Tài liệu Optimal portfolios with a loan dependent credit spread docx

Since there is obviously a connection between the default probability and the total percentage of wealth, which the investor is in debt, we study portfolio optimisation with a control de

M Krekel

Optimal portfolios with

a loan dependent credit spread

Berichte des Fraunhofer ITWM, Nr 32 (2002)

© Fraunhofer-Institut für Techno- und

Warennamen werden ohne Gewährleistung der freien Verwendbarkeitbenutzt

Die Veröffentlichungen in der Berichtsreihe des Fraunhofer ITWMkönnen bezogen werden über:

Fraunhofer-Institut für Techno- und

Das Tätigkeitsfeld des Fraunhofer Instituts für Techno- und Wirtschaftsmathematik ITWM umfasst anwendungsnahe Grundlagenforschung, angewandte Forschung sowie Beratung und kundenspezifische Lösungen auf allen Gebieten, die für Techno- und Wirtschaftsmathematik bedeutsam sind.

In der Reihe »Berichte des Fraunhofer ITWM« soll die Arbeit des Instituts ierlich einer interessierten Öffentlichkeit in Industrie, Wirtschaft und Wissenschaft vorgestellt werden Durch die enge Verzahnung mit dem Fachbereich Mathema- tik der Universität Kaiserslautern sowie durch zahlreiche Kooperationen mit inter- nationalen Institutionen und Hochschulen in den Bereichen Ausbildung und For- schung ist ein großes Potenzial für Forschungsberichte vorhanden In die Bericht- reihe sollen sowohl hervorragende Diplom- und Projektarbeiten und Dissertatio- nen als auch Forschungsberichte der Institutsmitarbeiter und Institutsgäste zu aktuellen Fragen der Techno- und Wirtschaftsmathematik aufgenommen werden Darüberhinaus bietet die Reihe ein Forum für die Berichterstattung über die zahlrei- chen Kooperationsprojekte des Instituts mit Partnern aus Industrie und Wirtschaft Berichterstattung heißt hier Dokumentation darüber, wie aktuelle Ergebnisse aus mathematischer Forschungs- und Entwicklungsarbeit in industrielle Anwendungen und Softwareprodukte transferiert werden, und wie umgekehrt Probleme der Pra- xis neue interessante mathematische Fragestellungen generieren.

kontinu-Prof Dr Dieter Prätzel-Wolters

Institutsleiter

Kaiserslautern, im Juni 2001

Optimal portfolios with a loan

dependent credit spread

This version January 18, 2002

Martin Krekel

Fraunhofer ITWM, Department of Financial Mathematics, 67653 Kaiserslautern, Germany Abstract: If an investor borrows money he generally has to pay higher interest rates than he would have received, if he had put his funds on a savings account The classical model of continuous time portfolio optimisation ignores this effect Since there is obviously

a connection between the default probability and the total percentage of wealth, which the investor is in debt, we study portfolio optimisation with a control dependent interest rate Assuming a logarithmic and a power utility function, respectively, we prove explicit formulae of the optimal control.

Keywords and phrases: Portfolio optimisation, stochastic control, HJB equation, credit

spread, log utility, power utility, non-linear wealth dynamics.

1 INTRODUCTION 1

1 Introduction

The continuous-time portfolio problem was first introduced by Merton in his pioneering works from 1969 and 1971 His goal is to find a suitable investment strategy which maximises the expected utility of the final wealth In the case of logarithmic and power utility this yields the result that it is optimal to invest a constant multiple of the total wealth in stocks With common market parameters this factor is mostly bigger than one In other words, the investor is advised to borrow a multiple of his own wealth to speculate in risky assets Of course in the presence of possible crashes no rational investor would do so, because this can result in immediate bankruptcy On the other hand, since the default probability of this particular credit is much higher, the counterpart who is lending the money will definitly claim higher yields than that for government bonds In addition, in a single stock setting, this yield should converge (w.r.t control) to the return of the stock, since the risk of the lender will be almost the same, as if he invests in the stock itself We introduce a control dependent interest rate, i.e credit spread, to take this credit risk into account.

We consider a security market consisting of an interest-bearing cash account and n risky assets The uncertainty is modelled by a probability space (Ω, F, {F}t∈[0,T ] , P ) The flow of

information is given by the natural filtration Ft, i.e the P -augmention of an n-dimensional

Brownian filtration Without loss of generality we set FT = F, so that all observable events

are eventually known In addition we make the assumptions that the market is frictionless except for the non-constant interest rate All traders are assumed to be price takers, and there are no transaction costs The cash account is modelled by the differential equation

dB(t) = B(t)R(t)dt, where R(t) is a bounded, strictly positive and progressively measurable process We will

in particular assume different interest rates for borrowing and lending This feature will be

modelled via a control dependent interest rate R(t) = r(πt), where r(.) : IRn → IR is a left

continuous and bounded function, which will be defined later on The price process of the

i-th, i = 1, , n, risky asset is given by

with σσ a strictly positive definite N × N -matrix The investor starts with an initial wealth

x0 > 0 at time t = 0 In the beginning this initial wealth is invested in different assets and he is allowed to adjust his holdings continuously up to a fixed planning horizon T His investment behavior is modelled by a portfolio process π(t) = (π1(t), , πn(t)) which is

progressively measurable and denotes the percentage of total wealth invested in the ular stocks If n

partic-i=1πi ≤ 1, 1 − n

i=1πi is the percentage invested in a savings account If

n

i=1πi > 1 the investor is actually borrowing money and the credit spread comes into the

game We are considering self-financing portfolio processes, thus the wealth process follows the stochastic differential equation

dX(t) = X(t)  

r(π(t))(1 − π(t) 1) + π(t)b 

dt + π(t)σdW (t) 

where U is the utility function of the investor The set A(0, x0) contains the admissible

controls with initial condition (0, x0), ”sufficiently” bounded and a corresponding wealth

process Xπ greater or equal to zero for all t in [0, T ] almost surely See Korn/Korn (2001) for an exact definition Note that the properties of r(.) ensure the existence of a solution of

the SDE (1) The term (3) raises the question, if the maximum exists Or in other words: Is

there a control π∗(.) ∈ A(0, x0), such that E(U (Xπ ∗(T ))) = supπ(.)∈A(0,x0)E(U (Xπ(T ))) ?

Via a verification theorem we will show that this is actually true.

We suggest three ways of modelling r(.) which should cover all practical needs, and also

prove to be quite usefull for numerical calculations Let ¯ r be the interest rate for a positive cash account and u1 = n

i=1ui the total percentage of wealth invested in stocks:

Figure 3: Logistic function

Simple dependencies, like r(u) = ¯ r for u1 ≤ 1 and r(u) = ¯r + λ for u1 > 1 can be

modelled with the help of the step function See Korn (1995) for the treatment of an option pricing problem in the presence of such a setting With the frequency polygon we are able

to model smoothly increasing credit spreads In the these cases, the optimisation problem (3) can be solved analytically, although we have to deal with some subcases separately The logistic function can be unterstood as a continuous approximation of a frequency polygon with just one triangle The main reason for its introduction is for numerical computations, because it is twice contiuously differentiable and can be handled without considering subcases separately An analytical solution is not available, but this does not matter with regard to the use in numerical context.

In section 3 we solve the optimisation problem for logarithmic utility (U (x) = ln(x)) and

in section 4 for power utility, that means U (x) = 1γxγ with γ ∈ (−∞, 0) ∪ (0, 1) Section 5

gives a conclusion.

Proposition 1 : Existence of the maximum

sup

x∈IRnM

θ(x) = Mθ(x∗ or x∗= arg max

x∈UMθ(x)

Existence: If r(x) is a frequency polygon or a logistic function, the existence of the

maximum follows by continuity of Mθ and compactness of U

Let r(x) be a step function as given in (4) Observe, that for all x ∈ Hi+1, i = 0, , m − 2,

we have MSθ

i (x) ≥ MSθ

i+1(x), since λi < λi+1 and x1 ≥ 1 in Hi+1 Because M Sθ

i is continuous we get:

Since π∗(.) is constant, it is an element of A(0, x0), thus the original problem (7) has been

solved too We summarize this in

Theorem 1 : Verification with logarithmic utility

The constant process π∗ defined by π∗(t) = u∗∀t ∈ [0, T ] as given in (14) is the optimal control and

or a frequency polygon as given in (4,5), we can determine the maximum explicitly, by using

{Di}i=0, ,m−1 the partition of IRn We investigate Mθ

i(x) separately on the sets Di Since

Mθ

i(x) are downwards opened parabolas (in both cases), we can determine the local maxima.

Then we compare these maxima to obtain the absolute maximum and the corresponding optimal control.

If r(u) is a logistic function, we have to calculate the maximum via numerical methods We

consider all these cases explicitly below:

3 LOGARITHMIC UTILITY 6

3.1 Step function

Theorem 2 : Optimal Portfolios with step functions and power utility

Let VS(t, x) be the value function given in (7) with r(u) a step function defined by (4) In addition, let MSΘ be the function to be maximised in Proposition 1 corresponding

to the step function r(u), i.e.

MSΘ(u) =



¯

r +m−1

i=0

λi1(α i ,α i+1](u1) (1 − u1) + ub − 1

2 Θuσσu, (15)

where λi and αi are given in (4) Then there exists an optimal (constant) control π∗(.) =

u∗= arg maxu∈IRnMS1(u) such that

VS(t, x) ≡ sup

π(.)∈A(t,x)Et,x(ln(Xπ(T ))) = Et,x(ln(Xπ ∗(T ))) The value u∗ is explicitly given below (with Θ = 1):

with Θ = 1 We include a real number Θ ∈ (0, ∞) in front of the quadratic term, because

we will use this Theorem in the next chapter As stated in the proof of Proposition 1:

3 LOGARITHMIC UTILITY 7 So:

arg max

u∈U Mθ(x) = arg max

u i Mi Sθ(ui) with ui = arg max

u∈D i

Mi Sθ(u)

As before mentioned, we achieve the local maxima and corresponding arguments on the sets

Di and then we compare them to obtain the absolute maximum Thus only the verification

of ui is left.

One-dimensional case

The MSθ

i (x) are downwards opening parabolas; so we just have to determine the apex

(ig-noring the domain Di) and check its position relative to Di If the apex is in Di= [αi , αi+1]

we have already found the maximum If it lies on the right(left) side of the intervall, the

maximum is achieved in αi+1 (αi )

2 Θuσσu 

(16)

Θ (σσ)−1(b − (¯ r + λi) 1)

Observe, that σσ is regular, as stated in Proposition 1 If vi ∈ Di, then we have found the

local maximum and so we can say ui = vi

If vi i, then the local maximum must lie in one of the hyperplanes Hirespectively Hi+1 , since −σσ is strictly negative definite and MSΘ

i therefore strictly concave If dist(Hi , vi) >(<) dist(Hi+1 , vi) then ui lies in Hi+1 (Hi ) Thus we have to calculate the maximum under

the constraint u1 = α, with α = αi resp α = αi+1, thus un = α − n−1

with b∗ k = bk − bn(k = 1, , n − 1) and σ ∗ ∈ IR(n−1)×(n−1) with

σki ∗ = σki − σni(k = 1, , n − 1) Observe, that rank(σ∗) = n − 1, since otherwise we would

 (1 − α)(¯r + λ) + αbn− 1

Note, that v does not depend on λ or ¯ r, because these quantities are fixed on the hyperplanes

Hi If the apex of the i-th parabola is achieved in the sets {Dj : j ≤ i}, then the absolute

maximum can not lie in one of the sets {Dj : j > i}, because arg max{x∈5j≥i D j }Mi Sθ∈ Diand MSθ

i (x) > MSθ(x)∀x ∈ 

j≥iDj (via λi < λi+1) So, if we are stepwise increasing

i (beginning at 0) we can stop the maximum-search, if the above condition is fullfilled Loosely speaking: The maximum can only be achieved on an apex of MSΘ

3 LOGARITHMIC UTILITY 9

-0.15-0.10-0.050.000.050.100.15

Figure 4: Parabolas MS1 with r step function and flat

In Figure 4 we plotted the corresponding function MS1 (to be maximised) with r modelled

as step function and with r flat Note, that there are generally jumps at αi, except for the

case when αi= 1.0 Since the coefficient of r(u) is (1-u), the parabola is continuous in 1.0, although r(u) jumps at that point.

3.2 Frequency polygon

The procedure is similar to the one for step functions, i.e we determine the maxima piecewise

on Di and then we compare them to obtain the absolute maximum In preparation for the

next section, we again include a parameter θ ∈ (0, ∞) in front of the square term.

Theorem 3 : Optimal Portfolios with polygons and logarithmic utility

Let VP(t, x) be the value function given in (7) with r(u) frequency polygon defined

by (5) In addition, let MP Θ be the corresponding function to be maximised in Proposition

1 with r(u) frequency polygon, i.e.

i=1σniσki ∗resp b∗d k = bk − bn− θαi

Mi P Θ(u) = (¯ r + ri− µiαi) + u 

b − 1(¯ r + ri− µi(1 + αi)) 

− 1

2 u(Θσσ+ 2µi 1 1)u Because MP θis continuous, the above procedure is valid More precisely, due to continuity,

we have supx∈UMθ(x) = maxi supx∈D iMSθ

Let Φi= ¯ r + ri− µiαi, Ψi= ¯ r + ri− µi(1 + αi)) and M P Θ

i be the parabola on Di , i.e.:

3 LOGARITHMIC UTILITY 11 The first step is to determine the apex without any restrictrictions on the domain.

= (Θσσ+ 2µi 1 1)−1(b − 1(¯ r − ri+ µi(1 + αi )))

If vi ∈ Di, then we have already found the local maximum and can define:

the constraint vi1 = α, with α = αi resp α = αi+1 again.

As before we have to use the components of the vector u and b to do our calculations Therefore, we will neglect the index i to avoid confusion:

with b∗ k = bk − bn and σ∗ ∈ IR(n−1)×(n−1) with σki ∗ = σki − σni.

Again we have rank(σ∗) = n − 1 Thus σ∗ is regular and

It is wortwhile to note that the maximum does neither depend on the interest rate ¯ r + ri,

nor on µi , and the calculation of the maximum is exactly the same as for step functions.

This is not suprising, since r is fixed on these hyperplanes.

Figure 5: Parabolas MP 1 with r frequency polygon and flat

Note that the parabola is generally not differentiable in the αi.

Since r(u) is bounded we get, very loosely speaking, a kind of downwards opened parabola.

Thus, surely there exists an absolute maximum In the one-dimensional case we have to solve the following equation:

(1 − u)λ αeαu+β

Example 3

Let r(u) be modelled as in Figure 3, i.e ¯ r = 5%, λ = 6%, α = 3 and β = 4 Then the

optimal control is 1.26.

-0.15-0.1-0.0500.050.10.15

4 POWER UTILITY 14

Again, this optimisation problem is solved by a pointwise maximization But, due to the non-linear structure of the above term, the correctness cannot be shown by some simple inequaltities as in the logarithmic case Instead of this, will show the correnctness via the verification theorem in Korn and Korn (2001):

Theorem 4 : Verification with power utility

The value function with power utility is given by:

As in the case with logarithmic utility the optimisation problem is reduced to the maximisation of downwards opening parabolas Hence, the further steps will be very similar.