European Journal of Operational Research 206 (2010) 609–613

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

No-arbitrage conditions, scenario trees, and multi-asset financial optimization

Alois Geyer a, Michael Hanke b,*, Alex Weissensteiner b

a WU (Vienna University of Economics and Business) and Vienna Graduate School of Finance, Heiligenstaedter Straße 46-48, 1190 Vienna, Austriab University of Innsbruck, Department of Banking and Finance, Universitaetsstr. 15, 6020 Innsbruck, Austria

a r t i c l e i n f o

Article history:Received 11 February 2009Accepted 10 March 2010Available online 15 March 2010

Keywords:Financial optimizationUncertainty modelingScenario treesSparse trees

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.03.022

* Corresponding author. Tel.: +435125077552; fax:E-mail address: Michael.Hanke@uibk.ac.at (M. Han

1 The simulated asset returns in the tree can be basedsuch as geometric Brownian motions, (vector) autoregrchoice among these models depends on the propertiesnot the focus of this paper.

a b s t r a c t

Many numerical optimization methods use scenario trees as a discrete approximation for the true (multi-dimensional) probability distributions of the problem’s random variables. Realistic specifications infinancial optimization models can lead to tree sizes that quickly become computationally intractable.In this paper we focus on the two main approaches proposed in the literature to deal with this problem:scenario reduction and state aggregation. We first state necessary conditions for the node structure of atree to rule out arbitrage. However, currently available scenario reduction algorithms do not take theseconditions explicitly into account. State aggregation excludes arbitrage opportunities by relying on therisk-neutral measure. This is, however, only appropriate for pricing purposes but not for optimization.Both limitations are illustrated by numerical examples. We conclude that neither of these methods issuitable to solve financial optimization models in asset–liability or portfolio management.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Beginning with Bradley and Crane (1972), there is a strand ofliterature on financial optimization models using numerical (sto-chastic) optimization methods. Further applications include,among many others, Kusy and Ziemba (1986); Cariño and Ziemba(1994); Gondzio and Kouwenberg (2001) and Geyer and Ziemba(2008). In these models, the uncertainty in asset prices (and possi-bly additional non-tradable state variables) is represented using afew mass points by a so-called scenario tree.1 For a general descrip-tion of stochastic programming techniques, see e.g., Birge and Louve-aux (1997), and for scenario generation methods, see Dupacová et al.(2000). Stochastic modeling based on scenario trees has also beenapplied successfully in many other areas (see, e.g., Ruszczinsky andShapiro, 2003).

The scenario trees serve as an approximation of the true multi-dimensional probability distributions of the problem’s randomvariables. Improving the accuracy of this approximation requireslarger trees. Larger trees, in turn, mean higher computational ef-fort, which may quickly become prohibitive. This leads to a desirefor a reduction in the size of such trees with as little degradation inapproximating accuracy as possible.

ll rights reserved.

+435125072846.ke).on various stochastic models

essions or GARCH models. Theof the underlying assets and is

Our focus in this paper is on multi-asset and multi-period finan-cial optimization models for which the absence of arbitrage in thetree is an essential requirement (cf. Klaassen, 1997). Even if onlybroad asset classes like stocks or bonds are considered, distinguish-ing geographical regions, industries or other characteristics canmake the number of such classes rather large. Further, allowingfor frequent portfolio re-balancing results in a large number ofdecision stages. Both aspects lead to tree sizes that quickly becomecomputationally intractable. This explains why methods for con-structing scenario trees that simultaneously avoid the curse ofdimensionality and ensure the absence of arbitrage are required.

In the literature on financial applications, two major approachesfor either generating small scenario trees or reducing the size ofscenario trees have emerged: Scenario reduction/generation basedon the minimization of distance functions between true andapproximating probability distributions (Pflug, 2001; Heitsch andRömisch, 2003; Dupacová et al., 2003) and state/time aggregation(Klaassen, 1998). The main strength of these methods is their abil-ity to generate trees with a comparatively large number of decisionstages, but a small number of nodes (so-called sparse trees). Wewill show that both methods have important limitations when ap-plied to financial optimization models. First, the absence of arbi-trage implies that there is a lower bound on the branching factor,i.e., the number of arcs emanating from each node in the tree. Thislimits the ability to reduce the size of scenario trees, but is not ta-ken into account in any of the scenario reduction algorithms we areaware of. Second, state aggregation using the risk-neutral measureto exclude arbitrage is only appropriate for pricing but not for opti-mization, for which the physical (or real-world) measure has to beused.

610 A. Geyer et al. / European Journal of Operational Research 206 (2010) 609–613

This paper is structured as follows: Section 2 briefly reviews theimportance of arbitrage-free trees together with necessary criteriafor the absence of arbitrage. Section 3 explores the problems asso-ciated with applying the aggregation method proposed by Klaassen(1998) in our context. Section 4 then shows why scenario reduc-tion methods have only limited potential for multi-asset financialoptimization models since necessary conditions for the node struc-ture of the scenario tree can be violated. Section 5 illustrates ourpoints using numerical examples, and Section 6 concludes.

2. Absence of arbitrage in scenario trees

An arbitrage opportunity may be described2 as a riskless invest-ment with a positive return. Such opportunities sometimes arise onfinancial markets but cannot prevail for long, since they will be rec-ognized and exploited by investors. Arbitrage is usually excluded inasset pricing models (see Cvitanic and Zapatero (2004, p. 84) for adiscussion of the role of no arbitrage in financial models as opposedto arbitrage in financial markets). However, in optimization modelsexcluding arbitrage is a necessity since any arbitrage opportunity inthe model would lead to unbounded solutions. If the problem is re-stricted, e.g., in the form of short-sale constraints, the problem is lessobvious since a solution will be found, but this solution will in gen-eral be biased. This has lead Klaassen (1997) to request that scenariotrees must be constructed in a way that rules out arbitrage.

We start by defining a necessary and sufficient condition for theabsence of arbitrage, and consider one of the stages in a multi-stage scenario tree. Given m non-redundant securities, we definean m-dimensional vector p of time t ¼ 0 security prices, a m� nprice matrix D of the m securities in the n different states at thenext stage. A portfolio h 2 Rm has a time t ¼ 0 market value h>p,and a value of h>D in the n different states at the next stage. Anarbitrage is a portfolio h satisfying either h>p 6 0 and h>D > 0, orh>p < 0 and h>D P 0. There is no-arbitrage if and only if there ex-ists an n-dimensional state price vector w > 0 such that the systemof equations

Dw ¼ p; ð1Þ

can be solved (see Duffie, 2001, p. 4). If Eq. (1) has a solution thereexists an equivalent martingale measure. This implies that the mod-el is viable (in the sense of Harrison and Kreps, 1979; Harrison andPliska, 1981), since all claims can be priced by the no-arbitrage prin-ciple, ensuring that h>p ¼ h>Dw. If the branching factor is lowerthan the number of assets ðm > nÞ, Eq. (1) has no solution, hencearbitrage opportunities must exist.

3. State aggregation and probability measures

Based on his findings in Klaassen (1997), Klaassen (1998) pro-poses a method to reduce the size of an arbitrage-free scenario treewhile maintaining the absence of arbitrage in the ‘‘downsized”tree. We will focus here only on the ‘‘state aggregation” definedin Klaassen (1998, p. 38f.).3 State aggregation replaces all successorsof a particular node in the tree by a singleton containing the ex-pected security prices under the risk-neutral measure. The arc to thissingleton is assigned a probability of 1.

Klaassen mentions two important aspects in this regard: Thefirst is consistency, which means that security prices calculatedfrom the tree should correspond to observed market prices. Thesecond is absence of arbitrage as defined in Section 2. Both aspectscan be guaranteed with the proposed aggregation procedure by

2 A formal definition of arbitrage will be stated below.3 For time aggregation, the same arguments apply. Moreover, a reduction in the

number of decision stages is not what is usually aimed for in multi-period financialoptimization modeling.

applying the equivalent martingale characterization of arbitrage-free asset prices. However, the risk-neutral measure is only appro-priate for pricing. This so-called risk-neutral valuation principle goesback to Harrison and Kreps (1979) and Harrison and Pliska (1981).Under certain assumptions they show that contingent claims willbe correctly priced if risk-neutrality is assumed, even if investorsare risk-averse. However, its applicability does not extend to port-folio optimization. Under the risk-neutral measure the expectedreturn of each security is the same. In the real-world however,investors require compensation for bearing risk. Therefore, portfo-lio optimization must be carried out under the real-world, or phys-ical or, using Klaassen’s terminology, subjective probabilitymeasure.4 These aspects are illustrated in a simple example in Sec-tion 5. As a consequence of the risk-neutral valuation principle ap-plied in the aggregation, we show that (i) the optimal portfolioweights derived under the risk-neutral measure differ from thosecomputed under the real-world measure, and (ii) the solutions fromthe aggregated and original trees differ.

4. Scenario reduction and the absence of arbitrage

The goal of scenario reduction methods is to represent a givenprobability distribution (which may be continuous) by a discretedistribution on a scenario tree, using a comparatively small num-ber of nodes. These methods start by simulating a large numberof sample paths – a so-called fan which represents the underlyingstochastic properties of assets. In the second step, only a few pathsare retained by minimizing distance functions between the originaland the approximating distributions (Pflug, 2001; Heitsch andRömisch, 2003; Dupacová et al., 2003). The reduction algorithmscan be controlled by specifying the accuracy of approximation, orthe number of nodes in the reduced tree. The branching factor inthe reduced tree is usually rather small, and may vary acrossnodes. As shown in Section 2, however, arbitrage opportunities willarise as soon as the branching factor n is lower than the number of(non-redundant) assets m. Eq. (1) captures the key consequences ofarbitrage opportunities in optimization models. If m > n, Eq. (1)has no solution, and therefore the model offers the possibility tobecome infinitely rich without the risk of a loss. Since this lowerbound on the branching factor applies to each (source) node inthe tree, the reduced tree must grow at least at a rate OðmtÞ(t . . . stage index) to exclude arbitrage opportunities.

The literature on scenario reduction reports reduced treeswhich grow at a (much) lower rate, however. At the same time,no problems associated with the violation of no-arbitrage condi-tions are mentioned. In order to explain this discrepancy, we lookat the derivation of the scenario reduction algorithm in Pflug(2001). The basic idea is to minimize the supremum of the distance� between the objective function evaluated using (a) the originalprobability distribution and (b) the approximating distributionon the tree. Pflug shows that this problem is equivalent to mini-mizing the Wasserstein distance between the original probabilitydistribution and its discrete approximation. The latter is easier tosolve and does not depend on the objective function of the under-lying problem. However, the equivalence of the two problems restson the (implicit) assumption that the sup-distance � is finite. Thisassumption is violated when there are arbitrage opportunities inthe approximated problem. As discussed in Section 2, there aretwo possible consequences: If arbitrage opportunities can beexploited without any limits, the objective value becomes infinite.

4 The need for optimization under the real-world measure has recently beenpointed out by e.g., Jobst and Zenios (2005) and Dempster et al. (2007). Ferstl andWeissensteiner (2010) show in detail how to account for the change of measure byadjusting the conditional probabilities in the scenario tree according to the estimatedmarket price of risk.

Fig. 1. Original binomial tree. The upper value in each node is the stock price, thelower value is the price of the riskless asset. The numbers at the arcs denoteconditional probabilities.

Fig. 2. Aggregated binomial tree. The upper value in each node is the stock price,the lower value is the price of the riskless asset. The numbers at the arcs denoteconditional probabilities.

A. Geyer et al. / European Journal of Operational Research 206 (2010) 609–613 611

In case there are constraints enforcing such limits, a solution willbe found, but it will generally be biased.

5. Numerical examples

In this section we present two numerical examples. The firstone shows that state aggregation leads to an incorrect representa-tion of the conditional return distribution under the real-worldmeasure. This, in turn, results in incorrect portfolio weights. Thesecond example illustrates how arbitrage opportunities arise afterapplying scenario reduction, and how they affect the solution.

5.1. State aggregation

Consider the following two-period portfolio optimization prob-lem. There is a riskless asset gaining rf ¼ 0:04 in value over eachtime period, and a risky asset with current price S0 ¼ 100, an ex-pected return of l ¼ 0:08 (both r and l with continuous com-pounding), and a return volatility of r ¼ 0:25 per period.5 Wefirst construct a binomial tree under the risk-neutral measure. Usingthe well-known Cox et al. (1979) parameterization, we arrive at thebinomial tree shown in Fig. 1. The up-factor is given byu ¼ expðrÞ ¼ 1:284 and the down-factor by d ¼ 1=u ¼ 0:779. Therisk-neutral probability of an up-move is

q ¼ expðrf Þ � du� d

¼ 0:5186: ð2Þ

The investor maximizes expected utility of terminal wealth fW2,

maxfh0 ;~h1g

E U fW2

� �h i; ð3Þ

subject to

h>0 p ¼W0; ð4Þh>0eD1 ¼ ~h>1

eD1; ð5Þ~h>1eD2 ¼ fW2: ð6Þ

5 We use typical annual values for l and r, consistent with empirical observationsfor equity indices.

h0; ~h1 are the decision variables (where � is used to indicatescenario-specific variables or values), p is the vector of asset pricesin t ¼ 0, and eDt denotes scenario-specific prices in t > 0, (e.g., thevalues in the boxes in Fig. 1).

Below we will also refer to the weights of the (two) assets in theportfolio w0 and ~w1.

In the special case of utility functions implying a myopic port-folio policy (e.g., log utility or power utility), state aggregation inlater stages has no effect on the optimal asset allocation in t ¼ 0as long as the two nodes at t ¼ 1 remain unaffected by the aggre-gation. For this reason, we chose quadratic utility in this numericalexample; i.e., E U fW2

� �h i¼ E 100fW2 � 2fW2

2

h i.6

Fig. 2 shows a modified tree where a state aggregation has beenperformed for the upper two end nodes, which have been mergedinto one. The consistency requirement is satisfied by setting the re-turn on the risky asset in this node equal to the risk-free rate. Thiscan be easily checked by pricing the stock using risk-neutral valu-ation, which shows that the state aggregation procedure workswell for pricing purposes. As already argued in Section 2, therisk-neutral measure must not be used for optimization. If we nev-ertheless solve problem (3)–(6), all wealth is allocated to the risk-free asset (for both the original and the aggregated tree). This is aconsequence of the risk-neutral measure which implies that theexpected return of all assets equals the risk-free rate, and thusthe risk-free asset (second order) dominates all other risky assetsfor risk-averse investors.

Now, to switch to the real-world probability measure, we keepthe stock prices in the original tree (Fig. 1) at their arbitrage-freeprices and change only the risk-neutral probabilities q to

p ¼ expðlÞ � du� d

¼ 0:6027: ð7Þ

Using this tree for portfolio optimization, we arrive at an initialasset allocation of w1

0 ¼ h10p1=h>0 p ¼ 92% to stocks and w2

0 ¼ 8% tothe risk-free asset (wi

t denotes the time t portfolio weight of asseti). In the modified tree from Fig. 2 (after a state aggregation hasbeen performed), the real-world probabilities p are substituted forthe risk-neutral probabilities q in the lower part of the tree. In theupper part, the single arc leading to an aggregated end node mustreceive a probability weight of 1, leaving no room for adjusting

6 We set initial wealth w0 ¼ 10 to make sure that terminal wealth lies in the regionof positive marginal utility. Thus, none of the results is due to implausibleimplications of quadratic utility.

8 Note the important differences between the nodes four, five and six in t ¼ 2: With

Table 1Numerical results for the portfolio optimization problem (3)–(6) based on scenarioreduction.

Bounds Asset weights for t ¼ 0

w �w w10 w2

0 w30

Unconstrained – – UnboundedNo short-sales 0 – 0.85 0.15 0Loose bounds 0.25 0.5 0.5 0.25 0.25Tight bounds 0.3 0.35 0.35 0.3 0.35

From top to bottom, we start by considering a problem without any constraints onthe asset allocation, then rule out short selling, and finally consider two sets of looseand tight bounds on the asset allocation.

612 A. Geyer et al. / European Journal of Operational Research 206 (2010) 609–613

the probability from q to p. This aggregated tree leads to a differentoptimal asset allocation, viz., w1

0 ¼ 98% stocks and w20 ¼ 2% risk-

free.This difference between the solutions from the original and the

aggregated problem (w vs. w) under the subjective measure iscaused by the reduction of the investment universe to a singleton.Using the same return for both assets effectively makes the assetsidentical and therefore redundant. As a consequence, the momentsof the conditional return distribution at the aggregated node (thehighlighted singleton in Fig. 2) no longer correspond to those ofthe real-world distribution: The (mean) return of the risky assetis equal to the risk-free rate, and the conditional distribution hasno volatility. All this leads to a bias in the portfolio weights ofthe ‘‘optimal” solution.

5.2. Scenario reduction

To illustrate potential problems with scenario reduction appliedto portfolio optimization problems, we replace the binomial modelof the previous section by a different setting. We consider threenormally distributed, uncorrelated securities with expected re-turns of 0.08 and standard deviation of 0.25 per stage, resp. Wesolve the problem (3)–(6) with the objective of maximizing powerutility of terminal wealth; i.e., E U fW2

� �h i¼ E fW1�c

2 =ð1� cÞh i

.Power utility has been chosen here for the convenience of a knownanalytical solution to this (multi-period) problem: The optimalweights are the same for each t and for each asseti : wi

t ¼ 1=3 ði ¼ 1;2;3Þ. For the special case of identical, uncorre-lated assets the same solution is obtained with quadratic (or anyother) utility as well. However, when we solve the problem numer-ically, using simulated returns which (may) violate no-arbitrageconditions, wealth levels will typically exceed the range of positivemarginal utility. Therefore, we use power utility with strictly posi-tive marginal utility. Since the focus of the present section is toillustrate the violation of no-arbitrage conditions, the myopicbehavior implied by power utility7 is no drawback.

We follow the approach by Høyland and Wallace (2001); Høy-land et al. (2003); Kaut (2003) to match the first four moments(including the correlations) of the simulated asset returns. This isachieved by an iterative procedure that combines simulation,Cholesky decomposition and various transformations to obtainthe correct correlations without changing the marginal moments.We use a constant branching factor of 6 to construct a total numberof 36 scenarios. We first match the moments of the three assets’ re-turns at t ¼ 1, and then – for each of the six nodes – those at t ¼ 2.To avoid arbitrage opportunities in the simulated returns (whichwould be exploited by the optimization algorithm) we apply theprocedure proposed by Klaassen (2002). We find that the analyticalsolution can be perfectly replicated. This agrees with the results inGeyer et al. (2009) which contains a more comprehensive analysisof the properties of moment matching for scenario generation inportfolio optimization.

To apply the scenario reduction algorithm based on Pflug (2001)we first simulate 5000 return paths for each asset and two periods.We match the first four moments and correlations (using Choleskyfactorization) with the specified properties. To make momentmatching and scenario reduction comparable in terms of the num-ber of scenarios, we apply Pflug’s algorithm to reduce the totalnumber of scenarios to 36 with six nodes at t ¼ 1. In our examplethis results in 6, 11, 9, 2, 4, and four successor nodes at t ¼ 2.

For one of the six nodes at t ¼ 1 (i.e., the fourth node with onlytwo successor nodes), arbitrage opportunities must arise. More

7 The choice of c is irrelevant for the analytical solution in this particular case ofthree identical assets. It becomes important, however, for the numerical results belowwhere we set c ¼ 2.

specifically, asset prices at node number four in t ¼ 1 (D1;4 denotesthe fourth column of eD1) are given by

D>1;4 ¼ 1:5427 1:4805 0:5303ð Þ:

Prices at the two successor nodes happen to be

eD2 ¼1:6105 2:75731:9171 1:15870:6340 0:3145

0B@

1CA:

Consider the following portfolio:

h>1 ¼ �0:0649 0:4258 �1:0000ð Þ:

For this portfolio, h>1 D1;4 ¼ 0 and h>1eD2 ¼ 0:0778 0:0000ð Þ, i.e., its

price at t ¼ 1 is 0, but its t ¼ 2 value is non-negative and strictlygreater than zero for one state – hence this portfolio is an arbitrage.

At the other nodes, arbitrage opportunities may also arise,depending on the returns in the respective successor nodes. Inour example, arbitrage happens to be present at node five with foursuccessors as well.8 Arbitrage opportunities at t ¼ 1 render the opti-mization problem unbounded, unless constraints on the asset alloca-tion are imposed. In the latter case, as already mentioned inSection 2, the problem is less obvious since a solution will be found,but this solution will be biased.

To illustrate this effect, we add constraints w 6 wt 6 �w to prob-lem (3)–(6) using the 36 scenarios obtained from the scenarioreduction algorithm. We restrict the asset allocation at each staget towards the analytical solution by introducing upper ð �wÞ andlower bounds ðwÞ. Adding short-sale constraints ðw ¼ 0Þ is suffi-cient to make the optimization problem bounded. However, theasset allocation is still far from the true optimum of wi

0 ¼ 1=3(see row ‘‘no short-sales” in Table 1). The addition of such con-straints only masks the problem arising from the presence of arbi-trage opportunities in the tree, but does not heal it. Tightening theupper and lower bounds succeeds in tilting the tree-based solutiontowards the correct optimum, but this can only be done effectivelywhen the solution is known ex ante, like in our simple example.Obviously, this is not a practicable alternative.

6. Conclusion

In multi-asset financial optimization models, small scenariotrees providing a good approximation to the true asset return dis-tributions are desirable. We have considered two approachesfound in the literature for this purpose, neither of which is foundto be suitable. Scenario reduction methods are conceptually

four successor nodes, there may exist returns for the following stage which excludearbitrage opportunities, as in node number six, though arbitrage may still arise for‘‘badly chosen” returns, as in node number five. With two successor nodes (as in thefourth node), however, arbitrage opportunities must exist for any set of returns, so itis impossible to exclude arbitrage for a tree containing such a node.

A. Geyer et al. / European Journal of Operational Research 206 (2010) 609–613 613

appealing and have been successfully applied in many areas. How-ever, when the tree is reduced too much (i.e., the branching factorof any source node falls below the number of non-redundant as-sets), no-arbitrage conditions are violated. State aggregation, onthe other hand, which has been explicitly designed to reduce treesize while maintaining absence of arbitrage, relies on the risk-neu-tral probability measure. Thus, it is only suitable for pricing pur-poses, but not for portfolio optimization. We conclude that aviable alternative is using the Høyland and Wallace (2001); Høy-land et al. (2003) algorithm to generate scenario trees with thesmallest possible branching factor that excludes arbitrage and al-lows for matching the first few (co-)moments under the real-worldmeasure. Any efforts to reduce the branching factor below thenumber of assets will lead to biased optimization results whenusing the real-world probability measure.

Acknowledgements

We acknowledge helpful comments by Georg Pflug, HermannElendner, and three anonymous referees. Financial support bythe Austrian National Bank (OENB-Jubilaeumsfondsprojekte No.11962 and 13054) and the Tyrolean Science Fund is gratefullyacknowledged.

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