938838223-MIT.pdf

Applications of optimal portfolio management
by
Dimitrios Bisias
Submitted to the Sloan School of Management
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Operations Research
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2015
c Massachusetts Institute of Technology 2015. All rights reserved.
Author .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sloan School of Management
June 22, 2015
Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Andrew W. Lo
Charles E. and Susan T. Harris Professor of Finance
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Patrick Jaillet
Dugald C. Jackson Professor, Department of Electrical Engineering
and Computer Science
Co-director, Operations Research Center

Applications of optimal portfolio management
by
Dimitrios Bisias
Submitted to the Sloan School of Management
on June 22, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Operations Research
Abstract
This thesis revolves around applications of optimal portfolio theory.
In the first essay, we study the optimal portfolio allocation among convergence
trades and mean reversion trading strategies for a risk averse investor who faces Value-
at-Risk and collateral constraints with and without fear of model misspecification.
We investigate the properties of the optimal trading strategy, when the investor fully
trusts his model dynamics. Subsequently, we investigate how the optimal trading
strategy of the investor changes when he mistrusts the model. In particular, we
assume that the investor believes that the data will come from an unknown member
of a set of unspecified alternative models near his approximating model. The investor
believes that his model is a pretty good approximation in the sense that the relative
entropy of the alternative models with respect to his nominal model is small. Concern
about model misspecification leads the investor to choose a robust optimal portfolio
allocation that works well over that set of alternative models.
In the second essay, we study how portfolio theory can be used as a framework for
making biomedical funding allocation decisions focusing on the National Institutes
of Health (NIH). Prioritizing research efforts is analogous to managing an invest-
ment portfolio. In both cases, there are competing opportunities to invest limited
resources, and expected returns, risk, correlations, and the cost of lost opportunities
are important factors in determining the return of those investments. Can we apply
portfolio theory as a systematic framework of making biomedical funding allocation
decisions? Does NIH manage its research risk in an efficient way? What are the
challenges and limitations of portfolio theory as a way of making biomedical funding
allocation decisions?
Finally in the third essay, we investigate how risk constraints in portfolio opti-
mization and fear of model misspecification affect the statistical properties of the
market returns. Risk sensitive regulation has become the cornerstone of international
financial regulations. How does this kind of regulation affect the statistical properties
of the financial market? Does it affect the risk premium of the market? What about
the volatility or the liquidity of the market?
3

Thesis Supervisor: Andrew W. Lo
Title: Charles E. and Susan T. Harris Professor of Finance
4

Acknowledgments
I would like to express my gratitude to my advisor and mentor, Professor Andrew
W. Lo, for his continuing support and advice over all the years I spent at MIT. His
immense knowledge in diverse research areas, enthusiasm, hard work, outstanding
leadership and motivation have been a source of inspiration. Working with him has
been an honor and privilege and I could not have imagined having a better advisor
and mentor for my Ph.D study.
I would also like to thank the rest of my thesis committee: Professor Dimitri P.
Bertsekas for comments that greatly improved this thesis and for his great books that
made me love the field of optimization in the first place and Professor Leonid Kogan
who provided his insight and expertise that greaty assisted this research.
In addition I would like to thank Dr. James F. Watkins, MD for his invaluable
help, insights and contribution to the second part of this research.
Moreover, I would like to thank Dr. Paul Mende, Dr. Saman Majd and Dr. Eric
Rosenfeld whom I had the fortune of being their teaching assistant in finance classes.
Paul’s experience in quantitative trading made me realize what career I would like to
follow and I am grateful for this.
Being part of MIT and in particular the ORC and LFE communities has been a
blessing and I consider myself very fortunate to be among very interesting and smart
people. I will always remember my years at MIT with nostalgia and joy and I hope
that I ’ll be able to express my gratitude in the future several times.
My life at MIT would not be so complete and joyful if I didn’t have good lifelong
friends to spend time and have productive discussions with. In particular, I would
like to thank Nick Trichakis and his wife Lena, Christos and Elli Nicolaides, Markos
and Sophia Trichas, Thomas and Anastasia Trikalinos, the golden coach George Pa-
pachristoudis and Gerry Tsoukalas.
Last but not least I would like to thank my parents Giorgo and Roula and my
sister Katerina for their unconditional love and support. I owe to them everything
and this thesis is dedicated to them.
5

Contents
1 Optimal trading of arbitrage opportunities under constraints 29
1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.2.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.2.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.2.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.2.4 Connection with Ridge and Lasso regression . . . . . . . . . . 46
1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3.1 Convergence trades . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3.2 Mean reversion trading opportunities . . . . . . . . . . . . . . 56
1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2 Optimal trading of arbitrage opportunities under model misspecifi-
cation 57
2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.2.1 Alternative models representation . . . . . . . . . . . . . . . . 61
2.2.2 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.3.1 No fear of model misspecification . . . . . . . . . . . . . . . . 65
2.3.2 Fear of model misspecification no constraints . . . . . . . . . . 67
2.3.3 Fear of model misspecification with VaR and margin constraints 70
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7

2.4.1 Convergence trades without constraints . . . . . . . . . . . . . 73
2.4.2 Mean reversion trading strategies without constraints . . . . . 78
2.4.3 Convergence trades with constraints . . . . . . . . . . . . . . . 92
2.4.4 Mean reversion trading strategies with constraints . . . . . . . 111
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3 Estimating the NIH Efficient Frontier 131
3.1 NIH Background and Literature Review . . . . . . . . . . . . . . . . 132
3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.2.1 Funding Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.2.2 Burden of Disease Data . . . . . . . . . . . . . . . . . . . . . 139
3.2.3 Applying Portfolio Theory . . . . . . . . . . . . . . . . . . . . 142
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.3.1 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . 147
3.3.2 Efficient Frontiers . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4 Impact of model misspecification and risk constraints on market 157
4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.2.1 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.2.2 Varying constraints . . . . . . . . . . . . . . . . . . . . . . . . 161
4.2.3 Varying risk aversions . . . . . . . . . . . . . . . . . . . . . . 165
4.2.4 Varying constraints and risk aversions . . . . . . . . . . . . . . 168
4.2.5 Varying fear of model misspecification . . . . . . . . . . . . . 168
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
A Technical Notes 173
8

List of Figures
1-1 Ellipsoids. Ellipsoids of poor investment opportunities for N=2 con-
vergence trades at times t = 0.3, 0.6, 0.9. . . . . . . . . . . . . . . . . 42
1-2 Weights for the case of uncorrelated spreads and collateral
constraint. Weights for the case of uncorrelated spreads. . . . . . . 45
1-3 VaR constraints, positive correlations. Wealth distribution at t =
0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated
(ρ = 0.5) convergence trades, while facing VaR constraints (K=1).
Initial wealth is $100. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1-4 VaR constraints, negative correlations. Wealth distribution at
t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively cor-
related (ρ = −0.5) convergence trades, while facing VaR constraints
(K=1). Initial wealth is $100. . . . . . . . . . . . . . . . . . . . . . . 48
1-5 VaR constraints, positive correlations, tight constraints. Wealth
distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two
positively correlated (ρ = 0.5) convergence trades, while facing VaR
constraints (K=0.25). Initial wealth is $100. . . . . . . . . . . . . . . 49
1-6 VaR constraints, negative correlations, tight constraints. Wealth
distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two
negatively correlated (ρ = −0.5) convergence trades, while facing VaR
constraints (K=0.25). Initial wealth is $100. . . . . . . . . . . . . . . 49
9

1-7 Wealth evolution under VaR constraint. Typical path of the
wealth evolution for an investor investing in two convergence trades
using the same noise process for positive and negative correlation under
the VaR constraint. Initial wealth is $100. . . . . . . . . . . . . . . . 50
1-8 Relation between final wealth and frequency the VaR con-
straint binds. Final wealth is negatively correlated to the percentage
of time the constraints bind when the initial values of the convergence
trades are low. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1-9 Margin constraints, positive correlations. Wealth distribution at
t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively cor-
related (ρ = 0.5) convergence trades, while facing margin constraints
(Collateral = 1). Initial wealth is $100. . . . . . . . . . . . . . . . . . 52
1-10 Margin constraints, negative correlations. Wealth distribution
at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively cor-
related (ρ = −0.5) convergence trades, while facing margin constraints
(Collateral = 1). Initial wealth is $100. . . . . . . . . . . . . . . . . . 52
1-11 Margin constraints, positive correlations, more collateral needed.
Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests
in two positively correlated (ρ = 0.5) convergence trades, while facing
margin constraints (Collateral = 2). Initial wealth is $100. . . . . . . 53
1-12 Margin constraints, negative correlations, more collateral needed.
Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests
in two negatively correlated (ρ = −0.5) convergence trades, while fac-
ing margin constraints (Collateral = 2). Initial wealth is $100. . . . . 53
1-13 Wealth evolution under margin constraint. Typical path of the
wealth evolution for an investor investing in two convergence trades
using the same noise process for positive and negative correlation under
the margin constraint. Initial wealth is $100. . . . . . . . . . . . . . . 54
10

1-14 Relation between final wealth and frequency the margin con-
straint binds. Final wealth is negatively correlated to the percentage
of time the constraints bind when the initial values of the convergence
trades are low. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1-15 Positions evolution under VaR constraints. Typical path of the
positions in two convergence trading opportunities under VaR con-
straints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1-16 Positions evolution under margin constraints. Typical path of
the positions in two convergence trading opportunities under margin
constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2-1 Partial derivative of the value function with respect to S for
a single convergence trade. VS as a function of time at S = 1
for different values of the robustness multiplier for a single convergence
trade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2-2 Distortion drift for a single convergence trade. Distortion drift
as a function of time at S = 1 for different values of the robustness
multiplier for a single convergence trade. . . . . . . . . . . . . . . . . 76
2-3 Distortion drift terms for a single convergence trade. Distor-
tion drift terms as a function of time at S = 1 for ν = 1 for a single
convergence trade. The first term corresponds to a positive distor-
tion drift that reduces the wealth of the investor since the investor is
shorting the spread, while the second term corresponds to a negative
distortion drift that points to worse investment opportunities. . . . . 77
2-4 Optimal weight of a single convergence trade. Weight of the
convergence trading strategy as a function of time at S = 1 for different
values of the robustness multiplier. . . . . . . . . . . . . . . . . . . . 77
a single mean reversion trading strategy. VS as a function of
time at S = 1 for different values of the robustness multiplier. . . . . 80
11

2-6 Distortion drift for a single mean reversion trading strategy.
Distortion drift as a function of time at S = 1 for different values of
the robustness multiplier. . . . . . . . . . . . . . . . . . . . . . . . . . 81
2-7 Distortion drift terms for a single mean reversion trading
strategy. Distortion drift terms as a function of time at S = 1
for ν = 1. The first term corresponds to a positive distortion drift
that reduces the wealth of the investor, since the investor is shorting
the spread, while the second term corresponds to a negative distortion
drift that points to worse investment opportunities. . . . . . . . . . . 81
2-8 Optimal weight of a single mean reversion trading strategy.
Weight of the mean reversion trading strategy as a function of time at
S = 1 for different values of the robustness multiplier. . . . . . . . . . 82
2-9 Optimal weights of two uncorrelated mean reversion trading
strategies for S1 = 1 and S2 = 2. Weights of the mean reversion
trading strategies as a function of time at S1 = 1 and S2 = 2 for
different values of the robustness multiplier. The correlation coefficient
is ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2-10 Ratio of the optimal weights. Ratio of the optimal weights of the
mean reversion trading strategies as a function of time at S1 = 1 and
S2 = 2 for different values of the robustness multiplier. The correlation
coefficient is ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2-11 Partial derivative of the value function with respect to S1 and
S2 at S1 = 1 and S2 = 2 when ρ = 0. Partial derivative of the value
function with respect to S1 and S2 as a function of time at S1 = 1 and
coefficient is ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
12

2-12 Optimal weights of two positively correlated mean reversion
trading strategies for S1 = 1 and S2 = 2. Weights of the mean
reversion trading strategies as a function of time at S1 = 1 and S2 =
2 for different values of the robustness multiplier. The correlation
coefficient is ρ = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2-13 Optimal weights of two negatively correlated mean reversion
coefficient is ρ = −0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 86
strategies for S1 = 1 and S2 = 1. Weights of the mean reversion
is ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2-15 Ratio of the optimal weights at S1 = 1 and S2 = 1 when ρ = 0.
Ratio of the optimal weights of the mean reversion trading strategies
as a function of time at S1 = 1 and S2 = 1 for different values of the
robustness multiplier. The correlation coefficient is ρ = 0. . . . . . . 88
S2 at S1 = 1 and S2 = 1 when ρ = 0. Partial derivative of the value
function with respect to S1 and S2 as a function of time at S1 = 1 and
coefficient is ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
coefficient is ρ = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
13

S2 at S1 = 1 and S2 = 1 when ρ = 0.9. Partial derivative of
the value function with respect to S1 and S2 as a function of time at
S1 = 1 and S2 = 1 for different values of the robustness multiplier.
The correlation coefficient is ρ = 0.9. . . . . . . . . . . . . . . . . . . 90
2-19 Ratio of the optimal weights at S1 = 1 and S2 = 1 when ρ = 0.9.
Ratio of the magnitude of the optimal weights of the mean reversion
is ρ = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
coefficient is ρ = −0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . 91
a single convergence trade when L = 0.1 and L = 100. VS
as a function of time at S = 1 for different values of the robustness
multiplier. The solid line is when L = 100 and the dotted line is for
L = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
a single convergence trade when L = 0.1. VS as a function of
time at S = 1 for different values of the robustness multiplier. The
collateral constraint is |F| ≤ 0.1. . . . . . . . . . . . . . . . . . . . . . 95
2-23 Optimal weight of a single convergence trade when L = 0.1.
Weight of the convergence trading strategy as a function of time at
S = 1 for different values of the robustness multiplier. The collateral
constraint is |F| ≤ 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 95
14

2-24 Optimal weight of a single convergence trade when L = 1.
Weight of the convergence trading strategy as a function of time at
S = 1 for different values of the robustness multiplier. The collateral
constraint is |F| ≤ 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2-25 Distortion drift for a single convergence trade when L = 0.1.
the robustness multiplier. The collateral constraint is |F| ≤ 0.1. . . . 96
2-26 Distortion drift for a single convergence trade when L = 1.
the robustness multiplier. The collateral constraint is |F| ≤ 1. . . . . 97
2-27 Distortion drift terms for a single convergence trade when
L = 0.1. Distortion drift terms as a function of time at S = 1 for
ν = 1 and L = 0.1. The first term corresponds to a positive distortion
drift that reduces the wealth of the investor and it is bounded above
due to the collateral constraint, while the second term corresponds to a
negative distortion drift that points to worse investment opportunities. 97
2-28 Distortion drift terms for a single convergence trade when
L = 1. Distortion drift terms as a function of time at S = 1 for ν = 1
and L = 1. The first term corresponds to a positive distortion drift
that reduces the wealth of the investor and it is bounded above due
to the collateral constraint, while the second term corresponds to a
negative distortion drift that points to worse investment opportunities. 98
2-29 Optimal weight of a single convergence trade when L = 0.1 and
L = 100. Weight of the convergence trading strategy as a function of
time at S = 1 for different values of the robustness multiplier. The
solid line is when L = 100 and the dotted line is for L = 0.1. . . . . . 98
15

2-30 Optimal weights of two uncorrelated convergence trades for
S1 = 1 and S2 = 2 when L = 0.5. Weights of the convergence trades
robustness multiplier. The correlation coefficient is ρ = 0 and the rhs
of the VaR constraint is L = 0.5. . . . . . . . . . . . . . . . . . . . . 100
2-31 Value of the normalized wealth variance for two uncorrelated
convergence trades at S1 = 1 and S2 = 2 when L = 0.5. Value
of the normalized wealth variance for two uncorrelated convergence
trades as a function of time at S1 = 1 and S2 = 2 for different values
of the robustness multiplier. The correlation coefficient is ρ = 0 and
the rhs of the VaR constraint is L = 0.5. . . . . . . . . . . . . . . . . 101
the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . 102
2-34 Optimal weights of two positively correlated convergence trades
for S1 = 1 and S2 = 2 when L = 0.05. Weights of the convergence
of the robustness multiplier. The correlation coefficient is ρ = 0.5 and
16

2-35 Value of the normalized wealth variance for two positively cor-
related convergence trades at S1 = 1 and S2 = 2 when L = 0.05.
Value of the normalized wealth variance for two positively correlated
convergence trades as a function of time at S1 = 1 and S2 = 2 for dif-
ferent values of the robustness multiplier. The correlation coefficient
is ρ = 0.5 and the rhs of the VaR constraint is L = 0.05. . . . . . . . 104
2-36 Optimal weights of two negatively correlated convergence trades
trades as a function of time at S1 = 1 and S2 = 2 for different values of
the robustness multiplier. The correlation coefficient is ρ = −0.5 and
2-37 Value of the normalized wealth variance for two negatively
correlated convergence trades at S1 = 1 and S2 = 2 when
L = 0.05. Value of the normalized wealth variance for two nega-
tively correlated convergence trades as a function of time at S1 = 1
and S2 = 2 for different values of the robustness multiplier. The cor-
relation coefficient is ρ = −0.5 and the rhs of the VaR constraint is
L = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
17

2-40 Optimal weights of two positively correlated convergence trades
of the robustness multiplier. The correlation coefficient is ρ = 0.8 and
2-41 Value of the normalized wealth variance for two positively cor-
related convergence trades at S1 = 1 and S2 = 1 when L = 0.05.
Value of the normalized wealth variance for two positively correlated
convergence trades as a function of time at S1 = 1 and S2 = 1 for dif-
ferent values of the robustness multiplier. The correlation coefficient
is ρ = 0.8 and the rhs of the VaR constraint is L = 0.05. . . . . . . . 109
2-42 Optimal weights of two negatively correlated convergence trades
for S1 = 1 and S2 = 1 when L = 8. Weights of the convergence
trades as a function of time at S1 = 1 and S2 = 1 for different values of
the robustness multiplier. The correlation coefficient is ρ = −0.8 and
correlated convergence trades at S1 = 1 and S2 = 1 when
L = 0.05. Value of the normalized wealth variance for two nega-
tively correlated convergence trades as a function of time at S1 = 1
and S2 = 1 for different values of the robustness multiplier. The cor-
relation coefficient is ρ = −0.8 and the rhs of the VaR constraint is
L = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
a single mean reversion trading strategy and a collateral con-
straint with L = 0.7. VS as a function of time at S = 1 for different
values of the robustness multiplier for L = 0.7. . . . . . . . . . . . . . 113
18

2-45 Distortion drift terms for a single mean reversion trading
strategy and a collateral constraint with L = 0.7. Distortion
drift terms as a function of time at S = 1 for ν = 2 and for L = 0.7.
The first term corresponds to a positive distortion drift that reduces
the wealth of the investor, since the investor is shorting the spread,
while the second term corresponds to a negative distortion drift that
points to worse investment opportunities. The first term is bounded
above due to the collateral constraint. . . . . . . . . . . . . . . . . . . 113
2-46 Optimal weight of a single mean reversion trading strategy
with a collateral constraint with L = 0.7. Weight of the mean
reversion trading strategy as a function of time at S = 1 for different
values of the robustness multiplier and for L = 0.7. . . . . . . . . . . 114
a single mean reversion trading strategy with different collat-
eral constraints. VS as a function of time at S = 1 for different
values of the robustness multiplier and different collateral constraints.
The solid line is for L = 70 and the dotted line for L = 0.7. . . . . . . 114
2-48 Optimal weight of a single mean reversion trading strategy
with different collateral constraints. Weight of the mean reversion
trading strategy as a function of time at S = 1 for different values of
the robustness multiplier and different collateral constraints. The solid
line is for L = 70 and the dotted line for L = 0.7. . . . . . . . . . . . 115
strategies for S1 = 1 and S2 = 2 when L = 3. Weights of the
coefficient is ρ = 0 and the rhs of the VaR constraint is L = 3. . . . . 117
19

mean reversion trading strategies at S1 = 1 and S2 = 2 when
L = 3. Value of the normalized wealth variance for two uncorrelated
20

trading strategies for S1 = 1 and S2 = 2 when L = 7. Weights
of the mean reversion trading strategies as a function of time at S1 =
1 and S2 = 2 for different values of the robustness multiplier. The
correlation coefficient is ρ = 0.5 and the rhs of the VaR constraint is
L = 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2-56 Value of the normalized wealth variance for two positively
correlated mean reversion trading strategies at S1 = 1 and
S2 = 2 when L = 7. Value of the normalized wealth variance for two
positively correlated mean reversion trading strategies as a function
of time at S1 = 1 and S2 = 2 for different values of the robustness
multiplier. The correlation coefficient is ρ = 0.5 and the rhs of the
VaR constraint is L = 7. . . . . . . . . . . . . . . . . . . . . . . . . . 122
correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is
L = 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
negatively correlated mean reversion trading strategies as a function
multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the
21

L = 2. Value of the normalized wealth variance for two negatively
correlated mean reversion trading strategies as a function of time at
S1 = 1 and S2 = 1 for different values of the robustness multiplier.
The correlation coefficient is ρ = 0 and the rhs of the VaR constraint
is L = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
correlation coefficient is ρ = 0.9 and the rhs of the VaR constraint is
L = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2-62 Value of the normalized wealth variance for two positively
positively correlated mean reversion trading strategies as a function
multiplier. The correlation coefficient is ρ = 0.9 and the rhs of the
22

correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is
L = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
negatively correlated mean reversion trading strategies as a function
multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the
3-1 NIH time series flowchart. Flowchart for the construction of NIH
appropriations time series. “NIH Approp.” denotes NIH appropria-
tions; “PHS Gaps” denotes Institute funding by the U.S. Public Health
Service; “Complete Approp.” denotes the union of these two series;
“FY Change” allows for the change in government fiscal years; “4Q
FY” time series refers to the resulting series in which all years are
treated as having four quarters of three months each. . . . . . . . . . 138
3-2 Appropriations data. NIH appropriations in real (2005) dollars,
categorized by disease group. . . . . . . . . . . . . . . . . . . . . . . 138
3-3 YLL time series flowchart. Flowchart for the construction of years
of life lost (YLL) time series. “WONDER Chapter Age Group” refers
to a query to the CDC WONDER database at the chapter level, strati-
fied by age group at death; “US Pop.” is the United States population
from census data as expressed in the WONDER dataset; and “US
GDP” denotes U.S. gross domestic product. . . . . . . . . . . . . . . 140
23

3-4 YLL data. Panel (a): Raw YLL categorized by disease group. Panel
(b): Population-normalized YLL (with base year of 2005), categorized
by disease group. Both panels are based on data from 1979 to 2007. 141
3-5 Efficient frontiers. Efficient frontiers for (a) all groups except HIV
and AMS, γ = 0; (b) all groups except HIV and AMS, γ = 5; (c) all
groups except HIV and AMS without the dementia effect, γ = 0; and
(d) all groups except HIV and AMS without the dementia effect, γ =5;
based on historical ROI from 1980 to 2003. . . . . . . . . . . . . . . . 148
4-1 Price of the risky asset as a function of the aggregate market
supply under varying constraints. We assume that we have 5
agents with the same risk aversion coefficients. The red plot assumes
the same L = 30 for all the agents, while the blue assumes L to be
different across the agents L1 = 10, L2 = 20, L3 = 30, L4 = 40, L5 = 50. 163
supply under tightening constraints. We assume that we have 5
agents with the same risk aversion coefficients. The blue plot assumes
L to be different across the agents L1 = 10, L2 = 20, L3 = 30, L4 =
40, L5 = 50 and the red assumes that each Li is reduced by 20%. . . . 164
supply with less variable constraints. We assume that we have 5
agents with the same risk aversion coefficients. The blue plot assumes
L to be different across the agents L1 = 10, L2 = 20, L3 = 30, L4 =
40, L5 = 50 and the red assumes that L1 = 20, L2 = 25, L3 = 30, L4 =
35, L5 = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
supply with constraints and varying risk aversions. We assume
that we have 5 agents with same constraints but different risk aversion
coefficients. The blue plot assumes L = 30 for each agent, while the
red line assumes that the agents are unconstrained. . . . . . . . . . . 166
24

supply with tightening constraints and varying risk aversions.
We assume that we have 5 agents with same constraints but different
risk aversion coefficients. The blue plot assumes L = 30 for each agent,
while the red line assumes that L = 20 for each agent. . . . . . . . . . 167
25

List of Tables
3.1 IoM recommendations. 12 major recommendations of the 1998
Institute of Medicine panel in four large areas for improving the process
of allocating research funds. . . . . . . . . . . . . . . . . . . . . . . . 133
3.2 ICD mapping. Classification of ICD-9 (1978–1998) and ICD-10 (1999–
2007) Chapters and NIH appropriations by Institute and Center to 7
disease groups: oncology (ONC); heart lung and blood (HLB); diges-
tive, renal and endocrine (DDK); central nervous system and sensory
(CNS) into which we placed dementia and unspecified psychoses to
create comparable series as there was a clear, ongoing migration noted
from NMH to CNS after the change to ICD-10 in 1999; psychiatric and
substance abuse (NMH); infectious disease, subdivided into estimated
HIV (HIV) and other (AID); maternal, fetal, congenital and pediatric
(CHD). The categories LAB and EXT are omitted from our analysis. 137
3.3 Return summary statistics. Summary statistics for the ROI of
disease groups, in units of years (for the lag length) and per-capita-
GDP-denominated reductions in YLL between years t and t+4 per
dollar of research funding in year t−q, based on historical ROI from
1980 to 2003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.4 ROI example. An example of the ROI calculation for HLB from 1986. 147
3.5 Portfolio weights. Benchmark, single- and dual-objective optimal
portfolio weights (in percent), based on historical ROI from 1980 to
2003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
27

Chapter 1
Optimal trading of arbitrage
opportunities under constraints
In financial economics, an arbitrage is an investment opportunity that is too good
to be true when there are no market frictions. In actual financial markets however,
there are frictions and even if there are arbitrage opportunities the investors may not
be able to fully exploit them due to the constraints they face.
We will explore two kinds of risky arbitrage opportunities when there are market
frictions. The first one is a case of a textbook arbitrage, a convergence trade strategy.
The second is a case of a statistical arbitrage, a mean reversion trading strategy. These
two strategies are two of the most popular trading strategies that hedge funds follow,
so studying them in detail when there are market frictions is a valuable exercise.
A convergence trade is a trading strategy consisting of long/short positions in two
similar assets, where we buy the cheap asset, we short the expensive asset and we wait
until the prices of the two assets to converge which we know it will happen for sure
some particular time in the future. An example of this trade involves the difference
in price between the on-the-run and the most recent off-the-run security. An on-
the-run security is the most recently issued, and hence most liquid, of a periodically
issued security. Since an on-the-run security is more liquid it trades at a premium
to off-the-run securities [29]. A convergence trade involves taking a long position in
the most recent off-the-run security and shorting the on-the-run security. The on-
29

the-run will become off-the-run upon the issue of a newer security and then there
will be almost no difference between the two securities in our trade so their prices
will converge. Another example involves investing in Treasury STRIPS with identical
maturity dates but different prices.
A mean reversion trading strategy involves investing in an asset or a portfolio of
assets whose value is a mean reverting process. Since most price series in the equity
space follow random walk, this strategy most commonly involves investing in a port-
folio of non-mean-reverting assets whose value is a stationary mean reverting series.
These price series that can be combined in such a way are called cointegrating. A
classic statistical arbitrage example is the pairs trading, which is the first type of
algorithmic mean reversion trading strategy invented by institutional traders, report-
edly by the trading desk of Nunzio Tartaglia at Morgan Stanley [64]. The statistical
arbitrage pairs trading strategy bets on the convergence of the prices of two similar
assets whose prices have diverged without a fundamental reason for this.
These arbitrage opportunities are risky under market frictions. In particular the
first case is exposed to the “divergence risk”, i.e. the fact that the pricing differential
between the two similar assets can diverge arbitrarily far from 0 prior to its conver-
gence at some particular time in the future. The second case is exposed both to the
“divergence risk” and to the “horizon risk”, in other words the fact that the times at
which the spread will converge to its long run mean are uncertain.
We will explore the optimal portfolio allocation of a risk averse investor who
invests in N convergence trades or mean reverting trading strategies, while facing
constraints. In particular, we will study the optimal trading strategy when he faces
VaR constraints or collateral constraints. Risk sensitive regulation, such as the VaR
constraint, has lately become a central component of international financial regu-
lations. Collateral or margin constraints, where the investor has to have sufficient
wealth to secure the liabilities taken by short positions, have been ubiquitous in the
financial transactions for centuries and margin calls have been behind several crises
including the LTCM debacle[54].
In the rest of this chapter, we will discuss the relevant literature review. Then we
30

will discuss about the setup of the model and the constraints, we will find the optimal
trading strategy of the investor and finally we will explore the characteristics of this
optimal strategy.
1.1 Literature review
Merton studied the problem of optimal portfolio allocation in a continuous time set-
ting without any market frictions [59]. The optimal portfolio involves two terms: a
market timing term and a hedging demand term. The first term is a myopic term that
represents the optimal allocation if you were interested at each time instant t only for
an horizon dt ahead. The second term represents the investor’s additional demand
due to the covariance of the wealth process with the attractiveness of the available
investment opportunities. Although Merton gives an analytical general solution this
is expressed in terms of the partial derivatives of the value function and additional
work is needed to derive the solution in terms of the model parameters. Additionally
it assumes that there are no market frictions.
Optimal trading of mean reversion trading strategies have been studied by both
Boguslavsky and Boguslavskaya [15] and Jurek and Yang [48]. They have found
analytical solutions for the optimal weight of a single mean reverting trading strategy
for risk averse CRRA investors. Their analysis is similar with the one in Kim and
Omberg [49], where they assume that there is a risk free asset with a constant risk-
free rate and a single risky asset with a mean reverting risk premium, which implies
a mean reverting instantaneous Sharpe ratio. In all the cases they have assumed that
there are no market frictions whatsoever.
Longstaff and Liu [53] have studied the problem of optimal trading of a single
convergence trade under a margin constraint. For the single convergence trade case
both VaR constraints and margin constraints collapse in the same constraint and the
problem is significantly easier. In addition by studying only convergence trades they
have taken out one important dimension of risk, the horizon risk, keeping only the
divergence risk. Brennan and Schwarz [17] have also studied the problem of optimal
31

trading of a single convergence trade including transaction costs when the arbitrage
potential is restricted by position limits.
The literature is rich with papers that study the existence of an equilibrium where
there exists mispricings. This persistence of mispricings is typically attributed to
agency problems, frictions or some kind of risk. Unlike textbook arbitrages, which
generate riskless profits and require no capital commitments, exploiting real-world
mispricings requires the assumption of some kind of risk. Shleifer and Vishny [74]
emphasized that risks such as the uncertainty about when the pricing differential
will converge to 0 and the possibility of a divergence of the mispricing prior to its
elimination may play a role in limiting the size of positions that arbitrageurs are
willing to take, contributing to the persistence of the arbitrage in equilibrium. Basak
and Choitoru [6] also showed that arbitrage can persist in equilibrium when there are
frictions. They study dynamic models with log utility and heterogeneous beliefs in
the presence of margin requirements and other portfolio constraints.
With respect to the constraints, Basak and Shapiro [7] study the problem of opti-
mal trading strategy of a risk averse investor who faces finite horizon VaR constraints
in a complete markets setting using the martingale representation approach [4]. Here
again there are no constraints in the optimal portfolio allocation at each time t but
there is only one constraint in the wealth at some finite horizon. Finally, Geanakoplos
[33] studies the collateral constraints, how these determine an equilibrium leverage
and how this leverage changes over time, the so-called leverage cycles.
Let us now discuss about the setup of the model and the constraints and find the
optimal trading strategy of the investor.
1.2 Analysis
We assume we have a risk averse investor maximizing the expected continuously
compounded rate of return or equivalently the expected logarithm of his final wealth
E(lnWT ). There are two cases to consider. In the first case, the investor can invest
in a risk-free asset and N non-redundant convergence trades, modeled as correlated
32

Brownian bridges. In the second case, the investor can invest in a risk-free asset
and N non-redundant mean reversion trading strategies, modeled as a multivariate
Ornstein-Uhlenbeck (OU) process. The investor faces two kinds of constraints: VaR
constraints or collateral constraints. We determine the optimal trading strategy and
its characteristics in both cases.
1.2.1 Models
As we mentioned already, a convergence trade is a trading strategy consisting of
long/short positions in two similar assets, where we buy the cheap asset, we short the
expensive asset and we wait until the prices of the two assets to converge which we
know it will happen for sure some time in the future. The spread of the convergence
trade can be modeled as a Brownian bridge driven by K Brownian motions, which
has the property that the spread will converge to 0 almost surely at some determined
time in the future. The stochastic differential equation governing the spread of the
trade is given by:
dSt = −
aSt
T − t
dt +
K
X
k=1
σkdZkt (1.1)
where St is the spread of the trade, a is a parameter controlling the rate of the mean
reversion to 0, T is the horizon of the investor which is also the time at which the
spread goes to 0 with probability 1 and Zt is a Brownian motion in RK
. We can
see that the reversion to 0 grows stronger as t → T. Therefore, the investment
opportunities get better as the spread gets larger and t → T, since then the drift
term pushing the spread towards 0 gets larger.
A mean reversion trading strategy involves investing in a stationary portfolio of
non-mean reverting assets, whose value is a mean reverting process. The value of
the portfolio can be modeled as an Ornstein-Uhlenbeck (OU) process. The stochastic
differential equation governing it is given by:
dSt = −φ(St − S̄)dt +
K
X
k=1
σkdZkt
33

In our case we have N of these mean reverting processes and we assume that they
are modeled as a multivariate Ornstein-Uhlenbeck process, which is defined by the
following stochastic differential equation:
dSt = −Φ(St − S̄)dt + σdZt (1.2)
Above Φ is a N-by-N square transition matrix that characterizes the deterministic
portion of the evolution of the process, S̄ is the vector representing the unconditional
mean of the process, σ is a N-by-K matrix that drives the dispersion of the process
and Zt is a Brownian motion in RK
.
The Ornstein-Uhlenbeck process has the nice property that its conditional distri-
bution is normal at all times, with mean equal to
Et[St+τ ] = S̄ + e−Φτ
(St − S̄)
and covariance matrix independent of St [60]. We assume that Φ has eigenvalues with
positive real part, so that the conditional expectation approaches to S̄ as t → ∞.
The Ornstein-Uhlenbeck process captures the two important dimensions of risk
in all relative value trades: the “horizon risk”, in other words the fact that the
times at which the spread will converge to its long run mean are uncertain and the
“divergence risk”, i.e. the fact that the pricing differential can diverge arbitrarily far
from its long run mean prior to its convergence. The Brownian bridge captures only
the “divergence” risk, since by its definition we assume that the investor has perfect
information about the magnitude of the mispricing at some future date T, i.e. we
assume that the date T on which the mispricing will be eliminated is known ahead
with certainty.
1.2.2 Constraints
We consider two kinds of constraints: VaR and collateral constraints. The VaR
constraint is a widely used statistical risk measure, adopted both by the regulators
34

and the private sector. It is the cornerstone of the capital regulations adopted by
Basel regulations. Both the 1996 market risk amendment of the original 1988 Basel
accord and the Basel II regulations have been built on the notion of Value-at-Risk
[47]. The Value at risk (VaR) at α-level is defined as the threshold value such that the
probability of losses greater than the threshold is less than α. In our case we consider
instantaneous VaR constraints which amount for determining an upper bound in
the wealth volatility, since locally the diffusion processes have normal distributions.
Therefore, the instantaneous VaR constraints are given by:
θT
Σθ ≤ LW2
where θ is a N by 1 vector of positions, Σ is the instantaneous covariance matrix of
the spreads, L is some proportionality constant that determines the tightness of the
constraint and W is the investor’s wealth.
Collateral or margin constraints have been ubiquitous in the financial transactions for
centuries. Even Shakespeare in the “Merchant of Venice” points out the importance
of the collateral, as Shylock charged Antonio no interest rate but he asked for a
pound of flesh as a collateral. The collateral constraints provide protection against
mark-to-market losses whenever an investor generates a liability by shorting an asset.
Therefore, they require that the investor’s wealth is bounded below by the collateral
necessary to secure the liabilities. They are given by:
N
X
i=1
λi|θi| ≤ W
where λi is the collateral necessary to secure the liability in spread i. In our work, each
unit of arbitrage should be understood as being relative to a fixed face or notional
amount and therefore each λi is a percentage of this fixed face value or notional
amount.
35

1.2.3 Solution
Let us now find the optimal trading strategy of a risk averse investor who maximizes
the expected logarithm of his final wealth E(lnWT ). We consider two cases:
• The investor invests in the risk free asset and in N correlated convergence trades.
• The investor invests in the risk free asset and in N correlated mean reversion
trading strategies.
For both cases our analysis is similar. For both cases we have:
Wt =
N
X
i=1
θitSit + θ0tB0t ∀t ∈ [0, T] (1.3)
where θit is the investor’s position in opportunity i for i = 1, · · · , N, θ0t is the in-
vestor’s position in the risk free asset, Sit is the spread of the convergence trade or
the value of the mean reverting portfolio and B0t is the price of the risk free asset.
The process θt is adapted to the filtration generated by the Brownian motion Zt.
The investor solves the following problem:
maximizeθ∈Θ E(lnWT )
subject to
dWt =
PN
i=1 θitdSit + θ0tdB0t
dSt = µ(S, t)dt + σ(S, t)dZt
(1.4)
where Θ is the set of admissible trading strategies. Let us first define ∀t ∈ [0, T] Ft =
θt/Wt ∈ RN
.
36

For the convergence trades case, investor’s wealth satisfies the following stochastic
differential equation:
dWt = Wtrdt +
N
X
i=1
θitSit(−
ai
T − t
− r)dt + θT
σdZt
dWt
Wt
= rdt +
N
X
i=1
FitSit(−
ai
T − t
− r)dt + FT
σdZt
By applying Ito’s Lemma we have that:
d(ln(Wt)) = rdt +
N
X
i=1
FitSit(−
ai
T − t
− r)dt − 1/2FT
t ΣFtdt + FT
t σdZt
Therefore it is:
ln(WT ) = ln(Wt) +
Z T
t
rs ds
+
Z T
t
N
X
i=1
FisSis(−
ai
T − t
− rs) −
1
2
FT
s ΣFs
!
ds
+
Z T
t
FT
s σdZs (1.5)
Assuming constant interest rate, we have:
Et(ln(WT )) = ln(Wt) + r(T − t)
+ Et
Z T
t
N
X
i=1
FisSis(−
ai
T − t
− rs) −
1
2
FT
s ΣFs
!
ds
!
+ Et(
Z T
t
FT
s σdZs) (1.6)
37

For the mean reversion trading strategies case, investor’s wealth satisfies the following
stochastic differential equation:
dWt = Wtrdt +
N
X
i=1
θit(−ΦT
i (St − S̄) − rSit)dt + θT
σdZt
dWt
Wt
= rdt +
N
X
i=1
Fit(−ΦT
i (St − S̄) − rSit)dt + FT
t σdZt
where Φi is the i’th row of the transition matrix Φ. By applying Ito’s Lemma we
have that:
d(ln(Wt)) = rdt +
N
X
i=1
Fit(−ΦT
i (St − S̄) − rSit)dt − 1/2FT
t ΣFtdt + FT
t σdZt
Therefore it is:
ln(WT ) = ln(Wt) +
Z T
t
rs ds
+
Z T
t
N
X
i=1
Fis(−ΦT
i (Ss − S̄) − rSis) −
1
2
FT
s ΣFs
!
ds
+
Z T
t
FT
s σdZs (1.7)
Assuming constant interest rate we have:
Et(ln(WT )) = ln(Wt) + r(T − t)
+ Et
Z T
t
N
X
i=1
Fis(−ΦT
i (Ss − S̄) − rSis) −
1
2
FT
s ΣFs
!
ds
!
+ Et(
Z T
t
FT
s σdZs) (1.8)
Under VaR constaints it is:
FT
t ΣFt ≤ L < ∞ ∀t
38

Under the margin constraints it is:
N
X
i=1
λi|Fit| ≤ 1 ∀t
FT
t ΣFt =
N
X
i=1
N
X
j=1
FitFjtσij
≤
N
X
i=1
N
X
j=1
λiλj|Fit||Fjt|
σij
λiλj
< C < ∞ ∀t
Therefore, for both the cases and both the constraints the integrand of the stochastic
integral belongs in H2
, which is a sufficient condition for the stochastic integral to be
a martingale. Consequently, Et(
R T
t
FT
s σdZs) is equal to 0.
Maximizing Et(ln(WT )) is equivalent to maximizing the third term is equations
(1.6), (1.8) for both the cases respectively. Let’s now stydy in detail the solution for
both cases for both the constraints.
VaR constraint
Maximizing Et(ln(WT )) under the VaR constraint is equivalent to solving ∀t the
following QCQP:
minimize FT
t µt +
1
2
FT
t ΣFt
subject to FT
t ΣFt ≤ L
(1.9)
where
µt =





S1t( a1
T−t
+ r)
.
.
.
SNt( aN
T−t
+ r)





(1.10)
for the convergence trades case and
µt =





ΦT
1 (St − S̄) + rS1t
.
.
.
ΦT
N (St − S̄) + rSNt





(1.11)
for the mean reversion trading strategies case.
39

We can easily solve the problem 1.9 by applying the KKT conditions or by ge-
ometry (see Appendix). Fopt
t , λopt
t are optimal iff they satisfy the following KKT
conditions ([10]):
• Primal feasibility: FT opt
t ΣFopt
t ≤ L
• Dual feasibility: λopt
t ≥ 0
• Complementary slackness: λopt
t (FT opt
t ΣFopt
t − L) = 0
• Minimization of the Lagrangean: Fopt
t = argmin L(Ft, λopt
t )
By solving the KKT conditions (see Appendix for details) we find that:
θopt
t =







−Σ−1
µtWt if µT
t Σ−1
µt ≤ L
− Σ−1µtWt
r
µT
t Σ−1µt
L
if µT
t Σ−1
µt ≥ L
This is equivalently written as:
θopt
t = −
Σ−1
µtWt
1 + λopt
t
where 1 + λopt
t = max 1,
r
µT
t Σ−1µt
L
!
Let’s now discuss more the properties of the solution. The investor has logarithmic
preferences. Therefore, he is a myopic optimizer - there is no hedging demand [59].
At each time t he looks dt ahead and decides how to trade in an optimal way. There
are two cases to consider:
• Case 1: At time t: µT
t Σ−1
µt ≤ L In this case, the optimal solution is the
unconstrained myopic optimal solution, since it satisfies the VaR constraint.
For the convergence trades case, this is equivalent to the spread St being in the
ellipsoid Et = {S | ST
(AtΣ−1
At)S ≤ L} where At = diag( a1
T−t
+ r, · · · , aN
T−t
+ r).
40

The volume of the ellipsoid Et is shrinking as t → T, since vol(E) =
QN
i=1( a1
T−t
+
r)−1
p
det(Σ)vol(B(0, 1)) where B(0, 1) is the unit sphere. Figure 1-1 shows this
shrinking ellipsoid at three time instants.
For the mean reversion trading strategies case, this is equivalent to the spread or
value of the trade being inside the convex set C = {S | (S−S̄)T
((Φ+rI)T
Σ−1
(Φ+
rI))(S − S̄) + 2rS̄T
Σ−1
(Φ + rI))(S − S̄) ≤ L − r2
S̄T
Σ−1
S̄}, which in the case
of r = 0 is the ellipsoid C = {S | (S − S̄)T
(ΦT
Σ−1
Φ)(S − S̄) ≤ L. If S̄ = 0 this
convex set is also an ellipsoid.
These ellipsoids characterize poor opportunities where the constraints are not
active. What constitutes poor investment opportunities changes over time for
the case of convergence trades, while it remains invariant for the mean reversion
trades case. For the case of convergence trades, the same spreads initially can be
considered poor investment opportunities, where the investor does not bind the
constraint, he is more conservative, but after some time they can be considered
good opportunities and the investor becomes more aggressive and binds the
constraint.
Informally, when the investment opportunities are poor, the spreads are more
likely to widen which then would lead to mark-to-market losses and the investor
would not have sufficient wealth to take advantage the better investment oppor-
tunities and simultaneously satisfy the VaR constraints. Therefore, the investor
is more conservative.
• Case 2: At time t: µT
t Σ−1
µt > L Now the unconstrained myopic optimal solu-
tion does not satisfy the VaR constraint. This case is equivalent to the spread
St being outside the shrinking ellipsoid Et for the convergence trades case or the
set C for the mean reversion trades case. Now the investment opportunities are
good. The investor wants to invest the unconstrained optimal trading strategy,
but due to the VaR constraint invests in the proportion of this optimal trading
strategy necessary to satisfy the VaR constraint.
41

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Ellipsoids of poor investment opportunities for t=0.3, 0.6, 0.9.
Spread 1
Spread
2
Figure 1-1: Ellipsoids. Ellipsoids of poor investment opportunities for N=2 conver-
gence trades at times t = 0.3, 0.6, 0.9.
Margin constraint
Maximizing Et(ln(WT )) under the margin constraint is equivalent to solving ∀t the
following convex program:
minimize FT
t µt +
1
2
FT
t ΣFt
subject to
PN
i=1 λi|Fit| ≤ 1
(1.12)
µt =





S1t( a1
T−t
+ r)
.
.
.
SNt( aN
T−t
+ r)





µt =





ΦT
1 (St − S̄) + rS1t
.
.
.
ΦT





42

Let’s apply the KKT conditions. Fopt
t , νopt
t are optimal iff they satisfy the KKT
conditions:
• Primal feasibility:
PN
i=1 λi|Fit| ≤ 1
• Dual feasibility: νopt
t ≥ 0
• Complementary slackness: νopt
t (
PN
i=1 λi|Fit| − 1) = 0
• Minimization of the Lagrangean Fopt
t = argmin L(Ft, λopt
t )
This program cannot be solved analytically in general. Again there are two cases to
consider.
• Case 1: At time t: kΛΣ−1
µtk1 ≤ 1 where Λ = diag(λ1, · · · , λN ) In this case, the
optimal solution is the unconstrained myopic optimal solution, since it satisfies
the margin constraint.
For the convergence trades, this is equivalent to having at time t: kΛΣ−1
AtSk1 ≤
1 where Λ = diag(λ1, · · · , λN ) and At = diag( a1
T−t
+ r, · · · , aN
T−t
+ r). In this case
we have that St is inside a “diamond” in N dimensional space, which shrinks
as t → T.
For the mean reversion trades, this is equivalent to having at time t: kΛΣ−1
(Φ(St−
S̄) + rSt)k1 ≤ 1 where Λ = diag(λ1, · · · , λN ).
Informally again, when the investment opportunities are poor, the spreads are
more likely to widen which then would lead to mark-to-market losses and the in-
vestor would not have sufficient wealth to take advantage the better investment
opportunities and have enough wealth for the collateral necessary to secure the
liabilities.
43

• Case 2: At time t: kΛΣ−1
µtk1 ≥ 1 where Λ = diag(λ1, · · · , λN ). Now the
investment opportunities are good, the unconstrained myopic optimal solution
does not satisfy the collateral constraint and the constraint binds at the optimal
solution.
Uncorrelated opportunities
There is a special case when the trading opportunities are uncorrelated, where we
can solve analytically the KKT conditions (see Appendix for details). In that case
the optimal positions are given by:
θopt
it =
sign(−µit)(|µit
λi
| − νopt
t )+
σ2
i
λi
Wt (1.13)
We observe the following:
• First of all for the convergence trades, in case the spread is positive we short the
spread as we would expect and in case it is negative we are long the spread. For
the mean reversion trades, the sign is the opposite of the sign of ΦT
i (St−S̄)+rSit.
• Second, if µt is high relative to the collateral then the magnitude of the position
is higher.
• Third, if the variability of the opportunity is high the magnitude of the corre-
sponding position is low.
• Finally the more interesting property of the solution is that it has a cutoff value,
the dual variable, and if the absolute value of µt over the collateral is greater
than the dual variable the position is different from zero otherwise the position
is 0.
It is:
νopt
t = 0 if
N
X
i=1
|λiµit|
σ2
i
≤ 1
44

and
νopt
t > 0 if
N
X
i=1
|λiµit|
σ2
i
> 1
The dual variable is 0 when the investment opportunities are poor. It is easy to see
that when the margin constraint binds we have:
F̃it
opt
=
sign(−µit)(|µit
λi
| − νopt
t )+
σ2
i
λ2
i
andkF̃k1 = 1 (1.14)
1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Weights in different arbitrage opportunities.
Weights
Figure 1-2: Weights for the case of uncorrelated spreads and collateral con-
straint. Weights for the case of uncorrelated spreads.
In Figure 1-2 we can see an example of how we invest in different convergence
trades when there is no correlation among them, with λ = 1 and volatilities equal
to 1 for all the opportunities. The height of each bar is the absolute value of µit
and we invest only in those spreads where the µit is larger than the dual variable.
If
PN
i=1 |µit| < 1, then the dual variable is 0, we invest in all the opportunities and
the collateral constraint does not bind. If
PN
i=1 |µit| > 1 as is in the figure then the
margin constraint binds, the dual variable is positive and we can find it as follows.
We start from the maximum µit and then reduce it until the sum of the weights is
equal to 1 where each weight is the distance between the absolute value of µit and ν.
45

1.2.4 Connection with Ridge and Lasso regression
Before we explore further the properties and the results of the optimal trading strate-
gies, it would be interesting to digress for a while and see what connection there is
between our problems and the regularized regressions.
In the basic form of regularized regression, the goal is not only to have a good fit,
but also regression coefficients that are “small”. Two of the most common forms of
regularized regressions are the Ridge and Lasso regression.
Ridge regression shrinks the regression coefficients by imposing a penalty on their
size [42]. Equation 1.15 is one of the ways to write the Ridge problem.
minimize
PN
i=1(yi − β0 −
Pp
j=1 xijβj)2
subject to
Pp
j=1 β2
j ≤ t
(1.15)
The Ridge regression coefficients solution is similar to the optimal trading strategy
followed by a risk averse investor with logarithmic preferences, who can choose among
N diffusion processes and faces VaR constraints. In both cases we have this propor-
tional shrinkage where we reduce all the weights by a constant.
Lasso regression is another common form of a regularized regression. It can be
used as a heuristic for finding a sparse solution. It does a kind of continuous subset
selection [16]. Equation 1.16 is one of the ways to write the Lasso problem.
minimize
PN
i=1(yi − β0 −
Pp
j=1 xijβj)2
subject to
Pp
j=1 kβjk ≤ t
(1.16)
The Lasso regression coefficients solution is similar to the optimal trading strategy
followed by a risk averse investor with logarithmic preferences, who can choose among
N diffusion processes and faces margin constraints. Therefore, we can expect that in
this case we will have a sparse solution where the weights of several of the opportu-
nities will be 0.
46

1.3 Results
Let us move on now to the results first for the convergence trades and then for the
mean reversion trading strategies.
1.3.1 Convergence trades
VaR constraints. We have simulated the optimal trading strategy for N = 2
correlated convergence trading opportunities under VaR constraints. We find the
following:
• It is often optimal for an investor to underinvest i.e. not to bind the constraint.
• The investor typically experiences losses early before locking at a profit as we
can see in Figures 1-3, 1-4, 1-5, 1-6.
• Tighter constraints lead to less variability and less skewness in the distribution
of wealth. They also lead to less final wealth as we can see in Figures 1-5, 1-6.
• The wealth is higher when the opportunities hedge each other, as we can see
in Figures 1-4, 1-6. This makes sense because when the constraints are binding
we care more about losing money which would then lead surely to liquidation
when the investment opportunities are better and therefore we prefer the op-
portunities to hedge each other. Figure 1-7 shows a typical path for the wealth
evolution using the same noise process for positive and negative correlation
under the VaR constraint. We see clearly this hedging effect where negative
correlation leads to more wealth.
• When the initial values of the convergence trades are low, the constraints bind
for a small percentage of time and final wealth is negatively correlated to the
percentage of time the constraints bind. Figure 1-8 shows this effect.
• The final portfolio wealth is highly positively skewed as it is obvious in Figures
1-3, 1-4, 1-5, 1-6
47

For all the simulations we used: σ1 = σ2 = 1, a1 = a2 = 1, S[0] = [1; 1], rf =
0.06, number of steps = 1000.
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Distribution of wealth Time 0.25 rho 0.5
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Distribution of wealth Time 1 rho 0.5
Figure 1-3: VaR constraints, positive correlations. Wealth distribution at t =
0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5)
convergence trades, while facing VaR constraints (K=1). Initial wealth is $100.
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.25 rho −0.5 K 1
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.5 rho −0.5 K 1
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.75 rho −0.5 K 1
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 1 rho −0.5 K 1
Figure 1-4: VaR constraints, negative correlations. Wealth distribution at t =
0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5)
convergence trades, while facing VaR constraints (K=1). Initial wealth is $100.
48

0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.25 rho 0.5 K 0.25
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.5 rho 0.5 K 0.25
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.75 rho 0.5 K 0.25
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 1 rho 0.5 K 0.25
Figure 1-5: VaR constraints, positive correlations, tight constraints. Wealth
distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively cor-
related (ρ = 0.5) convergence trades, while facing VaR constraints (K=0.25). Initial
wealth is $100.
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.25 rho −0.5 K 0.25
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.5 rho −0.5 K 0.25
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.75 rho −0.5 K 0.25
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 1 rho −0.5 K 0.25
Figure 1-6: VaR constraints, negative correlations, tight constraints. Wealth
distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively
correlated (ρ = −0.5) convergence trades, while facing VaR constraints (K=0.25).
Initial wealth is $100.
49

0 200 400 600 800 1000 1200
0
200
400
600
800
1000
1200
1400
Simulation step
Final
wealth
rho 0.5
rho −0.5
Figure 1-7: Wealth evolution under VaR constraint. Typical path of the wealth
evolution for an investor investing in two convergence trades using the same noise
process for positive and negative correlation under the VaR constraint. Initial wealth
is $100.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
200
400
600
800
1000
1200
1400
1600
Frequency the constraint binds
Final
wealth
Figure 1-8: Relation between final wealth and frequency the VaR constraint
binds. Final wealth is negatively correlated to the percentage of time the constraints
bind when the initial values of the convergence trades are low.
50

Margin constraints. We have also simulated the optimal trading strategy for N =
2 correlated convergence trading opportunities under margin constraints using the
same noise process as with the VaR constraints. We have similar results with the
case of VaR constraints as we see in Figures 1-9, 1-10, 1-11, 1-12, 1-13 with the
following important differences:
• When the constraints bind, it is often the case that the position in one of the
convergence trades is 0, i.e. we have less diversification, sparse solution. Figure
1-15 shows a typical path of the positions in two convergence trading opportu-
nities under VaR constraints, where we see that they tend to be different than
0. Figure 1-16 shows the evolutions of the positions in two convergence trading
opportunities under margin constraints for the same exactly asset processes as
before. We clearly see that often we invest only in one position, as we expected
due to the similarity of the positions with the Lasso regression coefficients.
• The final wealth is less skewed and smaller with respect to the case of VaR
constraints.
51

0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.25 rho 0.5 Collateral 1
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 1 rho 0.5 Collateral 1
Figure 1-9: Margin constraints, positive correlations. Wealth distribution at
t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5)
convergence trades, while facing margin constraints (Collateral = 1). Initial wealth
is $100.
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 0.25 rho −0.5 Collateral 1
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 1 rho −0.5 Collateral 1
Figure 1-10: Margin constraints, negative correlations. Wealth distribution at
t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5)
convergence trades, while facing margin constraints (Collateral = 1). Initial wealth
is $100.
52

0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 1 rho 0.5 Collateral 2
Figure 1-11: Margin constraints, positive correlations, more collateral
needed. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests
in two positively correlated (ρ = 0.5) convergence trades, while facing margin con-
straints (Collateral = 2). Initial wealth is $100.
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
0 500 1000 1500 2000 2500 3000 3500 4000
0
50
100
Time 1 rho −0.5 Collateral 2
Figure 1-12: Margin constraints, negative correlations, more collateral
needed. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests
in two negatively correlated (ρ = −0.5) convergence trades, while facing margin con-
straints (Collateral = 2). Initial wealth is $100.
53

0 200 400 600 800 1000 1200
0
100
200
300
400
500
600
700
Simulation step
Final
wealth
rho 0.5
rho −0.5
Figure 1-13: Wealth evolution under margin constraint. Typical path of the
wealth evolution for an investor investing in two convergence trades using the same
noise process for positive and negative correlation under the margin constraint. Initial
wealth is $100.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
200
400
600
800
1000
1200
1400
1600
Frequency the constraint binds
Final
wealth
Figure 1-14: Relation between final wealth and frequency the margin con-
straint binds. Final wealth is negatively correlated to the percentage of time the
constraints bind when the initial values of the convergence trades are low.
54

0 100 200 300 400 500 600 700 800 900 1000
−800
−600
−400
−200
0
200
400
600
800
Simulation step
Position
Convergence trade 1
Convergence trade 2
Figure 1-15: Positions evolution under VaR constraints. Typical path of the
positions in two convergence trading opportunities under VaR constraints.
0 100 200 300 400 500 600 700 800 900 1000
−400
−300
−200
−100
0
100
200
300
400
Simulation step
Position
Convergence trade 1
Convergence trade 2
Figure 1-16: Positions evolution under margin constraints. Typical path of
the positions in two convergence trading opportunities under margin constraints.
55

1.3.2 Mean reversion trading opportunities
Similar results we get also by simulating the optimal trading strategy for N = 2
correlated mean reversion trading opportunities under VaR and margin constraints.
This makes sense since without loss of generality, we have assumed that S̄ = 0 and
that Φ is a diagonal matrix, which makes the mean reversion trades case similar to the
convergence trades case. They are different in that the drift term of the convergence
trades for the same spreads S gets better and better as t → T, while for the mean
reversion trades it remains constant.
1.4 Conclusions
We explored the optimal portfolio allocation of a risk averse investor who invests in
N convergence trades or mean reverting trading strategies, while facing constraints.
In particular, we studied the optimal trading strategy when he faces VaR constraints
or collateral constraints. The optimal trading strategy is found by solving at each
time t a convex program for both cases, we characterized the solution of the convex
program and we found the properties of the optimal trading strategy. In all the
chapter, we have assumed that the investor completely trusts his model and he is
certain about the dynamics of the opportunities he faces. What happens if the model
is just an approximation? What happens if the investor believes that opportunities’
dynamics come from an unknown member of a set of unspecified models near an
approximating model? Concern about model misspecification will change the optimal
trading strategy of the investor and this is the topic of the next chapter.
56

Chapter 2
Optimal trading of arbitrage
opportunities under model
misspecification
A decision maker maximizes a utility function subject to a model. Standard control
theory helps a decision maker to make optimal decisions when his model is correct.
Robust control theory helps him to make good decisions when his model is approxi-
mately correct. In this chapter we will use methods of robust control theory to find
the optimal portfolio allocation of a risk averse investor, who invests in convergence
trades or mean reverting trading strategies, and is not completely confident about
the dynamics of his models.
In particular, we assume that the investor believes that the data comes from
an unknown member of a set of alternative models near his nominal model. These
alternative models are statistically difficult to distinguish from the nominal model.
The investor believes that his model is a pretty good approximation in the sense that
the discrepancies between the alternative models and his nominal model are small.
We will use the relative entropy to characterize the discrepancies between different
models. Concern about model misspecification leads the investor to choose a trading
strategy that is robust over the alternative models.
Three questions come naturally at this point:
57

• What does it mean to have a robust trading strategy?
A robust trading strategy is a strategy that works well over the set of alternative
probability models. We evaluate the worst performance of a given strategy over
that set of alternative probability models and we pick the one that maximizes
this worst case performance. It is essentially a “max-min” problem, a two-player
game in which a maximizing player chooses the best response to a malevolent
player who can disturb the stochastic model within limits.
• Why would we be interested in a robust decision rule over alternative models?
Why don’t we take a Bayesian approach, where we put a prior distribution over
the set of alternative probability models?
This could be another approach, but this set of alternative models may be too
large or too difficult for the investor to come up with a well behaved, plausible
prior distribution. In addition, we might want our solution to work well over
any kind of prior distribution [41].
• Why do we use the relative entropy to measure the discrepancy between an
alternative and the nominal model?
There are other ways to measure discrepancies between alternative probability
models, like Prokhorov distance [9] but the relative entropy with respect to a
measure P has nice properties and it is more tractable. It is given by:
D(Q) =
Z
log(
dQ
dP
)dQ
and it is a convex function of the measure Q.
In the rest of this chapter, we will review the relevant literature. Then we will
discuss about the set of the alternative models, the relative entropy and equivalent
ways to formulate our problem of the optimal robust portfolio allocation of the in-
vestor. Subsequently, we will find the optimal robust trading strategy of the investor
and finally we will explore the characteristics of this robust strategy.
58

2.1 Literature review
Whittle [79], [80] has studied mathematical methods for answering the question of
how to make decisions when you don’t fully trust your model.
Lars Hansen and Thomas Sargent in [41] have studied how to make economic
decisions in the face of model misspecification by modifying and extending aspects
of robust control theory. Their work revolves mostly around the linear-quadratic
regulator framework, where there is a certainty equivalence principle that allows a
deterministic presentation of the control theory.
Gilboa and Schmeidler in [34] have studied the max-min expected utility problem
where the decision maker has multiple priors and maximizes his expected utility as-
suming that nature chooses a probability measure to minimize his expected utility.
The minimization is over a closed and convex set of finitely additive probability mea-
sures. Their axiomatic treatment views this set of non-unique priors as an expression
of the agent’s preferences and the priors are not cast as distortions to a nominal
model.
Lars Hansen et al. [40] have studied robust decision rules when the agent fears that
the data are generated by a statistical perturbation of an approximating model that
is either a controlled diffusion process or a control measure over continuous functions
of time. They describe how stochastic formulations of robust control “constraint
problems” can be viewed in terms of Gilboa and Schmeidler’s max-min expected
utility model. They show the connection between the penalty robust control problem
and the constraint robust control problem, two closely related problems and formulate
the Hamilton Jacobi Bellman equations for various two-player zero sum continuous
time games that are defined in terms of a Markov diffusion process. We extend their
framework to the problem of optimal robust trading rules for a risk averse investor
who does not trust his model dynamics, believes that his nominal model is a good
approximation to the real model and invests in arbitrage opportunities.
Fleming and Souganidis [77] present how the Bellman-Isaacs condition defines a
Bellman equation for a two-player zero-sum game in which both players decide at time
59

0 or recursively. In other words, they show that the freedom to exchange orders of
maximization and minimization guarantees that equilibria of games where the choices
are done under mutual commitment at time 0 and of games where the choices are
done sequentially by both agents coincide.
Anderson et al. [3] show how the set of perturbed models in our formulations
is difficult to distinguish statistically from the approximating model given a finite
sample of timeseries observations.
Jacobson [44] and Whittle [78] studied risk sensitive optimal control in the context
of discrete-time linear quadratic regulator decision problems. They showed how the
risk-sensitive control law can be computed by equivalently solving a robust penalty
problem.
We will now discuss first how to represent the alternative probability models over
which we want our decision rules to be robust and how relative entropy can be used
to describe their discrepancies from the nominal model. We will formulate two closely
related nonsequential problems and the corresponding recursive HJB equations and
finally we will find the optimal robust portfolio allocation for a risk averse investor who
is not confident about the dynamics of his models and wants to invest in convergence
trades or mean reversion trading strategies.
2.2 Analysis
In Chapter 1 we saw that in the case when there is no model misspecification, the
investor wants to find the optimal portfolio allocation that solves the following prob-
lem:
maximizeθ∈Θ E(lnWT )
subject to
dWt = θtdSt + θ0tdBt
dSt = µ(S, t)dt + σ(S, t)dZt
dBt = rBdt
(2.1)
60

Here we have St ∈ RN
and we have studied the following special cases:
dSit = −
aiSit
T − t
dt +
K
X
k=1
σikdZkt
dSt = −Φ(St − S̄)dt + σdZt
and Θ is the set of admissible trading strategies:
Θ =

θ |θT
Σθ ≤ LW2
for the case of VaR constraints and
Θ =
(
θ |
N
X
i=1
λi|θi| ≤ W
)
for the case of the margin constraints.
In this Chapter, the investor doubts his model dSt = µ(S, t)dt + σ(S, t)dZt. To
capture this doubt of the investor, we surround the approximating model with a cloud
of models that are statistically difficult to distinguish and we add a malevolent agent
who picks the worst possible model. The investor wants to find the optimal trading
strategy that solves the following problem:
maxθ∈Θ minQ∈Q EQ(lnWT ) (2.2)
where Θ is the set of admissible trading strategies and Q is the set of alternative
probability models. Problem 2.2 fits the max-min expected utility model of Gilboa
and Schmeidler [34], where Q is a set of multiple different priors. Let’s now discuss
how we represent the set of alternative probability models.
2.2.1 Alternative models representation
We use martingales to represent perturbations to the probability models and relative
entropy to measure the discrepancy between our nominal model and the alternative
61

models. To understand better our continuous time formulations, we digress for a
while by borrowing an example from [41].
Let’s consider a discrete time approximating model and its innovations ǫt which are
i.i.d Gaussian shocks. An alternative model alters the distribution of these shocks.
We use martingales to represent distortions to the probabilities. Let π̂t(ǫ) be the
alternative density of the shock ǫt+1 based on date t information. Then the random
variable Mt =
Qt
j=1 mj, where mj =
π̂j−1(ǫ)
π(ǫ)
and M0 = 1, is a martingale and is a
ratio of the joint alternative density over the joint nominal density. We define the
entropy of the alternative distribution associated with Mt as the expected likelihood
ratio with respect to the distorted distribution E(Mtlog(Mt)). It has the property
that it is always non-negative and it is equal to 0, only when there is no distortion to
the nominal distribution.
Similarly, in our continuous time formulations we will use martingales to represent
distortions to the nominal probability model. We will construct an alternative model
by replacing Zt in our model by Ẑt +
R t
0
hsds, where Ẑt is a Brownian motion under the
alternative measure Q and ht is an adapted process that models the distortion, such
that the process ξt = e
R t
0
hsdZs− 1
2
R t
0
hT
s hsds
is a martingale. Therefore, the nominal model
is misspecified by allowing the conditional mean of the shock vector in the alternative
models to feed back arbitrarily on the history up to date t. Since ξ0 = 1 we have
that E(ξt) = 1. Since in addition, ξt 0, we can define a probability measure Q such
that Q(A) = E[1AξT ], in other words ξT = dQ
dP
is the Radon-Nikodym derivative of
Q with respect to P, where the measures Q and P are equivalent. In fact one can
always define a process ht so that for any measure Q the Radon-Nikodym derivative
of Q with respect to P, dQ
dP
, is given by the exponential martingale ξT . In this way,
our distorted models are:
• For the convergence trades,
dSt = −
aiSit
T − t
dt +
K
X
k=1
σik(dẐkt + hktdt) (2.3)
62

• For the mean reversion trades,
dSt = −Φ(St − S̄)dt + σ(dẐt + htdt) (2.4)
where Ẑt is a Brownian motion under Q. Why is it a Brownian motion under Q? The
answer lies with the Girsanov theorem [30] that states that if a process ht is such that
ξt is a martingale and ξT = dQ
dP
is the Radon-Nikodym derivative of Q with respect
to P, then the process ˆ
Z(t) = Zt −
R t
0
hsds is a Brownian motion under measure Q.
Therefore, we parameterize Q by the choice of the drift distortion adapted process
ht.
Similarly with the discrete time case, we measure the discrepancy between mea-
sures Q and P as the relative entropy D(Q) (see Appendix for derivation),
D(Q) =
Z T
0
1
2
EQ[hT
t ht]dt (2.5)
.
This is to be expected, since the relative entropy between a multivariate Gaussian
distribution N(µ, I) and the multivariate standard normal distribution is D(Q) =
1
2
µT
µ (See Appendix for derivation) and htdt is the conditional mean of the process
dZt under the alternative probability measure Q.
To express the notion that the nominal model is a good approximation to the real
model that generate the spread dynamics, we either restrain the alternative models
by D(Q) ≤ η or we penalize them with the magnitude of the entropy.
2.2.2 Model setup
Having described the set of alternative distributions, we are ready to formulate the
problem a risk averse investor faces who distrusts his model dynamics. As in [40] we
define two closely related problems:
63

• A multiplier robust control problem.
maxθ∈Θ minQ EQ(lnWT ) + νD(Q)
subject to dWt = θtdSt + θ0tdBt
dSt = µ(S, t)dt + σ(S, t)(dẐt + htdt)
dBt = rBdt
dξt = ξthtdZt
(2.6)
where ξT = dQ
dP
and D(Q) is given by equation 2.5. Here in essence there is an
implicit restriction manifested by the nonnegative penalty parameter ν.
• A constrained robust control problem.
maxθ∈Θ minQ EQ(lnWT )
subject to dWt = θtdSt + θ0tdBt
dSt = µ(S, t)dt + σ(S, t)(dẐt + htdt)
dBt = rBdt
dξt = ξthtdZt
D(Q) ≤ η
(2.7)
where ξT = dQ
dP
and D(Q) is given by equation 2.5.
In both cases the minimizing malevolent agent chooses the distortion process ht taken
θ as given and the maximizing investor chooses the optimal strategy taken ht as given.
We index the family of multiplier robust control problems by ν and the family of
constrained robust control problems by η. Obviously the two problems are related,
since the robustness parameter ν can be interpreted as the Lagrange multiplier on
the constraint D(Q) ≤ η. Actually we can show that if V (ν) is the optimal value
of the multiplier robust problem and K(η) is the optimal value of the constrained
robust problem then we have: K(η) = maxν≥0V (ν) − νη [40]. Therefore we will be
only interested in finding V (ν).
64

2.3 Solution
We will solve 2.6 by solving the corresponding Hamilton Jacobi Bellman (HJB) equa-
tion. We will solve the HJB equation for the case that there are no constraints in
the admissible trading strategies and when the trading strategies are constrained by
VaR or collateral considerations. But first let’s digress for a while and solve the HJB
equation for the case when there is no fear of model misspecification.
2.3.1 No fear of model misspecification
For now we assume that the investor completely trusts the dynamics of his models
dSt = µ(S, t)dt + σ(S, t)dZt. He chooses the trading strategy θ ∈ Θ that solves the
problem 2.1 where Θ is the set of admissible trading strategies:
Θ =

θ|θT
Σθ ≤ LW2
Θ =
(
θ|
N
X
i=1
λi|θi| ≤ W
)
for the case of the margin constraints. In this case the HJB equation is:
max
θ∈Θ
Vt + VW (Wr + θT
(µ(S, t) − rSt)) + V T
S µ(S, t)
+ 1/2VW W θT
Σθ + VW SΣθ + 1/2trace(ΣVSS) = 0
where Σ = σσT
and V (W, S, t) is the value function of the investor subject to the
terminal condition V (W, S, T) = ln(W).
Due to the logarithmic preferences of the investor it is: V (W, S, t) = ln(W)+H(S, t),
therefore VW S = 0, VW = 1
W
and VW W = − 1
W 2 . We also define ∀t ∈ [0, T] Ft = θt/Wt
∈ RN
.
65

The HJB equation becomes:
max
F ∈F
Vt + (r + FT
(µ(S, t) − rSt)) + V T
S µ(S, t)
− 1/2FT
ΣF + 1/2trace(ΣVSS) = 0
where F is the set of admissible trading strategies:
F =

F|FT
ΣF ≤ L
F =
(
F|
N
X
i=1
λi|Fi| ≤ 1
)
for the case of the margin constraints.
The optimal trading strategy is the solution to the following convex problem
min
F ∈F
FT
(−µ(S, t) + rSt) + 1/2FT
ΣF (2.8)
as we also proved with a different method in Chapter 1, where µt = −µ(S, t) + rSt
and in particular it is:
µt =





S1t( a1
T−t
+ r)
.
.
.
SNt( aN
T−t
+ r)





µt =





ΦT
1 (St − S̄) + rS1t
.
.
.
ΦT





66

2.3.2 Fear of model misspecification no constraints
In this section we assume that there are no constraints in the trading strategies fol-
lowed by the risk averse investor and the investor is not confident about the dynamics
of his models. The Hamilton Jacobi Bellman equation for the problem 2.6 is given
by:
max
θ
min
h
Vt + VW (Wr + θT
(µ(S, t) − rSt)) + V T
S µ(S, t)
+ 1/2VW W θT
Σθ + VW SΣθ + 1/2trace(ΣVSS) + VW θT
σh + V T
S σh +
ν
2
hT
h = 0
where Σ = σσT
and V (W, S, t) is the value function of the investor subject to the
terminal condition V (W, S, T) = ln(W). The malevolent agent picks the worst case
distortion drift process ht and the investor maximizes against the worst case scenario.
After defining ∀t ∈ [0, T] Ft = θt/Wt ∈ RN
the HJB equation becomes:
max
F
min
h
Vt + WVW (r + FT
(µ(S, t) − rSt)) + V T
S µ(S, t)
+1/2W2
VW W FT
ΣF+WVW SΣF+1/2trace(ΣVSS)+WVW FT
σh+V T
S σh+
ν
2
hT
h = 0
The inner minimization problem is a convex quadratic problem. The first order
conditions are:
WVW σT
F + σT
VS + νh = 0
h = −
σT
(WVW F + VS)
ν
The optimal value of the inner minimization problem is:
g(F) = −
W2
V 2
W Ft
ΣF + V T
S ΣVS + 2WVW FT
ΣVS
2ν
67

Plugging this back into the HJB equation we have:
max
F
Vt + WVW (r + FT
(µ(S, t) − rSt)) + V T
S µ(S, t)
+ 1/2W2
VW W FT
ΣF + WVW SΣF + 1/2trace(ΣVSS)
−
W2
V 2
W Ft
ΣF + V T
S ΣVS + 2WVW FT
ΣVS
2ν
= 0
Due to the logarithmic preferences of the investor it is: V (W, S, t) = lnW + H(S, t)
and in that case VW = 1
W
VW W = − 1
W 2 VW S = 0, VS(W, S, t) = HS(S, t) and the
minimizing drift distortion h = −σT (F +HS)
ν
independent of the wealth.
The HJB equation now becomes:
max
F
Vt + r + FT
(µ(S, t) − rSt) + V T
S µ(S, t)
− 1/2FT
ΣF + 1/2trace(ΣVSS) −
FT
ΣF + V T
S ΣVS + 2FT
ΣVS
2ν
= 0
The optimal trading strategy is the solution to the following convex quadratic prob-
lem:
maximize FT
(µ(S, t) − rSt −
ΣVS
ν
) −
1
2
(1 +
1
ν
)FT
t ΣFt (2.9)
The first order conditions are:
µ(S, t) − rSt −
ΣVS(St, t)
ν
= (1 +
1
ν
)ΣFopt
t
Fopt
t =
1
1 + 1
ν
Σ−1
(µ(S, t) − rSt −
ΣVS(St, t)
ν
)
Fopt
t =
ν
ν + 1
Σ−1
(µ(S, t) − rSt) −
VS(St, t)
ν + 1
We clearly see that as ν → ∞ the optimal trading strategy converges to the one where
we have no fear of model misspecification. This is to be expected since at this case the
problems 2.1 and 2.6 are equivalent. It is interesting to find the conditions under which
these weights are equal to the weights when there is no fear of model misspecification.
When there is a fear of model misspecification, the optimal weights are a convex
68

combination of Σ−1
(µ(S, t)−rSt), i.e.the weights without model misspecification and
−VS. Therefore these weights are equal to the weights when there is no fear of model
misspecification, when Vs + Fopt
= 0, which is equivalent to hmin = 0. Of course this
is expected since in that case there would be no distortion drift and the HJB equation
would be the same as the benchmark case of no model misspecification.
After plugging in the optimal trading strategy to the HJB equation, it becomes:
Vt + r + 1/2 trace(ΣVSS) + 1/2
1
1 + 1
ν
(µ(S, t) − rSt)T
Σ−1
(µ(S, t) − rSt)
+ V T
S

µ(S, t) −
µ(S, t) − rS
ν + 1

− 1/2
1
ν + 1
V T
S ΣVS = 0
We can plug in the optimal trading strategy to h = −σT (W VW F +VS)
ν
to find that:
hmin = −
(σT
Fopt
+ σT
VS)
ν
hmin = −
(σT 1
1+ 1
ν
Σ−1
(µ(S, t) − rSt − ΣVS
ν
) + σT
VS)
ν
hmin = −
σT
(Σ−1
(µ(S, t) − rSt) + VS)
ν + 1
We consider two cases:
• Convergence trades. The optimal trading strategy is given by:
θopt
t = −

ν
ν + 1
Σ−1
AtSt +
VS
ν + 1

Wt (2.10)
where At = diag( a1
T−t
+ r, · · · , aN
T−t
+ r). The optimal trading strategy is a
convex combination of the strategy without fear of model misspecification and
−VS with weights ν
ν+1
and 1
ν+1
. From the symmetry of the problem we have:
H(S, t) = H(−S, t) from which we get VS(St, t) = −VS(−St, t).
• Mean reversion trades. The optimal trading strategy is given by:
θopt
t = −

ν
ν + 1
Σ−1
(Φ(St − S̄) + rSt) +
VS
ν + 1

Wt (2.11)
69

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