Finite horizon trading strategy with transaction costs and exponential utility in a regime switching market

Finite Horizon Trading Strategy with Transaction Costs and Exponential Utilityin a Regime Switching Market XU Shanghua Supervisor : Prof.. This thesis studies the finite horizon optimal

Finite Horizon Trading Strategy with Transaction Costs and Exponential Utility

in a Regime Switching Market

XU Shanghua

Supervisor : Prof DAI Min

An academic exercise presented in partial fulfilment for

the degree of Master of Science in Mathematics

Department of Mathematics National University of Singapore

August 2010

This thesis studies the finite horizon optimal trading strategy withproportional transaction costs in a regime switching stock market Thisproblem is an extension of the classic investment strategy in a staticeconomic condition The exponential utility function is considered here.The study of this problem is mainly motivated by Dai et al (2010), inwhich the finite horizon optimal investment problem with proportionaltransaction costs under logarithm utility function in a regime switchingmarket is studied In this thesis, we use dynamic programming approach

to derive the Hamilton-Jacobi-Bellman (HJB) equations satisfied by thevalue functions For our exponential utility case, the transformation isdifferent from the one in the logarithm utility case, and we will get asystem of variational inequalities with gradient constraints For the powerutility case, there is also a similar system with gradient constraints Thedifference lies in that the case with exponential utility cannot lead to aself-contained system of double obstacle problems Due to the fact that noclosed-form solution exists, we employ two numerical methods, namely

the penalty method and the projected SOR method to solve the system ofvariational inequalities based on certain assumptions Finally we show theoptimal trading strategies.

List List of of of Author's Author's Author's Contributions Contributions

The author has proposed two numerical algorithms to solve the finitehorizon optimal investment problem with proportional transaction costs

in a regime switching market (under exponential utility), for which noanalytical solution exists yet The transformation from the original3-dimension problem to the 2-dimension problem is presented Althoughthe problem with gradient constraints is not easy to solve, we show thatthe system of variational inequalities cannot be transformed into aself-contained system of double obstacle problems So we need to befaced with the gradient constraints The results of two numericalalgorithms are equivalent The numerical results can also explain somephenomena in economics For example, we will see that youngerinvestors are more sensitive to changes in the rate of return of risky assetthan elder ones

I would like to thank my supervisor, Prof Dai Min, for his suggestionsand guidance over the past two years With his help and the teaching inthe modules of financial modeling & modeling and numerical simulations,

I have learnt many numerical algorithms through some projects whichmake me possible to do research in financial mathematics and to finish

my thesis What I have learnt from him will benefit me in the future

I would also like to express my thanks to my family, my lecturers and

my seniors Wang Shengyuan, Li Peifan and Zhao Kun for their guidancethroughout my life I sincerely appreciate all your help along the way

1.1 Historical work 71.2 Scope of this paper 9

2.1 The asset market 112.2 The investor's problem 132.3 HJB equation 14

3 Differences Differences between between between Exponential Exponential Exponential Utility Utility Utility and and and Logarithm Logarithm Logarithm Utility: Utility: The The Transformation Transformation Transformation Problem Problem 17

3.1 Dimension reduction: 3 to 2 173.2 Trading regions 20

4.1 The penalty method 224.2 The projected SOR method 25

5.1 The results of penalty method 28

5.2 The results of projected SOR method 32

5.3 Changes in transaction costs 34

5.4 Changes in rate of return of assets 36

5.5 Changes in switch intensities 38

5.6 Changes in risk aversion index 41

5.7 Exponential Utility vs Logarithm Utility 43

Chapter Chapter 1 1

Introduction

1.1 Historical Historical Work Work

In this paper, the optimal trading strategies for an exponential utilityinvestor who faces proportional transaction costs are studied This is anextension of the classic investment strategy in a static economiccondition

The study of portfolio optimization problems via stochastic processes

in continuous time was initiated by Merton (1969) He formulated theinvestment problem in infinite time horizon, and extended the model tofinite time horizon The investor chooses how to allocate his fundsbetween investment in a risk-free asset (' bank account ') and a risky asset(' stock ') in order to maximize the expected utility of terminal wealthover a finite horizon In the absence of transaction costs, the optimalstrategy would be time-independent under certain assumptions, and it is

to keep a constant fraction (' Merton proportion ') of total wealth in therisky asset However, such a strategy will lead to incessant trading, which

is impracticable in the real world

The proportional transaction costs model was first introduced byMagill and Constantinides (1976), and it leads to a stochastic singularcontrol problem They provided a heuristic argument that the optimalstrategy is described by a no-transaction region, which means the investordoes not buy or sell stocks unless his portion of wealth in stock moves out

of this region Since then, there have been a lot of papers studying theoptimal trading strategies for an investor facing proportional transactioncosts When the investor's horizon is infinite, the strategy is simplifiedsince it is time-independent

However, the finite horizon portfolio selection problem withproportional transaction costs has remained unsolved until recently Liuand Loewenstein (2002) approximated the strategy by a sequence ofanalytical solutions that converge to the real solution Dai and Yi (2009)characterized the strategy by PDE approach They proved that theoriginal HJB equation is equivalent to a double-obstacle problem.Uichanco (2006) used the penalty method to solve the obstacle problemand found it to be more efficient At the same time, researchers havestarted to consider the portfolio selection problem with regime switchingfeature, which means that the economic condition switches stochastically

between two market conditions Jang et al (2007) considered an infinitehorizon problem in a bull-bear switching market and explained the puzzle

of liquidity premium Dai et al (2010) considered a finite horizonportfolio selection problem with transaction costs in a regime switchingmarket in order to study the issue of leverage management This paper islargely motivated by the success of the approaches applied to the optimalinvestment problem in the regime switching market in the above twopapers

1.2 Scope Scope of of of this this this paper paper

In this paper, we propose numerical solutions to solve the finite horizonoptimal investment problem with proportional transaction costs in aregime switching market There is only one risky asset, the price of whichfollows the geometric Brownian motion Similar arguments as in Dai et

al (2010) will be used to derive the HJB equations satisfied by theinvestor's value function in each regime, and exponential utility functionwill be studied The HJB equation leads to a system of variationalinequalities with gradient constraints which correspond to the optimalbuying and selling boundaries The system of variational inequalitiescannot be transformed into a double-obstacle problem as in Dai et al.(2010) The penalty method and the projected SOR method will beemployed to numerically solve the variational inequalities To compare

the results, we plot the optimal buying and selling boundaries obtainedfrom both approaches We will also examine the effects of varyingparameters such as the transaction costs proportion.

The rest of the paper is organized as follows In Chapter 2, we presentthe formulation of the model Then in Chapter 3, we discuss thetransformation differences between exponential utility function andlogarithm utility function In Chapter 4, we propose numerical algorithms

to solve the problems raised in Chapter 2 and 3 We show the numericalresults and analyses in Chapter 5 The paper ends with a conclusion inChapter 6

Chapter Chapter 2 2

Model Model Formulation Formulation

In this chapter, we consider the finite horizon portfolio selection problemwith proportional transaction costs in a regime switching market Ourmodel formulation follows that of Dai et al (2010)

2.1 The The asset asset asset market market

The financial market under consideration consists of two assets: a risklessasset, referred to as the bank account, and a risky asset, referred to as astock Their price processes, denoted by Pt and Qt respectively, areassumed to satisfy:

dt)Pr(ε

])dBσ(ε)dt([Q

where εt∈{1,2} denotes the changing market condition that switches

between two regimes, "bull market" (regime 1) and "bear market"(regime 2), which is governed by a two-state Markov chain withgenerators

1k

k,

where k1,k2 >0 In other word, regime i switches into regime j at thefirst jump time of an independent Poisson process with intensity ki, for

on a filtered probability space (Ω,F,{Ft}t≥0,P) with B0 =0 almostsurely We denote ri =r(i), αi =α(i), and σi =σ(i) later on

Let Xt and Yt denote the monetary value of the investor's holdings

at time t, in the bank account and stock respectively With the assumption

of proportional transaction costs, Xt and Yt evolve according to thefollowing equations in regime i

t t

t i

t rX dt (1 λ)dL (1 μ)dM

t t

t t i t

buying and selling stock respectively The constants λ∈[0,+∞) and[0,1)

μ ∈ represent the proportional transaction costs incurred on buyingand selling of stock respectively We further assume λ+μ >0 to ensurethe presence of transaction costs From (2.1) and (2.2), it can be notedthat the purchase of dLt worth of stock involves a payment of

t

λ)dL

(1+ from the bank account while the sale of dMt worth of stockrealizes only (1−μ)dMt in cash

2.2 The The investor's investor's investor's problem problem

The investor's net wealth at time t, denoted by Wt , is defined as themonetary value of the holdings in the bank account after selling off allshares of the stock Notice the assumption α >i ri implies that it is neveroptimal for the investor to short sale the stock and as a result we alwayshave Yt ≥0 Due to transaction costs, we have:

t t

Given an initial position of (x0,y0)∈ , the investor's problem is toS

choose an admissible strategy so as to maximize the expected utility ofterminal wealth, that is, to maximize E0x0,y0[U(WT)] Here Ex,t y denotesthe conditional expectation at time t given the initial endowment Xt =x,

y

Yt = Moreover, we assume that the investor has a utility function givenby:

0β,e

The value function in regime i ∈{1,2} is defined to be

T)[0,tS,y)(x,)],[U(WE

supt)

y,

y) (x, A M) (L, i

in which Lt and Mt are constrained to be absolutely continuous withbounded derivatives, i.e

yσ2

1y

αx

r{

(k]μ)

m[(1− ∂x i−∂y i − i i− j =

Maximization with respect to l , m will produce a solution given by

⎨

⎧

=0

* κ l

if

if

0λ)

(1

0λ)

(1

i x i

y

i x i

y

<

∂+

−

∂

≥

∂+

−

∂

ϕ ϕ

ϕ ϕ

⎩

⎨

⎧

=0

* κ m

if

if

0μ)

(1

0μ)

(1

i y i x

i y i x

ϕ ϕ

The above solution is similar to the infinite horizon optimal portfolioselection problem studied by Davis and Norman (1990) This indicatesthat in each regime i, the optimal trading strategies are to buy or sell atthe maximum rate or not at all The solvency region S is divided intothree regions, "Buying" ( BRi ), "Selling" (SRi ) and "No Transaction"( NTi ) At the boundary between the BRi and NTi regions,

i x i

∂+

−

∂++

∂

− +

−

+ +

T)[0,tS,y)(x,,e

T)y,(x,

0]μ)

κ[(1]

λ)(1κ[

L

μ)y) (1 β(x i

i y i x i

x i

y i

i i

t

ϕ

ϕ ϕ

ϕ ϕ

ϕ ϕ

where

)(

kx

ry

αy

σ2

−

T)[0,tS,y)(x,,e

T)

y,

(x,

0}λ)

(1,μ)

(1,Lmin{

μ)y) (1 β(x i

i y i x i

y i x i

i i

t

ϕ

ϕ ϕ

ϕ ϕ

ϕ ϕ

, (2.5)

for i≠ j∈{1,2}

Chapter Chapter 3 3

Differences Differences between between between Exponential Exponential Exponential Utility Utility Utility and and and Logarithm Logarithm Utility: Utility: The The The Transformation Transformation Transformation Problem Problem

3.1 Dimension Dimension reduction: reduction: reduction: 3 3 3 to to to 2 2

Equation (2.5) is a 3-dimension problem, which is dimension-reducible.For the logarithm utility function U(W) =logW , we can use thehomogeneity to deduce that for any positive constant ρ ,

ρ ρ

y,(x,t)

,1,y

x(t)(z,

i t i

i y

∂

−

=

2 i z i

zz 2 i yy

y

1V2zV

=

=

−

∂+++

∂

−+

−

−

−∂

μ)1log(z

T)

(z,

V

01}

Vλ)1(z1,Vμ)1(z,VLV

min{

i

i z i

z i

* i i

where ΩT =(μ−1,+∞)×[0,T), and

)V(Vk)σ2

1(αV)zσr(αVzσ2

1V

We can see that the dimension cannot be reduced in the above way

In fact, for exponential utility function, we need to use a differenttransformation For simplicity, we assume r1=r2 =r The essence of thecorrect transformation is to rule out the dependence of x, and relies on thefact that x becomes er(T−t)x at maturity The transformation is asfollowing:

t) r(T

βye

t) (z, V βxe

t) r(T

i i

t) r(T i i

x = −β e −

i z t) r(T i i

y =−β e ∂ V

i zz t) 2r(T 2 i

2 i z t) r(T i i

yy = ( e ∂ V) − β e ∂ V

(z,

V

0}Vλ)

(1μ),(1V,VLV

min{

i

i z i

z i

' i i

where ΩT =(0,+∞)×[0,T), and

)e(1k)V(zσ2

1Vzσ2

1Vr)z(αV

i

2 i z 2 2 i i

zz 2 2 i i

z i

∂

−

∂+

3.2 Trading Trading regions regions

In each regime i, the "Buying" ( BRi ), "Selling" ( SRi ) and "NoTransaction" (NTi) regions are defined as following:

λ}

1t)(z,V:Ωt){(z,

μ}

1t)(z,V:Ωt){(z,

λ}

1t)(z,Vμ

1:Ωt){(z,

Comparing to the solutions of (3.2), we are more interested in thebuying and selling boundaries, which tell us how to trade at every timestep in practice But the above definition of the three regions does notshow obviously the properties of the buying and selling boundaries This

is due to the difficulty in dealing with the variational inequalities withgradient constraints

Dai et al (2010) has established an equivalence between (3.1) and adouble-obstacle problem in logarithm utility case Notice that (3.1) could

be written in the following form:

−++

<

∂

<

++

(z,V

μ1z

1orλ1z

1V

0,VLV

μ1z

1V

λ1z

10,VLV

i

i z i

* i i t

i z i

* i i t

(3.3)

in ΩT =(μ−1,+∞)×[0,T) Denote vi(z,t)=∂zVi(z,t), we have

)v(vk)vσr(αv)z2σr(αvzσ2

−

∈

−+

=

−+

<

<

++

1T)

(z,

v

μ1z

1v

0,vLv

λ1z

1v

0,vLv

μ1z

1v

λ1z

10,vLv

i

i i

'' i i

t

i i

'' i i

t

i i

'' i i

t)(z,V

t)

(z,

vi =∂z i Then we have

)v(v

e

k

)vv2z(2zvσ2

1)vzv(2zσ2

1)vzr)(v(α

2 i i

zz

2 i z

2 i i

z i i

−

∂+

∂+

∂+

−

=

∂

−

And we still cannot eliminate Vi and V So we need to solve (3.2)jdirectly We will propose numerical algorithms to solve it in the nextchapter.

Chapter Chapter 4 4

Numerical Numerical Schemes Schemes

In this chapter, we shall propose the numerical algorithms to solve (3.2).Due to the difficulty in dealing with these original variational inequalitieswith gradient constraints, we adopt both the penalty method and theprojected SOR method

4.1 The The penalty penalty penalty method method

Inspired by Uichanco (2006) and Dai and Zhong (2009), we use thepenalty method to deal with the system (3.2) Then we have the followingform:

(z,

V

λ)1Vm(

)Vμ

(1VLV

i

i z i

z i

' i i

(4.1)

where(z,t)∈(0,+∞)×[0,T), and L'iVi is given in (3.2), for i≠ j∈{1,2}

l , m are penalty parameters that can be chosen to be sufficiently large to

ensure that 1−μ−ε≤∂zVi ≤1+λ+ε, for any given ε > ,0 ε << 1

Boundary Boundary Conditions Conditions

In order to apply any implicit finite difference scheme, it is necessary toprescribe boundary conditions on the parabolic boundary For thistransformed problem, when z is large enough, the investor is in the sellingregion and hence should sell the stock So we need to impose an upperbound M for z In the following, we solve the problem in

T)[0,](0,

Denote the step size in space variable z and time variable t by dz and dtrespectively Then zn =ndz and tj= jdt Let 1k

j n,

j n,

V be thek-th step discrete solutions to (4.1) in Newton iteration at the point

+ +

+ +

− +

+

+ +

+ +

+ +

− +

+

+

− +

+

− +

λ)1dz

VV

m(

)dz

VV

μ

(1

)e

(1kV

Ldt

VV

λ)1dz

VV

m(

)dz

VV

μ

(1

)e

(1kV

Ldt

VV

1 k 2 1 k 2 1

k 2 1 k 2

V V 2

1 k 2

' z2

1 k 2 2

1 k 1 1 k 1 1

k 1 1 k 1

V V 1

1 k 1

' z1

1 k 1 1

j n, j

1, n j

1, n j n,

k j n, 1 k j n, 2 j

n, j

n, 1

j

n,

j n, j 1, n j

1, n j n,

k j n, 2 k j n, 1 j

n, j

n, 1

n 1

μ)z(1V

μ)z(1V

dt T n,

dt T n,

where

1 k 1 3 1 1 k 1 2 1 3 1 k 1 2 1 1 k 1

2 2 1

1=− ,

)V(V

2

nσ

2 2 1

2 =− n,j − n−1,j ,

r)n(α

and

1 k 2

' 3

' 1 1 k 2

' 2

' 1

' 3 1

k 2

' 2

' 1 1 k

2 2 2 '

1 =− ,

)V(V

nσ

2 2 2 '

−

−

μ(1

1 k 1 1 k

1n,j n 1,j

+ +

VV

(

1 k 1 1 k

1n 1,j n,j

and

+ + +

μ

(1

1 k 2 1

k

2n,j n 1,j

+ +

VV

(

1 k 2 1 k

2n 1,j n,j

, we use the generalizedNewton iteration which has been used by Forsyth and Vetzal (2002) tosolve the American option pricing problem We get the followinglinearizations:

−

)dz

VV

μ

(1

1 k 1 1 k

1n,j n1,j

} dz

V V μ - {1

1 k 1 1 k 1

k j 1, n k j n, 1

j 1, n j n,

)dz

VV

μ(1

+

+

λ)1dz

VV

(

1 k 1 1

k

1n 1,j n,j

λ}

1 dz

V V {

1 k 1 1 k 1

k j n, 1 k j 1, n

j n, j 1, n

λ)1dz

VV

−

)dz

VV

μ

(1

1 k 2 1 k

2n,j n 1,j

} dz

V V μ - {1

1 k 2 1 k 2

k j 1, n k j n, 2

j 1, n j n,

)dz

VV

μ(1

+

+

λ)1dz

VV

(

1 k 2 1

k

2n 1,j n,j

λ}

1 dz

V V {

1 k 2 1 k 2

k j n, 2 k j 1, n

j n, j 1, n

λ)1dz

VV

4.2 The The projected projected projected SOR SOR SOR method method

To deal with variational inequalities, we can also use the projected SOR

method (3.2) can be written as the following form:

0λ)1Vμ)(

1V](

)VL

0)V

L

(−∂t − 'i i ≥ , ∂zVi−1+μ≥0, ∂zVi −1−λ≤0, (4.3)

μ)z(1T)

(z,

Following the discretization in the penalty method above, we have

)dte

(1

k

V

)dtVβ(β)dtV

β(β)dt)V

β2β

(β

(1

k j n, 2 k j n, 1 1

j

n,

j 1, n j

1, n j

n,

V V 1

1

1 k 1 3 1 1 k 1 2 1 1 k 1 2

1 3

−

+ +

+

−+

≥

−++

(1k

V

)dtVβ(β)dtV

β(β)dt)V

β2β

(β

(1

k j n, 1 k j n, 2 1

j

n,

j 1, n j

1, n j

n,

V V 2

2

1 k 2

' 3

' 1 1

k 2

' 2

' 1 1 k 2

' 2

' 1

'

3

−

+ +

+

−+

≥

−++

(1kV)V

UL(D

k j 2 k j 1 1

j j

V V 1

1 1 k 1 j j j

− +

−+

(1kV

)VUL(D

k j 1 k j 2 1

j j

V V 2

2 1 k 2

' j

' j

' j

− +

−+

)dt]

e(1kVV[U)L(D

u

k j 2 k j 1 1

j j j

V V 1

1

k 1 j

1 j j

−+

Vμ)dz},(1

V),Vω(u

min{max{V

j 1, n j

1, n j n, j n, j

n, j

V[U)L(D

u

k j 1 k j 2 1

j j j

V V 2

2

k 2

' j 1 ' j

−+

Vμ)dz},(1

V),Vω(u

min{max{V

j 1, n j

1, n j n, j n, j

n, j

−

−

where ω ∈(1,2) is a constant.

Using the projected SOR method, we only need to do iterations at eachtime step, and the gradient constraints are put into the comparisonconditions So we do not have to consider the penalty terms, and we cansolve the linear equation systems by SOR approach The generaldiscretization scheme, boundary conditions and terminal conditions arethe same as those in the penalty method

Eventually, we will need to plot the buying and selling boundaries forproblem (3.2) The numerical results will be shown in the next chapter

Chapter Chapter 5 5

Numerical Numerical Results Results

5.1 The The results results results of of of penalty penalty penalty method method

We plot the optimal buying and selling boundaries in both bull marketand bear market as functions of t using the above discretization in penaltymethod Note that z here refers to the product of the parameter in theexponential utility function and what the current monetary value in therisky asset will be at maturity under risk free interest rate This isdifferent from the logarithm utility case, where z refers to the ratio ofbank account holdings to holdings in the risky asset In our case, sinceholdings in the risky asset are always assumed to be positive, the buyingand selling boundaries are also positive, implying that the investor shouldnever short sell stocks But we do not know whether the bank accountholdings are positive If the investor follows exponential utility, then he

or she does not need to consider whether to leverage, but just to followthe strategies which indicate the risky asset holdings Once z falls below

the buying boundary, the investor should buy stocks to bring the positionback into the no transaction region On the other hand, if z goes above theselling boundary, the investor should sell some of the stocks.

However, in both bull market and bear market, there exist a thresholdvalue of t beyond which no buying boundary exists This is consistentwith the observation in Liu and Loewenstein (2002) for the finite horizonoptimal investment problem They found that the optimal fraction ofwealth invested in stock decreases as time goes toward maturity because

of the finite time horizon and proportional transaction costs

We fix the following set of parameters: r =0.06, α1=0.2, σ1=0.2,