On the Individual Expectations of Non-Average Investors

An “average investor” is an investor who has “average risk aversion”, “average expectations” on the market returns and should invest in the “market portfolio” (this is, according to the Capital Asset Pricing Model, the best possible portfolio for such an investor). He is compared with a “non-average investor”. This—in our setting—is an investor who has the same “average risk aversion” but invests in other investment strategies, for example options. Such a “non-average investor” must consequently have expectations on the market return that are different from the average: the “non-average expectations”. In this paper we give an explicit formula for the “non-average expectations” in an arbitrary N-step model and for the extended concept in a BlackScholes model, in the path-independent case and in the path-dependent case. Further we explicitly classify all the investment strategies for which the resulting “non-average expectations” show this mean aversion property. Various examples are given in the paper. These investigations were part of more general investigations initiated by an investment company carrying out certain subtle option trading strategies.


Introduction
Often it is interesting for fonds-managers, asset managers, or consultants to know which kind of investor is appropriate to a certain strategy.So in this work we give an answer to the following question: "which sort of investor (differing from the average) is interested in trading a given (alternative) strategy?"This question occurs amongst others in the field of behavioral finance.It deals with the psychology of investors and the consequences of their expectations about the market which lead to investment decisions.We give an answer to this question with the help of certain mathematical model first introduced by H. Leland.
In two inspiring articles [1,2] Leland trys to identify the characteristics of investors who buy or sell European call options or other path-independent or path-dependent contingent claims.In both papers Leland considers investors who trade in those options just out of speculative reasons.In the first article he concentrates on investors who have the same return expectations as an average investor and he asks for their individual risk aversion.In the second article he considers investors with the same risk aversion as the average investor and he studies their differing expectations on the market return.
Our work will be based on this second article.In this article Leland considers an asset (market portfolio) in a binomial 3-step model and an "average investor" with given "average expectations" on the market returns, and with "average risk aversion", i.e. with a utility function U for which the above market portfolio maximizes the utility of the average investor.This "average investor" is compared with a "non-average investor" who has the same utility function U, but who follows investment strategies differing from just buying the market portfolio.Especially in [2] certain basic types of (path-dependent and of path-independent) option strategies in the binomial 3-step model are considered.As Leland points out the average investor will never purchase (or sell) fairly-priced options since options are in zero net supply.Thus investors holding options must differ from average, i.e., in our setting, their expectations on the market return must differ from the "average expectations".For more-step models Leland asserts that this mean-aversion can be found especially at nodes with stock-value close to the initial stock-value.For path-dependent (e.g.Asian or lookback call) contingent claim traders Leland detects "somewhat diffuse" return expectations.
It is the aim of this paper to fully discuss the above modelling in a general binomial N-step model and subsequent in a Black-Scholes model, and to give a complete answer to the following questions: 1) Can we give an explicit formula for the "non-average expectations" in an N-step model and extend the concept in a Black-Scholes model, as well in the pathindependent case as in the path-dependent case?arbitrary 2) Can we explicitly classify all the investment strategies for which the resulting "non-average expectations" show this mean-aversion property?
3) To what extent do the conclusions of Leland hold for N-step models, the Black-Scholes model and for arbitrary trading strategies?
The paper is organized as follows: In Section 2 we repeat Leland's setting and give all necessary definitions.Especially, we give an exact definition of a "strictly mean-averting trader".
In Section 3 and in Section 4 we discuss the binomial 2-step model for path-independent contingent claims, respectively for path-dependent contingent claims in full detail.(Later, the general cases can be reduced to the 2-step case to some extent.) In Section 5 we provide the explicit computation technique for the expectations of traders of path-independent as well as path-dependent contingent claims in the N-step model.
In Section 6 we show that these expectations imply a certain martingale property.
In Section 7 we explicitly characterize strictly meanaverting respectively mean-reverting investors in pathindependent contingent claims in a binomial N-step model.
In Section 8 we do the same in the continuous case (Black-Scholes model).
In Section 9 we consider as concrete example of (pathindependent) contingent claims, the case of call options.
Finally in Section 10 we consider a special example of path-dependent contingent claims.
Section 11 is devoted to conclusions and a final summary.

Lelands Approach. The Model
Leland [2] considers an average investor who has average expectations on market returns, average risk aversion (i.e., a common average utility function) and therefore invests in a market portfolio S.He assumes that S follows a binomial model.In our paper we will essentially also work in a binomial model as well.Later on, however, we will also consider the Black-Scholes model.
The parameters of the binomial model are given by : the number of steps N : the initial value of t S These parameters are fixed for all the investors.π u : is the probability of an up-move from the view of an average investor (the consensus probability) and is timeconstant.
π d : the consensus probability of a down move = 1 The risk aversion for the average investor is determined by the utility function U. We assume that this average investor shares the market expectations of the model for S, that he invests in S, and that he is a rational investor.So the above market portfolio S maximizes the utility of the average investor.From this assumption we can determine U as it is done in [2] (see also [3]) and we obtain: we are now interested in investtors with the same risk aversion, i.e. with the same utility function U, who, nevertheless, follow other trading strategies than the average investor.Since, by assumption, these investors also maximize their utility, they must have different expectations, i.e. different probabilities for up and down moves in the model.These individual probabilities will be in the center of our interest.
At every node 0 ; = 0, , ; = 0, , 1 we will have one but sometimes even several such individual probabilities for an up-move in the next time-step.This depends on whether the individual trading strategy is path-independent or path-dependent.In the latter case at every such node we have such individual probabilities.For each path leading to we have exactly one probability stands for a v i -move in step 1 i  .In the path-independent case we just use the notation Sometimes, if no confusion is possible, we use notations with reduced information for these individual probabilities for the sake of simplicity.It is the aim to com-pute these individual expectations from the individual trading strategies.Note that we carry out a certain "reverse engineering": in usual portfolio theory, once a choice of risk and investor measure are chosen, the optimal portfolio is derived.Here we assume an optimal portfolio under a specific risk measure, and we determine the investor expectations.
In [3] Leland for example considers the following concrete example set of parameters 0 2 = 100; = 1.2; = , 3  and an European call-option buyer who is long 1.5 options with strike and who holds an amount of 79.60 in cash.This particular choice is made because of certain norming reasons.The initial value of this portfolio is 100 like the initial value of the market portfolio.We illustrate the situation and Leland's results in Figures 1  and 2   Leland concludes that this investor is mean-averting in the sense that an up-move always implies a larger (or equal) probability of a further up-move than the probability for an up-move in the step before.The analogous property holds for down moves.
A further example in [2] with gives a similar result, i.e. again mean aversion of the investor.Leland moreover gives some informal remarks on the N-step case ("there seems to be mean aversion at nodes with stock values near to the initial stock value S 0 ") and two examples for path-dependent contingent claims in a 3-step model (here he detected rather diffuse individual expectations).

= 110 K
We felt that a more general discussion is necessary to obtain valid conclusions.So in the following we will try to explicitely determine all contingent claims (i.e.dynamic trading strategies) which can be considered by a strictly mean-averting, respectively by a strictly mean-reverting investor in a binomial N-step model and in the Black-Scholes model.Here we use the following definition of a strictly mean-averting investor (resp.mean-reverting investor) in a binomial model (the definition for the Black-Scholes model will be given later).
Definition 2.1.We call an investor strictly mean-averting if his trading strategy induces individual expectations with the following properties The investor will be called strictly mean-reverting if the "less or equal-sign" is replaced by the "larger or equal sign" in both inequalities.In the following we will call the corresponding strategies either mean-averting strategies resp.mean-reverting strategies.
For path-independent strategies the above properties reduce to    For our investigations we further need a suitable notion for contingent claims (i.e. for trading strategies) in our models.We denote trading strategies by denotes the payoff of the stra- happens.In the path-independent case this reduces to We will also use the notation .We restrict to "admissible" trading strategies, i.e. to strategies with In later sections we will proceed by induction, and it will turn out that much of the work is already contained in the full discussion of a 2-step model.This will be done in the next two sections.

Mean-Averting Investors in the Two-Step
Model: The Path-Independent Case We start with our "reverse engineering" in the 2-step model by assuming an optimal strategy (portfolio) W, the average utility function U and by calculating from this the investor expectations p.An arbitrary strategy in the 2-step case is given by The price at time zero of the strategy is Since we compare strategies W with the average strategy of buying the market portfolio S we have the budget constraint 1 (1) The trader following W is maximizing his utility (1 ( ;0)) ( ;1) ( ( , )) (1 ( ;0))(1 ( ;1)) ( ( , )).

p S p uS U W u u p S p uS U W u d p S p dS U W d u p S p dS U W d d
Hence by Lagrange we obtain the equations ).The sum of the right hand sides is 1, so that where E * denotes expectation with respect to the risk neutral measure.This of course easily generalizes in obvious form to higher step number.Since in this section we are interested in traders whose optimal contingent claim turns out to be path-independent, we have for simplicity we use the notation: It is easily checked that (1),( 2),(3) has a unique solution p 0 , p 1 , p 2 , namely (S i is the price of the market portfolio at time i).
In these relations W 0 , via the budget constraint (1), is uniquely determined by W 1 and W 2 .We only consider admissible strategies, i.e. 0 1 2 So the definition region for W is the triangle in Figure 3.This determines the region for strictly mean-averting investors A and for strictly mean-reverting investors R.
The boundary   belongs to both regions.g and the -axis are tangents to 1 W  , that is the projection of   onto the plane 1 2 (See Figures 4 and 5).Buying the market portfolio is mean-averting and mean-reverting, hence it is located on to check this once more).
The two mean-averting conditions now are not equivalent in general.Inserting ( 6) into (7) leads to the following two mean-aversion conditions For (i.e.path-independence) ( 8) and ( 9) are equivalent, of course.If then we easily check that (8) implies ( 9).If then ( 9) implies ( 8).
1 1 Hence, an investor is strictly mean-averting if and only if and (8) holds or and (9) holds.
Finally, an investor is neither mean-averting nor meanreverting if and only if and (9) holds but (8) does not hold or and (8) holds but (9) does not hold.
The conditions now (in the path-dependent case) depend on the model and (via α) on the expectations of the average investor.To obtain concrete explicit mean-averting strategies we still have to insert for W 0 from the budget constraint.The subsequent example should serve as an illustration.Following Leland we choose the parameters  and we set 1 = KW 1 with K = 1 (path-independent case, Figure 6), (Figure 7) and (Figure 8).We just show the W 1 , W 2 -plane (W 3 again then is uniquely determined by W 1 , W 2 and K).

Individual Investor Probabilities in a Binomial N-Step Model
We now consider the N-step binomial model.First (like   in Section 3) we again assume that a path-independent contingent claim is the optimal choice for the investor.We will give an explicit formula for the individual probabilities of an investor.As it is suggested from the results in the two-step case we will show the following Theorem 5.1.For a given path-independent strategy W the individual up-move probabilities   where 1 = U U   (Note that this relation in fact holds for any utility function U and not just for ).
Proof.We use induction on N. For we know that the result holds.Since we assume that the results hold for k-step models with , the formula (10) holds for all with (see Figure 9).
u S U W u u , the result also follows for .

 
0 ;0 p S We have stated and shown the result for the path-independent case first, because it is more intuitive.Following the proof we see that we can prove the path-dependent result in the same way (with the obvious notational adaptations).(Note that a path going through node N 1 (N 2 ) (see Figure 9) remains in R 1 (R 2 ) so we again can use the induction assumption).
Theorem 5.2.For a given path-dependent strategy W the individual up-move probabilities in the N-step case es expectation with respect to th es.)This fact easily can be obtaine section denot e individual probabiliti d for general path-independent strategies from Theorem 5.1, respectively from looking at the Lagrangean system (2) (in general form).
The assertion (as also noted by Leland) is not at all true for path-dependent strategies, as for example is immediately seen from the geometric Asian future in Section 10.Theorem 6.1.In the path-independent case we have Proof.Since the market portfolio has constant up-anddown-move factors u, d it suffices to show that (with respect to individual expectations).We show this property for = 1 j N  and .The method easily = 0 i extends to general i and j.We have Using the Lagrangean Equations ( 2) the last sum equals , , , Note that the equality denoted by does not hold in general for path-dependent strategies.

N-Step Model: The Path Ind pendent Case
Now we can explicitly classify the strictly mean-averting an-reverting strategies in the path-independent case.and me Theorem 7.1.The path-independent strategy is strictly mean-averting if and only if we have con nt individual probabilities if and r all = 1, , 1.
2) Trivially, for our special utility function U inequality nt to 2 i U  W W for all ( 11) is equivale and ( 12) is eq from Theor f both sides o 11) and of ( 12)) the following We will refer to this later. ( We conclude em 7.1 (taking the logarithm o f ( Proof of the Theorem: 7.1.Again we use induction on is true for N, and again we know that the assertion = 2 and assume that it is true for 1.

K N
  Since we have mean-aversion in regions R 1 and R 2 of for th .We show that ( d the proof is fin Then by Theorem 5.1 and by induction hypothesis we have mean-aversion also for R 3 and hence e whole strategy 11) implies (13) an ished.But this is just simple calculation (note that (11) ). Strictly mean-reverting strategies are treated quite analogously.be much averting or reverting strategies.
It seems to us to harder to explicitely classify the path-dependent strict mean-Let us consider now an investor (again in the path-independent case) who is neither mean-averting nor meanreverting and let us ask the question at which nodes of the corresponding model we have local mean-aversion or reversion.
By the above investigations (especially on the 2-step case) it is clear how to detect all strings of local monotonicity of expectations: just fix a certain point in time i.  ,0 < < 1, iff , , , N j i U U U U satisfy the meanaversion (reversion) property in analogy to remark 2) in this section.
We will illustrate this fact with an example in Section 9 (Figures 16 and 17).

Mean-Averting and Mean-Reverting
Investors in the Black-Scholes Model (again we set = 0 r ), implying for the utility exponent The Call Option Of course (from the definition, respectively from the model setting) investing in the market portfolio is as well a mean-averting as a mean-reverting strategy.Obviously as well in the binomial model as in the Black-Scholes model our conditions are satisfied: Binomial N-st Let us consider now buying call-options.To avoid un-fol gether with a positive (at least inimal) amount of cash c, i.e., a strictly positive trading interesting discussions of different cases we consider port ios of a call-option to m strategy.
The binomial N-step model: The values logW i lie on a curve of the form in the     Leland states that for larger N we recognize meanaversion (when buying a call option) for nodes near to the initial wealth of the market portfolio.Our above discussion (and the discussion at the end of Section 7) shows that this is the case, but not necessarily near to the initial wealth but rather near to the strike K and for nodes which lead only to end nodes out of the money (see the example in Figure 16 with the parameters of Leland's example 1 and with . Here you find the up-probability in each node.(we omit the substraction of S 0 as it would be usual).By Theorem 5.2 we now calculate the individual expectations where Hence the individual probabilities in this example are lue of the market portfolio, so we can write p(k) shorthand.For N large we have   In any case we do not have mean-aversion or meanreversion.
It is also easily checked (by using the definition of  g iff in Sect t p ion 2) tha (k) is monotonically increasin Based on the article [2] of Leland we have developed some techniques to calculate and to interpret the individual probabilities of an investor with average risk averon e investing strategies.It is possible with the help of these techniques to completely discuss path-independent strategies.T eby we could pa adapt these assertions.For example we can conclude that in general the strategy obtained by combining a call o tion and cash is not a mean-averting strategy during the lot of further fue work should be possible: for example a classification of mean-averting path-dependent strategies, the extension to arbitrary discrete market models or to American options and the inclusion of transaction costs would be a most interesting topic.
: the multiplicator for an up-move; u ; : the multiplicator for a down-move, gi d ven by = 1/ : the risk free interest rate in the model, assumed to be 0 r d u ;These parameters determine the risk-neutral probabilities of .
u u P PW u d P PW d u P W d d    2 .

Figure 3 .
Figure 3. Definition region for strategy W.
Easy calculation based on (4) shows that the two conditions are equivalent and reduce to the single vivid conany case there are no investors who are neither meanaverting nor mean-reverting).It is worth noting that the condition is in no way dependent on α or on the average expectations π u and π d .It is dependent on P u only via the dependence of W 0 on W 1 , W 2 through the budget constraint:

Figure
Figure 5. Curve   separating mean-averting (A) and mean-reverting (R) strategies W.4. Mean-Averting Investors in the Two-Step Model: The Path-Dependent CaseWe just have to return to the system (2), now with   , W d u and
So it remains to prove the formula for .To this end we use the first of the Lagrange Equations (2) in its N-step version
all i and W k = 0 e k, then we have mean-reversion if o 1 d always when there are at least three successive satisfy the mean-aversion (reversion) property in analogy to remark 2) in this section
, that the investor is strictly mean-averting in the Black-Scholes model, if he is strictly mean- .Conversely, a strategy, i.e. a contingent claim is strictly mean-reverting if and only if g(x) is concave.Note, that again the parameters  and  do not influen the mean-aversion property.ep ce 9. Example for the Path-Independent Case: Figur So in any case we have neither convexity nor concavity if there are at least three values for i with u 2i-N S 0 > K.If there are two values for i with u 2i-N S 0 > K then accidenta convexity.This indeed was the case in the two examples used by Leland with (see Fig- ures 12 and 13).If we however choose exactly Leland's para eters but then (see Figure14) we do not have convexity an Indeed the distribution of implied expectations in this example is given as in Figure15.

Figure 11 .
Figure 11.Logarithmic payoff of a call option portfolio.
Figure 17 shows the same situation as in Figure ependent Example: Geometric Asian Future In [2] Leland considers arithmetic Asian futures in a 3-step model.To illustrate the result of Theorem 5.2 we have calculated the implied probabilities for a geometric Asian future.For a path = 10 N 16 with a square in each node that is not mean-averting).thepayoff of this future is given by the geom the path-values: