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Applied Mathematics

Stochastic models for optimal investment

Ralf Korn

Fachbereich Mathematik,Universitat Kaiserslautern Fraunhofer ITWM Kaiser-slautern 67653 KaiserKaiser-slautern, Germany

e-mail:korn@mathematik.uni-kl.de

Received: November 05, 2001

Summary.

We review some stochastic approaches to the problem of optimally investing money at a securities market. Besides the sim-ple Markowitzone-period approach and the standard continuous-time solution in the Black-Scholes setting we also shortly highlight the portfolio optimization problem under transaction costs.

Mathematics Subject Classication (1991): 93E20

1. Introduction

The problem to invest money optimally at a securities market is a well-studied but from some practical point of view still unsolved prob-lem. The aim of this note is to review its modelling, solution and practical shortcomings. In particular, we will consider a typical sit-uation where the complete and structurally nice solution obtained by mathematicians is absolutely useless for practical purposes as it ignores the presence of transaction costs in real life. However, includ-ing transaction costs into the model will lead to a problem which is extremely hard to treat. For further aspects of portfolio optimiza-tion we refer the interested reader to the monographs Korn (1997) or Merton (1990).

The natural task of an investor who trades at a securities market is to become as rich as possible. However, this has to be made a bit more precise to formulate it as a mathematical problem: Given an initial capital of x the investor looks for a trading strategy 'which

yields an optimal nal wealth of X '(

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horizon T. To solve this problem one has to specify what optimal

really means. It cannot mean to become as rich as possible in the above sense. The reason for this is that the best possible strategy in this sense can only be found afterwards or, put it another way, otherwise one has to have the knowledge of the exact evolution of the security prices before choosing the trading strategy. As we do not have this knowledge, the natural modelling framework for the evolution of the security prices is a stochastic model. As we then cannot hope to obtain the best trading strategy in the above sense, we would try to look for the trading strategy that delivers the highest expected nal wealthE(X

'(

T)). Without any further specication

of the probabilistic model for the security prices the corresponding optimal strategy would be to invest all the money into the security with the highest expected return, E(P

i( T)=P

i(0)) (given that we

also do not allow for negative positions in any of the securities). However, here the form of the solution is indeed the problem as such a strategy to put all the money on just one security is a very risky one. To transform this obvious feeling into the model it is on one hand necessary to nd a measure for the riskiness of a trading strategy, and on the other hand to balance out the desire for a high (expected) return and for a low risk. We will present two standard approaches to this problem in the sequel.

2. Mean-variance approach in a one-period setting

The starting point of modern portfolio optimization theory (i.e. the theory of optimal investment) is the work of Markowitz (1952). His idea is to use the variance of the total return of the investments as a measure for the riskiness of the strategy. Then, he suggests to maximize the expected return of the investment over all those trading strategies that yield a return variance which is below a given upper bound.

More precisely: Assume that d securities with today's prices of p

1 :::p

d are traded. Their prices P

1(

T):::P d(

T) at the

invest-ment horizon T are modelled as random variables with nite rst

two moments. We then concentrate on the return of the dierent securities, R i = P i( T) p i

\Return of security i" and introduce the abbreviations

 i= E(R i)   ij = Cov(R i R j) :

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The investor is now allowed to choose his portfolio vector  = ( 1 ::: d) 0

(i.e. the vector of the fractions of his initial wealth of x which he

wants to invest into the dierent securities) at time 0. It is then straight forward that the portfolio return (i.e. the return on the whole investment) satises (1) R = X ( T) x = d X i=1  i R i E(R ) = d X i=1  i  i =  0  Var(R ) = d X ij=1  i  ij  j=  0  :

In this framework, one way to formulate Markowitz's idea is to con-sider the problem to maximize the mean return under a bounded variance, i.e. to solve

max  2R d  0 s:t::  i 0  d X i=1  i = 1   0 C :

This is a linear programming problem with just one additional quad-ratic constraint, the variance condition. The non-negativity condi-tions on the components of the portfolio vector prevent a negative nal wealth. The summation condition simply says that the whole wealth has to be invested into the dierent securities. This problem (and also the other variants of the Markowitz approach) is easy to understand and can be solved in a straight forward way.

However, it is not our objective to analyze the solution method in detail as there are numerous textbooks where this is done (see e.g. Sharpe (1970)). Instead of this we want to take a critical look at this approach. Although it is very appealing on both the theoretical and the practical side (a fact which earned Markowitz the Nobel prize for economics in 1990 !), it has some important drawbacks. First, there is no possibility to include other preferences than just the mean-variance one. Then, mean-variance is not necessarily a good measure for riskiness as it penalizes both high losses and high gains simultane-ously. Most serious, the whole model is an oversimplication of real life. There is only a static modelling of the security prices (they are simply identied with their rst two moments !) and the trading pos-sibilities of the investor. He is only allowed to trade once and then he is forced to watch the evolution of his wealth untilT with no chance

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The obvious question is how to overcome these shortcomings. An immediate suggestion is to consider multi-period models in dis-crete time, but due to combinatorial complexity a realistic mod-elling in discrete timeleads to nearly non-tractable models. Therefore, continuous-time modelling is introduced as an answer to the above criticism and as a solution out of the dilemma raised by the combina-torial complexity of multi-period discrete-time models. We thereby interchange combinatorial complexity of the simple model against the mathematical complexity of the continuous-time model and at the same time have a full tool box of sophisticated mathematics (in-cluding partial and stochastic dierential equations, Ito calculus and martingale theory) at our hands.

3. Optimal portfolios in the Black-Scholes setting

As a consequence of the criticisms and problems with the discrete-time formulation of the portfolio problem we now consider a new formulation in a continuous-time framework which is today known as the Black-Scholes framework although the origins of the model go back to work by Samuelson and others in the 1950s and 60s. Before specifying this model in detail we will give a more general description of the portfolio problem in the continuous-time framework:

The portfolio problem (General Task):

Find an optimal investment strategy, i.e.:

For a given initial wealth of x>0, determine how many shares of

whichsecurity an investor has to hold at every time instantto maxi-mize his utility of nal wealthX(T) at the time horizonT.

By specifying our model we can also give a more precise formula-tion of the portfolio problem:

The portfolio problem (Special Task in the Black-Scholes

setting):

In a Black-Scholes-type market as given by the security prices

(2) dP 0( t) =P 0( t)r dt P 0(0) = 1 (3) dP i( t) =P i( t) 0 @ b i dt+ n X j=1  ij dW j( t) 1 A P i(0) = p i i= 1::n

determine an admissible portfolio process ^(:) which solves the

prob-lem max E(U(X (

T))) where X (

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corresponding to the portfolio process (t) via dX ( t)=X ( t) ; (t) 0 b+(1;(t) 01) r  dt+(t) 0 dW(t)   X (0) = x:

Remarks:

a) While the price process in (2) describes a riskless investment with continuously compounded interest rates the process in (3) is the prototype model for an equity price (\stock price") and is nowadays referred to as the Black-Scholes price although its introduction goes back to the work of Samuelson in the fties. Its solution is given as

P i( t) =p iexp 0 @ 0 @ b i ; 1 2 n X j=1  2 ij 1 A t+ n X j=1  ij W j( t) 1 A

where the matrix is assumed to be regular and where the process W(t) is an n-dimensional Wiener process. As a consequence of this

the logarithm of the stock price is normally distributed.

b) The interpretation of the portfolio process (t) is the same

as the portfolio vector in the Markowitz model, but it can now be changed over time according to the information of the investor. How-ever, it is assumed that the only information available to the investor are the past and present security prices. Using the price equations (2) and (3), the equation for the wealth process exactly resembles equation (1) if one also realizes that the bond component - the one corresponding to P 0( t) - is given as 1;(t) 0 1 with 1 = (1:::1) 0 . Note also that the wealth of the investor is dened as the wealth of his holdings at timet, i.e. the money that the investor would get by

selling all his holdings immediately.

c) As a further ingredient which is new compared to the Markowitz approach the investor now maximizes the expected utility from his nal wealth. Here, the utility functionU(x) is dened to be a

strict-ly concave, monotonicalstrict-ly increasing and dierentiable function. For more details on all those denitions and their economic reasoning see e.g. Korn and Korn (2001).

There are two main approaches to solve the above problem. One is the so called stochastic control approach pioneered by Merton (for an overview on Merton's work see Merton (1990)) which we will de-scribe more detailed in a moment. The second and also more mod-ern approach is the martingale method developed by Pliska (1986), Karatzas e.a. (1987) and Cox and Huang (1989). As the martingale method is not suitable to deal with the above problem in the presence

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of transaction costs of the form that we will consider below we do not comment on it further (see Korn (1997) for a detailed treatment).

The basic idea underlying Merton's approach is to identify the wealth equation with a controlled stochastic dierential equation where the control parameter is the portfolio process. With this iden-tication, Merton is able to use standard tools and results of con-tinuous-time stochastic control theory such as the Hamilton-Jacobi-Bellman equation (see Fleming and Soner (1993) or Korn and Korn (2001) for such standard methods). In particular he derived the Ha-milton-Jacobi-Bellman equation for the value function

v(tx) = max  (:)2A(x) E tx( U(X ( T)))

(i.e. the optimal utility as a function of the starting timet and the

initial wealthxof the portfolio problem) which is a non-linear partial

dierential equation of the form (4)  2;]max n 1 2  0  0 x 2 v xx( tx) + (r+ 0( b;r))xv x( tx) +v t( tx)g= 0 (tx)20T](01) v(Tx) =U(x):

This equation is generally hard to solve and even its numerical treat-ment is not simple. To consider it is justied by so-called verica-tion theorems that state that a suciently smooth soluverica-tion of the Hamilton-Jacobi-Bellman equation coincides with the value function and that the argument of the maximization in (4) yields the optimal portfolio process (see Korn and Korn (2001)).

To give a complete treatment of the topic would be beyond the scope of that note and therefore we only present some well-known

Examples:

i) Logarithmic utility:U(x) = ln(x) )optimal portfolio process ^(t) = (

0) ;1

(b;r1)

ii) HARA utility:U(x) = 1  x



06=<1 )optimal portfolio process ^(t) =

1 1;(  0) ;1 (b;r1)

iii) Exponential utility:U(x) = 1;e ;x  >0 )optimalportfolio process ^(t) = 1  ^ X(t)(  0) ;1 (b;r1)e ;r ( T;t) :

Note that the above examples contain nice and structurally very sat-isfying solutions. In the rst two examples it is always optimal to keep the fractions of wealth invested in the dierent stocks constant while in the third example the amount of money invested in the stocks

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is kept constant if we assume r = 0. However, this does not mean

that there is no trading after the initial time. The reader should be aware that the above portfolio processes require trading at every time instant!

4. Real World's Problems with the Mathematician's

Solution

Although the above solutions of the portfolio problem have satisfying mathematical appeal, their direct application to the real world of investment causes some problems. We will just look at a view of them:

i) Continuous trading at every point in time is impossible

In a recent paper Rogers (1998) demonstrates that the use of the con-tinuous time results in a discrete fashion (i.e. use the concon-tinuous time solutions but implement them only at discrete time points) yields a suciently good approximation to continuous trading.

ii) Transaction costs (xed and proportional)

The presence of such transaction costs will lead to (immediate !) ruin of the investor if he trades at each point in time. We will consider this problem in greater detail in the next section.

iii)Explicit constraints on risk

There are no additional constraints on the risk of the investment strategies in the portfolio problem of Section 3. Computing an op-timal solution of the portfolio problem in the presence of such con-straints is by no means simple. An approach to such a problem is given in Emmer, Korn, and Kluppelberg (2001).

iv)Models with a crash possibility

As the stock prices in the Black-Scholes setting are continuous they do not allow for a sudden deep fall. Hua and Wilmott(1997) therefore introduce a \crash setting". A corresponding portfolio problem is set up and solved in Korn and Wilmott (1999).

There are many more aspects worth looking at such as trading with derivatives (see Korn and Trautmann (1999)) or the inclusion of so-called xed income products (see Korn and Kraft (2001)), but we will in the remaining section only concentrate on the treatment of transaction costs.

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5. Optimal Investment under Transaction Costs

As already indicated, trading at each time instant in the presence of xed and proportional transaction costs (even more: in the presence of every reasonable form of transaction costs) will result in ruin of the investor. As a consequence of this one has to consider dierent classes of trading strategies. Here, trading strategies are the number of shares of the dierent assets hold over time (in contrast to the fractions of wealth as modelled via the portfolio processes). As each change of the trading strategy will now result in losses due to transaction costs there is rst the decision about to transact or not and then in a second stage the decision about the form of the transaction, i.e. we look for a sequencef(

i S

i)

i2@g of transaction times and corresponding

transactions that solves the portfolio problem under transaction costs

(5) max ( i  S i )2Z E(U(X(T)))

whereZ denotes the set of all admissible such strategies, the so-called

admissible impulse control strategies (for a precise technical denition see Korn (1998)).

We will in the sequel restrict ourselves to the case of just one risky security and will also look at the case of a zero interest rate. It will prove to be convenient to consider the money invested in the bond and in the stock as basic processes for our decisions, i.e. we look at

B(t) = y 0(

t)P 0(

t) (the money invested in the bond at time t) and S(t) =y

1( t)S

1(

t) (the money invested in the stock at timet) where y

0( t)y

1(

t) denote the trading strategy (i.e. the number of shares

of bond and stock hold at time t. This will result in the following

dynamic behavior of bond and stock positions between transaction costs,

(6) dB(t) = 0

(7) dS(t) =S(t)(bdt+dW(t))

while at the ith transaction time we have the relations S(t) =S(t;) +S i  B(t) =B(t;);S i ;K;kjS i j

i.e. the transaction costs K;kjS i

jcorresponding to the

transac-tion S

i must be paid out of the bond holdings. This is the setting

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solution method for such impulse control problems lies in an imita-tion of the stochastic control method. To do so we dene the value function of problem (5) by v(tBS) = sup (i Si)2Z(tBS) E tBS( U(X(T)))

and compare it with the value of all the best of those strategies that contain an immediate transaction,

Mv(tBS) = max S

v(tB;K;S;kjSjS+S):

Note how the introduction of transaction costs leads to a curse of di-mensionality as now we have to consider each position in the dierent securities separately while in the model without transaction costs the wealth process was enough to describe our situation. With the two functions just introduced we can again prove a verication theorem (see Korn (1998) for its proof):

Theorem \Verication Theorem"

Let f be a classical solution to the quasi-variational inequalities

(\qvi") corresponding to problem (5),

Lf(tBS) := 1 = 2  2 S 2 f SS( tBS) +bSf S( tBS) +f t( tBS) 0 f(tBS) Mf(tBS) Lf(tBS)(f(tBS);Mf(tBS)) = 0 (8) f(TBS) =U(B+S)

then it coincides with value function, i.e. we havef(tBS) =v(tBS)

and an optimalstrategyf( i

S i)

i2@gis given by the \qvi-control" i = inf ft> i;1 jf(tBS) =Mf(tBS)g 0 := 0  S i:= argmax S fv(tB;K;S;kjSjS+S)g:

Remarks:

a) There remain a lot of problems with the above theorem. It is by far not easy to show existence of solutions of sucient regularity to the qvi (8). Even more, the regularity requirements in the above theorem have to be (and can be !) relaxed to be valid for other cases than just the one where the no transaction strategy is optimal. See the discussion in Korn (2000) for this problem. Also, the computation of

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solutions by numerical methods is so far not dealt with in a satisfying way which makes the problem a more or less open one until today! b) For a numerical example in the case of the exponential utility function see Korn (1998). There, an asymptotic expansion method is described to solve the problem approximately. The treatment of some special problems with two stocks and one bond is given in Korn and Laue (2000).

Although the impulse control method presented above is very at-tractive (still in continuous time, giving the investor the free choice of the time of actions), it has to be made more tractable and ecient to deal with portfolios of a realistic size. This will be a demanding task for future research.

References

1. Cox J., Huang C.F. (1989) Optimum consumption and portfolio policies when asset prices follow a diusion process, Journal of Economic Theory49,1989,

33-83.

2. Eastham J.E., Hastings K.J. (1988) Optimal impulse control of portfolios, Mathematics of Operations Research13(4), 588-605.

3. Emmer S., Kluppelberg C., Korn R. (2001) Optimal portfolios under bounded capital at risk, Mathematical Finance11(4), 365-384.

4. Fleming W.H., Soner M.H. (1993)Controlled Markov Processes and Viscosity Solutions, Springer , Berlin.

5. Hua P., Wilmott P. (1997) Crash course, working paper.

6. Karatzas I., Lehoczky J., Shreve S. (1987) Optimal portfolio and consumption decisions for a small investor on a nite horizon, SIAM Journal on Control and Optimization25, 1557-1586.

7. Korn R. (1997)Optimal Portfolios, World Scientic.

8. Korn R. (1998) Portfolio optimisation with strictly positive transaction costs, Finance and Stochastics2, 85-114.

9. Korn R. (1999) Some Applications of Impulse Control in Mathematical Fi-nance, Mathematical Methods of Operations Research50(3), 493-518 .

10. Korn R., Korn E. (2001)Option Pricing and Portfolio Optimisation, AMS. 11. Korn R., Kraft H. (2001) A stochastic control approach to optimal

portfo-lios with random interest rates, to appear in: SIAM Journal on Control and Optimization.

12. Korn R., Laue S. (2000) Portfolio optimisation with transaction costs and exponential utility, working paper.

13. Korn R., Trautmann S. (1999) Optimal control of option portfolio and appli-cations, OR-Spektrum11(1-2), 123-146.

14. Korn R., Wilmott P. (1999) Optimal investment under the threat of a crash, to appear in: International Journal on Theoretical and Applied Finance. 15. Markowitz H. (1952) Portfolio Selection, Journal of Finance 7, 77-91. 16. Merton R. (1990),Continuous-Time Finance, Blackwell.

17. Pliska S.R. (1986) A stochastic calculus model of continuous trading: Optimal portfolios, Mathematics of Operations Research11, 371-382.

18. Rogers L.C.G. (1998) The relaxed investor, working paper.

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