Master’s Thesis
M.Sc. Course “Actuarial Science"
Retirement wealth under fixed limits:
the optimal strategy for exponential utility

submitted by: supervised by: Lena Schütte Dr. Catherine Donnelly Matriculation
No.: 08-933-848 Dr. Michael Schmutz Email<EMAIL_ADDRESS>Submission date, place: 30.01.2017, Edinburgh

## Acknowledgements

All my gratitude and thanks go to Dr.Catherine Donnelly for accepting me as a
visiting scholar at Heriot-Watt University in Edinburgh and without whom this
thesis would not have been possible. Her valuable advice and constant guidance
as well as the friendly support in all matters related to my stay here, have
made the last four months a very exciting and enriching time, both
scientifically and personally.
I would also like to thank the Department of Actuarial Mathematics and
Statistics at Heriot-Watt for providing great facilities and my collegues from
the office for creating a fun and motivating atmosphere.
Further I would like to thank Dr. Michael Schmutz for agreeing to be my
supervisor from the part of the University of Basel, Jolanda Bucher for
assisting with the administrative procedures and Matthias Kohlbrenner for
reviewing the paper.
A grant from the Swiss-European Mobility Progamme via the Mobility Office of
the University of Basel is also gratefully acknowledged.

###### Contents

  1. Acknowledgements
  2. List of Symbols
  3. 1 Introduction
  4. 2 An Optimal Strategy for Exponential Utility
    1. 2.1 Derivation of the Optimal Strategy
      1. 2.1.1 Market Model and Hamilton-Jacobi-Bellman Equation
      2. 2.1.2 The Optimal Strategy for an Exponential Utility Function
    2. 2.2 Brief Analysis of the Optimal Strategy
      1. 2.2.1 Parameters: $\alpha$ and T
      2. 2.2.2 Initial Wealth and Terminal Wealth Distribution
    3. 2.3 Extended Analysis of the Optimal Strategy under a Restriction on Investment
      1. 2.3.1 Initial wealth
      2. 2.3.2 Parameter: $\mu$ -r
      3. 2.3.3 Parameter: $\sigma$
      4. 2.3.4 Parameter: r
  5. 3 An Optimal Strategy for Exponential Utility and Lower Constraint $K_{l}$
    1. 3.1 Derivation of the $K_{l}$-Strategy
    2. 3.2 Analysis of the $K_{l}$-Strategy
      1. 3.2.1 First Observations
      2. 3.2.2 Different Scenarios
    3. 3.3 Brief Analysis of the $K_{l}$-Strategy under a Restriction on Investment
    4. 3.4 Comparison of Terminal Wealth Distributions
      1. 3.4.1 Theoretical vs. Empirical Distribution
      2. 3.4.2 Impact of Restriction on Investment
  6. 4 An Optimal Strategy for Exponential Utility and Upper Constraint $K_{u}$
    1. 4.1 Derivation of the $K_{l}$-$K_{u}$-Strategy
      1. 4.1.1 Derivation of the $K_{u}$-Strategy
      2. 4.1.2 The $K_{l}$-$K_{u}$-Strategy
    2. 4.2 Analysis of the $K_{l}$-$K_{u}$-Strategy
      1. 4.2.1 Analysis of the $K_{l}$-Strategy
      2. 4.2.2 Analysis of the $K_{l}$-$K_{u}$-Strategy
    3. 4.3 Distribution of Terminal Wealth
  7. 5 Conclusion
  8. 6 Outlook
  9. Appendix A
    1. 6.1 Derivation of HJB
    2. 6.2 Solving the linear SDE
    3. 6.3 Expected Utility Theory: From Lotteries to Utility Functions
    4. 6.4 Exponential Utility and Risk Aversion
    5. 6.5 Analytical Approach: Effect of Investment-Restriction
    6. 6.6 Quantiles of Optimal Terminal Wealth Distribution
    7. 6.7 Behaviour of $\hat{\pi}_{l}$ and $\hat{\pi}_{u}$
    8. 6.8 $K_{l}$-$K_{u}$-Strategy: Illustration of the impact of $K_{l}$
  10. Appendix B
    1. 6.9 R-Code for Simulations

###### List of Figures

  1. 1 Impact of risk aversion on initial investment for $\hat{\pi}$ (T=5)
  2. 2 Impact of risk aversion on initial investment for $\hat{\pi}$ (varying T)
  3. 3 Absolute investment over time for $\hat{\pi}$ (varying T)
  4. 4 Investment and wealth processes for $\hat{\pi}$ and $\hat{\pi}_{m}$ (varying $X_{0}$) and terminal wealth distribution for $\hat{\pi}_{m}$
  5. 5 Performance of stock ($\mu$ -r = 0.01 and $\mu$ \- r = 0.05)
  6. 6 Wealth process for $\hat{\pi}$ and $\hat{\pi}_{m}$ ($\mu$-r = 0.01 and $\mu$-r = 0.05 )
  7. 7 Histogram of total return on initial wealth for $\hat{\pi}_{m}$ ($\mu$-r = 0.01 and $\mu$-r= 0.05)
  8. 8 Difference $\hat{\pi}$-$\hat{\pi}_{l}$ over time (amount invested)
  9. 9 Difference $\hat{\pi}$-$\hat{\pi}_{l}$ in function of current shadow wealth (amount invested at t=19)
  10. 10 Higher shadow wealth leads to higher sensitivity of $\hat{\pi}_{l}$.
  11. 11 For a decrease in stock value, $\hat{\pi}_{l}$ goes to zero.
  12. 12 If the stock performs well, the first years are more critical to the missed upside potential by the constrained strategy $\hat{\pi}_{l}$.
  13. 13 Initial investment as a function of $X_{0}$, and wealth process for fixed $X_{0}$
  14. 14 Empirical distribution of terminal wealth for $\hat{\pi}_{l}$
  15. 15 Difference $\hat{\pi}$-$\hat{\pi}_{u}$ in function of current shadow wealth (amount invested at t=19)
  16. 16 $\hat{\pi}_{u}$ becomes negative for higher wealth and nearing maturity ($K_{u}$ = 1’250).
  17. 17 Poor stock performance leads to higher investment ($K_{u}$ = 3’000).
  18. 18 Shadow initial wealth in function of $K_{l}$ and $K_{u}$ ($X_{0}$ = 1’000)
  19. (a) $K_{u}$=4’000
  20. 20 Amount invested over time and wealth processes for $\hat{\pi}_{l,u}$ (high $K_{l}$ and $K_{u}$)
  21. 21 Example of a Lottery [9]
  22. 22 An exponential utility function with parameter $\alpha$=0.0001
  23. 23 Histogram of terminal wealth for $X_{0}$ = 10; 1000; 10’000
  24. 24 Behaviour of $\tilde{\pi}_{p}p(\tilde{X}^{\hat{\pi}}_{t},t)$ (upper line) and $\tilde{\pi}_{c}c(\tilde{X}^{\hat{\pi}}_{t},t)$ over time
  25. 25 Behaviour of $\tilde{\pi}_{p}p(\tilde{X}^{\hat{\pi}}_{17},17)$ (upper line) and $\tilde{\pi}_{c}c(\tilde{X}^{\hat{\pi}}_{17},17)$ in function of current shadow wealth
  26. 26 Amount invested over time and wealth process for $\hat{\pi}_{l,u}$ (different $K_{l}$)

###### List of Tables

  1. 1 Empirical expected return and variance of terminal wealth for $\hat{\pi}$
  2. 2 Quantiles of $X^{\hat{\pi}_{m}}_{T}$ and comparison for $\hat{\pi}_{0}^{-1}$= 120$\%$, 100$\%$ and 80$\%$
  3. 3 Quantiles of $X^{\hat{\pi}_{m}}_{T}$ as total return and comparison for $\mu$-r = 5$\%$, 3$\%$ and 1$\%$
  4. (a)  $\sigma$ = 0.4
  5. (a)  r = 0$\%$
  6. 6 Theoretical quantiles of $X^{\hat{\pi}_{l}}$ for different $K_{l}$
  7. 7 Deviation of empirical from theoretical quantiles of $X^{\hat{\pi}_{l}}_{T}$ (different samplesizes)
  8. 8 Deviation of empirical from theoretical quantiles of $X^{\hat{\pi}_{l}}_{T}$ (different stepwidths)
  9. 9 Empirical, investment-constrained vs. theoretical quantiles of $X^{\hat{\pi}}_{T}$ (varying scenarios)
  10. (a) $K_{l}$ = -30’000
  11. 11 Quantiles of $X^{\hat{\pi}^{l,u}}_{T}$ (different $K_{u}$ )
  12. 12 Quantiles of $X^{\hat{\pi}}_{T}$ (varying $X_{0}$)
  13. (a)  $\sigma$ = 0.4
  14. (a)  $\sigma$ = 0.1

## List of Symbols

$X_{0}$ | initial wealth (independent of strategy)
---|---
$\hat{\pi}$ | optimal unconstrained strategy
$X^{\hat{\pi}}_{t}$ | wealth process under $\hat{\pi}$
$\hat{\pi}_{m}$ | optimal strategy modified by restriction on investment
$X^{\hat{\pi}_{m}}_{t}$ | wealth process under $\hat{\pi}_{m}$
$K_{l}$ | lower bound for terminal wealth
$\hat{\pi}_{l}$ | optimal strategy under a lower bound for terminal wealth
$X^{\hat{\pi}_{l}}_{t}$ | wealth process under $\hat{\pi_{l}}$
$\hat{\pi}_{l,m}$ |  $\hat{\pi}_{l}$ modified by restriction on investment
$\tilde{X}_{0}$ | shadow wealth
$\tilde{X}^{\hat{\pi}}_{t}$ | shadow process (wealth process under $\hat{\pi}$ for $\tilde{X}_{0}$
$p(t,\tilde{X}^{\hat{\pi}}_{t})$ | price of put option on shadow wealth (strike price $K_{l}$)
$\tilde{\pi}_{p}$ | strategy for replication of $p(t,\tilde{X}^{\hat{\pi}}_{t})$
$K_{u}$ | upper bound for terminal wealth
$\hat{\pi_{u}}$ | optimal strategy under an upper constraint for terminal wealth
$X^{\hat{\pi_{u}}}_{t}$ | wealth process under $\hat{\pi_{u}}$
$c(t,\tilde{X}^{\hat{\pi}}_{t})$ | price of call option on shadow wealth (strike price $K_{u}$)
$\tilde{\pi}_{c}$ | strategy for replication of $c(t,\tilde{X}^{\hat{\pi}}_{t})$

##  1 Introduction

What is the best way to invest money for retirement? This question might be
more relevant than ever, since private and institutional investors face a
challenging low-interest market environment and growing retirement needs. At
the same time, this question has been widely researched in financial
mathematics and economics and offers many interesting approaches. Among these,
using the utility of wealth for an investor instead of, for example, the
simple return, as a criterion seems to best reflect the investor’s needs. The
strategies that maximize the expected utility are commonly called optimal
strategies and they can only be derived analytically for few utility
functions. The exponential utility function is one of them, and in this thesis
it will be used to determine the optimal strategy in a simple Black-Scholes-
Setting.
Besides developing a good understanding of the resulting strategy and its
effects on the wealth at retirement, we are particularly interested in
improving its potential while still taking the investor’s needs into
consideration. The idea is therefore to introduce upper and lower constraints
on the resulting wealth: Utility theory suggests that investors are more
sensitive towards lower values of wealth, so they might be ready to give up
some investment potential in exchange for a garantee on a minimal return. From
another perspective, it could be favorable to constrain the wealth at
retirement to a maximum amount (for example, the present value of annuities)
and be compensated by higher probabilites for greater returns on the wealth
below this maximum value.
It is one aim of this thesis to explore the consequences on the optimal
strategy for exponential utility with retirement wealth facing upper and lower
constraints. In this sense, this paper can be seen as a complement to the
research done in [4], where a power utility function is considered.
The second modification of the optimal strategy that is investigated in this
thesis is introduced in order to avoid debts. Since the optimal strategy might
involve borrowing money or short-selling, there is a risk that the investor
ends up with a negative wealth. This is why we would like to limit the
investment to a maximum of 100$\%$ of wealth. This restriction will be
implemented on the pre-existing strategies we developed, its consequences will
therefore be assessed from empirical results only.

We will proceed step by step and gradually adapt the strategy.
As a basis, we will derive the optimal unconstrained strategy in Chapter 2,
using stochastic optimal control arguments. We will then analyse it briefly
with respect to investor-related parameters and focus an the analysis of the
optimal strategy with a restriction on investment.

In Chapter 3, the optimal strategy where terminal wealth faces only a lower
constraint will be developed. This will be done by formulating a dual problem
and solving this via risk-neutral valuation. We will then see that the
resulting optimal strategy corresponds to the optimal unconstrained strategy
combined with a put option. After some qualitative analysis, we will then
implement the investment-restriction and see how it affects the strategy.
Finally, an emphasis will be put on the theoretical distribution of the
resulting wealth (without constraints) and the error produced by the
implementation of the modified strategies.

In the last chapter, we will add an upper constraint to the problem. To do
this, we will first find the strategy for the isolated case of an upper
constraint by the similar methods as used before. Then we will combine it with
the results from the previous chapter. It turns out, that in addition to the
put option bought, the ’combined’ strategy requires to sell a call option. For
further analysis of this strategy, we will first briefly investigate the
isolated case of an upper constraint and then try to outline how the choice
for the upper and lower constraints affect the ’combined’ stategies, both
qualitatively and with respect to the distribution of terminal wealth.

Finally, the Appendix is thought to gather background information on the
theory used, as well as complementary analysis to validate the empirical
results.

##  2 An Optimal Strategy for Exponential Utility

We will start by finding the strategy that maximizes the expected exponential
utility of terminal wealth and briefly analyze its results. Further, the
strategy is modified by introducing a restriction on the amount invested. The
resulting strategy and terminal wealth distribution will then be investigated
in more detail.

###  2.1 Derivation of the Optimal Strategy

In this section, we will introduce the formal setting and derive the optimal
unconstrained strategy by solving an differential equation that characterises
optimal strategies. We will see that it requires a deterministic amount to be
invested in the risky asset, independently of the investor’s wealth or the
stock’s performance, but growing by the risk-free rate. The resulting terminal
wealth is normally distributed.

####  2.1.1 Market Model and Hamilton-Jacobi-Bellman Equation

We assume the Black-Scholes market model consisting of one risky stock and one
risk-free bond, available in the continous time interval [0,T]. The integer T
> 0 denotes the terminal time, for example the moment of retirement. The price
of the bond at time t is given by the deterministic price process {B(t), t
$\in$ [0,T]} with dynamics

$dB(t)=rB(t)dt,$ (2.1)

where r >0 is the risk-free interest rate and B(0) = 1 almost surely
(abbreviated a.s.).
The performance of the risky stock at time t is given by the stochastic price
process
{S(t), t $\in$ [0,T] } with dynamics

$dS(t)=\mu S(t)dt+\sigma S(t)dW(t),$ (2.2)

where $\sigma$ > 0 , S(0) = 1 a.s. , $\mu$ > r and W(t) is the 1-dimensional
standard Brownian motion defined on a complete probability space ($\Omega$,
$\mathcal{F},\mathbb{P}$).
The information available up to time t is represented by the filtration

$\mathcal{F}_{t}=\sigma\\{W(s),s\in[0,t]\\}\vee\mathcal{N}(P),$

where $\mathcal{N}(\mathbb{P})$ denotes the collection of all
$\mathbb{P}$-null events in the probability space. Further, call

$\pi=\\{\pi(t)$ is a $\mathbb{R}$-valued, $\mathcal{F}_{t}$ -progressively
measurable process and $\int_{0}^{t}\pi^{2}(s)ds$<$\infty$ $\forall
t\in[0,\textit{T}]\\}$

a portfolio, and $\pi(t)$ is the proportion of wealth invested in the risky
asset at time t.
We assume that the investor follows a self-financing strategy, which means
that wealth gains or losses arise solely from investment gains or losses.
Then, the corresponding wealth at time t, $X^{\pi}(t)$, can be described by
the dynamics

$dX^{\pi}(t)=\pi(t)X^{\pi}(t)\dfrac{dS(t)}{S(t)}+(1-\pi(t))X^{\pi}(t)\dfrac{dB(t)}{B(t)}$.

Assuming that the investor starts with a fixed, positive wealth x at time 0,
and substituting (2.1) and (2.2) into the dynamics, this gives the wealth
process defined by the wealth equation:

$dX^{\pi}(t)=(rX^{\pi}(t)+\pi(t)(\mu-r)X^{\pi}(t))dt+\sigma\pi(t)X^{\pi}(t)dW(t)\
\mathrm{and}\ X^{\pi}(0)=\mathrm{\textit{x}\ \textit{a.s.}},$ (2.3)

where x $\in$ $\mathbb{R^{+}}$. For better readability let
$X^{\pi}_{t}$:=$X^{\pi}$(t) and $\pi_{t}$:=$\pi$(t).
Let the set of admissible portfolios be defined as

$\mathcal{A}:=\\{\pi:\Omega\times[0,T]\rightarrow\mathbb{R}|X^{\pi}_{0}=x\;\;a.s.\;and\;\;\pi\;is\>\;a\;\;self\text{-}financing\;portfolio\\}$

and $\pi$ be called admissible if $\pi$ $\in$ $\mathcal{A}$.
Also, define the state price density process
$H(t):=e^{-(r+\dfrac{\theta^{2}}{2})t-\theta W_{t}}$.
To state the investor’s problem, we first let the utility of wealth be
described by a function (called utility function)

U: $\mathbb{R}^{+}\rightarrow\mathbb{R},\ x\longmapsto U(x)$.

The problem for the investor is then defined as follows:

######  Problem 1.

Find a strategy $\hat{\pi}\in\mathcal{A}$ such that

$\mathbb{E}[U(X^{\hat{\pi}}_{T})]=\sup\limits_{\pi\in\mathcal{A}}\mathbb{E}[U(X^{\pi}_{T})],$
(2.4)

holds.

It is not a priori clear that the solution $\hat{\pi}$ (called optimal
investment strategy) of (2.4) exists. However, if it does, the problem of
finding an optimal investment strategy is equivalent to finding a solution to
a stochastic differential equation known as Hamilton-Jacobi-Bellman Equation
(HJB). This is shown in detail in [2]. As the outline of its derivation can be
found in Appendix A, we will simply state
Hamilton-Jacobi-Bellman Equation (HJB)

$0=\dfrac{\partial V}{\partial t}(t,x)+\sup\limits_{\pi}\left\\{[rx+(\mu-r)\pi
x]\dfrac{\partial V}{\partial
x}(t,x)+\sigma^{2}\pi^{2}x^{2}\dfrac{1}{2}\dfrac{\partial^{2}V}{\partial
x^{2}}(t,x)\right\\}$ (2.5) $\text{with boundary condition}\ \ V(T,x)=U(x).$
(2.6)

For simplification, the notation $\dfrac{\partial V}{\partial
t}(t,x)=V_{t},\dfrac{\partial V}{\partial
x}(t,x)=V_{x},\dfrac{\partial^{2}V}{\partial x^{2}}(t,x)=V_{xx}$
is introduced.

####  2.1.2 The Optimal Strategy for an Exponential Utility Function

To find the solution of the HJB, it is necessary to know the utility function
as we need to set a boundary $V(T,x)=U(x)$. We will use the exponential
utility, which is convenient as it simplifies many calculations and is defined
by the function

U: $\mathbb{R}\rightarrow(-\infty,0]$, $x\longmapsto U(x)=-e^{-\alpha x}$, for
a constant $\alpha$ > 0.

U describes how an investor evaluates the wealth x, given an (individual)
parameter $\alpha$ that we call risk aversion. Note that its first derivative
$U^{\prime}(x)$ is converging to zero with increasing wealth, which means that
the contribution to utility decreases (i.e. the greater the wealth is, the
less an additional increase has an effect on the investor’s utility). This
property is called decreasing utility margin. Also, the risk aversion is
constant, which implies that the investor’s attitude to risk is independent of
his wealth. The properties of (exponential) utility functions will be further
discussed in the Appendix, so for now we will focus on the optimal strategy.

Since the supremum in (2.5) is identical to the maximum of the scalar function
$f(\pi)=$[$rx+(\mu-r)\pi x]V_{x}+\dfrac{1}{2}\sigma^{2}\pi^{2}x^{2}V_{xx}$,
the optimal value $\hat{\pi}$ needs to satisfy
$0=f^{\prime}(\hat{\pi})=(\mu-r)xV_{x}+\sigma^{2}\hat{\pi}x^{2}V_{xx}$
$\forall$ t $\in$ [0,T], hence

$\hat{\pi}=-\dfrac{(\mu-r)}{\sigma^{2}x}\dfrac{V_{x}}{V_{xx}}.$ (2.7)

Note that it also needs to be checked that $f^{\prime\prime}(\hat{\pi})$ =
$\sigma^{2}x^{2}V_{xx}<0$.
So with (2.7), (2.5) can be written as

$V_{t}+rxV_{x}-\dfrac{1}{2}\dfrac{(\mu-r)^{2}}{\sigma^{2}}\dfrac{(V_{x})^{2}}{V_{xx}}=0.$
(2.8)

For simplification, the notation $\theta:=\dfrac{\mu-r}{\sigma}$ (called the
market price of risk) is introduced. Based on existing results for similar
problems (for example [18]) we suggest the following:

######  Proposition 1.

The value function

$V(t,x)=-e^{-\alpha xe^{r(T-t)}-\frac{\theta^{2}}{2}(T-t)}\text{ is a solution
of the HJB.}$ (2.9)

###### Proof.

For simplification define $A(t,x):=\alpha
xe^{r(T-t)}+\dfrac{\theta^{2}}{2}(T-t)$ and write $V(t,x)=-e^{A(x,t)}$. Then
$V_{t}=\dfrac{dA(x,t)}{dt}e^{-A(x,t)}=-[\dfrac{\theta^{2}}{2}+\alpha
xre^{r(T-t)}]e^{-A(x,t)},\newline V_{x}=\dfrac{dA(x,t)}{dx}e^{-A(x,t)}=\alpha
e^{r(T-t)}e^{-A(x,t)}$ and$\newline V_{xx}=-\alpha^{2}e^{2r(T-t)}e^{-A(x,t)}$
, hence $f^{\prime\prime}(\hat{\pi}_{t})<0$.
Substituting $V_{xx}$ and $V_{x}$ from above into
$\dfrac{1}{2}\theta^{2}\dfrac{(V_{x})^{2}}{V_{xx}}\text{gives}\newline
\dfrac{1}{2}\theta^{2}\dfrac{\alpha^{2}e^{2r(T-t)}e^{-2A(x,t)}}{-\alpha^{2}e^{2r(T-t)}e^{-A(x,t)}}=-\dfrac{\theta^{2}}{2}e^{-A(x,t)},\text{hence}\
\ V_{t}+rxV_{x}-\dfrac{1}{2}\theta^{2}\dfrac{(V_{x})^{2}}{V_{xx}}=0$
So, V(t,x) satisfies equation (2.8) and since $V(T,x)=-e^{-\alpha x}$, it
satisfies the HJB (2.5)-(2.6) for exponential utility.∎

######  Proposition 2.

The optimal investment strategy is given by

$\hat{\pi}_{t}=\dfrac{\theta}{X^{\hat{\pi}}_{t}\alpha\sigma}e^{-r(T-t)}.$
(2.10)

###### Proof.

From (2.9):
$\hat{\pi}_{t}=-\dfrac{\theta}{x\sigma}\dfrac{V_{x}}{V_{xx}}=\dfrac{\theta}{x\alpha\sigma}e^{-r(T-t)}$.
Hence Proposition 1 holds and by Theorem 19.6 in [2] (verification theorem), V
is the optimal value function and $\hat{\pi}$ is the corresponding optimal
strategy. ∎

Note that $\hat{\pi}$ depends on time and wealth, so it is not constant.
However, the absolute amount invested
$\hat{\pi}_{t}X^{\hat{\pi}}_{t}=\dfrac{\theta}{\alpha\sigma}e^{-r(T-t)}$ does
not depend on the absolute wealth of the investor.
The optimal strategy gives us then

######  Proposition 3.

The optimal wealth process is given by

$X^{\hat{\pi}}_{t}=X^{\hat{\pi}}_{0}e^{rt}+t\dfrac{\theta^{2}}{\alpha}e^{r(t-T)}+\dfrac{\theta}{\alpha}e^{r(t-T)}W_{t}\
with\ X^{\hat{\pi}}_{0}=x.$ (2.11)

###### Proof.

Substituting (2.10) into (2.3) we get
$dX^{\hat{\pi}}_{t}=[rX^{\hat{\pi}}_{t}+\dfrac{\theta^{2}}{\alpha}e^{-r(T-t)}]dt+\dfrac{\theta}{\alpha}e^{-r(T-t)}dW_{t}$
.
This is a linear Stochastic Differential Equation (SDE) and the derivation of
its solution can be found in Appendix 6.2. ∎

In particular, for t=T it follows:
$X^{\hat{\pi}}_{T}=X_{0}e^{rT}+T\dfrac{\theta^{2}}{\alpha}+\dfrac{\theta}{\alpha}W_{T}$
, so the terminal wealth is normally distributed with
$\mathbb{E}_{0}[X^{\hat{\pi}}_{T}]=X_{0}e^{rT}+T\dfrac{\theta^{2}}{\alpha}$
and Var$(X^{\hat{\pi}}_{T})=\dfrac{\theta^{2}}{\alpha^{2}}T$.

###  2.2 Brief Analysis of the Optimal Strategy

We will start by analysing the optimal strategy with respect to factors that
can be influenced by the investor, namely the initial investment $X_{0}$, the
risk aversion $\alpha$ and the investment horizon T. The market parameters r,
$\mu$ and $\sigma$ will be considered as fixed values here, but their impact
will be investigated in detail in the next section.
Since the optimal strategy is inversely proportional to the investor’s risk
aversion parameter $\alpha$ and since the latter tends to take very low
values, we will see that it is particularly sensitive to it. Also the initial
wealth plays an important role, as low risk aversion can be compensated by
high initial wealth in order to reach the same expected return on initial
wealth.

####  2.2.1 Parameters: $\alpha$ and T

If we look at the formula in (2.10), it can be seen that the investment
strategy is inversely proportional and therefore highly sensitive to
$X^{\hat{\pi}}_{t}$ and to $\alpha$. This can be interpreted in the way that
the proportion invested in the stock is reduced with increasing wealth, which
makes sense in the context of decreasing marginal utility. Also, since
$\alpha$ is a measure for aversion to risk, it is intuitive that the
proportion of wealth put at risk (i.e. invested in the stock) is decreasing
with increasing $\alpha$.
In order to assess the impact of the parameter $\alpha$, we will from now on
consider the absolute amount invested at t, $\hat{\pi}_{t}X^{\hat{\pi}}_{t}$,
which is independent of $X^{\hat{\pi}}_{t}$. Since it is deterministic, it is
sufficient to look at the initial investment (t=0) to characterize the impact
of $\alpha$ , as it is done in Figure 1 for a ’standard’ setting (Note that T
is rather small here, but this will not make much difference, as we will see
later). Clearly, for low risk aversion, even small changes in $\alpha$ can
have a huge impact on the investment. It therefore needs to be accorded
special attention to the choice of $\alpha$.

Figure 1: Impact of risk aversion on initial investment for $\hat{\pi}$ (T=5)

What values for $\alpha$ are reasonable? Allthough the estimation of risk
aversion depends on the experimental setting and methods used, studies
indicate a similiar range: [6] find a median of 0.000708 and an average of
0.01978 for Italian households, [3] find a best estimate of 0.001 (with a
smallest value estimation of 0.000708) for Californian tomato growers, and
[14] find an average absolute risk aversion of 0.037 and a median of 0.0439
and refer to [7] for similiar values (0.003 average, 0.109 median) for
individual investors on a lending platform. Besides their impact on
investment, there are other reasons susggesting to focus on rather smaller
values of $\alpha$: Generally the setting is targeted at people that are
somewhat willing to invest, so extremly high risk aversion could be excluded.
Also, the distribution of risk aversion in a population seems to be right-
skewed (see [1]), hence the median 0.0007 might be a better reference value
than the average. However, values around 0.01 and 0.001 seem also to be
realistic options for $\alpha$ and should be considered as well.

Compared to $\alpha$, the impact of the (reasonable) time horizon T is rather
small. For instance, in the second diagram of Figure 2 the intersection with
$\hat{\pi}_{t}X^{\hat{\pi}}_{t}$-axis is at $\alpha$ = 0.0004. It results in a
difference of the initial investment in the stock of circa 250 for T between
10 and 30 years. This difference doesn’t change as much as the initial
investment changes with increasing $\alpha$. For example, if $\alpha$ =
0.0007, the difference of initial investment is 150 for the same range of T
(whereas the initial investment changed from around 1’400 to around 500).

Figure 2: Impact of risk aversion on initial investment for $\hat{\pi}$
(varying T)

####  2.2.2 Initial Wealth and Terminal Wealth Distribution

The absolute amount invested increases exponentially with time at the risk-
free interest rate r, independently of the performance of the stock (see
Fig.3).This means, that if wealth increases at a higher rate than r, the
proportion invested in stocks is decreasing, which in turn leads to a smaller
variance of terminal wealth, relative to its expected value. On the other
hand, for smaller wealth, we would expect a higher return, since a bigger
proportion is invested in stocks, which have a higher expected return ($\mu$ >
r).

Figure 3: Absolute investment over time for $\hat{\pi}$ (varying T)

$\alpha$ affects the proportion of wealth invested to the same extent as
$X_{0}$. This means, that for high initial wealth, but small risk aversion, an
investor would put the same percentage of wealth at risk, as a highly risk
averse person with a smaller amount of initial wealth. Consequently, the
distributions of terminal wealth are the ’same’ with respect to the return on
initial wealth and variance relative to the expected return. In other words:
Low wealth can be compensated by risk tolerance. This can be confirmed by the
empirical values, as shown in the table below for the case r = 0.01, $\mu$ =
0.03, $\sigma$ = 0.1, T = 20. Here,
$\sigma(X_{T})$ = $\sigma_{ML}(X_{T})$/$\mu_{ML}(X_{T})$, where
$\mu_{ML}(X_{T})=\mathbb{E}[X_{t}]$ and $\sigma_{ML}(X_{T})$ are the absolute
values for expected value and standard deviation of the terminal wealth
distribution, obtained from a maximum-likelihood fit for a sample of 1’000.
Note that, since $X^{\hat{\pi}}_{T}$ follows normal distribution, the
50$\%$-quantile is identical to the expected return. For more details on other
quantiles refer to the Appendix A.

(a) $\alpha$ = 0.01 $X_{0}$ | $\dfrac{E[X_{T}]}{X_{0}}$ |  $\sigma$($X_{T}$) | $\alpha X_{0}$
---|---|---|---
10 | 915$\%$ | 96$\%$ | 0.1
$10^{2}$ | 201$\%$ | 44$\%$ | 1
$10^{3}$ | 130$\%$ | 7$\%$ | 10
$10^{4}$ | 123$\%$ | 1$\%$ | 100
$10^{5}$ | 122$\%$ | 0$\%$ | 1000

(b) $\alpha$ = 0.001 $X_{0}$ | $\dfrac{E[X_{T}]}{X_{0}}$ |  $\sigma$($X_{T}$) | $\alpha X_{0}$
---|---|---|---
10 | 8’684$\%$ | 104$\%$ | 0.01
$10^{2}$ | 938$\%$ | 94$\%$ | 0.1
$10^{3}$ | 197$\%$ | 45$\%$ | 1
$10^{4}$ | 130$\%$ | 7$\%$ | 10
$10^{5}$ | 123$\%$ | 1$\%$ | 100

(c) $\alpha$ = 0.0001 $X_{0}$ | $\dfrac{E[X_{T}]}{X_{0}}$ |  $\sigma$($X_{T}$) | $\alpha X_{0}$
---|---|---|---
10 | 77’311$\%$ | 115$\%$ | 0.001
$10^{2}$ | 8’056$\%$ | 110$\%$ | 0.01
$10^{3}$ | 938$\%$ | 89$\%$ | 0.1
$10^{4}$ | 200$\%$ | 46$\%$ | 1
$10^{5}$ | 130$\%$ | 7$\%$ | 10

Table 1: Empirical expected return and variance of terminal wealth for
$\hat{\pi}$

Note that the expected return at T is a mix of interest on riskfree bond and
return on stock, depending on the proportion invested. Therefore, for bigger
values of $X_{0}$ the return converges to the deterministic return on the
riskfree bond ($e^{0.01\times 20}$ = 1.22) , and its variance converges to
zero. On the other hand, for small initial wealth, the amount invested is much
higher than the initial wealth itself (and generally higher for smaller
$\alpha$), hence the expected return is a large multiple of the initial value.
For example, for $\alpha$= 0.0001 and $X_{0}$ =10 the strategy requires to
invest 16’375 in stocks and -16’365 in the riskfree asset, which in turn then
leads to an expected theoretical absolute return of around 8’012. This means,
that for small wealth, the strategy is connected to investing more money than
provided by the investor (i.e. short-selling or borrowing), which one might
want to avoid.

###  2.3 Extended Analysis of the Optimal Strategy under a Restriction on
Investment

In this section we introduce a restriction on the investment in the risky
asset and investigate its effects on the terminal wealth distribution.
Alltough interactions of the parameters are complex, a pattern is indicated:
Generally, the restriction reduces smaller quantiles and has less effect on
higher ones, since it changes the distribution towards a log-normal
distribution. The impact generally seems to be higher for high initial wealth,
large $\mu$-r and large $\sigma$. For small r and small $\alpha$, the
advantages of the restriction on investment can be seen the best. In addition
to this empirical approach, a more theoretical one can be found in Appendix
6.4.

We start by defining the restriction to the values for
$\hat{\pi}_{t}X^{\hat{\pi}}_{t}$ in order to avoid that the amount invested in
the risky stock required by the optimal strategy exceeds the level of wealth:

######  Modification 1.

Restriction on Investment
Let the modified strategy $\hat{\pi}_{m}$ be defined for
$(t,X^{\hat{\pi}_{m}}_{t}$) $\in$ $[0,T)$ $\times$ $\mathbb{R}$ by

$\hat{\pi}_{m}(t,X^{\hat{\pi}_{m}}_{t})$ =
$\begin{cases}{\hat{\pi}_{t}}&\text{if
$X^{\hat{\pi}_{m}}_{t}\geq\hat{\pi}_{t}X^{\hat{\pi}_{m}}_{t}$}\\\ {1}&\text{if
$X^{\hat{\pi}_{m}}_{t}<\hat{\pi}_{t}X^{\hat{\pi}_{m}}_{t}$}\end{cases}$

where $X^{\hat{\pi}_{m}}_{t}$ is the corresponding wealth process with
$X^{\hat{\pi}_{m}}_{0}$ = $X_{0}$, and $\hat{\pi}_{t}$ is the optimal strategy
from Proposition 2.

####  2.3.1 Initial wealth

Let’s first look at the impact of the value of initial wealth on the strategy.
Intuitively, the higher this value, the less likely it is that wealth falls
under the investment amount required by the optimal strategy during the
process and hence the difference between modified and original strategy should
be small (or could be zero). From another perspective, it also means that the
proportion of wealth invested is small, so that the strategy (modified or not)
plays only a small part in the overall terminal wealth distribution and so
will changes in parameters that affect the strategy. Qualitatively, this can
be observed in the following plots, where the initial wealth is varied to be
120$\%$, 100$\%$ and 80$\%$ of the amount required by $\hat{\pi}_{0}$.

(a) $X_{0}$=120$\%\hat{\pi}_{0}X_{0}$

(b) $X_{0}$=100$\%\hat{\pi}_{0}X_{0}$

(c) $X_{0}$=80$\%\hat{\pi}_{0}X_{0}$

Figure 4: Investment and wealth processes for $\hat{\pi}$ and $\hat{\pi}_{m}$
(varying $X_{0}$) and terminal wealth distribution for $\hat{\pi}_{m}$

One can observe a shift from higher to lower values of terminal wealth for
decreasing $X_{0}$ and increasing effect of the investment-restriction (the
red line in the histograms is a normal distribution fit for reference). This
is also reflected in the quantiles of the empirical terminal wealth
distribution, where $\mathcal{Q}_{p}(X_{0})$ is the p-quantile resulting from
initial wealth $X_{0}$,$\hat{\pi}_{0}^{-1}$ = $X_{0}$/($\hat{\pi}_{0}X_{0}$)
is the initial wealth as percentage of the required initial investment and
$\Delta(\mathcal{Q}_{p})=\mathcal{Q}_{p}(X_{0})/\mathcal{Q}_{p}(\hat{\pi}_{0}X_{0})$
is the change of quantiles relative to $\mathcal{Q}_{p}(\hat{\pi}_{0}X_{0})$=
$\mathcal{Q}_{p}(100$%$\hat{\pi}_{0}X_{0})$ :

$X_{0}$ | $\mathcal{Q}_{0.25}$($X_{0}$) | $\mathcal{Q}_{0.50}$($X_{0}$) | $\mathcal{Q}_{0.75}$($X_{0}$) |  $\mathcal{Q}_{0.95}$($X_{0}$) | $\Delta(\mathcal{Q}_{0.25})$ | $\Delta(\mathcal{Q}_{0.5})$ | $\Delta(\mathcal{Q}_{0.75})$ | $\Delta(\mathcal{Q}_{0.95}$)
---|---|---|---|---|---|---|---|---
4’912 | 4’037 | 7’549 | 10’848 | 15’043 | 134 $\%$ | 117$\%$ | 112$\%$ | 107$\%$
4’094 | 3’010 | 6’450 | 9’653 | 14’064 |  |  |  |
3’275 | 2’292 | 4’660 | 8’067 | 12’673 | 76$\%$ | 72$\%$ | 84$\%$ | 90$\%$
Table 2: Quantiles of $X^{\hat{\pi}_{m}}_{T}$ and comparison for
$\hat{\pi}_{0}^{-1}$= 120$\%$, 100$\%$ and 80$\%$

It is remarkable that the change of $X_{0}$ affects lower quantiles more than
higher quantiles: While an increase of $X_{0}$ by 20$\%$ leads to an increase
greater than 20$\%$ of the 25$\%$-quantile, the 95$\%$-quantile only increases
by 7$\%$. The same happens when $X_{0}$ is reduced by 20$\%$. It can also be
observed an asymmetry between increase and decrease of $X_{0}$ in terms of its
effect on quantiles. For instance, at $\mathcal{Q}_{0.5}$, we observe a
decrease by 28$\%$ versus an increase of only 17$\%$. Similar effects can also
be observed for a lower expected return $\mu$ and higher volatility $\sigma$.
In particular, a higher volatility seems to lead to stronger changes, for
example with $\sigma$ = 40$\%$, we reach an increase of $\mathcal{Q}_{0.25}$
to 140$\%$ and a decrease to 72$\%$.

The reason for the shift of the quantiles lies in the missed upside potential
of the paths of wealth that fall under the amount required to follow the
optimal strategy. Instead of investing the full amount required by the optimal
strategy, only the wealth available can be invested, this results generally in
lower values for the terminal wealth. The lower the initial wealth is, the
more paths are limited by the modification of the strategy and lead to smaller
wealth, which results in lower quantiles. As extremely well performing paths
are more rare and less likely to be constrained by the modification, the
change of $X_{0}$ affects them less, and so the upper quantiles (i.e.
$\mathcal{Q}_{0.95}$) are less affected.
Another reason could be the fact, that the proportion invested is smaller for
larger $X_{0}$ and therefore the modification of the strategy has less impact
on the distribution of higher values of terminal wealth. In that case, even if
the part invested in the stock performs less well due to the modification,
this will have limited effect, since the outcome of the strategy is dominated
by the high proportion invested in riskfree bond.
The asymmetry in the changes of quantiles might be explained by the following
interpretation: When wealth falls and stays under the amount required by the
optimal strategy, it is 100$\%$ invested in stocks, hence its distribution is
identical to the one of the stocks, which is a log-normal distribution. So,
for lower $X_{0}$ the distribution is closer to log-normal, whereas for
increasing $X_{0}$ it converges to the normal distribution of the optimal
terminal wealth.

####  2.3.2 Parameter: $\mu$ -r

In contrary to the original strategy, where the amount invested was
independent of the performance of the stock, the restriction on investment
establishes a link to the stock market. Therefore market-parameters as
expected return $\mu$, risk free interest rate r and volatility $\sigma$ will
influence the resulting terminal wealth distribution to another extent. Let us
first look at the difference of the market rate and the riskfree rate. Since
the initial investment amount required by the optimal strategy changes with
variation of $\mu$ \- r, we set $X_{0}=\hat{\pi}_{0}X_{0}$ and r = 0.01 and
compare the resulting output for different $\mu$ and fixed $\alpha$ = 0.001.

Figure 5: Performance of stock ($\mu$ -r = 0.01 and $\mu$ \- r = 0.05)

Figure 6: Wealth process for $\hat{\pi}$ and $\hat{\pi}_{m}$ ($\mu$-r = 0.01
and $\mu$-r = 0.05 )

Since the absolute initial wealth varies for the different scenarios, we look
at the quantiles of the total return instead of the absolute terminal wealth
to make a better comparison.
In the diagram below, the x-axis shows the total return in 100$\%$ and the red
line represents a normal distribution fit.

Figure 7: Histogram of total return on initial wealth for $\hat{\pi}_{m}$
($\mu$-r = 0.01 and $\mu$-r= 0.05)

Again, we see an increasing weight of lower quantiles with decreasing gap,
indicating a change in the type of the distribution rather than a simple shift
of values. For the empirical quantiles, let $\mathcal{Q}_{p}$($\mu$-r) be the
p-quantile of the return on initial investment at T= 20 for a gap $\mu$-r and
$\Delta(\mathcal{Q}_{p})$=$\mathcal{Q}_{p}$($\mu$)/$\mathcal{Q}_{p}$(0.03) be
the change in quantiles in comparison to a gap $\mu$-r of 3$\%$.

$\mu$-r | $\mathcal{Q}_{0.25}$($X_{0}$) | $\mathcal{Q}_{0.50}$($X_{0}$) | $\mathcal{Q}_{0.75}$($X_{0}$) |  $\mathcal{Q}_{0.95}$($X_{0}$) | $\Delta(\mathcal{Q}_{0.25})$ | $\Delta(\mathcal{Q}_{0.5})$ | $\Delta(\mathcal{Q}_{0.75})$ | $\Delta(\mathcal{Q}_{0.95}$)
---|---|---|---|---|---|---|---|---
5$\%$ | 149$\%$ | 239$\%$ | 314$\%$ | 419$\%$ | 240 $\%$ | 213$\%$ | 195$\%$ | 182$\%$
3$\%$ | 103$\%$ | 187$\%$ | 268$\%$ | 383$\%$ |  |  |  |
1 $\%$ | 58$\%$ | 122$\%$ | 211 $\%$ | 322$\%$ | 19$\%$ | 22$\%$ | 26$\%$ | 28$\%$
Table 3: Quantiles of $X^{\hat{\pi}_{m}}_{T}$ as total return and comparison
for $\mu$-r = 5$\%$, 3$\%$ and 1$\%$

As before, we see the effect of change of parameters more pronounced in lower
quantiles, for example with $\mu$= 5$\%$, $\mathcal{Q}_{0.25}$ increases by
+140$\%$ while $\mathcal{Q}_{0.95}$ only increases by +82$\%$. This pattern
can be observed for different combinations of the other parameters as well,
allthough the effect on quantiles is generally stronger for smaller initial
values and large variance. At the same time, this is also where the difference
to the results for the optimal unconstrained strategy are most obvious (for
instance, at $\mu-\textit{r}$ = 5$\%$, $\hat{\pi}$ leads to an increase by
+116$\%$ of $\mathcal{Q}_{0.25}$ and by +82$\%$ of $\mathcal{Q}_{0.95}$, the
latter being the same as for $\hat{\pi}_{m}$.)
Then again, an asymmetry between the effects of reduction and increase of
$\mu$-r can be observed. For example, if the gap is reduced to 1/3, the
respective quantiles are reduced to values between 20$\%$ and 30$\%$, whereas
an increase by 2/3 leads to in increase of lower quantiles to more than
200$\%$ (the proportional increase would lead to 166$\%$). For other paramters
$\sigma$ and $\alpha$ the behaviour is similar.

The reason for this asymmetry lies in the application of the investment-
constraint.
Generally, for the basic secenario $\mu$-r = 0.03, the majority of the wealth
paths seems to be affected by the constraint (see Fig. 7) and so the terminal
return’s distribution is close to log-normal. Since a reduction of $\mu$-r
does not have much effect on the application of the constraint and the
terminal distribution, the effect on the quantiles is rather low. In contrary,
an increase of $\mu$-r might lead to a few more paths performing better and
not being affected by the investment-constraint and so the terminal wealth
distribution will be slightly closer to the normal distribution of the optimal
terminal wealth. This change of the distribution can be observed in the
quantiles as an (overproportional) increase of the lower quantiles and in the
histograms as a shift of values to the left for low $\mu$. Another way of
interpretation, as before, is the following: the more paths of wealth follow
the constrained part of the strategy, the less upside and downside potential
is used, which leads to more concentration of lower values and ultimately to a
right-skewed log-normal-distribution. Also note, that if $\hat{\pi}_{t}X_{t}$
grows at the lower rate r < $\mu$, the absolute gap between stocks (and thus
wealth) and $\hat{\pi}_{t}X_{t}$ grows with time -leading to a falling
probability to switch to the modified strategy. We therefore observe most of
the differences between original and modified strategy in the first years of
investment.
Another effect, as seen before, could be the linked to the fact, that high
quantiles indicate high wealth, which in turn are connected to a low
proportion invested and so they are less affected by the restriction.
Note, that the amount required to be invested in the risky asset by the
optimal strategy is proportional to $\mu$-r. Changes here have an equivalent
effect on the modified strategy as ‘inverse‘ changes of the initial wealth,
which we considered in the section above (in other words: whether or not the
modification of the strategy has an impact depends on how high $X_{0}$ is in
comparison to $\hat{\pi}_{0}X_{0}$). However, this can not be observed in the
plots and quantiles as we had set $X_{0}=\hat{\pi}_{0}X_{0}$.

####  2.3.3 Parameter: $\sigma$

For now, we have seen that the stronger the impact of the constraint on
investment is, the more the terminal wealth distribution changes towards a
log-normal distribution. We will assume that the same effect will happen, when
the impact of the constraint changes because of different parameters $\sigma$
or r. This is why, instead of how, we are now interested in assessing how much
the effect of this restriction can be. We will therefore compare the empirical
quantiles of the terminal wealth distribution from this modified strategy with
those from the original one. Since the values of the investment-constrained
strategy might be of more interest, they are listed here, but those from the
optimal unconstrained strategy can be found in Appendix 6.6.
We look at the quantiles of a small and high volatility for different expected
return-interest gaps, having set $\alpha$ = 0.001, r = 0.01 and
$X_{0}=\hat{\pi}_{0}X_{0}$.

$\mu$-r |  $\mathcal{Q}_{0.25}$($X_{0}$) |  $\mathcal{Q}_{0.50}$($X_{0}$) |  $\mathcal{Q}_{0.75}$($X_{0}$) |  $\mathcal{Q}_{0.95}$($X_{0}$)
---|---|---|---|---
5$\%$ | 28$\%$ | 172$\%$ | 354$\%$ | 585$\%$
3$\%$ | 17$\%$ | 96$\%$ | 296$\%$ | 532$\%$
1 $\%$ | 11$\%$ | 52$\%$ | 231$\%$ | 481$\%$

(a) $\sigma$ = 0.4

$\mu$-r |  $\mathcal{Q}_{0.25}$($X_{0}$) |  $\mathcal{Q}_{0.50}$($X_{0}$) |  $\mathcal{Q}_{0.75}$($X_{0}$) |  $\mathcal{Q}_{0.95}$($X_{0}$)
---|---|---|---|---
5$\%$ | 209$\%$ | 245$\%$ | 284$\%$ | 340$\%$
3$\%$ | 155$\%$ | 195$\%$ | 231$\%$ | 284$\%$
1 $\%$ | 103$\%$ | 144$\%$ | 181$\%$ | 234$\%$

(b) $\sigma$ = 0.1 Table 4: Quantiles of $X^{\hat{\pi}_{m}}_{T}$ as total
return for $\sigma$ = 0.1 and 0.4 (and varying $\mu$-r)

Unsurprisingly, the spread of the terminal distribution is bigger for a high
variance and more narrow for a small one. More striking seem to be the rather
disappointing results for ${\sigma=0.4}$, where $\hat{\pi}_{m}$ yields worse
results than the unconstrained strategy for all the returns shown in the
table: the peak being at $\mathcal{Q}_{0.25}$ and $\mu$ -r = 0.05 with 28$\%$
instead of 94$\%$ and the only exception being for a particularly ’bad’
performance (i.e. having 11$\%$ instead of -3$\%$ at $\mathcal{Q}_{0.25}$ for
$\mu$-r = 0.01). In contrary to scenarios with high volatility that seem to be
very much affected by the investment-constraint, a rather low volatility of
10$\%$ yields almost the same results as the optimal strategy. Of course, in
that case, the range is generally much smaller and the results are lower.
Overall, this could be good news anyway, because we would expect small $\mu$-r
to be connected to smaller risk, for which $\sigma$ is an indicator, and a
higher gap to be linked to a higher volatility. So, $\mu$-r = 0.01 and
$\sigma$ = 0.1 resp. $\mu$-r = 0.05 and $\sigma$ = 0.4 might be realistic
situations and $\mathcal{Q}_{0.5}$ = 144$\%$ resp. $\mathcal{Q}_{0.5}$=
172$\%$ are acceptable results.
Also note, that the variance does have a rather big impact on the initial
investment amount as well, for $\mu$-r = 5$\%$ we get $\hat{\pi}_{0}X_{0}$ =
256 when $\sigma$ = 0.1, compared to $\hat{\pi}_{0}X_{0}$ = 4’094 for
${\sigma=0.4.}$

####  2.3.4 Parameter: r

Let us consider here the cases of zero and negative interest rates, a small
volatility $\sigma$ = 0.1, T=20 and $\alpha$ = 0.001. As before, we set
$X_{0}=\hat{\pi}_{0}X_{0}$. For comparison, the median of terminal wealth of a
100 $\%$ investment in stocks and of a 100 $\%$ investment in the bond are
also listed. Note that the stock’s distribution is right-skewed and the median
is smaller than the mean.

$\mu$ | $X_{0}$ | $\mathcal{Q}_{0.25}$ | $\mathcal{Q}_{0.50}$ | $\mathcal{Q}_{0.75}$ | $\mathcal{Q}_{0.95}$ | $\mathcal{Q}_{0.50}(\pi_{t}\equiv 1)$ | $\mathcal{Q}_{0.50}(\pi_{t}\equiv 0)$
---|---|---|---|---|---|---|---
10$\%$ | 10’000 | 270$\%$ | 302$\%$ | 331$\%$ | 374 $\%$ | 739$\%$ | 100$\%$
6$\%$ | 6’000 | 189$\%$ | 221$\%$ | 250$\%$ | 293$\%$ | 332$\%$ |
1$\%$ | 1000 | 84$\%$ | 118$\%$ | 148 $\%$ | 191$\%$ | 122$\%$ |

(a) r = 0$\%$

$\mu$ | $X_{0}$ | $\mathcal{Q}_{0.25}$ | $\mathcal{Q}_{0.50}$ | $\mathcal{Q}_{0.75}$ | $\mathcal{Q}_{0.95}$ | $\mathcal{Q}_{0.50}(\pi_{t}\equiv 1)$ |  $\mathcal{Q}_{0.50}$($\pi_{t}\equiv 0)$
---|---|---|---|---|---|---|---
4$\%$ | 6’107 | 138$\%$ | 164$\%$ | 188$\%$ | 223 $\%$ | 223$\%$ | 82$\%$
2$\%$ | 3’664 | 104$\%$ | 131 $\%$ | 155$\%$ | 190$\%$ | 149$\%$ |
0$\%$ | 1’221 | 69$\%$ | 97$\%$ | 122 $\%$ | 157$\%$ | 100$\%$ |

(b) r = -1$\%$ Table 5: Quantiles of $X^{\hat{\pi}_{m}}_{T}$ as total return
for r= 0$\%$ and r= -1$\%$ (and varying $\mu$)

Naturally, the return on initial wealth is lower for lower risk-free rates.
However, in most of the cases it is still better than a 100$\%$ investment in
bonds. At the same time, the more $\mu$ exceeds r, the more the missed upside
potential is expressed in the return on terminal wealth, compared to a 100$\%$
investment in stocks. In the example above we set $X_{0}=\hat{\pi}_{0}X_{0}$,
but if we fix the initial wealth and consider a less risk averse person, the
initial amount to be invested would be larger and therefore the proportion
invested in the stock smaller. The terminal wealth in terms of return will
therefore be closer to the 100$\%$ stock investment, than to the return on a
100$\%$ bond investment. For example, for $\alpha=0.0001$, we get
$\mathcal{Q}_{0.50}$ = 675$\%$ if r= 0 and $\mu$ = 10$\%$ and
$\mathcal{Q}_{0.50}$ = 203$\%$ if $\mu$ = 4 $\%$ and r =-1$\%$.
This means that the quantiles above also reflect the risk aversion preferences
of the investor and a comparison to alternatives (like investing entirely in
bonds or stocks) should take this into consideration.
Comparing these results with the unrestricted optimal strategy shows almost no
difference for $\alpha$ =0.001: Here, the maximum difference is +6$\%$ at
$\mathcal{Q}_{0.95}$ = 90$\%$ for $\hat{\pi}$ and $\mu$ = 0.01, r =0. But, for
$\alpha$ =0.0001, which implies much more investment in the risky asset, we
find a great difference: For almost all the quantiles, the missed upside
potential is huge. For example, for r=0$\%$ it lies between 200$\%$ and
1’510$\%$ for all $\mu$ considered. The interesting exception to this are the
lower quantiles with $\mu$=0.01. In these cases, 25$\%$ of optimal terminal
wealth paths resulting from the unrestricted strategy take values below
-2$\%$. This is where the restriction kicks in and shows a great advantage:
For $\mu$ = 1$\%$ and r=0$\%$, $\mathcal{Q}_{0.25}$ is at 82$\%$, and for
$\mu$=0$\%$, r= -1$\%$ it still is at 67$\%$ (see Table 14 in Appendix 6.5).

##  3 An Optimal Strategy for Exponential Utility and Lower Constraint $K_{l}$

###  3.1 Derivation of the $K_{l}$-Strategy

We now introduce a lower constraint to limit the terminal wealth and derive
the optimal strategy for this setting. We will find, that it involves
investing one part of the initial wealth following the optimal strategy
(yielding a shadow wealth process) and using the other part to buy a put
option to hedge this process. For simplicity, we will sometimes refer to it as
$K_{l}$-Strategy.
We begin by modifiying Problem 1 by a constraint $K_{l}$ $\in$
$\mathbb{R}_{(-\infty,X_{0}e^{rT})}$.

######  Problem 2.

Find an optimal strategy $\hat{\pi}_{l}\in\mathcal{A}$ such that

$\mathbb{E}[U(X^{\hat{\pi}_{l}}_{T})]=\sup\limits_{\pi\in\mathcal{A}}\mathbb{E}[U(X^{\pi}_{T})]\
\text{and}\ X^{\hat{\pi}_{l}}_{T}\geq K_{l}\text{ holds a.s.}$ (3.1)

In order to solve this problem, we first determine the optimal terminal
wealth.

######  Proposition 4.

For Problem 2, the optimal terminal wealth is of the form

$X^{\hat{\pi}_{l}}_{T}=\tilde{X}^{\hat{\pi}}_{T}+\max\\{K_{l}-\tilde{X}^{\hat{\pi}}_{T},0\\},$
(3.2)

where $\tilde{X}^{\hat{\pi}}_{t}$ is the optimal wealth process from (2.15),
with
$\tilde{X}_{0}=(-$ln$(\dfrac{y}{\alpha})+rT-\dfrac{\theta^{2}}{2}T)\dfrac{1}{\alpha}e^{-rT}$
(called shadow value) for y > 0 such that
$\mathbb{E}[H_{T}X^{\hat{\pi_{l}}}_{T}]=X_{0}$ and $\hat{\pi}_{t}$ the
corresponding optimal strategy .

###### Proof.

The statement follows directly from Lemma 2 in [5].
To see this, let
$I(y)=U^{\prime-1}(y)=-\dfrac{1}{\alpha}\ln(\dfrac{y}{\alpha})$ be the inverse
of $U^{\prime}(x)=\alpha e^{-\alpha x}$.
Note that U is strictly increasing and concave.
By Lemma 2,
$X^{\hat{\pi}_{l}}_{T}=\max\\{K,I(yH_{T})\\}=I(yH_{T})+\max\\{K-I(yH_{T}),0\\}$
for y >0 such that $\mathbb{E}[H_{T}X^{\hat{\pi_{l}}}_{T}]=X_{0}$ and where
$H_{T}=H(T)$ is the state price density at T.
Determine $\tilde{X}_{0}$ such that $I(yH_{T})=\tilde{X}^{\hat{\pi}}_{T}$.
This is the case if
$I(yH_{T})=\dfrac{-1}{\alpha}[\ln(\dfrac{y}{\alpha})-(r+\dfrac{\theta^{2}}{2})T-\theta
W_{T})]=\tilde{X}_{0}e^{rT}+T\dfrac{\theta^{2}}{\alpha}+\dfrac{\theta}{\alpha}W_{T}$
Hence we find
$\tilde{X}_{0}=(rT-\dfrac{\theta^{2}}{2}T-\ln(\dfrac{y}{\alpha}))\dfrac{1}{\alpha}e^{-rT}$.
∎

Note that the optimal terminal wealth has the structure of the optimal
terminal value for an unconstrained wealth process plus a put option (on this
unconstrained wealth process). We can thus find the optimal strategy by
determining a replicating portfolio that yields
$\max\\{K_{l}-\tilde{X}^{\hat{\pi}}_{T},0\\}$ at T. This will be done via
risk-neutral valuation arguments, in analogy to [4].

######  Proposition 5.

The price at time t $\in$ [0,T] of a put option with payoff
$\max\\{K_{l}-\tilde{X}^{\hat{\pi}}_{T},0\\}$ is given by

${p(t,\tilde{X}^{\hat{\pi}}_{t})=\Phi(d_{l}(t,\tilde{X}^{\hat{\pi}}_{t}))(e^{-r(T-t)}K_{l}-\tilde{X}^{\hat{\pi}}_{t})+\dfrac{\theta\sqrt{T-t}}{\alpha}e^{-r(T-t)}\phi(d_{l}(t,\tilde{X}^{\hat{\pi}}_{t}))}$
(3.3)

where $\Phi(x)$ is the cumulative normal distribution, $\phi(x)$ its density
and
$d_{l}(t,\tilde{X}^{\hat{\pi}}_{t})=(K_{l}-\tilde{X}^{\hat{\pi}}_{t}e^{r(T-t)})\dfrac{\alpha}{\sqrt{T-t}\theta}$.

###### Proof.

Assume that the market is free of arbitrage and complete. Then there exists a
risk-neutral measure $\mathbb{Q}$ such that $W^{\mathbb{Q}}_{t}:=W_{t}+\theta
t$ is a standard Brownian motion. Hence, under $\mathbb{Q}$ the discounted
wealth process is a martingale:
$E_{\mathbb{Q}}[e^{-rt}\tilde{X}^{\pi}_{t}|\mathcal{F}_{t-1}]=E_{\mathbb{Q}}[e^{-rt}(e^{rt}\tilde{X}^{\pi}_{0}+e^{r(t-T)}\dfrac{\theta}{\alpha}W_{t}^{\mathbb{Q}})|\mathcal{F}_{t-1}]=E_{\mathbb{Q}}[\tilde{X}^{\pi}_{0}+e^{-rT}\dfrac{\theta}{\alpha}W_{t}^{\mathbb{Q}}|\mathcal{F}_{t-1}]\newline
=\tilde{X}^{\pi}_{0}+e^{-rT}\dfrac{\theta}{\alpha}W^{\mathbb{Q}}_{t-1}=e^{-r(t-1)}\tilde{X}^{\pi}_{t-1}\
\forall\pi\in\mathcal{A},\ \forall t\in[1,T].$
So we can evaluate the put option by risk-neutral pricing. Before this is
done, note that:
$\tilde{X}^{\hat{\pi}}_{T}<K_{l}\newline
\iff\tilde{X}_{0}e^{rT}+\dfrac{\theta}{\alpha}W_{T}^{\mathbb{Q}}<K_{l}\newline
\iff
W_{T}^{\mathbb{Q}}<(K_{l}-\tilde{X}_{0}e^{rT})\dfrac{\alpha}{\theta}\newline
\iff Z<(K_{l}-\tilde{X}_{0}e^{rT})\dfrac{\alpha}{\sqrt{T}\theta}=:d_{0}$
for the substitution $W_{T}^{\mathbb{Q}}:=Z\sqrt{T}$ and Z
$\sim\mathcal{N}(0,1)$.
Then the price of the put option at time 0 is given by:
$p(0,\tilde{X}_{0})=e^{-rT}\mathbb{E}_{\mathbb{Q}}[\max\\{K_{l}-\tilde{X}^{\hat{\pi}}_{T},0\\}]=e^{-rT}\mathbb{E}_{\mathbb{Q}}[(K_{l}-\tilde{X}^{\hat{\pi}}_{T})\mathds{1}_{\\{\tilde{X}^{\hat{\pi}}_{T}<K_{l}\\}}]\newline
=e^{-rT}\mathbb{E}_{\mathbb{Q}}[K_{l}\mathds{1}_{\\{Z<d_{0}\\}}]-e^{-rT}\mathbb{E}_{\mathbb{Q}}[(\tilde{X}_{0}e^{rT}+\dfrac{\theta}{\alpha}Z\sqrt{T})\mathds{1}_{\\{Z<d_{0}\\}}]\newline
=e^{-rT}K_{l}\Phi(d_{0})-\tilde{X}_{0}\Phi(d_{0})-\dfrac{\theta\sqrt{T}}{\alpha}e^{-rT}\mathbb{E}_{\mathbb{Q}}[Z\mathds{1}_{\\{Z<d_{0}\\}}]\newline
=\Phi(d_{0})(e^{-rT}K_{l}-\tilde{X}_{0})-\dfrac{\theta\sqrt{T}}{\alpha}e^{-rT}\dfrac{1}{\sqrt{2\pi}}\bigintsss_{-\infty}^{d_{0}}Ze^{-{Z^{2}}/2}dZ\newline
=\Phi(d_{0})(e^{-rT}K_{l}-\tilde{X}_{0})+\dfrac{\theta\sqrt{T}}{\alpha}e^{-rT}\dfrac{1}{\sqrt{2\pi}}e^{\small{-\dfrac{d_{0}^{2}}{2}}}$
Similarly, for any t $\in$ [0,T] this gives:
$p(t,\tilde{X}^{\hat{\pi}}_{t})=\Phi(d_{l}(t,\tilde{X}^{\hat{\pi}}_{t}))(e^{-r(T-t)}K_{l}-\tilde{X}^{\hat{\pi}}_{t})+\dfrac{\theta\sqrt{T-t}}{\alpha\sqrt{2\pi}}e^{-r(T-t)}e^{\small{-\dfrac{d_{l}(t,\tilde{X}^{\hat{\pi}}_{t})^{2}}{2}}}\newline
=\Phi(d_{l}(t,\tilde{X}^{\hat{\pi}}_{t}))(e^{-r(T-t)}K_{l}-\tilde{X}^{\hat{\pi}}_{t})+\dfrac{\theta\sqrt{T-t}}{\alpha}e^{-r(T-t)}\phi(d_{l}(t,\tilde{X}^{\hat{\pi}}_{t}))$
for the notation defined in the Proposition.∎

As a replicating portfolio $\tilde{\pi}_{p}$ for the pricing function
$p(t,\tilde{X}^{\hat{\pi}}_{t})$ we suggest

######  Proposition 6.

The replicating portfolio of the put option in (3.2) is given by

$\tilde{\pi}_{p}(t,\tilde{X}^{\hat{\pi}}_{t})=\dfrac{-\Phi(d_{l})}{\sigma\sqrt{T-t}(\Phi(d_{l})d_{l}+\phi(d_{l}))}$
(3.4)

for
$d_{l}=d_{l}(t,\tilde{X}^{\hat{\pi}}_{t})=(K_{l}-\tilde{X}^{\hat{\pi}}_{t}e^{r(T-t)})\dfrac{\alpha}{\sqrt{T-t}\theta}$
as in Proposition 5.

###### Proof.

It needs to be shown that $p(t,\tilde{X}^{\hat{\pi}}_{t})$ satisfies the
wealth equation for $\tilde{\pi}_{p}$ from (3.4), because then it is a wealth
process that replicates the price function $p(t,\tilde{X}^{\tilde{\pi}}_{t})$
and reaches the terminal value $\max\\{K_{l}-\tilde{X}^{\hat{\pi}}_{T},0\\}$,
so $\tilde{\pi}_{p}$ would be a suitable strategy.
To do this, form the partial derivatives via the substitution

$d:=d_{l}(t,\tilde{X}^{\hat{\pi}}_{t})=(K_{l}-\tilde{X}^{\hat{\pi}}_{t}e^{r(T-t)})\dfrac{\alpha}{\sqrt{T-t}\theta}$
and $d_{x}=\dfrac{\alpha}{\theta\sqrt{T-t}}(-e^{r(t-t)})$:
$p_{x}=\dfrac{d}{d\tilde{X}^{\hat{\pi}}_{t}}p(t,\tilde{X}^{\hat{\pi}}_{t})=-\Phi(d)+\phi(d)d_{x}\\{K_{l}e^{-r(T-t)}-\tilde{X}^{\hat{\pi}}_{t}\\}+\dfrac{\theta\sqrt{T-t}}{\alpha}e^{-r(T-t)}\phi(d)(-d)d_{x}\newline
=-\Phi(d)+\phi(d)d_{x}(K_{l}e^{-r(T-t)}-\tilde{X}^{\hat{\pi}}_{t}-e^{-r(T-t)}(K_{l}-\tilde{X}^{\hat{\pi}}_{t}e^{r(T-t)})=-\Phi(d),\newline
p_{xx}=\dfrac{d^{2}}{d(\tilde{X}^{\hat{\pi}}_{t})^{2}}p(t,\tilde{X}^{\hat{\pi}}_{t})=\phi(d)\dfrac{\alpha}{\theta\sqrt{T-t}}e^{r(T-t)}$
and
$p_{t}=\dfrac{d}{dt}p(t,\tilde{X}^{\hat{\pi}}_{t})=\phi(d)d_{t}(e^{-r(T-t)}K_{l}-\tilde{X}^{\hat{\pi}}_{t})+\Phi(d)rK_{l}e^{-r(T-t)}$
$+\dfrac{\theta\phi(d)}{\alpha}(\sqrt{T-t}re^{-r(T-t)}-\dfrac{1}{2\sqrt{T-t}}e^{-r(T-t)})+\dfrac{\theta}{\alpha}\sqrt{T-t}e^{-r(T-t)}\phi(d)(-d)d_{t}$
which simplifies (first and last term cancel) to
$p_{t}=\Phi(d)rK_{l}e^{-r(T-t)}+\dfrac{\theta\phi(d)}{\alpha}e^{-r(T-t)}(\sqrt{T-t}r-\dfrac{1}{2\sqrt{T-t}})$
.
Note that $p(t,\tilde{X}^{\hat{\pi}}_{t})$ is twice differentiable and
$\tilde{X}^{\hat{\pi}}_{t}$ satisfies the differential equation from the proof
of (2.15), hence is an Ito drift diffusion process. So, Ito’s Lemma can be
applied, and
$dp(t,\tilde{X}^{\hat{\pi}}_{t})=\\{\Phi(d)rK_{l}e^{-r(T-t)}+\dfrac{\theta}{\alpha}\phi(d)e^{-r(T-t)}r\sqrt{T-t}+(r\tilde{X}^{\hat{\pi}}_{t}+\dfrac{\theta^{2}}{\alpha}e^{-r(T-t)})(-\Phi(\theta))\\}dt+\dfrac{\theta}{\alpha}e^{-r(T-t)}(-\Phi(d))dW_{t}$.
Then $p(t,\tilde{X}^{\hat{\pi}}_{t})$ satisfies the wealth equation
$dp(t,\tilde{X}^{\hat{\pi}}_{t})=(r+\tilde{\pi}_{p}\theta\sigma)p(t,\tilde{X}^{\hat{\pi}}_{t})dt+\sigma\tilde{\pi}_{p}p(t,\tilde{X}^{\hat{\pi}}_{t})dW_{t}$
iff
$\dfrac{\theta}{\alpha}e^{-r(T-t)}(-\Phi(d))=\sigma\tilde{\pi}_{p}p(t,\tilde{X}^{\hat{\pi}}_{t})$,
which is the case for
$\tilde{\pi}_{p}(t,\tilde{X}^{\hat{\pi}}_{t})=\dfrac{-\Phi(d)}{\sigma\sqrt{T-t}(\Phi(d)d+\phi(d))}$.

∎

Combining the optimal strategy of the modified unconstrained problem and of
the replicating portfolio gives the overall strategy.

######  Proposition 7.

An optimal strategy for Problem 2 is given by the amount to be invested in the
risky asset at t

$\hat{\pi}_{l}(t,\tilde{X}^{\hat{\pi}}_{t})=\dfrac{\theta}{\alpha\sigma}e^{-r(T-t)}+p(t,\tilde{X}^{\hat{\pi}}_{t})\dfrac{-\Phi(d_{l})}{\sigma\sqrt{T-t}(\Phi(d_{l})d_{l}+\phi(d_{l}))}$
(3.5)

where $\tilde{X}^{\hat{\pi}}_{t}$ is an optimal wealth process with initial
wealth $\tilde{X}_{0}$ (the shadow value from Proposition 4) and $d_{l}$ as in
Proposition 6.

We will sometimes refer to $\tilde{X}^{\hat{\pi}}_{t}$ as the shadow wealth
process.

###### Proof.

Note that by definition of
$\hat{\pi}_{l}(t,\tilde{X}^{\hat{\pi}}_{t})=\hat{\pi}_{t}\tilde{X}^{\hat{\pi}}_{t}+\tilde{\pi}_{p}(t,\tilde{X}^{\hat{\pi}}_{t})p(t,\tilde{X}^{\hat{\pi}}_{t})$
and (2.3), the resulting wealth process is
$X^{\hat{\pi}_{l}}_{t}=\tilde{X}^{\hat{\pi}}_{t}+p(t,\tilde{X}^{\pi}_{t})$.
In particular, at t=T:
$X^{\pi_{l}}_{T}=\tilde{X}^{\hat{\pi}}_{T}+\max\\{K-\tilde{X}^{\hat{\pi}}_{T},0\\}$,
so it yields the optimal terminal value under lower constraint K, hence it is
an optimal strategy. ∎

###  3.2 Analysis of the $K_{l}$-Strategy

In order to understand the consequences of a lower constraint on terminal
wealth on the optimal strategy and the performance of wealth, we will first
look at the formal structure of $\hat{\pi}_{l}$. From this, we can deduce that
the investement is generally lower than the one required by the optimal
unconstained strategy, and goes to zero when nearing terminal time. Also, it
is most sensitive for values of shadow wealth around $K_{l}e^{-r(T-t)}$.
We then look at the qualitative behaviour in different scenarios, where we can
observe that the $K_{l}$-strategy is more successful than the unconstrained
one for a continued decrease of the stock value, but underperforms it for
increasing stock values, in which case the difference is amplified by the
expected return rate.

####  3.2.1 First Observations

To start, consider the formula for the $K_{l}$-strategy given by Proposition
7. Note that the first term, $\hat{\pi}_{t}\tilde{X}^{\hat{\pi}}_{t}$ is
identical to the optimal unconstrained strategy, since the amount invested is
independent of the initial wealth (so, substituting $X_{0}$ by the shadow
value $\tilde{X_{0}}$ has no effect on the strategy). Hence, the difference of
this new strategy to the previous optimal one is determined by the second
term, $p(t,\tilde{X}^{\hat{\pi}}_{t}$)$\tilde{\pi}_{p}$.
As $p(t,\tilde{X}^{\hat{\pi}}_{t}$) is the price of a put option, it is always
positive. Then, $p(t,\tilde{X}^{\hat{\pi}}_{t}$)$\tilde{\pi}_{p}$ > 0 is the
case if and only if $f(d):=\Phi(d)d+\phi(d)<$ 0 for a suitable $d$. But $f$ is
a strictly positive function, therefore the optimal strategy under lower
constraint $K_{l}$ requires always less investment than the optimal
unconstrained strategy.
The parameter time does have an effect on the strategy, as the formula
suggests, and as can be seen in the example of Figure 9. Again, we focus on
the difference $-\tilde{\pi}_{p}p(t,\tilde{X}^{\hat{\pi}})$.

Figure 8: Difference $\hat{\pi}$-$\hat{\pi}_{l}$ over time (amount invested)

The closer we get to the terminal time T (here 20 years), the greater the
difference to the optimal strategy becomes, and so the smaller the optimal
constrained strategy
$\hat{\pi}_{l}$ = $\hat{\pi}-(-\tilde{\pi}_{p}p(t,\tilde{X}^{\hat{\pi}}))$
gets and so the less is invested in the risky stock.
Since we are rather interested in the maximum difference to the optimal
strategy, we now fix t=19 and look at the behaviour of
$-\tilde{\pi}_{p}p(t,\tilde{X}^{\hat{\pi}})$ with respect to the shadow
wealth.

Figure 9: Difference $\hat{\pi}$-$\hat{\pi}_{l}$ in function of current shadow
wealth (amount invested at t=19)

Here, one can see that for a good performance of the stock (i.e. one that
leads to a shadow wealth > 4000), the part of the strategy related to the put-
option is very close to zero. So, the constrained and unconstrained strategies
are practically identical. However, these high values of wealth are presumably
rarely reached, as we will start the process with very small or negative
initial shadow wealth.
On the other hand, a bad performance implies an investment amount for
$\tilde{\pi}_{p}p(t,\tilde{X}^{\hat{\pi}}_{t})$ close to the one required by
the optimal unconstrained strategy. Overall, in those cases, the investment
for the constrained wealth drops to values close to zero.
Note, that between those two scenarios, the strategy is very sensitive to
changes of shadow wealth, as indicated by the steep slope in Figure 10. As a
point of orientation, fix
$\tilde{X}^{\hat{\pi}}_{t}=K_{l}e^{-r(T-t)}$. By the formula of the strategy,
one can easily check that this implies ${d_{l}=0}$ and therefore
$\hat{\pi}_{l}=\dfrac{1}{2}\hat{\pi}$, hence the midpoint between the two
scenarios is reached. This is why we would generally expect the constrained
strategy to adapt fast to changes and be fluctuating, when the shadow wealth
is near $K_{l}e^{-r(T-t)}$.
Also note that $p(t,X^{\hat{\pi}}_{t})\tilde{\pi}_{p}$ is inversely
proportional to the variance of the stock. Accordingly, the difference between
optimal constrained strategy and optimal unconstrained strategy is reduced for
higher $\sigma$.
To see how the strategy behaves in practice and how it affects the (terminal)
wealth, we look at three different scenarios.

####  3.2.2 Different Scenarios

Scenario 1: $X^{\hat{\pi}}_{T}>K_{l}$, $X^{\pi_{l}}_{T}=K_{l}$

Figure 10: Higher shadow wealth leads to higher sensitivity of
$\hat{\pi}_{l}$.

In this case, we can see an increase of the stock value from t=15 on, with a
peak around t=18. This increase leads to an increase of the shadow wealth,
which implies a decrease of the price of the associated put option. Following
the optimal $K_{l}$-constrained strategy, the investment goes up. Since the
values of shadow wealth are in the sensitive zone (around 0 from t=15 on, see
Fig.11), the strategy shows a clear peak and is very volatile. However, the
amount invested remains at a rather low level both compared to the optimal
strategy, and also proportionally to the overall wealth. Also, the ’jumps’ in
the investment are a reflection of past movements of the stock. As these big
movements are often followed by smaller movements of stock (e.g. at t=18.6 we
have an increase of +30 vs. at the next time step we have a decrease of -10 of
value of stock), the consequences on the optimal lower constrained wealth are
limited. These two effects could be the reasons why the constrained optimal
wealth process is barely affected by the volatility of the stock. Note that we
set ${K_{l}=X_{0}}$=1’000, which is rather high, so the upside and downside
potential of the new strategy is limited and most of the wealth is invested in
the riskfree bond.

Scenario 2. $X^{\hat{\pi}}_{T}<K_{l}$, $X^{\pi_{l}}_{T}=K_{l}$

Figure 11: For a decrease in stock value, $\hat{\pi}_{l}$ goes to zero.

In the second scenario, one can see the case in which the constrained strategy
outperforms the unconstrained one. Here, we set ${K_{l}=200}$, to better
observe how the new strategy behaves. As the optimal unconstrained strategy
consists of investing a deterministic amount, independently from the actual
value of wealth (or stock), it fails in the scenario of a decreasing value of
stock. This is where the advantage of the constrained strategy kicks in: it
reacts to a decrease in wealth by reducing the amount invested. If the
downward trend of the stock is continued, this strategy is therefore more
successful.

Scenario 3: $X^{\pi_{l}}_{T}>K_{l}$

Figure 12: If the stock performs well, the first years are more critical to
the missed upside potential by the constrained strategy $\hat{\pi}_{l}$.

The last scenario shows what happens when we have a continued upward trend. As
with higher shadow wealth the strategies converge, the investment processes
are nearly the same, so the wealth performs in a parallel way. The gap between
the resulting terminal wealths, that is a realisation of the missed upside
potential, is therefore characterised by what happens for lower shadow wealth
values. Since the change of wealth depends on the amount invested multiplied
by the change in value of the stock, one should consider both the initial
difference between the strategies and the increase of the stock. Both are
highly dependant on the expected rate $\mu$, while the other parameters can be
neglected. For example, in the case above, the initial difference between the
investment is 2’692, which increases for $K_{l}$>200 (to 3’668 for $K_{l}$
=1’000), decreases for other reasonable values of $\alpha$ >0.0001, r >0.01,
$\sigma$>0.2, but for $\mu$=0.06 it is 8’486. This difference in investments
is then expected to be multplied by the same high $\mu$, which results in an
even higher difference of wealth. This effect can also be observed in Fig.13,
where, even though the difference of investments is decreasing, the gap
between the corresponding wealth is still growing until circa t=10.

Note that only for a continued positive trend of the stock (as we have seen in
the last scenario) the optimal constrained strategy results in a terminal
wealth larger that $K_{l}$. For the more frequent scenarios, the terminal
wealth is exactly $K_{l}$. We therefore would expect a probability mass point
at $K_{l}$ in the terminal wealth distribution.

###  3.3 Brief Analysis of the $K_{l}$-Strategy under a Restriction on
Investment

In analogy to the optimal strategy without lower bound, we restrict the
investment to be maximum 100$\%$ of the actual wealth.

######  Modification 2.

Let the modified strategy $\hat{\pi}_{l,m}$ be defined for
$(t,X^{\hat{\pi}_{l,m}}_{t})\in$ [0,T]$\times\mathbb{R}$ by

$\hat{\pi}_{l,m}(t,X^{\hat{\pi}_{l,m}}_{t})$ =
$\begin{cases}{\hat{\pi}_{l}(t,X^{\hat{\pi}_{l,m}}_{t})}&\text{if
$X^{\hat{\pi}_{l,m}}_{t}\geq\hat{\pi}_{l}(t,X^{\hat{\pi}_{l,m}}_{t})X^{\hat{\pi}_{l,m}}_{t}$}\\\
{1}&\text{if
$X^{\hat{\pi}_{l,m}}_{t}<\hat{\pi}_{l}(t,X^{\hat{\pi}_{l,m}}_{t})X^{\hat{\pi}_{l,m}}_{t}$}\end{cases}$

and $\hat{\pi}_{l}(t,X_{t})$ is the optimal $K_{l}$-constrained strategy from
Proposition 7.

Since the investment in the risky stock required by the $K_{l}$-restricted
strategy is always smaller than the one of the unconstrained strategy, the
absolute effect of restriction of the investment will generally be lower. But
since the new strategy adapts in function of the shadow wealth, which in turn
depends on the stock performance, we can observe that a lower value of wealth
is connected to a lower investment amount. The difference between this
investment-restricted $K_{l}$-strategy and the unrestricted one in terms of
the amount invested will therefore depend less on the movements of stock.
However, as seen before, once the investment is limited to 100$\%$ of wealth,
it falls below the optimal one and so the wealth is likely to grow less since
the process can not fully profit from the upside potential. This results in a
higher chance of staying under the required optimal investment amount, so the
wealth process is more likely to be ’trapped’ at low values. The earlier in
the process it happens, the more effect it will have, so we will look on the
starting conditions more closely. It is not surprising that we will see a
connection between the resulting terminal wealth and the initial setting of
wealth and optimal investment.

Two parameters can be identified to have the most impact on the performance of
the strategies and the restriction: the variance $\sigma$ and the lower bound
$K_{l}$.
Generally speaking, the standard situation is that the investment required at
t=0 is already exceeding the initial wealth $X_{0}$. This can not easily be
seen and has to do with the structure of the $K_{l}$-constrained strategy,
being that $X_{0}$ is the basis for the calculation of the shadow value
$\tilde{X_{0}}$, which in turn determines the initial investment. In the
diagrams of Figure 14 we see the initial investment as a function of $X_{0}$
and the line y= $X_{0}$ for comparison (having set r=0.01, $\mu$=0.03, T =20)
for different combinations of those two parameters. The corresponding wealth
processes (for a sample of 30) for a fixed $X_{0}$= 1000 are also displayed.

$K_{l}$=100, $\sigma$=0.1

$K_{l}$=1000, $\sigma$=0.1

$K_{l}$=500, $\sigma$=0.3

Figure 13: Initial investment as a function of $X_{0}$, and wealth process for
fixed $X_{0}$

One can observe that for small $K_{l}$ the difference between those lines is
rather big (except for small values $X_{0}$, which mostly can be excluded as
we had set $X_{0}$ >$K_{l}e^{-rT}$ $\approx$ 80$\%$ $K_{l}$), and so the
actual investment will be much smaller for the strategy with the investment
constraint. This will lead to fewer movements in the wealth, hence the
terminal wealth will be more concentrated. For a downward trend of stocks, the
investment will be close to 0, which will be an advantage, as the investment-
constrained strategy will yield a concentration of terminal wealth around
values higher than $K_{l}$ (where the wealth of the strategy without
investment-constraint would end up in those cases). Also note, that even if
the wealth processes are identical for some time, a decrease in value of the
stock will reduce the investment amount faster for the investment-constrained
strategy, so possible loss is minimised. At the same time, of course, for an
upwards trend of stocks, the strategy without constraints on investment
benefits from higher investments and therefore higher returns.
Another relevant parameter for the modified strategy seems to be the market
volatility $\sigma$. For example, using $\sigma$=0.3 instead of $\sigma$=0.1,
generally leads to an initial investment lower than the initial wealth, hence
less difference between the investment-restricted and unrestricted strategies.

###  3.4 Comparison of Terminal Wealth Distributions

After the qualitative observations, we will now quantify the resulting
terminal wealth distribution. We are particularly interested in the difference
to the distribution of the unconstrained optimal terminal wealth and the
impact of the investment constraint.
Let us first look at the theoretical distribution of the optimal terminal
wealth under the lower constraint, and then compare it to the empirical
results.

####  3.4.1 Theoretical vs. Empirical Distribution

We begin with the derivation of the theoretical distribution of terminal
wealth for an investor that follows the optimal unconstrained strategy from
the last chapter. From Proposition 3, we have
$\mathbb{P}[X^{\hat{\pi}}_{T}\leq
x]=\mathbb{P}[X^{\hat{\pi}}_{0}e^{rT}+T\dfrac{\sigma^{2}}{\alpha}+\dfrac{\theta}{\alpha}W_{T}\leq
x]$
=
$\mathbb{P}[W_{T}\leq(x-X_{0}e^{rT}-T\dfrac{\theta^{2}}{\alpha})\dfrac{\alpha}{\theta}]\newline
=\Phi(d_{T})$, where
$d_{T}:=(x-X_{0}e^{rT}-T\dfrac{\theta^{2}}{\alpha})\dfrac{\alpha}{\theta\sqrt{T}}$.
Now consider the the optimal strategy, where the terminal wealth is subject to
a lower constraint.
$\mathbb{P}[X^{\hat{\pi}_{l}}_{T}\leq x]\newline
=\mathbb{P}[\tilde{X}^{\hat{\pi}}_{T}+\max\\{K_{l}-\tilde{X}^{\hat{\pi}}_{T},0\\}\leq
x]$ by Proposition 4,$\newline =\mathbb{P}[K_{l}\leq
x|\tilde{X}^{\hat{\pi}}_{T}<K_{l}]\mathbb{P}[\tilde{X}^{\hat{\pi}}_{T}<K_{l}]+\mathbb{P}[X^{\hat{\pi}_{l}}_{T}\leq
x|\tilde{X}^{\hat{\pi}}_{T}\geq K_{l}]P[\tilde{X}^{\hat{\pi}}_{T}\geq
K_{l}]\newline =\mathbb{P}[K_{l}\leq
x]\mathbb{P}[\tilde{X}^{\hat{\pi}}_{T}<K_{l}]+\mathbb{P}[K_{l}\leq
X^{\hat{\pi}}_{T}\leq x]$
= $\begin{cases}{\mathbb{P}[\tilde{X}^{\hat{\pi}}_{T}<x]}&\text{if $x\geq
K_{l}$}\\\ {0}&\text{if $x<K_{l}.$}\end{cases}\newline \newline $So, the
cumulative distribution of the optimal terminal wealth under the
$K_{l}$-constrained strategy has a probability mass point at $K_{l}$ and
follows the distribution of the terminal wealth of an unconstrained optimal
strategy with a shadow value for initial wealth for $x>K_{l}$. This is
expressed as a jump from zero to a positive value at $K_{l}$ in the CDF, as it
is shown below for a standard example ($X_{0}$= 1’000, $K_{l}$=800,
$\alpha$=0.0001, r=0.01, $\mu$=0.03, $\sigma$=0.1 and T = 20, samplesize
1’000) .

Figure 14: Empirical distribution of terminal wealth for $\hat{\pi}_{l}$

In this example, the probability, that the optimal terminal wealth lands
exactly on the lower constraint, is circa 60$\%$ and with higher $K_{l}$ this
probability increases even more.
Consequently, as $K_{l}$ $\rightarrow$ \- $\infty$, the jump is shifted to the
left, and the distribution converges to the distribution of the shadow
terminal wealth, which is normally distributed. At the same time, the shadow
value $\tilde{X}_{0}$ converges to $X_{0}$, so the shadow wealth process
converges to the optimal unconstrained process. This can also be
quantitatively observed in the theoretical quantiles below, where the
unconstrained strategy is highlighted as a reference.

$K_{l}$ | $\tilde{X_{0}}$ |  $\mathcal{Q}_{0.25}$($X_{0}$) |  $\mathcal{Q}_{0.50}$($X_{0}$) |  $\mathcal{Q}_{0.75}$($X_{0}$) |  $\mathcal{Q}_{0.95}$($X_{0}$)
---|---|---|---|---|---
-$\infty$ | - | 3’188.6 | 9’221.4 | 15’254.2 | 23’933.4
1’000 | -10’701.41 | 1’000 | 1’000 | 1’000 | 9’641.3
-1’000 | -3’377.14 | -1’000 | 3’875.2 | 9’908.0 | 18’587.2
-30’000 | 999.5621 | 3’188.0 | 9’220.9 | 15’253.7 | 23’932.9
Table 6: Theoretical quantiles of $X^{\hat{\pi}_{l}}$ for different $K_{l}$

Before we look at the impact the investment-constraint has on the terminal
$K_{l}$-bounded wealth distribution, we want to get a sense of the errors
between the theoretical distribution and the empirical one (i.e. the one we
obtain from the simulation implemented in the code). Here, there are mainly
two sources of error: One being the sample size that allows only an
approximation of the quantiles, the other one being the step width h, which
describes the frequency of re-balancing the portfolio. The latter does also
play a role in real-life situations, as it is unrealistic to expect continuous
trading (as assumed in the derivation of the theoretical strategy) because of
technical, time and cost constraints.
To gain an idea of the dimension of the error, we consider a standard case
($X_{0}$ = 1’000, $K_{l}$=100, T = 20, $\alpha$ =0.0001, $\mu$ = 0.03, r =
0.01, $\sigma$ = 0.1) with a fixed h = 0.1 and different sample sizes. As a
reference, the theoretical quantiles of $X^{\hat{\pi}_{l}}_{T}$ are also
listed in terms of total return.

Samplesize s | $\mathcal{Q}_{0.25}(X^{\hat{\pi}_{l}}_{T})$ | $\mathcal{Q}_{0.50}(X^{\hat{\pi}_{l}}_{T})$ | $\mathcal{Q}_{0.75}(X^{\hat{\pi}_{l}}_{T})$ | $\mathcal{Q}_{0.95}(X^{\hat{\pi}_{l}}_{T})$
---|---|---|---|---
Theoretical return | 10$\%$ | 116$\%$ | 719$\%$ | 1’587$\%$
s = 1’000 | 0$\%$ | 17 $\%$ | 0 $\%$ | -1 $\%$
s = 3’000 | 0$\%$ | 5$\%$ | -2$\%$ | 1 $\%$
s = 5’000 | 0$\%$ | 12$\%$ | -1 $\%$ | 0$\%$
Table 7: Deviation of empirical from theoretical quantiles of
$X^{\hat{\pi}_{l}}_{T}$ (different samplesizes)

The deviation at $\mathcal{Q}_{p}(X^{\hat{\pi}_{l}}_{T})$ is given by
$[\mathcal{Q}^{emp,s}_{p}(X^{\hat{\pi}_{l}}_{T})-\mathcal{Q}^{theor}_{p}(X^{\hat{\pi}_{l}}_{T})]/{\mathcal{Q}^{theor}_{p}(X^{\hat{\pi}_{l}}_{T}})$,
where $\mathcal{Q}^{emp,s}_{p}(X^{\hat{\pi}_{l}}_{T})$ is the p-quantile from
the empirical distribution obtained from a sample size s.

Now we fix s = 3’000 and look at different step widths h. For the reader’s
orientation, the concrete interpretation of h would be: h=1/10 means ’once a
month’, h=1/49 means ’once a week’, h=1/100 means ’twice a week’.
Again, the deviation is
$[\mathcal{Q}^{emp,h}_{p}(X^{\hat{\pi}_{l}}_{T})-\mathcal{Q}^{theor}_{p}(X^{\hat{\pi}_{l}}_{T})]/{\mathcal{Q}^{theor}_{p}(X^{\hat{\pi}_{l}}_{T}})$,
where $\mathcal{Q}^{emp,h}_{p}(X^{\hat{\pi}_{l}}_{T})$ is the p-quantile from
the empirical distribution obtained from a step width h.

step width h | $\mathcal{Q}_{0.25}(X^{\hat{\pi}_{l}}_{T})$ | $\mathcal{Q}_{0.50}(X^{\hat{\pi}_{l}}_{T})$ | $\mathcal{Q}_{0.75}(X^{\hat{\pi}_{l}}_{T})$ | $\mathcal{Q}_{0.95}(X^{\hat{\pi}_{l}}_{T})$
---|---|---|---|---
h = 1/10 | 0$\%$ | 12 $\%$ | -5 $\%$ | -2 $\%$
h = 1/49 | 0$\%$ | -4$\%$ | 4$\%$ | 0 $\%$
h = 1/100 | 0$\%$ | -6$\%$ | -1 $\%$ | -1$\%$
Table 8: Deviation of empirical from theoretical quantiles of
$X^{\hat{\pi}_{l}}_{T}$ (different stepwidths)

First note, that the standard deviation of the theoretical normal distribution
is very high (770$\%$) and so the standard deviation of the sample mean also
is: ${770\%/\sqrt{3^{\prime}000}\approx 14\%}$. Therefore the values of the
deviation at $\mathcal{Q}_{0.50}$ are high, but lie within reasonable range.
For this thesis, the computational and time ressources were limited, but it
would be interesting to further investigate the tradeoff between error
reduction by using a greater sample size and the additional computing
resources needed. Compared to the sample size, the size of step width seems to
have less impact on the error, indicating that the focus should be on the
sample size if one wished to achieve more confidence.
The error of 0$\%$ at $\mathcal{Q}_{0.25}$ can also be explained, because this
quantile lies below the lower constraint $K_{l}$= 100 that is equivalent to a
total return of 10$\%$.

In addition, these results do not change significantly for lower $K_{l}$. For
example if ${K_{l}=-30^{\prime}000}$, the range of errors is similar. This
means in particular, that these observations are also true for implementation
of the optimal unconstrained strategy.
It also needs to be added that the calculation of the shadow value in the code
produces a slight error, but since it is usually <$10^{-9}$, we consider it to
be negligible. Furthermore, the normal random generator implemented in R might
produce inaccuracy of the normal distribution in the tails. However, we would
not see this here in the quantiles.

####  3.4.2 Impact of Restriction on Investment

In order to evaluate the difference to the theoretical distribution produced
by the limitation of investment to 100$\%$, we look at the three scenarios
from Section 3.3 (having fixed ${h=1/49}$, s = 3’000). Here, the quantiles are
given as a total return on initial wealth, and
$\Delta\mathcal{Q}_{p}:=[\mathcal{Q}^{emp}_{p}(X^{\hat{\pi}_{l,m}}_{T})-\mathcal{Q}^{theor}_{p}(X^{\hat{\pi}_{l}}_{T})]/{\mathcal{Q}^{theor}_{p}(X^{\hat{\pi}_{l}}_{T}})$
( for $\mathcal{Q}^{emp}_{p}$ the empirical and $\mathcal{Q}^{theor}_{p}$ the
theoretical p-quantile) measures the deviation.

Scenario | Distr. |  $\mathcal{Q}_{0.25}$($X_{0}$) |  $\mathcal{Q}_{0.50}$($X_{0}$) |  $\mathcal{Q}_{0.75}$($X_{0}$) |  $\mathcal{Q}_{0.95}$($X_{0}$)
---|---|---|---|---|---
$K_{l}$= 100, | Theor. | 10$\%$ | 116$\%$ | 719$\%$ | 1’587$\%$
$\sigma$ = 0.1 | $\hat{\pi}_{l,m}$ | 110 $\%$ | 162 $\%$ | 225 $\%$ | 343 $\%$
| $\Delta\mathcal{Q}_{p}$ | 1004 $\%$ | 40 $\%$ | -69 $\%$ | -78 $\%$
$K_{l}$= 1000, | Theor. | 100$\%$ | 100$\%$ | 100 $\%$ | 964 $\%$
$\sigma$ = 0.1 | $\hat{\pi}_{l,m}$ | 106$\%$ | 128$\%$ | 194 $\%$ | 340 $\%$
| $\Delta\mathcal{Q}_{p}$ | 6$\%$ | 28$\%$ | 94 $\%$ | -65 $\%$
$K_{l}$ = 500, | Theor. | 50$\%$ | 50$\%$ | 231 $\%$ | 520$\%$
$\sigma$ = 0.3 | $\hat{\pi}_{l,m}$ | 51$\%$ | 69$\%$ | 223 $\%$ | 519 $\%$
| $\Delta\mathcal{Q}_{p}$ | 2$\%$ | 39 $\%$ | -3$\%$ | 0 $\%$
Table 9: Empirical, investment-constrained vs. theoretical quantiles of
$X^{\hat{\pi}}_{T}$ (varying scenarios)

These results reflect the behaviour we have seen in the qualitative analysis:
For low $K_{l}$, the effect of the constraint on investment is generally
stronger. Also, the positive effect on lower quantiles can be seen clearly,
which is linked to the concentration around values larger than $K_{l}$. For
$K_{l}$ = 1’000, the uplift of small quantiles is not that strong, the impact
of the investment-constraint can rather be detected around values in the
middle, which is also consistent with Figure 14. Note, that in these two cases
the ’price’ for the uplift of lower quantiles is a strong reduction of
$\mathcal{Q}_{0.95}$. This result is even worse if we consider the relative
loss in utility instead of wealth, which might be a more consistent way to
evaluate the return. For example in the first scenario, the theoretical
utility at $\mathcal{Q}_{0.5}$ of -0.89 increases by 6$\%$, whereas at
$\mathcal{Q}_{0.95}$ it declines from -0.2 by 255$\%$. In more volatile market
conditions, however, the optimal $K_{l}$-constrained strategy will change much
less if a restriction on the investment in the stock is introduced.

To see if the lower-bounded terminal wealth distribution is more affected by a
constraint on investment to 100$\%$ than the optimal unbounded terminal wealth
distribution, we look at the situation from Section 2.3 for comparison. Here,
the initial wealth was set to be 120$\%$ , 100$\%$ and 80$\%$ of the amount
required by the optimal strategy for an initial investment in stocks. Because
of the convergence of the strategies, for $K_{l}$ = -30’000, the results from
$\hat{\pi}_{l,m}$ are identical to the ones from the optimal (unbounded)
strategy $\hat{\pi}_{m}$ .

(a) $K_{l}$ = -30’000

$X_{0}$ | Distr. |  $\mathcal{Q}_{0.25}$($X_{0}$) |  $\mathcal{Q}_{0.50}$($X_{0}$) |  $\mathcal{Q}_{0.75}$($X_{0}$) |  $\mathcal{Q}_{0.95}$($X_{0}$)
---|---|---|---|---|---
4’912 | Theor. | 101$\%$ | 163$\%$ | 224$\%$ | 313$\%$
| $\hat{\pi}_{l,m}$ | 80 $\%$ | 160 $\%$ | 215$\%$ | 302 $\%$
| $\Delta\mathcal{Q}_{p}$ | -21 $\%$ | -2 $\%$ | -4 $\%$ | -3 $\%$
4’094 | Theor. | 97$\%$ | 171$\%$ | 245 $\%$ | 351 $\%$
| $\hat{\pi}_{l,m}$ | 74$\%$ | 158$\%$ | 227 $\%$ | 336 $\%$
| $\Delta\mathcal{Q}_{p}$ | -24$\%$ | -7$\%$ | -7 $\%$ | -4 $\%$
3’275 | Theor | 91$\%$ | 183$\%$ | 275 $\%$ | 408$\%$
| $\hat{\pi}_{l,m}$ | 70$\%$ | 141$\%$ | 238 $\%$ | 378 $\%$
| $\Delta\mathcal{Q}_{p}$ | -24$\%$ | -23$\%$ | -13$\%$ | -7 $\%$

(b) $K_{l}$ = 3’000

$X_{0}$ | Distr. |  $\mathcal{Q}_{0.25}$($X_{0}$) |  $\mathcal{Q}_{0.50}$($X_{0}$) |  $\mathcal{Q}_{0.75}$($X_{0}$) |  $\mathcal{Q}_{0.95}$($X_{0}$)
---|---|---|---|---|---
4’912 | Theor. | 82$\%$ | 144$\%$ | 205$\%$ | 293$\%$
| $\hat{\pi}_{l,m}$ | 82 $\%$ | 145 $\%$ | 205$\%$ | 292 $\%$
| $\Delta\mathcal{Q}_{p}$ | -0 $\%$ | 1 $\%$ | 0 $\%$ | -1 $\%$
4’094 | Theor. | 73$\%$ | 132$\%$ | 206 $\%$ | 312 $\%$
| $\hat{\pi}_{l,m}$ | 76$\%$ | 133$\%$ | 205 $\%$ | 310 $\%$
| $\Delta\mathcal{Q}_{p}$ | 4$\%$ | 1$\%$ | 0 $\%$ | -1 $\%$
3’275 | Theor. | 92$\%$ | 95$\%$ | 187 $\%$ | 320 $\%$
| $\hat{\pi}_{l,m}$ | 91$\%$ | 183$\%$ | 185 $\%$ | 309$\%$
| $\Delta\mathcal{Q}_{p}$ | 0$\%$ | 2$\%$ | -1$\%$ | -3 $\%$

Table 10: Empirical, investment-constrained vs. theoretical quantiles of
$X^{\hat{\pi}}_{T}$ (varying $K_{l}$)

As expected, the impact of the investment constraint is lower for the
$K_{l}$-bounded strategy $\hat{{\pi}_{l}}$, simply because it generally
requires less investment. In particular, $X_{0}$ is a percentage of the
initial investment $\hat{{\pi}}_{0}X_{0}$, which is generally larger than
$\hat{{\pi}_{l}}(0)X_{0}$. Another difference is the observation, that the
constraint on the investment has a negative effect on the optimal
unconstrained strategy $\hat{{\pi}}$, contrary to the positive effect on
$\hat{{\pi}}_{l}$. This is, because the higher investment in stocks required
by $\hat{{\pi}}$ comes with a stronger upside potential in the first place.
Limiting this investment (which is more likely in case of bad performance,
i.e. at lower quantiles) also reduces the possibility for high returns, which
is expressed in lower values of terminal wealth. Finally, the effect on
$\mathcal{Q}_{0.25}$ for initial value $X_{0}$ = 3’275 is particularly
interesting, because 92$\%$ is the lower constraint $K_{l}$. So, even though
the difference is small here, it indicates that the terminal wealth can fall
below the lower constraint if a restriction on investment is introduced, and
in other cases the effect might be stronger.

##  4 An Optimal Strategy for Exponential Utility and Upper Constraint $K_{u}$

We will now expand the previous problem by adding an upper bound for the
terminal wealth. The resulting strategy will then be qualitatively analysed.
To understand the resulting terminal wealth distribution we will take a more
theoretical approach.

###  4.1 Derivation of the $K_{l}$-$K_{u}$-Strategy

The optimal strategy under upper and lower constraints for terminal wealth
will be derived by first deriving the optimal strategy for the isolated case
of only an upper constraint. We will find, that it involves selling a call
option and using the extra income to follow the optimal strategy. Combining
this with the lower constraint will then require to use a part of the new
initial shadow wealth to buy a put option, which then will reduce its value.
Again, for simpliciy, we will sometimes call this strategy the
$K_{u}$-Strategy and the one combined with the lower constraint the $K_{l}$
-$K_{u}$-Strategy.

####  4.1.1 Derivation of the $K_{u}$-Strategy

We now look at a setting where the terminal wealth faces an upper constraint
$K_{u}\in\mathbb{R}_{[X_{0}e^{rT},\infty)}$ and use similar arguments as
before to determine the optimal strategy. We begin by stating the modified
problem.

######  Problem 3.

Find an optimal strategy $\hat{\pi}_{u}\in\mathcal{A}$ such that

$\mathbb{E}[U(X^{\hat{\pi}_{u}}_{T})]=\sup\limits_{\pi\in\mathcal{A}}\mathbb{E}[U(X^{\pi}_{T})]\
\text{and}\ X^{\hat{\pi}_{u}}_{T}\leq K_{u}\text{ a.s. holds.}$ (4.1)

This is solved in a similar way to the lower constraint, by determining the
structure of the optimal terminal wealth and constructing a replicating
portfolio.
However, we first need to establish a statement analogous to Lemma 2 from [5].

######  Lemma 1.

The optimal terminal wealth corresponding to the solution of Problem 3 is
given by

$X^{\hat{\pi}_{u}}_{T}=\min\\{K_{u},I(yH_{T})\\},$ (4.2)

where U is concave, I is the inverse of $U^{\prime}$ and y is a positive
number such that the budget constraint
$\mathbb{E}[H_{T}X^{\hat{\pi}}_{T}]=X_{0}$ holds.

###### Proof.

The proof is similar to the one from [5].
Since U is concave, we have $U(a)-U(b)\leq U^{\prime}(b)(a-b)\ \forall\
a,b\in\ \mathbb{R}$.
From this follows in particular for any admissible strategy $\pi$ such that
$X^{\pi}_{T}\leq K_{u}$:
$\mathbb{E}[U(X^{\pi}_{T})]-\mathbb{E}[U(X^{\hat{\pi}_{u}}_{T})]\newline
\leq\mathbb{E}[U^{\prime}(X^{\hat{\pi}_{u}}_{T})(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})]\newline
=\mathbb{E}[U^{\prime}(X^{\hat{\pi}_{u}}_{T})(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})|X^{\hat{\pi}}_{T}\geq
K_{u}]\mathbb{P}[X^{\hat{\pi}_{u}}_{T}\geq K_{u}]\newline
+\mathbb{E}[U^{\prime}(X^{\hat{\pi}_{u}}_{T})(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})|X^{\hat{\pi}_{u}}_{T}<K_{u}]\mathbb{P}[X^{\pi_{u}}_{T}<K_{u}]$.
Evaluating the second term:
$X^{\hat{\pi}_{u}}_{T}<K_{u}\implies X^{\hat{\pi}_{u}}_{T}=I(yH_{T})\text{ and
since }U^{\prime}(I(yH_{T}))=yH_{T},\text{we have}\newline
\mathbb{E}[U^{\prime}(X^{\hat{\pi}_{u}}_{T})(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})|X^{\hat{\pi}_{u}}_{T}<K_{u}]=\mathbb{E}[yH_{T}(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})|X^{\hat{\pi}_{u}}_{T}<K_{u}]$
Evaluating the first term:
From $X^{\hat{\pi}_{u}}_{T}=\min\\{K_{u},I(yH_{T})\\}\leq K_{u}\implies
X^{\hat{\pi}_{u}}_{T}=K_{u}$ follows
$(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})=(X^{\pi}_{T}-K_{u})\leq 0$.
Also, since $U^{\prime}$ is decreasing and $X^{\hat{\pi}_{u}}_{T}\leq
I(yH_{T})$ it follows $U^{\prime}(X^{\hat{\pi}_{u}}_{T})\geq
U^{\prime}(I(yH_{T}))=yH_{T}$. This leads to
$\mathbb{E}[U^{\prime}(X^{\hat{\pi}_{u}}_{T})(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})|X^{\hat{\pi}_{u}}_{T}\geq
K_{u}]=-\mathbb{E}[U^{\prime}(X^{\hat{\pi}_{u}}_{T})(X^{\hat{\pi}_{u}}_{T}-X^{\pi}_{T})|X^{\hat{\pi}_{u}}_{T}\geq
K_{u}]\newline
\leq-\mathbb{E}[yH_{T}(X^{\hat{\pi}_{u}}_{T}-X^{\pi}_{T})|X^{\hat{\pi}_{u}}_{T}\geq
K_{u}]=\mathbb{E}[yH_{T}(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})|X^{\hat{\pi}_{u}}_{T}\geq
K_{u}]$.
So in summary we have
$\mathbb{E}[U(X^{\pi}_{T})]-\mathbb{E}[U(X^{\hat{\pi}_{u}}_{T})]\newline \leq
y\mathbb{E}[H_{T}(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})|X^{\hat{\pi}_{u}}_{T}\geq
K_{u}]\mathbb{P}[X^{\hat{\pi}_{u}}_{T}\geq
K_{u}]+y\mathbb{E}[H_{T}(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})|X^{\hat{\pi}_{u}}_{T}<K_{u}]\mathbb{P}[X^{\hat{\pi}_{u}}_{T}<K_{u}]\newline
=y\mathbb{E}[H_{T}(X^{\pi}_{T}-X^{\hat{\pi}_{u}}_{T})]=y(X_{0}-X_{0})=0$,
because the budget constraint holds for both strategies. So,
$\mathbb{E}[U(X^{\pi}_{T})]\leq\mathbb{E}[U(X^{\hat{\pi}_{u}}_{T})]$ for all
admissible strategies $\pi$, from which follows statement (4.1)∎

In order to solve Problem 3 for the exponential utility function, we state:

######  Proposition 8.

The optimal terminal wealth for Problem 3 under the utility function is given
by

$X^{\hat{\pi}_{u}}_{T}=\tilde{X}^{\hat{\pi}}_{T}-\max\\{\tilde{X}^{\hat{\pi}}_{T}-K_{u},0\\}$
(4.3)

where $\tilde{X}^{\pi}_{t}$ is the optimal unconstrained wealth process from
(2.15), with shadow value
$\tilde{X}^{\hat{\pi}}_{0}=(-ln(\dfrac{y}{\alpha})+rT-\dfrac{\theta^{2}}{2}T)\dfrac{1}{\alpha}e^{-rT}$
for y >0 such that $\mathbb{E}[H_{T}X^{\hat{\pi}}_{T}]={X}_{0}$.

###### Proof.

With Lemma 1 and the notation and results from proof of Proposition 4, we get
for the exponential function:
$X^{\hat{\pi}_{u}}_{T}=\min\\{K_{u},I(yH_{T})\\}=I(yH_{T})-\max\\{I(yH_{T})-K_{u},0\\}$
with the same formula for the shadow value. ∎

So, in this case the optimal terminal wealth is identical to the wealth
resulting from the optimal strategy and starting value $\tilde{X}_{0}$ minus
the payoff of a call option with strike price $K_{u}$. This means, that the
optimal wealth process can be replicated by the replicating strategy of the
call option plus the optimal strategy from the first chapter. Again, we first
determine the pricing function corresponding to the payoff of the call
function $\max\\{\tilde{X}^{\hat{\pi}}_{T}-K_{u},0\\}$:

######  Proposition 9.

The pricing function corresponding to the call option with payoff
$\max\\{\tilde{X}^{\hat{\pi}}_{T}-K_{u},0\\}$ is given by

${c(t,\tilde{X}^{\hat{\pi}}_{t})=\Phi(-d_{u})(\tilde{X}^{\hat{\pi}}_{t}-K_{u}e^{-r(T-t)})+\dfrac{\theta\sqrt{T-t}}{\alpha}e^{-r(T-t)}\phi(d_{u})}$
(4.4)

with $\Phi(x)$ the cumulative normal distribution function, $\phi(x)$ its
density, and
$d_{u}=d_{u}(t,\tilde{X}^{\hat{\pi}}_{t})=(K_{u}-\tilde{X}^{\hat{\pi}}_{t}e^{r(T-t)})\dfrac{\alpha}{\sqrt{T-t}\theta}$.

###### Proof.

By using the same risk neutral valuation arguments and notation as for the
proof of Proposition 5, we get for t= 0
$c(0,\tilde{X}^{\hat{\pi}}_{0})=e^{-rT}\mathbb{E}[(\tilde{X}^{\hat{\pi}}_{T}-K_{u})\mathds{1}_{\\{Z>d_{0}\\}}]\newline
=\tilde{X}^{\hat{\pi}}_{0}\Phi(-d_{0})-K_{u}e^{-rT}\Phi(-d_{0})+\dfrac{\theta\sqrt{T}}{\alpha}e^{-rT}\dfrac{1}{\sqrt{2\pi}}\bigintsss_{d_{0}}^{\infty}Ze^{-Z^{2}/2}dZ$
=
$\Phi(-d_{0})[\tilde{X}^{\hat{\pi}}_{0}-K_{l}e^{-rT}]+\dfrac{\theta\sqrt{T}}{\alpha}e^{-rT}\dfrac{1}{\sqrt{2\pi}}e^{-d_{0}^{2}/2}$,
using that $1-\Phi(d_{0})=\Phi(-d_{0})$. The statement then follows from
expanding for any $t\in$ [0,$T$].∎

Next, we determine the replicating portfolio. We state

######  Proposition 10.

The replicating portfolio of the pricing function (4.4) is given by the
strategy

$\tilde{\pi}_{c}(t,\tilde{X}^{\hat{\pi}}_{t})=\dfrac{\Phi(-d_{u})}{\sigma\sqrt{T-t}(\phi(d_{u})-\Phi(-d_{u})d_{u})},$
(4.5)

with $d_{u}$ as in Proposition 9.

###### Proof.

First take the partial derivatives of $c(t,\tilde{X}^{\hat{\pi}}_{t})$:
$c_{t}=e^{-r(T-t)}[-\Phi(-d_{u})rK_{u}+\dfrac{\theta}{\alpha}\sqrt{T-t}\phi(d)(r+\dfrac{1}{2(T-t)})],\newline
c_{x}=\Phi(-d_{u}),\newline
c_{xx}=\phi(-d_{u})\dfrac{\alpha}{\sqrt{T-t}\theta}e^{r(T-t)}$ using
$\tilde{X}^{\hat{\pi}}_{t}-K_{u}e^{-r(T-t)}=-d_{u}e^{-r(T-t)}\sqrt{T-t}\dfrac{\theta}{\alpha}$
and cancellations.
Since $\tilde{X}^{\hat{\pi}}_{t}$ is an Ito drift diffusion process, we can
apply Ito’s Lemma and get
$dc=\\{c_{t}+c_{x}(r\tilde{X}^{\hat{\pi}}_{t}+\dfrac{\theta^{2}}{\alpha}e^{-r(T-t)})+c_{xx}\dfrac{\theta^{2}}{2\alpha^{2}}e^{-2r(T-t)}\\}dt+c_{x}\dfrac{\theta}{\alpha}e^{-r(T-t)}dW_{t}$.
In order to satisfy the dynamics of the wealth equation,
$dc=(rc+\tilde{\pi}_{c}\theta\sigma)dt+\sigma\tilde{\pi}_{c}cdW(t)$ , it needs
to hold
$c_{x}\dfrac{\theta}{\alpha}e^{-r(T-t)}$ = $\sigma\tilde{\pi}_{c}c$, from
which follows equation (4.5). ∎

Again, combining (4.5) with the optimal strategy gives the wanted result.

######  Proposition 11.

An optimal strategy for Problem 3 is given by the amount to be invested at t

$\hat{\pi}_{u}(t,\tilde{X}^{\hat{\pi}}_{t})=\dfrac{\theta}{\alpha\sigma}e^{-r(T-t)}-c(t,\tilde{X}^{\hat{\pi}}_{t})\dfrac{\Phi(-d_{u})}{\sigma\sqrt{T-t}(\phi(d_{u})-\Phi(-d_{u})d_{u})}$
(4.6)

for the shadow wealth process $\tilde{X}^{\hat{\pi}}_{t}$ and
$d_{u}=d_{u}(t,\tilde{X}^{\hat{\pi}}_{t})\text{ as in Proposition 9}$.

###### Proof.

By definition of
$\hat{\pi}_{u}(t,\tilde{X}^{\hat{\pi}}_{t})=\hat{\pi}_{t}\tilde{X}^{\hat{\pi}}_{t}-\tilde{\pi}_{c}(t,\tilde{X}^{\hat{\pi}}_{t})c(t,\tilde{X}^{\hat{\pi}}_{t})$
the resulting wealth process is
$X^{\hat{\pi}_{u}}_{t}=\tilde{X}^{\hat{\pi}}_{t}-c(t,\tilde{X}^{\hat{\pi}}_{t})$
with
$X^{\hat{\pi}_{u}}_{T}=\tilde{X}^{\hat{\pi}}_{T}-\max\\{\tilde{X}^{\hat{\pi}}_{T}-K_{u},0\\}$,
so (4.3) holds. ∎

####  4.1.2 The $K_{l}$-$K_{u}$-Strategy

If we combine the upper and lower constraints on terminal wealth, we get an
intuitive result, which is summarized in the following Proposition (using the
previous notation).

######  Proposition 12.

The strategy determined by the absolute investment in the stock at t,

$\hat{\pi}_{l,u}(t,\tilde{X}^{\hat{\pi}}_{t})=\hat{\pi}_{t}\tilde{X}^{\hat{\pi}}_{t}+\tilde{\pi}_{p}(t,\tilde{X}^{\hat{\pi}}_{t})p(t,\tilde{X}^{\hat{\pi}}_{t})-\tilde{\pi}_{c}(t,\tilde{X}^{\hat{\pi}}_{t})c(t,\tilde{X}^{\hat{\pi}}_{t}),$
(4.7)

provided that ${X}_{0}$ = $\mathbb{E}[H_{T}X^{\hat{\pi}_{l,u}}_{T}]$, is a
solution to

$\mathbb{E}[U(X^{\hat{\pi}_{l,u}}_{T})]=\sup\limits_{\pi\in\mathcal{A}}\mathbb{E}[U(X^{\pi}_{T})]\
\text{and}\ K_{l}\leq X^{\hat{\pi}_{u}}_{T}\leq K_{u}\text{ a.s. for
}K_{l}<K_{u}.$ (4.8)

###### Proof.

By definition of $\hat{\pi}$, $\tilde{\pi}_{p}$ and $\tilde{\pi}_{c}$, the
strategy (4.7) results in the terminal wealth
$\tilde{X}^{\hat{\pi}}_{T}+\max\\{K_{l}-\tilde{X}^{\hat{\pi}}_{T},0\\}-\max\\{\tilde{X}^{\hat{\pi}}_{T}-K_{u},0\\}$
, which both satisfies (3.2) and(4.2) (note that $K_{u}>K_{l}$) and hence
solves both Problem 2 and Problem 3. ∎

This implies that the shadow initial wealth $\tilde{X}_{0}$ will increase if
we add an upper constraint to the (pre-existing) lower constraint on terminal
wealth, because an additional amount of ’money’ is available due to selling a
call-option. The effect of the increase will depend on the price of the call-
option, which in turn is determined by the strike price $K_{u}$. Before we
look at this effect more closely, we analyse the behaviour of the
$K_{u}$-strategy, the optimal strategy that only faces an upper constraint,
separately.

###  4.2 Analysis of the $K_{l}$-$K_{u}$-Strategy

In contrary to the case of a lower constraint, now a call option is involved
in the optimal strategy, whose value is increasing with wealth. At the same
time it also becomes very sensitive and as a consequence this will require to
short-sell large amounts of stocks in very short time. The good news is that
with increasing $K_{l}$ and $K_{u}$ an almost sure convergence to
$\hat{\pi}_{l}$ can be observed, since the frequency of these extremes
diminishes. We will also find a criterion for a ’balance’ between $K_{u}$ and
$K_{l}$.

####  4.2.1 Analysis of the $K_{l}$-Strategy

Since the payoffs of put and call options are antagonistic, so will be the
behaviour of the strategies linked to them. This means for example, if the
value of the shadow wealth rises, the put-option loses value and the
replicating strategy would increase investment, equivalently the call-option
would gain value, so its replicating strategy would reduce investment. It is
therefore not surprising that the difference to the optimal strategy due to an
upper constraint looks like the ’inverted’ difference to the optimal strategy
due to a lower constraint (below, an illustration of the case $K_{u}$ = 2’000
and $\mu$ = 0.03, $r$ = 0.01, $\sigma$ = 0.1 and $T$=20).

Figure 15: Difference $\hat{\pi}$-$\hat{\pi}_{u}$ in function of current
shadow wealth (amount invested at t=19)

This can also be seen in the formula, for example it holds that:

-$\tilde{\pi}_{c}(d)$=$\dfrac{-\Phi(-d)}{\sigma\sqrt{T-t}(\phi(d)-\Phi(-d)d)}=\dfrac{-\Phi(-d)}{\sigma\sqrt{T-t}(\phi(-d)+(-d)\Phi(d)}$ =$-\tilde{\pi}_{p}(-d)\ \forall\ d\in\mathbb{R}.$

A more detailed illustration and comparison of these processes can be found in
the Appendix. Here, we will only briefly outline the consequences of this
strategy on the wealth process. It can be observed that the behaviour over
time is similar for both constrained strategies and it shows a peak on a high
values of t (in the case of the Appendix it was at t=17, in the one from the
previous chapter it was at t= 19). Combining this with the increase of the
strategy for high values of $\tilde{X}^{\hat{\pi}}_{t}$ as seen in Figure 16,
these effects add up for high $\tilde{X}^{\hat{\pi}}_{t}$ and t. This leads to
very small (i.e. negative) values for $\hat{\pi}_{l}$, since the positive
investment from the optimal strategy can’t compensate this extreme effect.
In the example below it can be seen particularely well how the time and the
shadow wealth influence the strategy and the wealth resulting from it.

Figure 16: $\hat{\pi}_{u}$ becomes negative for higher wealth and nearing
maturity ($K_{u}$ = 1’250).

In the first time sequence from t=0 to circa t= 10 the stock price, the shadow
wealth and the associated call option are moderately increasing. However, the
value of shadow wealth is increasing from circa 1’000 to circa 5’000, which
has an overproportional effect on the strategy, as seen in Figure 16. This is
why the moderate fluctations of the shadow wealth produce movements in the
strategy that are increasingly more accentuated. To this adds the effect of
time, which can be seen later on. Even though the movements in the shadow
wealth are not increasingly extreme, the strategy $\hat{\pi}_{l}$ is
increasingly sensitive (even around low values of $\tilde{X}^{\hat{\pi}}_{t}$
at circa $t$=17). This being said, the strategy seems to work well, the strong
negative investments in stocks towards maturity pay off when the stock is even
just slightly decreasing. In fact, since the goal was to reach a target wealth
$K_{u}$ or below, it might perform too well. This is could be the reason why
investments are drastically reduced to zero in the last periods before
maturity.
In practice, the strong negative investments can be interpreted as short-
selling and borrowing and could lead to a few problems. First, transaction
costs might become a considerable factor, as the strategy requires to sell and
buy a large number of stocks within short time. Then, the rebalancing will
happen in discrete time, opposite to the assumptions of the theory (in the
simulation, the step width was $h$=1/10). Since the strategy is very
sensitive, the output might differ more from the theoretical results (or
possibly less for shorter $h$) and further investigations of the impact of $h$
would be useful before implementing this strategy in practice. Finally, in the
case of poor stock performance and loss of value, the terminal wealth itself
will be negative. Without a lower constraint, this seems not only to lead to
an increased probability of debts, but in particular an increased probability
of very hight depts. However, this problem could be easily avoided by setting
a lower constraint for terminal wealth. Looking at the effect of the upper
constraint, the impact of the call-option-related part of $\hat{\pi}_{l,u}$
can be reduced by setting a higher $K_{u}$. However, it might be difficult to
explain the choice of a very high constraint, as the missed potential might be
minimal. A reasonable choice of the constraints will also be discussed in the
next section. For a first assertion, we have set $K_{u}$ = 3’000 in the next
scenario.

Figure 17: Poor stock performance leads to higher investment ($K_{u}$ =
3’000).

Here, the effects we have seen before are also playing a role, but overall
their impact is reduced. Towards the end of the investment horizon, we even
observe a convergence of the optimal $K_{u}$-constrained strategy to the
optimal unconstrained one. This is, because of two reasons. First, a higher
$K_{u}$ generally leads to a higher probability that the shadow wealth falls
under it, which would result in a payoff of zero for the call-option. Second,
the performance of the stock in this case is rather poor, and so is the
performance of shadow wealth. Hence, the price of the option is near zero,
especially for times near to maturity. This again shows the importance of the
value of shadow wealth to the strategy: looking back at Figure 16, which
illustrated a similar case, we see that the difference of investment to the
optimal unconstrained strategy for a shadow wealth around 0 is a factor 10
less than the one from wealth around 5’000. Equivalently, a ’bad’ stock
performance leads to an increased investment by $\hat{\pi}_{u}$, which makes
sense, because in this scenario the target wealth then would be less likely to
be surpassed.
It would be interesting to further investigate the behaviour of
$\hat{\pi}_{u}$. But since the investment-constrained $K_{u}$-strategy, as
well as the combination with the lower constraint, the $K_{l}$\-
$K_{u}$-strategy, might reduce its sensitivity and since ultimately these
strategies are of more practical interest to us, we would rather have a closer
look at these two.

####  4.2.2 Analysis of the $K_{l}$-$K_{u}$-Strategy

We now combine the upper and the lower bound for terminal wealth and measure
the impact of these constraints on the strategy.
First, note that $K_{l}$ and $K_{u}$ determine the weight of
$\tilde{\pi}_{c}c(\tilde{X}^{\hat{\pi}}_{t},t)$ and
$\tilde{\pi}_{p}p(\tilde{X}^{\hat{\pi}}_{t},t)$ on the total optimal strategy
$\hat{\pi}_{l,u}$. Also, they are critical for the computation of
$\tilde{X}_{0}$. We therefore first establish a point of orientation for
comparison of $K_{l}$ and $K_{u}$.

######  Proposition 13.

For an upper bound $K_{u}$ and a lower bound $K_{l}$ for terminal wealth, the
initial wealth $X_{0}$ and a solution for shadow wealth for the optimal
constrained strategy from Proposition 12 $\tilde{X_{0}}$, it holds:

$K_{u}+K_{l}=2X_{0}e^{rT}\implies\tilde{X}_{0}=X_{0}.$ (4.9)

.

###### Proof.

First note that $K_{u}=2X_{0}e^{rT}-K_{l}$ implies $d_{u}$ =
$K_{u}-X_{0}e^{rT}$ = -$(K_{l}-X_{0}e^{rT})$ = -$d_{l}$.
Plugging this in the formula of the call option price gives
c(0, $X_{0}$)
=
$\Phi(-d_{u})(X_{0}-K_{u}e^{rT})+\dfrac{\theta}{\alpha}\sqrt{T}e^{-rT}\phi(d_{u})$
=
$\Phi(d_{l})(X_{0}-(2X_{0}e^{rT}-K_{l})e^{rT})+\dfrac{\theta}{\alpha}\sqrt{T}e^{-rT}\phi(-d_{l})$
=
$\Phi(d_{l})(K_{l}e^{-rT}-X_{0})+\dfrac{\theta}{\alpha}\sqrt{T}e^{-rT}\phi(d_{l})$
= p(0,$X_{0}$)
So, the prices at t=0 of the options are the same, hence they set each other
off and it holds:
$X_{0}$ \+ p (0,$X_{0}$) - c(0,$X_{0}$) = $X_{0}$, hence $X_{0}$ is a solution
for shadow wealth. ∎

We will see later that if the described relation between $K_{u}$ and $K_{l}$
holds, the distribution of terminal wealth on [$K_{l}$,$K_{u}$] will be
identical to the distribution of the optimal unconstrained terminal wealth.
Further, it can easily be seen that a decrease of $K_{l}$ and an increase of
$K_{u}$ at the same time by the same amount will have no impact on the
distribution (except for the limits of its definition area).
However, if $K_{l}+K_{u}>2X_{0}e^{rT}$, then $\tilde{X_{0}}$< $X_{0}$ and the
same holds for the inequality in the other sense. To see how $\tilde{X}_{0}$
changes for different ($K_{l},K_{u}$)-combinations, refer to the illustration
below, where $X_{0}$=1’000 (in a setting with $r$=0.01, $\mu$=0.03,
$\sigma$=0.2).

Figure 18: Shadow initial wealth in function of $K_{l}$ and $K_{u}$ ($X_{0}$ =
1’000)

We will now have a closer look on the qualitative implications on the
strategies if $K_{l}$ and $K_{u}$ vary and are particularly interested in
reducing the sensitivity of $\hat{\pi}_{l,u}$ towards
$\tilde{X}^{\hat{\pi}}_{t}$. The following scenarios are simulated for a
sample of 20 paths and $T$=20, $r$=0.01, $\sigma$ = 0.2 and $\mu$ =0.03.
$K_{l}$ is fixed at 0 and for a better comparison, the scales in the diagrams
weren’t adapted.

(a) $K_{u}$=4’000

(b) $K_{u}$=2’443

(c) $K_{u}$=1’500

Figure 19: Amount invested over time and wealth processes for
$\hat{\pi}_{l,u}$ (different $K_{u}$)

It can be observed, that the impact of the part of the strategy related to the
upper constraint, $\hat{\pi}_{u}c(0,\tilde{X}^{\hat{\pi}}_{t})$ is still
strong, and its extremely low investments dominate the total strategy and the
wealth process. However, a difference between different upper constraints can
be found. In the first scenario, $K_{u}$ = 4’000 is rather large (but it lies
between the 50$\%$ and 75$\%$ quantile of the optimal strategy under the lower
bound $K_{l}$=0, so it still represents a significant reduction of upside
potential) and so the weight of the upper constrained-part of the strategy is
smaller. This can be seen, since fewer strategy-paths follow extremely low
investments and there is a concentration on an upper line converging to the
optimal strategy, which seems to be an upper boundary for the investments.
Also, looking at the terminal wealth, two points of concentration can be
spotted: $K_{l}$ and $K_{u}$, indicating that the terminal wealth distribution
now has two probability mass points.
The scenario in the middle represents the setting from Proposition 13, where
$X_{0}$ = $\tilde{X}_{0}$. It is, in a sense, a balanced mix of upper and
lower constraints, and so the negative excesses of investment are more
pronounced. From the paths of the optimal wealth processes it can be guessed
that the stock performance is rather worse than in the first scenario. Since
it is known that $\hat{\pi}_{u}$ is sensitive especially to high values of
$X_{t}$, the investments of this scenario would be expected to be even smaller
for other simulations.
In the third case, the upper constraint is very low, which leads, as expected,
to an even higher impact of $\tilde{\pi}_{c}c(0,\tilde{X}^{\hat{\pi}}_{t}$) on
$\hat{\pi}_{l,u}$. Here, from $t$=0 onwards, most of the investments are
negative.

The case for different $K_{l}$ is similar: For lower $K_{l}$ the
$\hat{\pi}_{l,u}$ is closer to $\hat{\pi}_{u}$ and for higher $K_{l}$ it is
closer to $\hat{\pi}_{u}$. An illustration of this can be found in the
Appendix. If we combine these two results, and set a high lower constraint and
a high upper constraint, the total strategy indeed looks more acceptable. For
example, in the case below, we have set $K_{l}$ = 800 and $K_{u}$ = 5’000 for
an initial investment of 1’000.

Figure 20: Amount invested over time and wealth processes for
$\hat{\pi}_{l,u}$ (high $K_{l}$ and $K_{u}$)

###  4.3 Distribution of Terminal Wealth

In this section we will see, that the effect of adding lower and upper bounds
to the optimal strategy on terminal wealth distribution can be described as a
shift of the quantiles by the future value of the difference of initial wealth
and initial shadow wealth.

Let us first calculate the theoretical distribution of terminal wealth,
following the optimal strategy subject to both a lower constraint $K_{l}$ and
an upper constraint $K_{u}$.
$\mathbb{P}[X^{\hat{\pi}_{l,u}}_{T}\leq x]\newline
=\mathbb{P}[\tilde{X}^{\hat{\pi}}_{T}-\max\\{\tilde{X}^{\hat{\pi}}_{T}-K_{u},0\\}+\max\\{K_{l}-\tilde{X}^{\hat{\pi}}_{T},0\\}\leq
x]$ by Proposition (12)
= $\mathbb{P}[\\{\tilde{X}^{\hat{\pi}}_{T}|\tilde{X}^{\hat{\pi}}_{T}\geq
K_{u}\\}\cap\\{\tilde{X}^{\hat{\pi}}_{T}|K_{u}\leq
x\\}]+\mathbb{P}[\\{\tilde{X}^{\hat{\pi}}_{T}|K_{l}\leq\tilde{X}^{\hat{\pi}}_{T}<K_{u}\\}\cap\\{\tilde{X}^{\hat{\pi}}_{T}|\tilde{X}^{\hat{\pi}}_{T}\leq
x\\}]$
\+
$\mathbb{P}[\\{\tilde{X}^{\hat{\pi}}_{T}|\tilde{X}^{\hat{\pi}}_{T}<K_{l}\\}\cap\\{\tilde{X}^{\hat{\pi}}_{T}|K_{l}\leq
x\\}]$,
since $K_{l}<K_{u}$ and
$\\{\tilde{X}^{\hat{\pi}}_{T}|\tilde{X}^{\hat{\pi}}_{T}\geq
K_{u}\\}\cup\\{\tilde{X}^{\hat{\pi}}_{T}|K_{l}\leq\tilde{X}^{\hat{\pi}}_{T}<K_{u}\\}\cup\\{\tilde{X}^{\hat{\pi}}_{T}|\tilde{X}^{\hat{\pi}}_{T}<K_{l}\\}$
is the union of disjoint sets and has probability 1.
For $K_{l}\leq x<K_{u}$ we then get:
$\mathbb{P}[X^{\hat{\pi}_{l,u}}_{T}\leq x]$
=
$\mathbb{P}[\\{\tilde{X}^{\hat{\pi}}_{T}|K_{l}\leq\tilde{X}^{\hat{\pi}}_{T}\leq
x\\}]+\mathbb{P}[\\{\tilde{X}^{\hat{\pi}}_{T}|\tilde{X}^{\hat{\pi}}_{T}<K_{l}\\}]$
= $\mathbb{P}[\tilde{X}^{\hat{\pi}}_{T}\leq
x\\}]-\mathbb{P}[\tilde{X}^{\hat{\pi}}_{T}<K_{l}]+\mathbb{P}[\tilde{X}^{\hat{\pi}}_{T}<K_{l}]$
= $\mathbb{P}[\tilde{X}^{\hat{\pi}}_{T}\leq x]$ ,
and it is easy to see that $\mathbb{P}[X^{\hat{\pi}_{l,u}}_{T}\leq x]=1$ for
$x\geq K_{u}$ and $\mathbb{P}[X^{\hat{\pi}_{l,u}}_{T}\leq x]=0$ for $x<K_{u}$
.
In summary this leads to:

$\mathbb{P}[X^{\hat{\pi}_{l,u}}_{T}\leq x]$ =
$\begin{cases}{\mathbb{P}[\tilde{X}^{\hat{\pi}}_{T}\leq x]}&\text{if
$K_{l}\leq x<K_{u}$}\\\ {1}&\text{if x $\geq K_{u}$}\\\ {0}&\text{if x <
$K_{l}$}\end{cases}$

So, again, we will find probability mass points at the boundaries $K_{l}$ and
$K_{u}$ and the probability of reaching values outside these limits is zero.
In between, the terminal wealth distribution follows the cumulative
distribution of the shadow terminal wealth, i.e normal distribution with
$\mathbb{E}_{0}[\tilde{X}^{\hat{\pi}}_{T}]=\tilde{X}^{\hat{\pi}}_{0}e^{rT}+T\dfrac{\theta^{2}}{\alpha}$
and Var$(\tilde{X}^{\hat{\pi}}_{T})=\dfrac{\theta^{2}}{\alpha^{2}}T$.
From this follows:

######  Proposition 14.

The shift of quantiles of terminal wealth that results from introducing upper
and lower constaints on terminal wealth is given by:

$\tilde{\mathcal{Q}}_{p}=\mathcal{Q}_{p}+(\tilde{X}_{0}-X_{0})e^{rT},$ (4.10)

where $\mathcal{Q}_{p}$ is the p-quantile of the distribution of
$X^{\hat{\pi}}_{T}$, $\tilde{\mathcal{Q}}_{p}$ is the p-quantile of the
distribution of $X^{\hat{\pi}_{l,u}}_{T}$, $X_{0}$ is the initial wealth, and
$\tilde{X_{0}}$ the initial shadow wealth from Proposition 12.

###### Proof.

To see the statement, first recall that the original unconstrained terminal
wealth distribution has normal distribution. For better readability we
introduce the notation
$E:=\mathbb{E}[X^{\hat{\pi}}_{t}]=X_{0}e^{rT}+T\dfrac{T\theta^{2}}{\alpha}$
and $V:=\dfrac{\theta}{\alpha}\sqrt{T}$.
We then have by the definition of the quantiles:
$\mathcal{Q}_{p}$
= $\inf\\{z|\mathbb{P}[X^{\hat{\pi}}_{T}\leq z]\geq p\\}$
= $\inf\\{z|\mathbb{P}[\dfrac{X^{\hat{\pi}}_{T}-E}{V}\leq\dfrac{z-E}{V}]\geq
p\\}$
= $\inf\\{z|\mathbb{P}[Z_{T}\leq\dfrac{z-E}{V}]\geq p\\}$ with
$Z_{T}:=\dfrac{X^{\hat{\pi}}_{T}-E}{V}\sim\mathcal{N}$(0,1)
= $\inf\\{z|\Phi(\dfrac{z-E}{V})\geq p\\}$ and so
$\Phi(\dfrac{\mathcal{Q}_{p}-E}{V})=p$, because $\Phi$ is (right) continuous.
We then have ${\mathcal{Q}_{p}}$ = $\Phi^{-1}(p)V+E$ , where $\Phi^{-1}(p)$ is
the p-quantile of the normal distribution.
Applying the same procedure to
$X^{\hat{\pi}_{l,u}}_{T}\sim\mathcal{N}(\tilde{E},\tilde{V})$ with
$\tilde{E}=\tilde{X}_{0}e^{rT}+T\dfrac{\theta^{2}}{\alpha}$ and $\tilde{V}=V$
leads to:
$\tilde{\mathcal{Q}}_{p}=\Phi^{-1}(p)\tilde{V}+\tilde{E}$
= $\Phi^{-1}(p)V+\tilde{X}_{0}e^{rT}+T\dfrac{\theta^{2}}{\alpha}$ =
$\Phi^{-1}(p)V+\tilde{X}_{0}e^{rT}+X_{0}e^{rT}+T\dfrac{\theta^{2}}{\alpha}-X_{0}e^{rT}$
= $\mathcal{Q}_{p}+(\tilde{X}_{0}-X_{0})e^{rT}$ . ∎

This result has an intuitive interpretation. For example, if we only consider
an upper constraint on terminal wealth, then
$\tilde{X}_{0}=X_{0}+c(0,\tilde{X}_{0})$ and so the quantiles are shifted
exactly by the future value of the price of the call option at t=0. In other
words: the results of this strategy are equivalent to selling a call option
(on an optimal shadow process) and putting the received money in a bank
account. For the lower constraint, the effect is analogous. In this sense, the
constained strategy is somewhat trivial.
Combining this result with Figure 19, where the effect of the upper and lower
constraints on the shadow wealth is illustrated, we can assess their impact on
the terminal wealth distribution.
In particular, giving up significant upside potential by lowering $K_{u}$
leads to a great positive shift of quantiles. This is also reflected in Table
13, which shows the resulting empirical quantiles for a fixed lower constraint
$K_{l}$ = 0, a market given by $\sigma$= 0.2, r = 0.01, $\mu$ = 0.03 and
varying upper constraints(sample size 400). For example, setting $K_{u}$ as
low as 1’500 leads to a shadow wealth almost 4 times higher than the original
initial wealth. Hence, the quantiles are lifted significantly, by
$(3^{\prime}901-1^{\prime}000)e^{0.01\cdot 20}=3^{\prime}543$. However, since
the upper constraint is so low, this can only by seen in
$\mathcal{Q}_{0.10}(X_{T})$. Note that $K_{u}$ = 2’443 is the case where the
constraints set each other off in the sense that the distribution of terminal
wealth between the boundaries is identical to the unconstrained one (here,
this can be seen at $\mathcal{Q}_{0.25}(X_{T}))$.

$K_{u}$ | $\tilde{X_{0}}$ |  $\mathcal{Q}_{0.10}$($X_{T}$) |  $\mathcal{Q}_{0.25}$($X_{T}$) |  $\mathcal{Q}_{0.50}$($X_{T}$) |  $\mathcal{Q}_{0.75}$($X_{T}$) |  $\mathcal{Q}_{0.95}$($X_{T}$)
---|---|---|---|---|---|---
unconstrained | 1’000 | -2’444 | 164 | 3’067 | 6’158 | 10’649
$\infty$ | -1’038.1 | 0 | 0 | 578 | 3’669 | 8’160
4’000 | -288.7 | 0 | 0 | 1’493 | 4’000 | 4’000
2’443 | 999.7 | 0 | 164 | 2’443 | 2’443 | 2’443
1500 | 3’901.0 | 1’100 | 1’500 | 1’500 | 1’500 | 1’500
Table 11: Quantiles of $X^{\hat{\pi}^{l,u}}_{T}$ (different $K_{u}$ )

##  5 Conclusion

When implementing an optimal strategy using exponential utility, special
attention should be accorded to risk aversion. First, in practice, assuming
constant risk aversion seems to be problematic. Second, the strategy is very
sensitive to the risk aversion parameter, so the latter should be fitted
carefully. A characteristic of this optimal unconstrained strategy seems to be
the deterministic amount that is invested in the stock. As a consequence, it
is particularly attractive for a small investors with lower initial wealth
around 4’000, but has no noticeable effect on higher wealth (around 10’000).
Also, the terminal wealth is normally distributed, which could be an
advantage, because it is a well-known concept, but it can also lead to
negative retirement wealth.
Introducing constraints on the retirement wealth can be an effective tool to
control downside risk and improve the return between the limits, but their
impact depends on the choice of their values. A high lower constraint will
reduce the investment close to zero and a small lower constraint might be of
little meaning. Besides this, the lower constrained optimal strategy seems to
be suitable for implementation. Concerning the upper constraint, it might be
easier to set an (individual) value, since also larger values show a relevant
increase of quantiles. However, this strategy involves short-selling large
amounts of stocks and, at least in practice, seems to be very sensitive. This
effect can be reduced by setting a lower constraint, which is one of the
reasons why it might be best to use the constraints in combination. Another
could be the possibility of financing the upper by the lower constraint,
yielding the same distribution as the optimal unconstrained strategy between
the boundaries.
Restricting the investment to 100$\%$ of wealth does avoid negative terminal
wealth, but changes its distribution towards a log-normal, i.e. lower
quantiles decrease. In addition, retirement wealth can surpass the upper and
lower constraints. The extent of the effect of this restriction depends on
many factors and can vary between no effect and full effect even for realistic
scenarios.
As this is a first approach to the optimal strategy under constraints, the
model was chosen to be rather simple. On one hand, the simulation of the
market with constant volatility and expected return and only one risky asset
might not reflect the complexity of real markets in an accurate way. On the
other hand, the investment process does not take into consideration additional
requirements by the investor such as a saving process, value loss by inflation
or transaction costs. Especially the latter could have a considerable impact
and should therefore be taken into account.

##  6 Outlook

A basis for future research could be the consideration of other utility
functions, that are more realistic, and compare the results. Since the effect
of constraints on the terminal wealth distribution could be quantified in this
thesis and an equivalent result is known for power utility, a direct
comparison between these strategies could be a start.
Further, it might be a difficult task for investors to estimate their future
needs for retirement and fix a constraint at the beginning of a (rather long)
investment horizon. Because the developed strategy involves trading with
options, it might offer enough flexibility to modify the constraints at some
point within the investment period and it could be interesting to further
investigate this possibility.
To further develop the optimal constrained strategy for exponential utility,
it also seems necessary to consider the investor’s saving process, transaction
costs and effects of inflation.

## Appendix A

###  6.1 Derivation of HJB

The idea is to expand the problem by not only considering a wealth process
starting from $t_{0}=0$ as defined in (2.3), but for any fixed time $t$ $\in$
[0,T]. The optimal strategy then depends on time $t$ and the wealth at time
$t$ and yields the expected utility of terminal wealth given by the optimal
value function

$V:[0,T]\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{+},\newline
(t,y)\longmapsto
V(t,y):=\sup\limits_{\pi\in\mathcal{A}}\\{\mathbb{E}[U(X^{\pi}_{T})|X^{\pi}_{t}=y]\\}.$
(6.1)

For simplicity we write
$\mathbb{E}[U(X^{\pi}_{T})|X^{\pi}_{t}=y]=:\mathbb{E}_{t,y}[U(X^{\pi}_{T})]$.
Let $\hat{\pi}$ and $\tilde{\pi}$ be two strategies (called control laws)
defined on [t,T] $\times\mathbb{R}_{[y,\infty)}$, for a fixed starting point
($t$,y)$\in$ [0,T] $\times\mathbb{R}^{+}$, such that $\hat{\pi}$ is the
optimal control law and $\tilde{\pi}$ is a control law that switches to the
optimal control law after a short period of time h:

$\tilde{\pi}(s,y)$= $\begin{cases}{\hat{\pi}(s)}&\text{if t $\in$ [t +
h,T]}\\\ {\pi(s)}&\text{if s $\in$ [t,t+h)}\end{cases}$
for a fixed arbitrary control law $\pi\in\mathcal{A}$ and $h>0$ such that
$t+h$ < T.
The value function is defined as follows:

$J:[0,T]\times\mathbb{R}^{+}\times\mathcal{A}\rightarrow\mathbb{R}^{+},\
(t,y,\pi)\longmapsto J(t,y,\pi)=\mathbb{E}_{t,y}[U(X^{\pi}_{T})]$ (6.2)

Of course, for the optimal strategy the value function is identical with the
optimal value function. Comparing the two strategies, it is clear that the
value function of the optimal strategy should by definition be larger or equal
to the value function of any other strategy, in particular $V(t,y)\geq
J(t,y,\tilde{\pi})$ $\forall$ (t,y) $\in[0,T]\times\mathbb{R}^{+}.$
If $\tilde{\pi}$ takes $X^{\pi}_{t}$ at time $t$ to $X^{\pi}_{t+h}$ at time
t+h, the expected utility at terminal time is
$\mathbb{E}[U(X^{\hat{\pi}}_{T})|X^{\hat{\pi}}_{t+h}=X^{\pi}_{t+h}]=V(t+h,X^{\pi}_{t+h})$.
Since $X^{\pi}_{t+h}$ is stochastic and $X^{\pi}_{t}=y$ is fixed, it follows
$J(t,y,\tilde{\pi})$ =
$\mathbb{E}[V(t+h,X^{\pi}_{t+h})|X^{\pi}_{t}=y]=\mathbb{E}_{t,y}[V(t+h,X^{\pi}_{t+h})]$.
From this results the inequality

$V(t,y)\geq\mathbb{E}_{t,y}[V(t+h,X^{\pi}_{t+h}]$ (6.3)

Using Ito’s formula, $V(t+h,X^{\pi}_{t+h})$ can be expanded:
$\mathbb{E}[V(t+h,X^{\pi}_{t+h})]=V(t,y)+\newline
\mathbb{E}_{t,y}[$$\int_{t}^{t+h}\\{\dfrac{\partial V}{\partial
t}(s,X^{\pi}_{s})+[(rX^{\pi}_{s}+\pi_{s}(\mu-r)X^{\pi}_{s}]{\dfrac{\partial
V}{\partial
x}}(s,X^{\pi}_{s})+(\sigma\pi_{s}X^{\pi}_{s})^{2}\dfrac{\partial^{2}V}{\partial
x^{2}}\\}ds$$]+E_{t,y}[\sigma\pi_{s}X^{\pi}_{s}dW(s)]$. If we assume enough
integrability, the stochastic part vanishes. With equation (2.7) follows

$0\geq E[\int_{t}^{t+h}\\{\dfrac{\partial V}{\partial
t}(s,X^{\pi}_{s})+[(rX^{\pi}_{s}+\pi_{s}(\mu-r)X^{\pi}_{s}]{\dfrac{\partial
V}{\partial x}}(s,X^{\pi}_{s})|X^{\pi}_{t}=y]$ (6.4)

Dividing both sides by h, finding the limit for h$\downarrow$ 0 and setting
$x=y$ then gives the Hamilton-Jacobi-Bellmann Equation (HJB) as in Chapter 1.

###  6.2 Solving the linear SDE

Having set a boundary condition $X(0)=X_{0}$, we want to solve the
differential equation

>
> $dX^{\hat{\pi}}_{t}=[rX^{\hat{\pi}}_{t}+\dfrac{\theta^{2}}{\alpha}e^{-r(T-t)}]dt+\dfrac{\theta}{\alpha}e^{-r(T-t)}dW_{t}$

Via the substitutions $dZ_{t}:=rdt$ and
$dH_{t}:=\dfrac{\theta^{2}}{\alpha}e^{-r(T-t)}dt+\dfrac{\theta}{\alpha}e^{-r(T-t)}dW_{t}$
this transforms to

> $dX^{\hat{\pi}}_{t}=X^{\hat{\pi}}_{t}dZ_{t}+dH_{t}$ and $X(0)=X_{0}$

and can be written as $X^{\hat{\pi}}_{t}=\bigintsss_{0}^{t}X_{s}dZs+H_{t}$.
By Theorem 52 in Chapter 5 of [17], it has a (unique) solution

>
> $\epsilon_{H}(Z)_{t}=\epsilon(Z)_{t}\\{H_{0}+\bigintsss_{0}^{t}\epsilon(Z)_{s}^{-1}d(H_{s}-[H,Z]_{s})\\},\
> \ \ \ \ \ (\star$)

where [H,Z] is the quadratic covariation defined in Section 6, Chapter 2.
First, note that$\bigintsss_{Z_{0}}^{Z_{t}}dZ(t)=\bigintsss_{0}^{t}rdt$ , and
hence

$Z_{t}=Z_{0}+rt.$ (6.5)

Then, by Theorem 37 in Chapter 2 of [17], $\epsilon(Z)_{t}$ has the form
$\epsilon(Z)_{t}=e^{Z_{t}-[Z,Z]_{t}/2}$
and since $[Z,Z]_{t}=0$, this gives with (6.5)

$\epsilon(Z)_{t}=e^{Z_{0}+rt}$ (6.6)

Further, from the second substitution, we get:

$H_{t}=H_{0}+\bigintsss_{0}^{t}\dfrac{\theta^{2}}{\alpha}e^{-r(T-s)}ds+\bigintsss_{0}^{t}\dfrac{\theta}{\alpha}e^{-r(T-s)}dW_{s}.$
(6.7)

Plugging (6.6), (6.5), (6.4) into ($\star$) and having $d[H,Z]_{t}=0$ gives a
solution

> $X^{\hat{\pi}}_{t}$ =
> $e^{Z_{0}+rt}\left\\{H_{0}+\bigintsss_{0}^{t}e^{-Z_{0}-rs}(\dfrac{\theta^{2}}{\alpha}e^{-r(T-s)}ds+\dfrac{\theta}{\alpha}e^{-r(T-s)}dW_{s})\right\\}$
> .

At t= 0 we have $X_{0}=e^{Z_{0}}(H_{0}+0)$, so we can set $Z_{0}=0$ and
$H_{0}=X_{0}$ and get

>
> $X^{\hat{\pi}}_{t}=e^{rt}X_{0}+e^{rt}e^{-rT}\left\\{\bigintsss_{0}^{t}e^{-rs}(\dfrac{\theta^{2}}{\alpha}e^{rs}ds+\dfrac{\theta}{\alpha}e^{rs}dW_{s})\right\\}$
> =
> $e^{rt}X_{0}+e^{t-T}\left\\{\bigintsss_{0}^{t}(\dfrac{\theta^{2}}{\alpha}ds+\dfrac{\theta}{\alpha}dW_{s})\right\\}$
> =
> $e^{rt}X_{0}+e^{t-T}(\dfrac{\theta^{2}}{\alpha}t+\dfrac{\theta}{\alpha}W_{t})$,

which is the optimal wealth process from Proposition 3.
Note that for a completion of the proof it also needs to be checked that
$H_{t}$ is a semimartingale and $Z_{t}$ a continuous semimartingale.

###  6.3 Expected Utility Theory: From Lotteries to Utility Functions

Utility theory is an approach to describe the decisions of people in
situations when the outcome is not certain and is widely used in financial
economics. Since it also is the basis for the results used in this thesis,
this section is thought to give a brief outline of its main ideas. However,
they can not be reflected in all completeness and rigour. This summary is
based on [13], [10],[12] and [8], except for the examples.

Figure 21: Example of a Lottery [9]

The basic construction of a situation facing uncertainty is called a lottery:
It has two possible outcomes, each with a probability, and the probabilites
add up to one. To stay close to the model of this thesis, this could be the
value of a stock moving up to a value $S_{u}$ with a probability p and moving
down to a value $S_{d}$ with a probability 1-p.
Of course, the decision-maker doesn’t need to participiate in the lottery -so
the question arises, when he would be willing to do so. A key observation here
is that the simple expected value of the outcome is not a suitable criterion
to describe the decisions of most people. In fact, they prefer a deterministic
outcome over an uncertain one to a certain degree, even when the expected
value of the lottery is higher than the deterministic alternative. This
observation is commonly called risk aversion.
To assess the decision maker’s risk aversion in more detail, it is possible to
compare different lotteries. For example, the decision-maker could be in a
situation where he can invest in two different stocks with different upward
and downward possibilites and different resulting values. He then would prefer
one option over the other or be indifferent. These preferences are described
by preference relations $\succeq$ and are assumed to have some properties to
ensure consistency (completeness, transitivity, monotony, continuity).
Furthermore it is assumed, that they satisfy the so-called independence axiom,
which states that if one prefers a lottery A over the lottery B he would also
prefer another lottery, that leads to A with a certain probability p (and with
probability 1-p to another lottery C ) over one that leads to the lottery B
with p (and also with probability 1-p to another lottery C). In the previous
example, this could mean that if the investor prefers stock A over stock B, he
will also prefer a portfolio with one stock A and a stock C over a portfolio
with stock B and stock C.
This axiom implies one main result of utility theory, the Representation
Theorem, which quantifies preferences between situations of uncertainty. Since
the axiom of independence allows to ’continously mix’ lotteries, every lottery
can be traced down to the comparison with a value, which is the lottery that
has a certain output with probability 1. In our example, this would be some
value accorded to the investment in a riskfree asset (it is not the expected
value). The theorem states that the number accorded to any lottery has the
form of a linear combination of some function on the outputs of the lottery,
weighted by their probabilites. In other words: The preferences of a decision-
maker with respect to scenarios with uncertain outcome can be described by the