2]dτ

(3.64)

with

η =√b2 − 4ac (3.65)

and parameters a, b, c, d, g defined in (3.60).

3.3.2 Mixture Portfolio: Multiple OU Assets and GBM As-

sets

In this section, we further extend our model to multiple correlated OU assets and

GBM assets. Suppose there are n risky assets jointly following multi-dimensional OU

process and m = N − n(N ≥ n) risky assets jointly following the multi-dimensional

GBM process:

dX = µX(X)dt+ σX(X)dZ , i = 1, 2, ..., N (3.66)

40

We first define the following parameters:

Λ =

λ1

. . .

λn

Σn =

σ1

. . .

σn

Σm =

σn+1

. . .

σN

ΣN =

σ1

. . .

σN

M1 = (µ1, ..., µn)T M2 = (µn+1, ..., µN)T

Xn = (X1, ..., Xn)T Xm = (Xn+1, ..., XN)T

(3.67)

The drift coefficient µX(X) can be written in block-matrix format as:

µX(X) =[λ1(µ1 −X1), ..., λn(µn −Xn), µn+1Xn+1, ..., µNXN

]T=([

Λ(M1 −Xn)]T

(diag(M2)Xm)T) (3.68)

and the diffusion coefficient σX(X) is the N -by-N diagonal matrix that can also be

written in the block-matrix form:

σX(X) =

σ1

. . .

σn

σn+1Xn+1

. . .

σNXN

=

Σn 00 Σmdiag(Xm)

(3.69)

41

Z is a vector of N correlated Brownian Motions. The correlation matrix of the risk

factors is characterized by:

dZdZT = σρdt =

1 ρ12 ... ρ1N

ρ21 1 ρ2N

.... . .

ρN1 ρN2 1

dt (3.70)

We further define Σ and its inverse as:

Σ = ΣNσρΣTN =

Σ11 Σ12

Σ21 Σ22

Σ = Σ−1 =

Σ11 Σ12

Σ21 Σ22

(3.71)

where Σ11(Σ11),Σ12(Σ12),Σ21(Σ21),Σ22(Σ22) are n-by-n,n-by-m,m-by-n,m-by-m sub-

matrices respectively. Y is the value of risk-free money market account with a deter-

ministic growth rate r:

dY = rY dt (3.72)

Let π = (π1, π2, ..., πN)T ∈ RN be the portfolio weights invested in X =

(X1, X2, ..., XN)T and the rest of the wealth is invested in the risk free account.

Then the total wealth dynamics satisfies :

dW

W=[πTµW + r(1− πT1)

]dt+ πTσWdZ (3.73)

with

µW (X) =[ λ1

X1(µ1 −X1), ...,

λn

Xn

(µn −Xn), µn+1, ..., µN

]T=([

Λ(M1 −Xn)diag(Xn)−1]T

MT2

) (3.74)

42

σW (X) =

σ1/X1

. . .

σn/Xn

σn+1

. . .

σN

=

Σndiag(Xn)−1 00 Σm

(3.75)

where µW (X) is a N -by-1 vector, σW (X) is a n-by-n diagonal matrix and 1 represents

the N -by-1 all-ones vector. For simplicity, we use µX , σX , µW , σW in the following

text.

Following the same stochastic control method to maximize the expected CRRA

utility of terminal wealth E[U(WT )], assume J(W,X, τ) to be the indirect utility

function where τ is the horizon of the investment period, we can derive the following

HJB equation:

maxπ

− Jτ + 1

2W2πTΣWπJWW +W

[πT (µW − r1) + r

]JW

+WπTΣWXJWX + µXTJX + 1

2Tr(ΣXJXXT )

= 0(3.76)


J(W,X, 0) = U(W ) (3.77)

whereΣW = σWσρσ

TW

ΣWX = σWσρσTX

ΣX = σXσρσTX

(3.78)

43

And the optimal trading strategy is:


ΣW−1(µW − r1)− 1

WJWW

ΣW−1ΣWXJWX (3.79)

Similar to the ansazt we used in single mixture portfolio model, we assume the solution

of J(W,X, τ) takes the form:

J(W,X, τ) =(Wφ(τ,X))1−γ − 1

1− γ (3.80)

with

φ(τ,X) = expA(τ) +B(τ)TX + 1

2XTC(τ)X

(3.81)

and boundary condition:

A(0) = 0, B(0) = 0, C(0) = 0 (3.82)

where A(τ) is a scalar, B(τ) is a N -by-1 vector with last m entries being zeros:

B(τ) =(B(τ) 0

)(3.83)

Where B(τ) denotes a n-by-1 vector. C(τ) is a N -by-N matrix with C(τ), a n-by-n

matrix C(τ) on the top left and rest entires being zeros :

C(τ) =

C(τ) 00 0

(3.84)

44

substituting the ansatz of J(W,X, τ) and optimal allocation π∗ into the HJB equation

(3.76), we can obtain the following PDE contains only Xn :

−[A′(τ) + B′(τ)TXn + 1

2XTn C′(τ)Xn

]+ 1

2γ

(M2 − r1)T Σ22(M2 − r1)

+(M2 − r1)T Σ21[ΛM1 − (Λ + rIn)Xn

]+[ΛM1 − (Λ + rIn)Xn

]TΣ12(M2 − r1)

+[ΛM1 − (Λ + rIn)Xn

]TΣ11

[ΛM1 − (Λ + rIn)Xn

]+ r

+1− γγ

[ΛM1 − (Λ + rIn)Xn)

]T [B(τ) + C(τ)Xn

]+[Λ(M1 −Xn)

]T [B(τ) + C(τ)Xn

]+1− γ

2γ[B(τ) + C(τ)Xn

]TΣ11

[B(τ) + C(τ)Xn

]+ 1

2Tr

Σ11C(τ)

= 0

(3.85)

Hence, we can obtain the ODE systems for A(τ), B(τ) and C(τ). The results are

stated in the following theorem.

Theorem 3.3.2. Given risky asset prices X follow (3.66)-(3.70), fixed risk free rate r,

and investment horizon τ , the optimal asset allocation to risky assets that maximizes

the expected CRRA utility of terminal wealth E[U(WT )] is:

π∗(τ,X) = 1γ

ΣW−1(µW − r1) + 1− γ

γdiag(Xn)

[B(τ) + C(τ)Xn

](3.86)

whereC ′(τ) =1− γ

γC(τ)TΣ11C(τ)− 1

γ

[ΛC(τ) + C(τ)TΛ

]−2r(1− γ)

γC(τ) + 1

γ(Λ + rIn)T Σ11(Λ + rIn)

(3.87)

B′(τ) =[1− γ

γΣ11C(τ)− 1

γΛ− 1− γ

γrIn

]TB(τ) + 1

γC(τ)TΛM1

− 12γ

(Λ + rIn)T (ΣT

21 + Σ12)(M1 − r1)− 2(Λ + rIn)Σ11ΛM1

(3.88)

45

A′(τ) = 12γ

(ΛM1)T Σ11(ΛM1) + (M2 − r1)T Σ22(M2 − r1)

+(ΛM1)T (Σ12 + ΣT21(M2 − r1)

+ 1γMT

1 ΛB(τ)

+1− γ2γ B(τ)TΣ11B(τ) + r + 1

2Tr[Σ11C(τ)

](3.89)


A(0) = 0, B(0) = 0, C(0) = 0 (3.90)

where Λ,M1,M2,Σ11, Σ11, Σ12, Σ21, Σ22, µW ,ΣW are defined in (3.67), (3.71), (3.74)

and (3.78) respectively. 1 is the n-by-1 all-ones vector and In is the n-by-n identity

matrix.

Similar to ODE system (3.42)-(3.45), we can solve (3.87)-(3.90) numerically by

discretizing the ODEs and simulating them using the initial conditions. To be specific,

at tn+1, we approximate Ctn+1 using Ctn and (3.87), we approximate Btn+1 using Btn

and Ctn and (3.88), we approximate Atn+1 using Btn and Ctn and (3.89). Also notice,

if n = 1, i.e. there are only 1 OU assets but multiple GBM assets, the ODE system

(3.87)-(3.90) reduced to 1-dim ODE system (3.56) - (3.58), which can be solved in

closed form.

46

3.4 Properties of Optimal Trading Strategy

In this section, we analyze some properties of the optimal trading strategy for single

risky asset case1. As we have mentioned in the previous section, the optimal trading

strategy in E.q.(3.17) are comprised of two parts: the first part 1γ

[λ(µ−X)/X−r

(σ/X)2

], is

the myopic demand which captures the asset demand induced exclusively by the

current risk premium. It is proportional to the portfolio risk premium, inversely

proportional to the individual’s relative risk aversion and does not depend on horizon

τ , hence called “myopic”. The second part 1−γγ

[C(τ)X+B(τ)

]X, is the inter-temporal

hedging demand. It reflects the strategic behavior of the investor who wishes to hedge

against future adverse changes in investment opportunities. The sign and magnitude

of the inter-temporal hedging demand are horizon-dependent and are determined by

the coefficient functions, C(τ) and B(τ). With the explicit form of C(τ) and B(τ),

we can analyze certain properties (sign, magnitude, monotonicity) of the hedging

demand, which are stated as follows. Throughout this section, we assume the risk

averse parameter γ > 1, long term mean reverting level µ > 0 and risky asset price

X > 0. Proofs are given in Appendix A.3.

Property 1. The inter-temporal hedging demand is concave in asset price X. More-

over, the coefficient for the quadratic term X2, 1−γγC(τ), is non-positive for all τ , and

with a plausible choice of λ and r, the coefficient for the linear term X, 1−γγB(τ), is

non-negative for all τ .

Property 1 tells us the sign and magnitude of the inter-temporal hedging demand.

We also like to point out it is possible to show that the B(τ) term represents an

adjustment for a non-zero long-run mean in the risky asset (see [50, 19]). from

Property 1, we can further summarize the sign, magnitude and monotonicity of inter-

temporal hedging demand with respect to asset price X.1Jured and Yand (2007) did a similar analysis for one asset case, but they specialize to the case

of µ = 0. In contrast, we provide analysis for all µ > 0 and γ > 1

47

Property 2. Given fixed horizon τ , the inter-temporal hedging demand is positive

and increasing when X increase from 0 to −B(τ)/(2C(τ)), positive and decreasing

when X increase from −B(τ)/(2C(τ)) to −B(τ)/C(τ), finally negative and decreasing

when X increase from −B(τ)/C(τ) to infinity.

Property 2 directly follows from Property 1. Intuitively, because the asset price is

mean-reverting, a relatively low asset price will cause a risk-averse investor (γ > 1) to

desire a greater allocation to the arbitrage opportunity than implied by the myopic

demand, hence the inter-temporal hedging demand is positive. But when asset price

goes way above certain level, the investor should bet on subsequent drop of the asset

price, hence decrease his holdings or even short the position.

Except for the the sign and magnitude, it’s also interesting to explore the mono-

tonicity of inter-temporal hedging demand as a function of horizon τ . It is a pop-

ular recommendation that investors with longer investing horizon should invest a

higher fraction of wealth into risky assets. However, when asset price follows mean-

reverting process, the situation is much more complicated. By taking derivative of

inter-temporal hedging demand with respect to τ , we have ∂HD/∂τ = 1−γγ

[C ′(τ)X +

B′(τ)]X, where HD represents the inter-temporal hedging demand, C ′(τ) and B′(τ)

are derivatives of C(τ) and B(τ) with respect to τ respectively. We summarize the

results of ∂HD/∂τ in the following property.

Property 3. ∂HD/∂τ , the derivative of inter-temporal hedging demand HD with

respect to horizon τ , is concave in asset price X. The coefficient for the quadratic

term X2, 1−γγC ′(τ) is non-positive for all τ . ∂HD/∂τ ≥ 0 if X ≤ −B′(τ)/C ′(τ) and

∂HD/∂τ < 0 if X > −B′(τ)/C ′(τ).

From Property 1, Property 2 and Property 3, we know that −B(τ)/C(τ) deter-

mines the sign of the inter-temporal hedging demand HD and −B′(τ)/C ′(τ) deter-

mines the sign of ∂HD/∂τ , hence the monotonicity of inter-temporal hedging demand48

with respect to horizon. Therefore, −B(τ)/C(τ) and −B′(τ)/C ′(τ) divide the Asset

Price-Horizon Space ([0,∞] × [0, T ]) into different regions. Within each region, the

sign and monotonicity of inter-temporal hedging demand can be completely deter-

mined, which we summarize in the following property.

Property 4. With a plausible choice of λ and r, −B(τ)/C(τ) > 0 for all τ and

−B(τ)/C(τ) ≥ −B′(τ)/C ′(τ) for all τ with equality achieved only when τ = 0.

Hence, curve X = −B(τ)/C(τ) and X = −B′(τ)/C ′(τ) divide the Asset Price-

Horizon Space ([0,∞] × [0, T ]) into three regions. The sign of HD and ∂HD/∂τ in

each region are shown in Figure.3.1.

0.0 0.2 0.4 0.6 0.8 1.0

τ

0

5

10

15

20

25

Ass

etP

rice

Region 3: HD > 0 ,∂HD

∂τ> 0 Region 2: HD > 0 ,

∂HD

∂τ< 0

Region 1: HD < 0 ,∂HD

∂τ< 0

−B(τ )/C(τ )

−B′(τ )/C ′(τ )

Figure 3.1: Sign and Horizon Dynamics of Inter-temporal Hedging Demand

We now dig into more details of Property 4 in each region shown in Figure.3.1.

First, we fix the risky asset price at X. If X > −B(τ)/C(τ), which automatically

leads to X > −B′(τ)/C ′(τ) as shown in Region 1, the inter-temporal hedging49

demand is negative as the price is expect to drop to long-term mean level, and the

inter-temporal hedging demand will monotonically gradually decay to 0 as horizon

shrinks to 0, which is the same as in Merton’s problem. If X < −B(τ)/C(τ), then

there are two cases. If the horizon is long enough, −B′(τ)/C ′(τ) will eventually go

below zero as τ → T , then X > −B′(τ)/C ′(τ) as shown in Region 2. In this case,

inter-temporal hedging demand is positive as the price is expected to rise to long-term

mean level, but also since horizon is long enough, the inter-temporal hedging demand

can continue to increase even if the horizon shrinks. Finally as the horizon τ goes

to 0, −B′(τ)/C ′(τ) will converge to −B(τ)/C(τ), hence X < −B′(τ)/C ′(τ). In this

situation, the inter-temporal hedging demand is positive and decrease to 0 as τ → 0.

Finally, we analyze the relationship between the myopic demand (MD) and the

inter-temporal hedging demand (HD). The sign of the myopic demand, 1γ

[λ(µ−X)/X−r

(σ/X)2

], is solely depend on the asset price X, but not τ . With simple computation, one

can find whether X exceeds level −g/c determines the sign of MD. Also, notice

−B(0)/C(0) = −g/c (proved in Lemma.A.3.4). Together with Property 4, we can

also divide the Asset Price-Horizon Space ([0,∞]× [0, T ]) into three regions, within

each region, the sign of MD and HD are fully determined. And we summarize the

results in the following property.

Property 5. −B(τ)/C(τ) is decreasing with respect to τ with −B(0)/C(0) = −g/c.

Hence, curve X = −B(τ)/C(τ) and horizontal line X = −g/c divide the Asset Price-

Horizon Space ([0,∞]× [0, T ]) into three regions. The signs of MD and HD in each

region are shown in Figure.3.2.

50

0.0 0.2 0.4 0.6 0.8 1.0

τ

15.36

15.38

15.40

15.42

15.44A

sset

Pri

ce

Region 1: MD < 0, HD < 0

Region 2: MD > 0, HD < 0

Region 3: MD > 0, HD > 0

−B(τ )/C(τ )

−g/c

Figure 3.2: Signs of Myopic Demand and Inter-temporal Hedging Demand

Property 5 tells us that if the risky asset X is high (Region 1) or low (Region

3) enough, the hedging demand will reinforce the myopic demand as they have the

same sign. However, within range X ∈ [−B(τ)/C(τ),−g/c], i.e, when risky asset is

mis-priced, but the price is not high enough (Region 2), MD and HD have opposite

sign, meaning the hedging demand will mitigate the myopic demand. also this range

is shrinking to 0 as τ → 0.

51

3.5 Simulations of Optimal Trading Strategies

3.5.1 Single OU Asset simulation

In this section, we present the simulation results for portfolio optimization strategy

with single risky asset. In particular, we provide the optimal strategy corresponding

to a sample price path, monotonicity of inter-temporal hedging demand at different

price level, average wealth/utility growth path and terminal wealth distribution over

10,000 Monte Carlo simulations.

In Figure.3.3, we simulate one sample path of a single 252-period OU process

using the following parameters: the horizon T = 1, the long-term mean revering level

of the OU process µ is 15.4463, the rate of mean-reversion of the OU process λ is

0.113, which implies a shock half-life of 6.13 days, the daily standard deviation of

the OU shocks σ is 0.606 and the risk-free rate r is fixed at 5% per annum. These

process parameters are calibrated using historical data of VIX front month future

price in year 2013, and scaled such that each period corresponds to one business day.

We use the same set of parameters for all simulations throughout section 3.5.1. The

blue curve (corresponding to y-axis on the right) represents the sample price path of

the risky asset starting from X0 = 12. The red curve (corresponding to y-axis on the

left) represents the optimal strategy, in other words, the optimal proportion of total

wealth to invest in the risky asset, for a CRRA investor with risk averse parameter

γ = 2. The optimal position can be further decomposed into myopic demand and

inter-temporal hedging demand, which are indicated by orange and green dashed

curves (both corresponding to y-axis on the left) respectively. As we can see from the

plot, the optimal strategy roughly behaves like buy-low-sell-high. The inter-temporal

hedging demand has smaller magnitude than the myopic demand and vanishes as

horizon shrinks to zero.

52

050

100

150

200

250

time

151050510

proportion of wealth invested in risky asset

tota

l pos

ition

myo

pic

dem

and

inte

r-tem

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e1213141516171819

asset price

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.

53

0.0 0.2 0.4 0.6 0.8 1.0

τ

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0.00

prop

orti

onof

wea

lth

inve

sted

inri

sky

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t

(a) Asset Price X above −B(τ )/C(τ )

15.0

15.1

15.2

15.3

15.4

15.5

15.6

asse

tpr

ice

0.0 0.2 0.4 0.6 0.8 1.0

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0.00

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0.08

prop

orti

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lth

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sted

inri

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asse

t

(b) Asset Price X below −B(τ )/C(τ )

15.20

15.25

15.30

15.35

15.40

15.45

15.50

15.55

15.60

asse

tpr

ice

0.0 0.2 0.4 0.6 0.8 1.0

τ

−0.020

−0.015

−0.010

−0.005

0.000

0.005

prop

orti

onof

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inve

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inri

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asse

t

(c) Asset Price X cross −B(τ )/C(τ )

Inter-temporal Hedging Demand Fixed Asset Price −B(τ)/C(τ) −B′(τ)/C ′(τ)

15.20

15.25

15.30

15.35

15.40

15.45

15.50

15.55

15.60

asse

tpr

ice

Figure 3.4: Sign and Horizon Dynamics of Inter-temporal Hedging Demand. a) upper:fixed price level at 15, higher than −B(τ)/C(τ) and −B′(τ)/C ′(τ). b) lower left, fixedprice level at 15.3, lower than −B(τ)/C(τ), cross −B′(τ)/C ′(τ). c) lower right, fixedprice level at 15.45, cross −B(τ)/C(τ) and −B′(τ)/C ′(τ).

It is also of interest to study the behavior of inter-temporal hedging demand as

a function of horizon τ if we fix asset price X. As we have discussed in previous

section about the properties of optimal strategy, the relationships between asset

price X and −B(τ)/C(τ) as well as between X and −B′(τ)/C ′(τ) determine the

shape of inter-temporal hedging demand. Figure.3.4 illustrates the inter-temporal

hedging demand with different fixed price levels. In sub-figure (a), the blue and

green curves (corresponding to right-hand-side y-axis) represent −B(τ)/C(τ) and

−B′(τ)/C ′(τ) respectively. Notice they start from the same level, then −B′(τ)/C ′(τ)54

decreases to below zero, whereas −B(τ)/C(τ) also decrease with τ , but remains

positive and above −B′(τ)/C ′(τ). The asset price, represented by the black dashed

line, is fixed at X = 15.45, and above both −B(τ)/C(τ) and −B′(τ)/C ′(τ). The

red curve (corresponding to the left-hand-side y-axis) represents the inter-temporal

hedging demand which is negative because X > −B(τ)/C(τ) and decreasing with

τ because X > −B′(τ)/C ′(τ). In sub-figure (b), the fixed asset price is X = 15.3,

which is lower than −B(τ)/C(τ) for all τ , leading to inter-temporal hedging demand

being positive, but will cross −B′(τ)/C ′(τ) as τ goes from 0 to T , In sub-figure (c),

the fixed asset price is hold at X = 15.4, which will cross both −B(τ)/C(τ) and

−B′(τ)/C ′(τ). When price cross −B(τ)/C(τ), the hedging component changes sign

and when price cross −B′(τ)/C ′(τ), the hedging component changes from increasing

to decreasing with τ .

Figure.3.5 illustrates the average wealth growth path and average utility growth

path over 10,000 Monte Carlo Simulations of the 252-period OU processes using

the same set of parameters adopted in the previous figures. The initial wealth are

normalized to 1 and the initial risky asset price X0 randomly drawn from its station-

ary distribution. The average wealth, represented by the left-hand-side sub-figure,

grows from 1 to 919.32 over 252 periods. The average utility, represented by the

right-hand-side sub figure, grows from 0 to 0.9941 over the same time period.

Figure.3.6 shows the terminal wealth distribution over the aforementioned 10,000

path simulations. The mean (median) terminal wealth is 919.32 (581.46) and the

standard deviation of the terminal wealth is 1398.73; the return distribution has a

skewness (kurtosis) coefficient of 8.31 (139.32).

55

0 50 100 150 200 250time

0

200

400

600

800av

erag

e we

alth

wealth0 50 100 150 200 250

time

0.0

0.2

0.4

0.6

0.8

aver

age

utilit

y

utility

Figure 3.5: Average Wealth Growth and Average Utility Growth over 10,000 MonteCarlo Simulations

0 100000 200000 300000 400000 500000Terminal Wealth

0

2500

5000

7500

10000

12500

15000

17500

20000

Figure 3.6: Terminal Wealth Distribution over 10,000 Monte Carlo Simulations

56

3.5.2 Multiple OU Assets simulation

In this section, we present the simulation results for portfolio optimization strategy

with multiple correlated OU assets. In particular, we simulate three OU assets with

the following parameters:

M = (20, 15, 10)T Λ =

0.1

0.05

0.01

σX =

0.8

0.5

0.3

σρ =

1 0.05 −0.5

0.05 1 0.5

−0.5 0.5 1

(3.91)

The risk free rate remains 5% per annum and risk aversion is increased to γ = 6 due

to limitation of leverage.

Similar to single OU asset simulation, Figure.3.7 presents the optimal proportion

(red) and its two component: myopic demand (orange) and inter-temporal hedging

demand (green) for each asset over a 252-period Monte Carlo simulation, along with

the asset prices(blue).

Figure.3.8 illustrates the average wealth growth path and average utility growth

path over 10,000 Monte Carlo Simulations of 252-period OU processes. The initial

wealth are normalized to 1 and the initial risky asset prices randomly drawn from

the stationary distribution. The average wealth, grows from 1 to 1976.03 over 252

periods. And the average utility, grows from 0 to 0.2 over the same time period.

57

Figure.3.9 shows the terminal wealth distribution over the 10,000 simulations.

The mean (median) terminal wealth is 1976.03 (1579.55) and the standard deviation

of the terminal wealth is 1486.76; the return distribution has a skewness (kurtosis)

coefficient of 2.82 (16.14).

58

050

100

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1510505

proportion invested in risky assetAs

set O

ne

050

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8642024

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Asse

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642024

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Asse

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14161820222426

asset price

101112131415161718

asset price

78910111213

asset price

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59

0 50 100 150 200 250time

0

250

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750

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1250

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1750

2000

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wealth0 50 100 150 200 250

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0.00

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aver

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utility

Figure 3.8: Average Wealth Path and Average Utility Path

0 5000 10000 15000 20000Terminal Wealth

0

200

400

600

800

1000

Figure 3.9: Average Wealth Path and Average Utility Path

60

Chapter 4

Optimal Strategies with

Transaction Cost - Deep Neural

Network Approach

4.1 Learning The No-Trade Zone Using Deep Neu-

ral Network

Taking advantage of mean-reverting dynamics, we aim to simultaneously long and

short several correlated assets that are tradable on the market. However, one has

to pay transaction costs for all the underlying tradables. When transaction costs

are significant, the analytic strategy we derived in the previous section is no longer

optimal.

Most existing studies on optimal asset allocation with transaction costs (see

[8, 9, 68]) assume that asset prices follow the geometric Brownian motion (GBM),

for one-asset case the problem comes down to solving a HJB equation with two

unknown boundaries (free boundaries). The region between the two boundaries is

61

called no-trade (NT) zone. If the current holding position is in NT zone then no

action is required, otherwise one should rebalance positions to the nearest point on

the NT zone.

In order to compute the NT zone when there is transaction cost, we will use deep

neural network (DNN) to approximate the upper and lower boundaries of the NT

zone. We formulate the problem as a supervised learning problem and pick the loss

function to be the difference between expected terminal utility under the policy we

derived analytically in previous section and policy output by the DNN. By minimizing

loss we are able to train the neural network to better approximate boundaries of the

NT zone. When there is transaction cost, the policy learned by DNN significantly

outperforms the analytical policy we derived in Chapter 3.

62

4.2 Single Asset No-Trade Zone Parameterization

When asset price follows the geometric Brownian motion and the investor pick the

CRRA utility function, Liu and Lowestein [68] showed that the lower boundary rl

and upper boundary ru of NT zone are functions of time, i.e., they take the form

rl(t) and ru(t). The values of rl and ru denote the proportion of total wealth to

invest in the risky asset. However, when the asset price is mean reverting, these

two boundaries should also be functions of asset price, hence take the form rl(t,Xt),

ru(t,Xt).

The exact formula of rl(t,Xt) and ru(t,Xt) are unknown. We use neural network

to parametrize these two functions. Neural network can be regarded as a non-linear

parameterization of a function. It has been shown that a feedforward neural network

with a single hidden layer could approximate arbitrary functions to high accuracy

as long as the number of neurons is large enough (see Hornik [37]). It is used as a

non-parametric method in statistics and time series analysis (see Tsay [110]). More

recently, deep neural network has been used to approximate derivatives of unknown

functions and for solving high dimensional non-linear partial differential equations

(see E. et al. [5, 25, 36] and Sirignano and Spiliopoulos [103]). For more details of

feedforward neural network, please refer to Chapter 6 of Goodfellow et al.[30]. When

used for function approximation, we usually feed neural network with training data

xi, yi where yi is the target function value corresponding to independent variable

xi. Iterative methods such as stochastic gradient descent is employed to minimize

the error (for example, least square error) between the target function value yi and

the output of neural network fθ(xi). The goal is to find the best set of parameter θ

that minimizes the loss associated with the training data. An example of a single

layer neural network is given in Figure 4.1.

63

Figure 4.1: A single layer feedforward neural network fθ(x). The input is a m-dimensional vector (x1, ...xm). The parameters θ to be estimated are w1...wm and b.The activation function is f which is usually taken as a Sigmoid function or RectifiedLinear Unit(ReLU). The output of the network is f(b+∑

wixi).

Suppose the investment horizon is [0, T ]. We discretize time horizon into N sub-

intervals with equal length ∆t = tk − tk−1 = TN

. At time tk, the lower and upper

boundaries of NT zone is only a function of Xtk because t is hold fixed at t = tk.

Denote the analytic policy we derived for no transaction cost by π∗(t, x), we can

parametrize the lower and upper boundaries at tk by neural networks:

rtku (x) ≈ π∗(tk, x) + fθutk(x) (4.1)

rtkl (x) ≈ π∗(tk, x)− fθltk (x) (4.2)

Importantly, the parameterization of rtku (x) and rtkl (x) makes use of the prior

knowledge from the analytic policy π∗(tk, x) derived under zero transaction cost as-

sumption. Intuitively, when there is no transaction cost, we should always rebalance

our position to the curve defined by π∗(tk, x) and this means rtku (x) = rtkl (x) =

π∗(tk, x). By increasing transaction cost, the NT zone will widen and π∗(tk, x) should

fall into the NT zone, this amounts to say fθutk (x) and fθltk (x) should take small posi-64

tive values. When implementing the algorithm, we heuristically initialize the output

of fθutk (x) and fθltk(x) to be positive but also allow them to take negative values,

this is achieved by using leaky ReLU function (Xu et al. [115]) on the output layer.

Such parameterization guides the neural network to search solutions with a relatively

small region where the output takes small positive numbers. In contrast, if one does

not incorporate prior knowledge π∗(tk, x) and directly parameterize lower and upper

boundary by

rtku (x) ≈ fθutk(x),

rtkl (x) ≈ fθltk(x),

We have observed in our experiments that it would take much more time to train the

neural network, and it is highly possible that the neural network will not converge.

With parameterization (4.1)-(4.2) via a warm start and prior knowledge, we narrow

down the search region and reduce the likelihood of getting trapped in a local optimal

solution.

65

4.3 One-asset Trading Strategy Parameterization

In the remainder of the paper we constrain the range of π hence rtkl and rtku to

[−1, 1] because rebalancing happens on discrete time points. This constraint could

be relaxed as rebalancing becomes more frequent, in particular there is no constraint

on π in Chapter 3 for continuous rebalancing.

When we arrive at time tk, the proportion we invested in risky asset is πtk−, we

rebalance the proportion to πtk+ with the following policy:

πtk+ =

rtkl (Xtk) πtk− < rtkl (Xtk)

πtk− rtkl (Xtk) ≤ πtk− ≤ rtku (Xtk)

rtku (Xtk) πtk− > rtku (Xtk)

(4.3)

We require πT+ = 0, i.e. we need to clear all the risky asset position at the

end of investment horizon. Yet, it is easy to change the end-horizon position to

any target asset mix. See Grinold [31] for more details.

We explain the policy (4.3) in more details:

• When the proportion πtk− invested in risky asset is below the lower boundary

rtkl (Xtk) of NT zone, we buy more risky asset to rebalance our position πtk+ to

the lower boundary of the NT zone.

• When rtkl (Xtk) ≤ πtk− ≤ rtku (Xtk), our position πtk− is above the lower boundary

rtkl (Xtk) and below the upper boundary rtku (Xtk). We are in the NT zone hence

no action is needed and πtk+ = πtk−.

66

• When πtk− > rtku (Xtk), our position in the risky asset is above the upper bound-

ary therefore we need to sell some risky asset to rebalance our position πtk+ to

the upper boundary of the NT zone.

At a fixed time tk, a typical NT zone and its induced policy is given in Figure 4.2:

Figure 4.2: A typical NT zone and corresponding rebalance operations. The meanreverting level is 15.4463 and transaction cost is 2.0%. i) From point a to a’: pointa is outside the NT zone, we rebalance it to point a’. We convert the position frominvesting 50% of wealth in shorting asset to investing 32% of wealth in longing theasset because the current observed asset price is 14.73 which is way below the meanreverting level 15.4463. ii) From b to b’: b is outside the NT zone, we rebalance it tob’. We convert the position from investing 75% of wealth in longing asset to investing71% of wealth in shorting the asset because the current observed asset price is 16.13which is way above the mean reverting level 15.4463. iii) At c: because point c iswithin NT zone, no rebalance operation is needed.

The NT zone in Figure 4.2 is simulated with parameters estimated from VIX

future in year 2013-2014 and the assumption that there is a 2% transaction cost. The

67

NT zone is for tk = 0.4 when investment horizon is [0, T ] = [0, 1]. The blue curve

denotes the lower boundary of NT zone and orange curve is the upper boundary of

the NT zone. The region between two curves is the NT zone. The mean reverting

level is 15.4463, three scattered points a, b and c correspond to 3 cases we discussed

following policy in (4.3)

• Red point a = (14.73,−0.5). This point means Xtk = 14.73 and the proportion

πtk− we invested in risky asset is -0.5, i.e. we are using 50% of our wealth to

short the asset. However, 14.73 is below the mean reverting level 15.4463 hence

we should switch our position to long the asset, we rebalance our position to

blue point a′ = (14.73, 0.32) on the lower boundary of the NT zone, this means

we clear our short position and put πtk+ = 32% of our wealth to long the asset.

• Orange point c = (15.5, 0.1). This point means Xtk = 15.5 and πtk− = 0.1.

This point lies in the NT zone hence no action is needed. Because 15.5 is close

to mean reverting level, the price change in near future will be mainly driven

by noise, there is no deterministic trend hence we should not rebalance to avoid

transaction cost.

• Green point b = (16.13, 0.75). This point means Xtk = 16.73 and πtk− = 0.75,

we are using 75% of our wealth to long the asset. Because 16.73 is well above

the mean reverting level, we should switch our long position to short position

πtk+ = −0.71 i.e. putting 71% of our wealth to short the asset. We move from

the green point b to the purple point b′ on the upper boundary of the NT zone.

68

4.4 Training DNN for One Asset

Assuming in the deep neural network there is linear transaction cost rate α, then the

transaction cost from each rebalancing operation is:

ctk = αWtk |πtk+ − πtk−| (4.4)

According to continuous time dynamics formula (3.1) and (3.2) for X, Y and W

together with transaction cost formula (4.4), the system dynamics from tk−1 to tk is

given by:

∆tk−1X = Xtk −Xtk−1 = (e−λh)Xtk−1 + µ(1− e−λh) +N(t, h) (4.5)

∆tk−1Y = Ytk − Ytk−1 = ert(erh − 1) (4.6)

∆tk−1W = Wtk −Wtk−1 =πt+k−1Wtk−1

Xtk−1

∆tk−1X +(1− π+

tk−1)Wtk−1

Ytk−1

∆tk−1Y − ctk−1

(4.7)

where

N(t, h) := σe−λ(t+h)∫ t+h

teλudZu ∼ N(0, σ

2(1− e−2λh)2λ )

h := tk − tk−1 = T

N

At each training step, we start at t0 = 0 with n sample paths initialized as:

X0 = [x0, ..., x0]T1×n

W0 = [w0, ..., w0]T1×n

π0− = [0, ..., 0]T1×n

69

At each time step tk our position rebalances from the output of the neural network

rtku (x) and rtkl (x) together with the rebalancing rule (4.3). We then move to the next

time step tk+1 using equations (4.5)-(4.7). At terminal time T , the n× 1 vector WT

represents the terminal wealth over n sample paths, and the empirical loss over n

sample paths is defined as:

loss∆= 1n

n∑i=1

(J(w0, x0, T )− U(W (i)

T ))

(4.8)

where J(W0, X0, T ) is the value function given by equation (3.8)- (3.9) and equation

(3.18)- (3.20).

The computational graph defines the data-flow of the deep neural network (Fig-

ure. 4.3). In the computational graph, ‘Rebalance Rule’ follows equation (4.3), ‘Sys-

tem Dynamics’ follows equations (4.5)-(4.7).

70

Figu

re4.

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71

4.5 Parameterizing Multi-assets NT Zone and

Trading Strategy

Suppose there are d risky assets, the asset prices at time tk is Xtk ∈ Rd, then the

upper and lower boundaries of NT zone is characterized by two d-dimensional vectors

rtku (Xtk) and rtkl (Xtk) ∈ Rd. We parameterize the two boundaries by neural networks

like we did for one-asset case, except the output of the neural network is d-dimensional:

rtku (X) ≈ π∗(tk,Xtk) + fθutk(X) ∈ Rd (4.9)

rtkl (X) ≈ π∗(tk,Xtk)− fθltk (X) ∈ Rd (4.10)

The rebalance rule is similar the rule (4.3) for one-asset case, here subscript (i) denotes

the i-th component:

π(i)tk+ =

rtkl (Xtk)(i) π(i)tk− < rtkl (Xtk)(i)

π(i)tk− rtkl (Xtk)(i) ≤ π

(i)tk− ≤ rtku (Xtk)(i)

rtku (Xtk)(i) π(i)tk− > rtku (Xtk)(i)

(4.11)

Similar to the one-asset case, we add a leverage ratio constraint : ||πt+||∞ ≤ 1,

i.e. the leverage ratio used on each individual asset should not exceed the ±100%

limit.

72

4.6 Training DNN for Multiple Assets

The computational graph for multi-assets case is exactly the same as the one-asset case

in Figure 4.3. We next give the environment dynamics in the computational graph.

We follow Wan [114] to calibrate the parameters of the mean reverting process. The

form of OU process in equation (3.22) and (3.27) simplifies the procedure to derive

the (3.42)-(3.44), but it is hard to calibrate σX and σρ separately. In practice, we

calibrate another matrix ΣOU which contains the information from σX and σρ. We

rewrite the dynamics of the asset price dX as:

dX = µX(X)dt+ ΣOUdW (4.12)

where dW is the differential of a d-dimensional Brownian motion starts at 0 and

satisfies:

dWdW = Id×d

The following lemma proves the equivalence between (4.12) and (3.22), and how to

rewrite the optimal policy in terms of ΣOU . Then it is sufficient to calibrate ΣOU rather

than calibrating σX and σρ. The introduction of σX and σρ simplifies derivation of

the ODEs (3.42)-(3.44). Similarly, the environment dynamics could be written in

ΣOU via equation (4.12) rather than (3.73).

Lemma 4.6.1. Let A be the Cholesky decomposition of σρ, if ΣOU = σXA, then SDE

of (4.12) and (3.22) has the same distribution. The optimal policy still follows (3.86)

73

provided we can rewrite the ΣW , ΣWX and ΣX in terms of ΣOU :

ΣW =

1X1

.

1Xd

ΣOUΣTOU

1X1

.

1Xd

(4.13)

ΣWX =

1X1

.

1Xd

ΣOUΣTOU (4.14)

ΣX = σXAATσX = ΣOUΣT

OU (4.15)

Next, let Nidi=1 be d independent standard normal variables, and σij be the

entries of ΣOU , then by solving the SDE (4.12) together with (3.72) and (3.73), the

dynamics of the system is:

∆tk−1X(i) = X(i)

tk −X(i)tk−1 = (µi −Xtk−1)(1− e−λih) +

d∑j=1

σij

√1− e−2λih

2λiNj (4.16)

∆tk−1Y = Ytk − Ytk−1 = ertk−1(erh − 1) (4.17)

∆tk−1W = Wtk −Wtk−1 = Wtk−1πTtk−1+

∆tk−1XXtk−1

+(1− πTtk−1+1)Wtk−1

Ytk−1

∆tk−1Y − ctk−1

(4.18)

where the transaction cost is

ctk :=d∑i=1

αWtk |π(i)tk+ − π

(i)tk−|

We employ the same method as in Section 3.3 to initialize and generate n sample

paths for training the deep neural network, the only differences are the sizes of Xtk ,

Wtk and πtk± should be n × d rather than n × 1, and the environment dynamics is

given by (4.16)-(4.18) rather than (4.5)-(4.7). The empirical loss is the same as that

74

defined by (4.8):

loss∆= 1n

n∑i=1

(J(w0, x0, T )− U(W (i)

T ))

(4.19)

75

4.7 Neural Network Structure and Training Pro-

cedure

For each time step tk, we build two neural networks for representing functions fθutk (x),

fθltk(x) and the lower/upper boundaries as given by (4.1)-(4.2) or (4.9)-(4.10). Each

neural network is a feedforward neural network comprised of 3 hidden layers, with

[20,40,80] neurons in each hidden layer. Batch Normalization is employed to speed

up the training (Ioffe and Szegedy [48]). We use leaky ReLU (Xu et al. [115])

activation function with leaky coefficient α = 0.2 in each hidden layer, and on the

output layer with leaky coefficient α = 0.05. The reason we are not using identity

function in the last layer is because intuitively lower boundary should below π∗ and

upper boundary should be above π∗, hence the output fθutk (x) and fθltk(x) must be

positive. One way to ensure positive output is to employ ReLU on the last layer, but

this will prevent the gradient backpropogation when the output is negative. Hence

we use leaky ReLU with a small coefficient which allows gradient backpropagation.

The Adam optimizer (Kingma and Ba [51]) is implemented with a decaying learning

rate that starts at 1 and decays by 0.5 every 500 steps. The total number of training

steps is 3000, and for each training step, we generate a batch of 100000 sample paths

to compute the stochastic gradient decent. The neural network is implemented and

trained using Tensorflow, a widely used deep learning tool developed by Google.

At each training step we generate a batch of new training data using (4.5)-(4.7) or

(4.16)-(4.18). More specifically, we generate a batch of independent normal random

variables and feed them into equation (4.5) or (4.16). One could employ an “early

stopping” rule and terminate the algorithm within 3000 training steps, by using a

validation data set. The validation set could be a data set generated before the

training starts, or it could be generated online independent to the training set. If the

76

loss on validation set is stable for several steps or begins to increase, then the training

terminates. After training, one could generate a new batch of sample paths as the

test set. In this dissertation the size of test data equals 10000 sample paths.

77

4.8 Relations Between No-Trade Zone and Trans-

action Cost

For these empirical tests, the parameters are estimated from the data of VIX front

month future in the year 2013 following maximum likelihood estimation [47]:

µ = 15.4463 λ = 0.113× 252 σ = 0.606×√

252

T = 1 N = 50 r = 0.05

x0 = 13.950 w0 = 1000 γ = 2

As shown in Figure 4.4, the NT zone at time tk = 0.4 illustrates how transaction cost

affects the NT zone:

Figure 4.4: No trade zone at t=0.4 under different transaction cost. Larger transac-tion costs leads to wider NT zone.

In Figure 4.4, ‘lb’ denotes the lower boundary and ‘ub’ denotes the upper

boundary of the NT zone, percentage number denotes the transaction cost rate α in

(4.4) varying from 0.7% to 2.8%. As expected, the NT zone becomes wider as the

78

transaction cost increases.

Another way to characterize the NT zone is to consider the 4 points where low-

er/upper boundary hits the +1/-1 limit. These are four vertices of NT zone. By

plotting the price level at which lower/upper boundary hits the limit, we gain insight

of the NT zone evolution over time with different transaction cost. We make several

observations:

• In each subplot, as transaction cost increases, the divergence between the curve

and the 0-transaction-cost curve becomes larger. This means that the NT zone

becomes wider as the transaction cost increases at every time step.

• The price at which lower boundary reaches +1 decreases as time approaches the

end of the horizon, while the price at which upper boundary hits -1 increases.

This phenomenon becomes more obvious as transaction cost gets higher. This

suggests that the investor should be more cautious and less aggressive as the

horizon approaches.

• When the transaction cost increases, the computed curves become more jagged.

This phenomenon could be overcome by expanding the batch size or increase

the training steps. These small spikes happen at prices that deviate much from

the mean reverting level. Because these prices are rarely seen in training hence

the values for the NT zone boundaries at these prices are less accurate as the

values for the NT zone boundaries around the mean reverting level.

79

Figu

re4.

5:Pr

ice

leve

lat

whi

chbo

unda

ryhi

tsth

ele

vera

gera

tiolim

it.a)

Upp

erle

ft:Pr

ice

leve

lat

whi

chlo

wer

boun

dary

hits

+1

limit.

b)U

pper

right

:Pr

ice

leve

lat

whi

chup

per

boun

dary

hits

-1.

c)Lo

wer

left:

Pric

ele

vela

tw

hich

lowe

rbo

unda

ryhi

ts-1

.d)

Lowe

rrig

ht:

Pric

ele

vela

tw

hich

uppe

rbo

unda

ryhi

ts+

1.

80

4.9 Performance Based on Simulation

Assuming there is 2.0% transaction cost, the utility is scaled by subtracting 0.999 and

multiplied by 106 to speed up training. We plot the portfolio performance of the policy

given by neural network and the optimal policy we derived under zero transaction

cost. We can see the policy learned by neural network significantly outperforms the

policy we derived under zero transaction cost.

Figure 4.6: Mean utility over 10000 sample paths. The blue curve is mean utilityover time following the optimal strategy we derived under 0 transaction cost. Theorange curve is the mean utility overtime following the NT zone policy.

81

Chapter 5

Numerical Experiments With

Market Data

In this chapter, we provide numerical experiments with real market data. We first

describe the method of constructing mean-reverting tradables. Although the discov-

ery of pairs is not the focus of this paper, the construction process is important for us

to estimate the transaction cost. Followed by that, we present the backtest results of

a single and multiple risky assets portfolio for year 2014-2017. In particular, we are

able to solve the multi-asset portfolio choice problem with 48 OU assets, 1 risk-free

asset and 50 time steps in 3 hours GPU time, which is considerable faster than tra-

ditional numerical methods. Finally, we summarize computational cost of portfolios

containing different number of assets and conjecture the approximate the complexity

of DNN method.

82

5.1 Creating Mean Reverting Asset

In practice, we slightly modified the method employed by Avellaneda and Lee [2] to

create a mean reverting assets by longing and shorting correlated tradable assets on

the market. We regress the stock price on the ETF price and model the residual as a

mean-reverting process. The stock and ETF daily prices are obtained from Thomson

Reuters Datastream. Suppose the stock price is St, the relationship between stock

price and ETF follows:

St = β × ETFt + εt

By calculating β using past 60 days data on St and ETFt, we found that the residual

εt is mean reverting around 0 for a variety of stocks. Then one share of mean reverting

asset X is created by longing one share of stock and short β share of ETF. To avoid

negative value of X, we pretend its value is actually St − β × ETFt + C where

C is the constant amount of extra cash we have to put into one share of X, i.e.

Xt = St−β×ETFt +C. As long as C is large enough, Xt will never go below 0. We

let C = 20 in our experiment. Assume the transaction cost rate for 1 share of ETF

or 1 share of stock is 40bps.

In Section 5.2, we use stock JPM and XLF ETF to construct the mean reverting

asset. There are two sources of transaction cost:

• The carry cost of risky asset X. The is the transaction cost associated with

changing β. Each day we recalibrate β using past 60 days’ data, the recalibrated

β changes from βt− to βt+ hence for each share of X that we are holding, we

need to pay |βt− − βt+| ×ETFt × 0.004 amount of transaction fee. Usually the

daily change |βt−− βt+| is around 0.02, the ETF price is around $20. Then the

daily transaction cost for maintaining 1 share of X is around 20×0.02×0.0004 =

0.0016. Most of the time our wealth takes value in [1000, 2000] hence the number83

of shares of X we are longing or shorting is between [50, 75], the total carry cost

of X should not exceed $0.0016× 75 = $0.12 per day.

• The transaction cost to long or short one share of X. To long or short one

share of X, we need to long (respectively, short) 1 share of stock and short

(respectively, long) β share of ETF, the total transaction cost of this operation

will be St×0.004+|β|×ETFt×0.004. Recently St ≈ 100 and ETFt ≈ 20, hence

the transaction cost of creating 1 share of X is around (100 + 20 ∗ 3)× 0.004 ≈

0.064, given the price Xt ≈ 20, this amounts to 0.06420 ≈ 3.2% to 1 share of X.

In our experiments, we take into account two kinds of transaction cost discussed

above. In particular, we assume the daily carry cost of each share of X is $0.00016

and the transaction cost for each share of X is 3.2% for all mean reverting assets.

84

5.2 Experiment with One Asset

As an illustrative example, we combine stock JPM and XLF ETF to create the mean

reverting asset, and backtest the approach with data for the year 2014-2017. At the

start of each year, the parameters are calibrated using historical data from 2009 to the

previous year. The investment horizon is one year, with daily closing price and time

step to be one trading day. The starting wealth of each year is 1000. The risk-free

rate is r = 0.05 and the risk aversion coefficient is γ = 2. Parameters µ, λ and σ for

the OU process are estimated using maximum likelihood estimation (MLE) [47].

85

Figu

re5.

1:Ba

ckte

stre

sult

for

year

2014

.T

hem

ean

reve

rtin

gpa

iris

crea

ted

usin

glis

ted

stoc

kJP

Mor

gan

Cha

se(J

PM)

and

XFL

ETF.

a)C

umul

ativ

ewe

alth

.b)

Mea

nre

vert

ing

leve

lest

imat

edus

ing

hist

oric

alda

taan

das

set

pric

e.c)

Perc

enta

gein

vest

edin

risky

asse

t.

The out of sample results for the year of 2014 is plotted in Figure 5.1. By com-

paring b) and c) in Figure 5.1, we have following observations:

86

• At the beginning we short the asset because its price is more than $1 above its

mean level.

• Starting in April we shift our position from fully short the asset to fully long

the asset. This change is due to asset price dropping from about $1 above mean

level to $3 below mean level.

• In July we change our position from longing the asset to shorting roughly about

50% of our wealth in the asset. This is because asset price goes above the mean

reverting level in July, however the price is not high enough for us to fully

short the asset. From July onwards, the asset price oscillates around the mean-

reverting level hence we only rebalance the position slightly.

The backtest results for 2015-2017 have similar behaviours:

87

Figu

re5.

2:Ba

ckte

stre

sult

for

year

2015

.T

hem

ean

reve

rtin

gpa

iris

crea

ted

usin

glis

ted

stoc

kJP

Mor

gan

Cha

se(J

PM)

and

XFL

ETF.

a)C

umul

ativ

ewe

alth

.b)

Mea

nre

vert

ing

leve

lest

imat

edus

ing

hist

oric

alda

taan

das

set

pric

e.c)

Perc

enta

gein

vest

edin

risky

asse

t.

88

Figu

re5.

3:Ba

ckte

stre

sult

for

year

2016

.T

hem

ean

reve

rtin

gpa

iris

crea

ted

usin

glis

ted

stoc

kJP

Mor

gan

Cha

se(J

PM)

and

XFL

ETF.

a)C

umul

ativ

ewe

alth

.b)

Mea

nre

vert

ing

leve

lest

imat

edus

ing

hist

oric

alda

taan

das

set

pric

e.c)

Perc

enta

gein

vest

edin

risky

asse

t.

89

Figu

re5.

4:Ba

ckte

stre

sult

for

year

2017

.T

hem

ean

reve

rtin

gpa

iris

crea

ted

usin

glis

ted

stoc

kJP

Mor

gan

Cha

se(J

PM)

and

XFL

ETF.

a)C

umul

ativ

ewe

alth

.b)

Mea

nre

vert

ing

leve

lest

imat

edus

ing

hist

oric

alda

taan

das

set

pric

e.c)

Perc

enta

gein

vest

edin

risky

asse

t.

In the backtest, despite the carry cost and high transaction cost, the strategy

steadily generates positive returns over the last 4 years. Rebalancings are mostly

90

taken at “reasonable time”, i.e. fully short around the local high price and fully short

at the local low price. Rebalancings are sparse due to the NT zone is derived from

the deep neural network. These decisions are refined by means of the ODE system

without transaction costs.

91

5.3 Experiment with Multi-Asset Portfolios

We form a portfolio of 48 stocks together with their corresponding ETFs (16 ETFs

in total) to construct the mean reverting assets as we did in previous section. The

information of stock-ETF pairs are summarized in Appendix C. Assuming a 3.2%

transaction cost for each asset, and the same parameters of r = 0.05 and γ = 2 as we

did in one-asset case, we employ a fully connected feed-forward neural network with

three hidden layers, where the number of neurons are [64, 128, 256]. Note that the

number of parameters to simulate the computations in each neural network is about

8 times more than for single-asset case. Instead of building one neural network for

each trading day, we build one neural network for every 5 trading days, with the same

NT zone values for each day between weekly rebalancing date. The initial wealth is

W0 = 1000. The output of the neural network is a 16-dimensional vector. Moreover,

leverage constraints are added so that ||π||∞ ≤ 0.5, i.e. the long or short position in

each asset should not exceed 50% of the total wealth, so the maximum leverage should

not exceed 8 times of the current wealth. In practice the total leverage is much less

than 800% because the short and long positions in different assets offset each other.

We estimate Λ and ΣOU in the OU process (4.12) according to the method in Wan

[114]. Similar with the single asset case, parameters for each year are estimated from

the historical data of previous year with except that parameters used for year 2014

backtest are estimated from historical data of 2010-2013. The estimated parameters

are summarized in Appendix C. From Figure 5.5, we can see the strategy learned by

deep neural network steadily generates profits.

92

Figu

re5.

5:C

umul

ativ

ewe

alth

whe

nba

ckte

stus

ing

48m

ean

reve

rtin

gas

sets

from

2014

to20

17.

93

5.4 Computation Time

We carry out backtest on different number of assets and record the total computation

time needed. The experiment is implemented with TensorFlow-GPU and run on

Princeton University’s TigerGPU Server with one NVDIA P100 GPU node.

# of Assets Layer 1 Layer 2 Layer 3 Time Used (HH:MM:SS)

16 64 128 256 3:24:35

32 128 256 512 5:37:05

48 128 256 512 6:34:44

Table 5.1: Number of parameters and computation time used for different number ofassets.

The dimension of input and output layer of each Neural Network is the same as

number of assets. In the above table, Layer i means the i-th hidden layer. Table. 1

shows the computation time scales polynomially (if not linear) to the dimension of

the problem and our approach could be applied to high dimensional problem.

94

Chapter 6

Conclusion

In this paper, we present a deep-learning method for solving the portfolio optimiza-

tion problem where the underlying assets follow Ornstein-Uhlenbeck processes and

transaction costs are not negligible. We formulated this as a supervised learning

problem using deep neural network to approximate the boundaries of the NT zones

and based on the ODE solutions for the no transaction cost case.

Backtest with real data for year 2014-2017 shows the strategy performs well

in both one-asset and multi-asset cases, yielding about 35% annual return in the

single asset case and doubling the initial wealth each year in the multi-asset case.

Importantly, our method based on DNN enjoys superb run-time efficiency. It does

not suffer from curse of dimensionality as other conventional numerical methods.

Hence the trading strategy can be extended to portfolios with a large number of

assets. This opens a door for future research opportunities as one can further test out

other promising neural network architectures or to combine with the reinforcement

learning to explore better dynamic trading strategies as we will discuss further in

details in the next chapter.

95

We would like to emphasize the importance of combining analytical solutions with

deep learning techniques. Without the theoretical results as a proper “guidance”, the

neural network will stuck in a local optimum far from the true optimal solution.

Without neural network’s high flexibility in functional approximation, traditional

numerical methods cannot solve for a function in high-dimensions in reasonable time.

To link them together is the key to the successful results in this dissertation.

96

Chapter 7

Future Work

The general framework we developed in Chapter 2 establishes a powerful founda-

tion for studying dynamic asset allocation problems with transaction costs in high-

dimensions. In this section, we discuss future works will be pursued from the following

three perspectives:

7.1 Refining and Extending the Current Model

There are several ways we can refine the current DNN model to achieve higher time

efficiency and better prediction results. First, we can accelerate the training process

by using the trained n-th NN’s parameters as a warm start for training n+ 1-th NN,

as intuitively the two consecutive NNs should have similar parameters as the NTZ

boundaries for consecutive days will not change drastically. We can also develop

more efficient NN architectures. The current DNN architecture consists of a series

of m 3-layer feed-forward NNs, with each NN predicting trading strategies for one

or a few consecutive days. For a high-dim k-assets portfolio strategy with long

horizon, training m DNN, when m or k is large, will be extremely costly in time and

space. One possibility is to use Recurrent Neural Networks (RNNs) or Long Short

Term Memory networks (LSTMs) to simplify the current feed-forward architecture,97

improve the time efficiency while keep the predicability, Finally, we can also employ

robust estimation of the parameters. The profitability of the analytical portfolio

strategy under no transaction cost highly rely on the accuracy of the estimated

parameters (see [50]), hence the strategy will incur loss due to inaccurate estimation

of the parameters of the price dynamics.

Besides refining the current model, the framework of combing analytical solutions

with deep learning is very generic and can be easily extended. In Chapter 3, we

specifically solve a portfolio optimization problem consisting OU assets. In fact,

the analytical solutions of the portfolio optimization problem under no transaction

cost assumption can be attained for a broad class of price dynamics, including but

not limited to asset returns governed by factors following OU process, asset prices

following exponential OU process and portfolios consisting inflation factor etc. All

those problems becomes analytically intractable once transaction cost is included.

Hence, a similar approach can be utilized in those problems.

Further, to take into account sudden changes in the asset dynamics such as a

financial crisis, regime shifting can be added as well. In general, there are two types of

regimes shifting models. The first one assumes observable market states. Valdez and

Vargiolu (2013) derive the solution of the classical Merton problem when the risky

assets follow geometric Brownian motions with switching coefficients. Their results

are straightforward: the optimal allocation is a fix mix allocation depending on the

current observed regime. The other one, which is also more realistic, assumes the

investor does not know the actual market state but can only observe the noisy output

such as asset prices. In this case, regimes are often modeled by Hidden Markov Model

(HMM). Elliott et al. (2010) considers the Markowitz mean-variance problem under

a hidden Markovian regime-switching Black-Scholes-Merton economy. In particular,

98

the drift of the risky asset is modulated by a continuous-time, finite-state hidden

Markov chain whose states represent different states of an economy. They provide

the explicit solution to the mean-variance problem using the stochastic maximum

principle. Honda (2003) studies the Merton problem with the drift of the risky asset

modeled as a continuous-time Markov chain. He solved the problem for log utility

U(W ) = log(W ) and power utility U(W ) = Wα

αwith parameter α = 1

2 . Interest-

ingly for the latter case, the optimal allocation not only contains the single-period

mean-variance allocation, but also an additional hedging demand depending on the

difference of expected returns and the probability of regimes. Yet, adding linear

transaction cost makes those aforementioned problems again intractable analytically

and numerically. With deep learning, we can numerically solve these problems in the

same way we did in chapter 4.

99

7.2 Applying the Core Concept to Reinforcement

Learning.

The portfolio optimization problem we studied in this dissertation can be naturally

formulated as a reinforcement learning problem. Take the algorithm of Q-learning

as an example. We can construct a deep Q-network (DQN) to learn the optimal

action-value function Q∗(S, a) (also called Q-function), where state S is a tuple of

horizon, asset price, wealth and current proportion of wealth in the risky assets;

and action a is the proportion of wealth invested in risky asset in the next period.

Intuitively, one can parameterize an approximate value function Q(S, a; θ), where

θ are the parameters of DQN, simulate thousands of price paths, and then let

the DQN learn the optimal Q-function. (For the details of the DQN architecture

and Q-learning algorithm, see [79]) However, we have found out that with random

initialization, DQN cannot converge to a solution close to optimum, hence a more

sophisticated initialization is needed. This is where the analytical solution comes

into play.

To apply the concept, first notice that the Q-function under no transaction cost

can be derived from the analytical solution:

Q(S, a) = Q(Xt,Wt, at?1, at)

= E(J(τ − δ,Xt+δ,Wt+δ))(7.1)

where J(τ − δ,Xt+δ,Wt+δ) is the indirect utility function. Then, we train a NN with

the known Q-function (7.1) via supervised learning. If we increase the transaction

cost from zero by a small amount, one can conjecture that the Q function will not

change much. Hence, we can initialize the DQN Q(S, a; θ) as the trained NN from

100

the previous step and then begin the Q-learning. By adding a pre-training step, the

DQN is more likely to converge to the optimum.

101

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110

Appendix A

Technical Details: Proofs and

Computations

A.1 Solution to single-asset HJB equation

As mentioned in section 2, we assume indirect utility J takes the following ansatz:

J = (Wφ(τ,X))1−γ − 11− γ

φ(τ,X) = expA(τ) +B(τ)TX + 12C(τ)X2

(A.1)

we first derive the partial derivatives:

JW = φ(τ,X)(Wφ(τ,X))−γ

JWW = −γφ(τ,X)2(Wφ(τ,X))−γ−1

JX = (Wφ(τ,X))1−γ[B(τ) + C(τ)X]

JXX = (Wφ(τ,X))1−γC(τ) + (1− γ)[B(τ) + C(τ)X]2

JWX = (1− γ)φ(τ,X)(Wφ(τ,X))−γ[B(τ) + C(τ)X]

Jτ = (Wφ(τ,X))1−γ[A′(τ) +B′(τ)X + 12C′(τ)X2]

(A.2)

111

The optimal allocation π∗ becomes:

π∗ = (−λ+ r

γσ2 X2 + λµ

γσ2X) + 1− γγ

[B(τ) + C(τ)X]X (A.3)

Substitute (A.2) and (A.3) into (3.5), we can obtain the following partial differential

equation after simplification:

−[A′(τ) +B′(τ)X + 1

2C′(τ)X2

]+[λ(µ−X)− rX

][(−λ+ r

γσ2 X + λµ

γσ2 ) + 1− γγ

(B(τ) + C(τ)X)]

+ r

−σ2γ

2[(−λ+ r

γσ2 X + λµ

γσ2 ) + 1− γγ

(B(τ) + C(τ)X)]2

+ λ(µ−X)[B(τ) + C(τ)X

]+σ

2

2

C(τ) + (1− γ)

[B(τ) + C(τ)X

]2+σ2(1− γ)

[(−λ+ r

γσ2 X + λµ

γσ2 ) + 1− γγ

(B(τ) + C(τ)X)][B(τ) + C(τ)X

]= 0

(A.4)

Left-hand-side of Eq.(A.4) is a quadratic equation of X. Right-hand-side being 0

forces the coefficients of quadratic term, first term and the constant term to be 0. To

be specific, the quadratic term coefficient is:

−12C′(τ)− (λ+ r)

[− λ+ r

γσ2 + 1− γγ

C(τ)]− σ2γ

2 (−λ+ r

γσ2 + 1− γγ

C(τ))2

−λC(τ) + (1− γ)σ2

2 C(τ) + σ2(1− γ)[− λ+ r

γσ2 + 1− γγ

C(τ)]C(τ) = 0

(A.5)

the first term coefficient is:

−B′(τ) +[− (λ+ r)

( λµγσ2 + 1− γ

γB(τ)

)+ λµ

(λ+ r

γσ2 + 1− γγ

C(τ))]

−σ2γ[( λµγσ2 + 1− γ

γB(τ)

)(λ+ r

γσ2 + 1− γγ

C(τ))]

+λ(µC(τ)−B(τ)

)+ σ2(1− γ)B(τ)C(τ)

+σ2(1− γ)[(λ+ r

γσ2 + 1− γγ

C(τ))B(τ) +

( λµγσ2 + 1− γ

γB(τ)

)C(τ)

]= 0

(A.6)

112

The constant term is:

−A′(τ) +[λµ( λµγσ2 + 1− γ

γB(τ)

)+ r

]− σ2γ

2( λµγσ2 + 1− γ

γB(τ)

)2+ λµB(τ)

+σ2

2[C(τ) + (1− γ)B(τ)2

]+ σ2(1− γ)

[B(τ)

( λµγσ2 + 1− γ

γB(τ)

)]= 0

(A.7)

By rearranging Eq. (A.5), (A.6) and (A.7), we can obtain the following ODE system:

C ′(τ) = aC2(τ) + bC(τ) + c

B′(τ) = aB(τ)C(τ) + b

2B(τ) + dC(τ) + g

A′(τ) = a

2B(τ)2 + dB(τ) + σ2

2 C(τ) + (λµ)2

2γσ2 + r

(A.8)

with boundary condition A(0) = B(0) = C(0) = 0 and parameters:

a =1− γγ

σ2, b =2(γr − r − λ)

γ

c =(λ+ r)2

γσ2 , d =λµ

γ, g = −λµ(λ+ r)

γσ2

(A.9)

We start solving the ODE system (A.8) from the first equation which only contains

C(τ). Notice it can be solved by solving the following integral:

∫ τ

0

dC

aC2(τ) + bC(τ) + c= τ (A.10)

Following Kim and Omberg (1996), there are three sets of solutions for A(τ),

B(τ) and C(τ) based on the sign of q = b2 − 4ac and b.

113

Normal Solution

q = b2−4ac > 0 , which is equivalent to γ > 1−λ2/r2, Notice we only consider γ > 1

in the main text, which falls into this category. Define:

η =√b2 − 4ac (A.11)

Then rewrite Eq. (A.10) as:

1η

∫ τ

0

1C − −b−η2a

− 1C − −b+η2a

dC = τ (A.12)

Together with boundary condition C(0) = 0, we can solve for C(τ):

C(τ) = 2c(1− e−ητ )2η − (b+ η)(1− e−ητ ) (A.13)

Then we continue to solve for B(τ) in the second equation in the ODE system (A.8),

which can be written as:

B′(τ)−[aC(τ) + b

2]B(τ) = dC(τ) + g (A.14)

We denote:p(τ) = aC(τ) + b

2f(τ) = dC(τ)

(A.15)

Then we compute:

µ(τ) = e∫p(τ)dτ = eητ/2

[2η − (b+ η)(1− e−ητ )

]∫µ(τ)f(τ)dτ =

[−4grη

+ 2g]eητ/2 +

[−4grη− 2g

]e−ητ/2

(A.16)

114

Hence, B(τ) can be derived as:

B(τ) = 1µ(τ)

( ∫µ(τ)f(τ)dτ + constant

)(A.17)

Together with boundary condition B(0) = 0, we can solve for B(τ):

B(τ) = −4gr(1− e−ητ/2)2 + 2gη(1− eητ )η[2η − (b+ η)(1− e−ητ )] (A.18)

Finally, A(τ) can be easily solve by integrating the last equation in (A.8):

A(τ) =∫ a

2B(τ)2 + dB(τ) + σ2

2 C(τ) + (λµ)2

2γσ2 + rdτ (A.19)

Hyperbolic Solution

q = b2 − 4ac = 0 and b 6= 0, which is equivalent to γ = 1 −λ2

r2 and λ < r. The

hyperbolic solutions are:

C(τ) =−1

a(τ − 2/b)−b

2a

B(τ) =1

τ − 2/b

[2ae− bd4a τ 2 − 2e

bτ]

A(τ) =∫ a

2B(τ)2 + dB(τ) + σ2

2 C(τ) + (λµ)2

γσ2 + rdτ

(A.20)

115

Tangent Solution

q = b2 − 4ac < 0, which is equivalent to γ < 1−λ2

r2 and λ < r. Define:

η =√−q

ψ = η

2φ = tan−1( b

η)

(A.21)

The tangent solutions are:

C(τ) = η

2a tan (ψτ + φ)− b

2aB(τ) = 1

cos (ψτ + φ)

[− bη

2aψ (cos (ψτ + φ)− cosφ) + 2ae− bd2aψ (sin (ψτ + φ)− sinφ)

]

A(τ) =∫ a

2B(τ)2 + dB(τ) + σ2

2 C(τ) + (λµ)2

γσ2 + rdτ

(A.22)

116

A.2 Solution to multi-asset HJB equation

Similar to single asset case, we assume indirect utility J has the following format:

J = (Wφ(τ,X))1−γ − 11− γ

φ(τ,X) = expA(τ) +B(τ)TX + 12XTC(τ)X

(A.23)

The partial derivatives of J of are:

JW = φ(τ,X)(Wφ(τ,X))−γ

JWW = −γφ(τ,X)2(Wφ(τ,X))−γ−1

JX = (Wφ(τ,X))1−γ[B(τ) + C(τ)X]

JXXT = (Wφ(τ,X))1−γC(τ) + (1− γ)

[B(τ) + C(τ)X

][B(τ) + C(τ)X

]TJWX = (1− γ)φ(τ,X)(Wφ(τ,X))−γ[B(τ) + C(τ)X]

Jτ = (Wφ(τ,X))1−γ[A′(τ) +B′(τ)TX + 12XTC ′(τ)X]

(A.24)

Hence, the optimal allocation π∗ becomes:

π∗ = 1γ

Σ−1W (µW − r1) + 1− γ

γΣ−1W ΣWX [B(τ) + C(τ)X] (A.25)

Substitute (A.24) and (A.25) into (3.33), we can obtain the following partial differ-

ential equation after simplification:

−[A′(τ) +B′(τ)TX + 1

2XTC ′(τ)X]

+ 12γ (µW − r1)TΣ−1

W (µW − r1) + r

+1− γγ

(µW − r1)TΣTWXΣ−1

W

[B(τ) + C(τ)X

]+ µTX

[B(τ) + C(τ)X

]+1− γ

2γ[B(τ) + C(τ)X

]TΣX

[B(τ) + C(τ)X

]+ 1

2Tr

ΣXC(τ)

= 0

(A.26)

117

Again, left-hand-side of the equation is a quadratic equation of X, right-hand-side is

0, which means the coefficients of quadratic term, first term and the constant term

have to vanish. The quadratic term is:

−12XTC ′(τ)X + 1

2γ

XT (Λ + r1)TΣ−1

X (Λ + r1)X− 1− γ

γXT (Λ + r1)TC(τ)X

+(1− γ)2

2γ XC(τ)TΣXC(τ)X−XΛCX + 1− γ2 XC(τ)TΣXC(τ)X = 0

(A.27)

The first term is:

−B′(τ)X− 1γ

(ΛM)TΣ−1X (Λ + r1)X + 1− γ

γ

MTΛC(τ)X−B(τ)T (Λ + r1)TX

+(1− γ)2

γB(τ)TΣXC(τ) +MTΛTC(τ)X−B(τ)TΛX + (1− γ)XC(τ)TΣXB(τ) = 0

(A.28)

The constant term is:

−A′(τ) + 12γ (ΛM)TΣ−1

X (ΛM) + 1− γγ

MTΛB(τ) + (1− γ)2

2γ B(τ)TΣXB(τ)

+r +MTΛB(τ) + 1− γ2 B(τ)TΣXB(τ) + 1

2Tr(ΣXC(τ)) = 0(A.29)

After further simplification, we can obtain the ODE system:

C ′(τ) =1− γγ

C(τ)TΣXC(τ)− 1γ

[ΛC(τ) + C(τ)TΛ

]−2r(1− γ)

γC(τ) + 1

γ(Λ + rI)TΣ−1

X (Λ + rI)(A.30)

B′(τ) =[1− γ

γΣXC(τ)− 1

γΛ− 1− γ

γrI]TB(τ)

−1γ

(Λ + rI)TΣ−1X ΛM + 1

γC(τ)TΛM

(A.31)

A′(τ) = 12γ (ΛM)TΣ−1

X (ΛM) + 1γMTΛB(τ)

+1− γ2γ B(τ)TΣXB(τ) + r + 1

2Tr[ΣXC(τ)

] (A.32)

118


A(0) = 0, B(0) = 0, C(0) = 0 (A.33)

119

A.3 Proof of Optimal Strategy Properties

To prove Property 1, we only need to prove the following two lemmas and the rest

will follow.

Lemma A.3.1. Assume γ > 1 and µ > 0, the coefficient function C(τ) ≥ 0 for all

τ ≥ 0 with equal sign only attained at τ = 0.

Proof. Since we havec = (λ+ r)2

γ> 0

1− e−ητ ≥ 0 ∀ τ ≥ 0(A.34)

with equal sign only attained at τ = 0. Therefore the numerator of C(τ) is :

2c(1− e−ητ ) ≥ 0 ∀ τ ≥ 0 (A.35)

with equal sign only attained at τ = 0. With choice of γ > 1, we can easily see that:

η =√b2 − 4ac > |b| (A.36)

Hence the denominator of C(τ):

2η − (b+ η)(1− e−ητ ) ≥ 2η − (b+ η)

= η − b > 0(A.37)

which completes the proof.

Lemma A.3.2. Assume γ > 1, µ > 0 and λ > r, the coefficient function B(τ) ≤ 0

for all τ ≥ 0 with equal sign only attained at τ = 0.

120

Proof. First, rewrite B(τ) as:

B(τ) = −2gη

2r(1− eητ/2)2 − η(1− e−ητ )2η − (b+ η)(1− e−ητ )

= −2gηB(τ)

(A.38)

where −2g/η > 0 and the denominator of B(τ) is positive for all τ ≥ 0 from

Lemma.A.3.1. The numerator of B(τ) can be rearranged as:

2r(1− eητ/2)2 − η(1− e−ητ )

= 2r(1− eητ/2)2 − η(1− e−ητ/2)(1 + e−ητ/2)

≤ (2r − η)(1− eητ/2)2

(A.39)

where:η =√b2 − 4ac

= 2γ

√r2γ2 + γ(λ2 − r2)

>2γrγ = 2r

(A.40)

The last strict inequality follows from assumption λ > r. Therefore, the numerator

of B(τ) is non-positive for all τ ≥ 0 and only equals 0 when τ = 0.

Following Lemma A.3.1 and Lemma A.3.2, the concavity of inter-temporal hedg-

ing demand in asset price X easily holds. as (1 − γ)/γ < 0 for risk averse (γ > 1)

investors. We also would like to point out that the assumption of λ > r is reasonable.

To see this, assume the annualized risk-free rate r is 5%. If the mean-reverting speed

λ equals r, it would imply that the half-life of shocks to the OU process would have

to roughly be equal to L = ln(2)/0.05 ≈ 13.86 years. With this half-life, however,

it would be essentially impossible to distinguish a mean-reverting process from a

random walk given a plausible value for the standard deviation of the shocks.

121

Property 2 directly follows from Property 1, except that we still need to prove

the existence of B(0)/C(0), as the quantity is a zero-divided-by-zero, it equals

B′(0)/C ′(0) by L’Hospital’s Rule, and we put related results in Lemma.A.3.4.

To establish Property 3, the concavity of ∂HD/∂τ , we only need to prove the

following lemma:

Lemma A.3.3. Assume γ > 1, µ > 0, the coefficient function C ′(τ) > 0 for all

τ ≥ 0.

Proof.

C ′(τ) = 4cη2e−ητ[2η − (b+ η)(1− e−ητ )

]2 > 0 (A.41)

In order to establish Property 4, we need to prove the following two lemmas:

Lemma A.3.4. Assume γ > 1, µ > 0, then −B(τ)/C(τ) > 0 for all τ ≥ 0.

Proof. If τ > 0, by Lemma A.3.1 and Lemma A.3.2,

C(τ) > 0

B(τ) < 0(A.42)

therefore,

−B(τ)/C(τ) > 0 (A.43)

If τ = 0, then C(0) = B(0) = 0. By L’Hospital’s Rule,

−B(0)C(0) = −B

′(0)C ′(0)

= − g

cη

[− 2( b

η+ 1) + 3( b

η+ 1) + η

r− 1 + 1− b

η

]= −g

c> 0

(A.44)

122

which completes the proof.

Lemma A.3.5. Assume γ > 1, µ > 0 and λ > r, then −B(τ)/C(τ) ≥ −B′(τ)/C ′(τ)

for all τ ≥ 0 with equality achieved only when τ = 0.

Proof. let x denote e−ητ/2.

B(τ)C(τ) −

B′(τ)C ′(τ)

= − g

cη

[η − 2r(1− x)

1 + x

]+ g

cη2

[r(b+ η)x+ (η2 − 2br) + r(b− η)

x

]= − g

cη

[2brη− r( b

η+ 1)x− 2r(1− x)

1 + x− r(b/η − 1)

x

]= − g

cη

[r(1− b

η)(−2 + x+ 1

x) + 2x(1− x)

1 + x

](A.45)

Then we analyze the sign of each term in Eq. (A.45):

− g

cη> 0 (A.46)

By Lemma A.3.1,

η > |b| ⇒ 1− b

η> 0 (A.47)

Finally, since x ∈ (0, 1], we have:

x+ 1x− 2 ≥ 0

2x(1− x)1 + x

≥ 0(A.48)

Together with Eq.(A.46), (A.47) and (A.48), we have:

B(τ)C(τ) −

B′(τ)C ′(τ)

=− g

cη

[r(1− b

η)(−2 + x+ 1

x) + 2x(1− x)

1 + x

]≥ 0

(A.49)

123

It is easy to see that in Eq.(A.48), the equalities are achieved when x = 1, leading to

Eq.(A.49) being tight only when x = 1, in other words, τ = 0.

To establish Property 5, notice the we only need to prove the decreasing property

of −B(τ)/C(τ) (stated in Lemma.A.3.6). The rest will follow from Lemma.A.3.4 and

Lemma.A.3.6.

Lemma A.3.6. −B(τ)/C(τ) ≥ −B′(τ)/C ′(τ) for all τ ≥ 0 with equality achieved

only when τ = 0.

Proof. We can rewrite −B(τ)/C(τ) as:

−B(τ)C(τ) = −g

c+ 2gr

cη

1− x1 + x

= −gc

+ 2grcη− 4gr

cη

11 + eητ

(A.50)

Notice,−4grcη

> 0 and 11+eητ is a decreasing function of τ , which completes the proof.

124

A.4 Proof of Multi-Asset Environment Dynamics

Lemma A.4.1. Let A be the Cholesky decomposition of σρ, if ΣOU = σXA, then

SDE of (4.12) and (3.22) has same distribution. The optimal policy still follows

(3.86) provided we can rewrite the ΣW , ΣWX and ΣX in terms of ΣOU :

ΣW =

1X1

.

1Xd

ΣOUΣTOU

1X1

.

1Xd

(A.51)

ΣWX =

1X1

.

1Xd

ΣOUΣTOU (A.52)

ΣX = σXAATσX = ΣOUΣT

OU (A.53)

Proof. By equation 3.27 we have:

dZdZ = σρdt

We also have:

dWdW = Id×ddt

Hence if we let A be the Cholesky decomposition of σρ, AAT = σρ, we have:

dZ d= AdW

where d= means equal in distribution. Matching the term ΣOUdW term in (4.12) and

σXdZ in (3.22) becomes:

ΣOUdW = σXdZ = σXAdW

125

which is equivalent to:

ΣOU = σXA (A.54)

To prove (A.51)-(A.53), we use (??) and (A.54):

ΣW = σWσρσTW

= σWAATσTW

=

1X1

.

1Xd

σXAATσTX

1X1

.

1Xd

=

1X1

.

1Xd

ΣOUΣTOU

1X1

.

1Xd

ΣWX = σWσρσ

TX

=

1X1

.

1Xd

σXAATσX

=

1X1

.

1Xd

ΣOUΣTOU

ΣX = σXσρσTX

= σXAATσTX

= ΣOUΣTOU

126

Appendix B

Parameter Estimation

B.1 Parameter Estimation of univariate mean-

reverting process

In this section, we state maximum likelihood estimation the parameters of the

Ornstein-Uhlenbeck model. This process solves the stochastic differential equation

dXt = λ(µ−Xt)dt+ σdWt, X0 = x (B.1)

with θ = (λ, µ, σ) ∈ R × R × R+ and it is ergodic for λ > 0. We can easily compute

the conditional distribution pθ(Xt|X0) Given X0 = x, at time t, the process Xt is

normally distributed with with mean m(t, x) and variance v(t,x) respectively:

m(t, x) = Eθ(Xt|X0 = x) = µ+ (x− µ) exp(−λt)

v(t, x) = Varθ(Xt|X0 = x) = σ2

2λ[1− exp(−2λt)

] (B.2)

We assume that the process is observed at discrete times ti = i∆i, i = 0, 1, ..., n,

the sampling rate is constant ∆i = ∆ and T = n∆. Given n + 1 observations

127

X = (X0, X1, ..., Xn) of the process, the likelihood function can be derived as:

Ln(X; θ) =n∏i=1

pθ(Xi|Xi−1)pθ(X0) (B.3)

the log-likelihood function `n(X; θ) = logLn(X; θ) is :

`n(X; θ) =n∑i=1

log pθ(Xi|Xi−1) + log(pθ(X0))

= −n2 log(σ2

2λ)− 12

n∑i=1

log(1− e−2λ∆)

− λ

σ2

n∑i=1

(Xi − µ− (Xi−1 − µ)e−λ∆)2

1− e−2λ∆

(B.4)

In the last equality, we omit the constant term. The maximum likelihood estimates

(MLE) µ, λ, σ can be solved numerically by optimizing the log-likelihood function.

For implementation in R, see [47].

128

B.2 Parameter Estimation of multivariate mean-

reverting process

The parameter estimation for multivariate OU processes is less strait forward as

the univariate one. However, notice that the OU process can also be considered as

the continuous-time analogue of the discrete-time first-order autoregression(AR(1))

process. We conduct the parameters estimation procedure of multivariate OU

processes in two steps: first, we use the collected data to fit an AR(1) model by least

square estimation (see [93] for complete explanation); then, we follow the method

descried in Wan [114] to convert the AR(1) parameters we estimated from first step

to parameters of its covariance equivalent multivariate OU process.

First, consider the following AR(1) process:

Xt =

X1,t

X2,t

...

Xm,t

=

φ1,1 φ1,2 · · · φ1,m

φ2,1 φ2,2 · · · φ2,m

... ... . . . ...

φm,1 φm,2 · · · φm,m

X1,t−1

X2,t−1

...

Xm,t−1

+

ε1,t

ε2,t

...

εm,t

= ΦXt−1 + εt, t = 1, ..., n

(B.5)

where Xt is a time series of m-dimensional state vectors, observed at equally spaced

instance t, εt is the uncorrelated m-dimensional normal random vector with mean 0

and covariance matrix Σε. Introducing the following matrices:

U =n∑t=1

Xt−1XTt−1 V =

n∑t=1

XtXTt W =

n∑t=1

XtXTt−1 (B.6)

129

To view (B.5) as a linear regression model with fixed predictors Xt−1, the least square

estimate for the coefficient matrix Φ can be written as:

Φ = WU−1 (B.7)

The estimate for the covariance matrix Σε is:

Σε = 1n−m

n∑t=1

εtεTt with εt = Xt − ˆPhiXt−1

= 1n−m

(V −WU−1W T )(B.8)

Next, we consider the following general multivariate OU process,

dXt = d

X1,t

X2,t

...

Xm,t

=

α1,1 α1,2 · · · α1,m

α2,1 α2,2 · · · α2,m

... ... . . . ...

αm,1 αm,2 · · · αm,m

X1,t

X2,t

...

Xn,t

dt+

σ1,1 σ1,2 · · · σ1,m

σ2,1 σ2,2 · · · σ2,m

... ... . . . ...

σm,1 σm,2 · · · σm,m

d

W1,t

W2,t

...

Wm,t

= AXdt+ ΣOUdWt

(B.9)

where Σ is the diffusion matrix and Wt is a vector of m independent standard Brow-

nian Motions. We need to determine the parametric relations between Φ, Σε in the

AR(1) process and A, ΣOU in OU process. By matching the first moments, we must

have the following equation for all t:

E(Xt|X0) = eAtX0 = ΦtX0 ∀t (B.10)

130

which implies that:

eA = Φ (B.11)

By eigenvalue decomposition, we can rewrite Φ as:

Φ = V ΛV −1

= V

λ1

. . .

λn

V−1

(B.12)

where Λ is a diagonal matrix with eigenvalues of Φ and each column in V is the

corresponding eigenvectors. Then

A = V ΛV −1

= V

log(λ1)

. . .

log(λn)

V−1

(B.13)

where Λ is the diagonal matrix with each element being the log of the eigenvalues of Φ.

To further determine ΣOU , we first solve for Σ = ΣOUΣTOU , then ΣOU can be found

by Cholesky decomposition. One way to solve for Σ is by matching the variances of

AR(1) and OU processes when time t goes to infinity. For the AR(1) process,

limt→∞

Var(Xt) =∞∑i=0

ΦiΣa(Φi)T (B.14)

131

From the eigenvalue decomposition Φ = V ΛV −1, we have Φi = V ΛiV −1, hence Eq.

(B.14) can be written as:

limt→∞

Var(Xt) = V[ ∞∑i=0

ΛiV −1Σa(V −1)TΛi]V T (B.15)

We further define:

F = V −1Σa(V −1)T

R =∞∑i=0

ΛiV −1Σa(V −1)TΛi =∞∑i=0

ΛiFΛi(B.16)

Let Fkl and Rkl denote the element of k-th row and l-th column of F and R respec-

tively, For stationary process, λk < 1 for all k. Hence,

Rkl =∞∑i=0

λikFklλil = Fkl

1− λkλl(B.17)

which implies that:

limt→∞

Var(Xt) = V RV T (B.18)

On the other hand, the multivariate OU process, limt→∞ Var(Xt) can be calculated

as:limt→∞

Var(Xt) = limt→∞

V∫ t

0eyΛ

[V −1Σ(V −1)T

]eyΛdyV T

= limt→∞

V∫ t

0eyΛFOUe

yΛdyV T

(B.19)

where FOU is defined as FOU = V −1Σ(V −1)T . By making limt→∞ Var(Xt) in AR(1)

and OU process equal, we can derive the following equation:

R = limt→∞

∫ t

0eyΛFOUe

yΛdy (B.20)

Denote limt→∞∫ t

0 eyΛFOUe

yΛdy by M , the element of k-th row and l-th column of FOU

and M by (FOU)kl and Mkl respectively. By the fact that |λk| < 1 and log(λk|) < 0

132

for stationary process, we can compute Mkl as:

Mkl = (FOU)kllog(λk) + log(λl)

(B.21)

In order for Eq.B.20 to hold, we must have:

Rkl = − (FOU)kllog(λk) + log(λl)

= Fkl1− λkλl

(B.22)

which implies:

(FOU)kl = − log(λk) + log(λl)1− λkλl

Fkl (B.23)

Finally, we can obtain Σ from FOU = V −1Σ(V −1)T :

Σ = V FOUVT (B.24)

To connect the estimation results to the parameters in Eq. (3.22) , we observe the

off diagonal elements of A are small enough to be negligible, hence we can safely take

the diagonal elements of A and use it as parameter estimation of Λ. We also observe

the long term mean level is small for each pair asset, hence M is assumed to be 0.

133

Appendix C

Summary of Data

C.1 Information of ETF and Stocks

This section contains the information of the stocks and ETFs we used to construct

the mean-reverting pairs. We select 16 ETFs from different sectors and industries, of

which the basic information are summarized in Table.C.1. From each ETF, we select

several large cap stocks and computing the residuals based on method described in

section 6.1. In total, we constructed 48 stock-ETF pairs, which are summarized in

Table. C.2.

134

Table C.1: Information of ETF universe as of July 31st, 2017

ETF Ticker Industry/Sector num of constituents

XLY Consumer Discretionary Select Sector SPDR Fund 85

XLV Health Care Select Sector SPDR Fund 61

XLU Utilities Select Sector SPDR Fund 28

XLP Consumer Staples Select Sector SPDR Fund 26

XLK Technology Select Sector SPDR Fund 72

XLI Industrial Select Sector SPDR Fund 67

XLF Financial Select Sector SPDR Fund 66

XLB Materials Select Sector SPDR Fund 25

IYR US Real Estate ETF 126

KBE SPDR S&P Bank ETF 71

KCE SPDR S&P Capital Markets ETF 54

XHB SPDR S&P Homebuilders ETF 35

XRT SPDR S&P Retail ETF 96

XSD SPDR S&P Semiconductor ETF 35

XME SPDR S&P Metals & Mining ETF 27

XES SPDR S&P Oil & Gas Equipment & Services ETF 36

135

Table C.2: Information of Stocks as of July 31st, 2017

Stock Ticker Company Name Corresponding ETF Ticker Weight in ETF(%) Market Value (USD)CMCSA COMCAST CORP XLY 7.21 887,753,697.60SBUX STARBUCKS CORP XLY 3.28 404,206,659.50

F FORD MOTOR CO XLY 1.78 219,767,939.60MDT METTLER TOLEDO INTERNATIONAL XLV 3.88 687,739,023.40DUK DUKE ENERGY CORP XLU 8.24 603,470,688.30

D DOMINION ENERGY INC XLU 7.32 536,096,118.50AEP AMERICAN ELECTRIC POWER XLU 5.14 376,677,451.50PG PROCTER & GAMBLE CO XLP 11.94 1,130,736,003.00KO COCA COLA CO XLP 9.17 868,605,563.10PEP PEPSICO INC XLP 5.02 475,444,377.80CL COLGATE PALMOLIVE CO XLP 3.54 335,114,420.80

MSFT MICROSOFT CORP XLK 10.70 1,801,492,210.00INTC INTEL CORP XLK 3.08 518,421,285.50CSCO CISCO SYSTEMS INC XLK 2.99 503,136,364.50ORCL ORACLE CORP XLK 2.88 484,276,558.50

T AT&T INC XLK 4.23 711,705,321.60VZ VERIZON COMMUNICATIONS INC XLK 2.78 468,726,337.80

HON HONEYWELL INTERNATIONAL INC XLI 4.78 534,065,047.60JPM JPMORGAN CHASE & CO XLF 10.66 2,721,152,969.10WFC WELLS FARGO & CO XLF 8.08 2,062,197,598.10

IP INTERNATIONAL PAPER CO XLB 4.01 146,747,145.90WY WEYERHAEUSER CO IYR 2.48 110,970,397.80PLD PROLOGIS INC IYR 3.06 137,234,842.40VTR VENTAS INC IYR 2.41 107,973,688.00BK BANK OF NEW YORK MELLON CORP KBE 2.04 70,123,026.60

NTRS NORTHERN TRUST CORP KBE 2.02 69,378,281.75FITB FIFTH THIRD BANCORP KBE 1.97 67,783,463.76

C CITIGROUP INC KBE 1.96 67,327,950.33PNC PNC FINANCIAL SERVICES GROUP KBE 1.95 67,145,720.32RF REGIONS FINANCIAL CORP KBE 1.93 66,285,504.60FII FEDERATED INVESTORS INC KCE 2.19 2,310,153.33

ETFC E TRADE FINANCIAL CORP KCE 2.18 2,298,720.32STT STATE STREET CORP KCE 2.16 2,278,086.72IVZ INVESCO LTD KCE 2.15 2,266,990.92LAZ LAZARD LTD KCE 2.14 2,259,915.32TOL TOLL BROTHERS INC XHB 4.64 48,374,789.31JCI JOHNSON CONTROLS INTERNATION XHB 4.61 48,117,356.96

MAS MASCO CORP XHB 4.58 47,788,269.36PHM PULTEGROUP INC XHB 4.57 47,712,714.00LEN LENNAR CORP XHB 4.54 47,331,464.85ODP OFFICE DEPOT INC XRT 1.19 1,939,990.80

URBN URBAN OUTFITTERS INC XRT 1.19 1,939,036.01CY CYPRESS SEMICONDUCTOR CORP XSD 3.28 10,786,392.09ADI ANALOG DEVICES INC XSD 3.22 10,582,650.45ATI ALLEGHENY TECHNOLOGIES INC XME 4.54 33,491,984.90HAL HALLIBURTON CO XES 3.23 8,381,826.31

PTEN PATTERSON UTI ENERGY INC XES 3.20 8,309,311.23SPN SUPERIOR ENERGY SERVICES INC XES 3.39 8,821,779.90

136

C.2 Estimated Parameters

In this section, we summarize the estimated parameters for the multi-variate OU pro-

cesses according Equation.(4.12). Notice, the parameters for year 2014 are estimated

from historical data of year 2010-2013, parameters for year 2015 are estimated from

historical data of year 2016 and so on so forth.

137

Table C.3: Λ for Year 2014-2017

Ticker 2014 2015 2016 2017CMCSA 0.1512044 0.09903023 0.088121957 0.079624755SBUX 0.119861566 0.106692767 0.098077297 0.089953586

F 0.066364963 0.050982492 0.04489592 0.043963961MDT 0.107307167 0.102353912 0.086329823 0.08685961DUK 0.121337593 0.08755698 0.058748914 0.061531854

D 0.089492081 0.098878838 0.067219952 0.066434508AEP 0.113546658 0.09884012 0.078860184 0.072107818PG 0.126228638 0.096886326 0.069082293 0.05817882KO 0.115512178 0.096943248 0.081430882 0.069477828PEP 0.149839963 0.12689585 0.111955847 0.087106367CL 0.174192155 0.135077399 0.105763485 0.082967234

MSFT 0.14835767 0.13526843 0.085187717 0.080934768INTC 0.085382411 0.076182585 0.0530736 0.045079178CSCO 0.10030578 0.083682861 0.081974905 0.07533517ORCL 0.078216727 0.072394562 0.062725916 0.058971221

T 0.082766055 0.045339141 0.05609628 0.022236044VZ 0.096536174 0.090379316 0.078847628 0.073606135

HON 0.194247732 0.128088189 0.12284851 0.08929841JPM 0.077781933 0.082246427 0.062812423 0.058392144WFC 0.140411733 0.101826513 0.08560014 0.069309928

IP 0.110442126 0.094170262 0.077487284 0.064448131WY 0.148304805 0.111528619 0.084138711 0.059737781PLD 0.127881484 0.124128489 0.10390303 0.083838078VTR 0.123566047 0.103444615 0.094639029 0.092862104BK 0.152665573 0.086988846 0.066685143 0.06256798

NTRS 0.085238352 0.09623448 0.080502361 0.078105738FITB 0.185831425 0.146666291 0.119258936 0.087743548

C 0.121257087 0.097341215 0.085707487 0.080808048PNC 0.111797979 0.076926986 0.07132905 0.068030345RF 0.13340155 0.113899191 0.066644951 0.065864507FII 0.070298788 0.0692512 0.063582922 0.048911198

ETFC 0.106017484 0.076158733 0.08130912 0.07331601STT 0.069168964 0.067691624 0.076441998 0.068969932IVZ 0.084301846 0.079766503 0.067302767 0.060173509LAZ 0.165662314 0.162293273 0.148215258 0.122143918TOL 0.120344576 0.101937009 0.101144192 0.092829882JCI 0.134428205 0.12877133 0.121046752 0.077376392

MAS 0.124948355 0.101403999 0.085898027 0.068374814PHM 0.105976914 0.093875333 0.069488354 0.058985726LEN 0.102457074 0.06532253 0.05861495 0.054436747ODP 0.130774618 0.069522574 0.069515129 0.05851729

URBN 0.099900322 0.07510007 0.054682685 0.044866865CY 0.066689969 0.043098203 0.03907061 0.043387069ADI 0.085303172 0.08190453 0.089768265 0.077714042ATI 0.127979297 0.089665411 0.071413592 0.068926531HAL 0.111331436 0.106928803 0.087892309 0.092583281

PTEN 0.087740809 0.08052275 0.072652146 0.069694441SPN 0.120296202 0.102141582 0.091339108 0.069543253

138

Tabl

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34

139

Tabl

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Portfolio Optimization with Mean-reverting Assets ... · Jing Ye A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy