Journal of Mathematical Finance, 2011, 1, 72-82
doi:10.4236/jmf.2011.13010 Published Online November 2011 (http://www.SciRP.org/journal/jmf)
On the Individual Expectations of Non-Average Investors
Lucia Del Chicca, Gerhard Larcher
Institute of Financial Mathematics, University of Linz, Altenberger Strasse Linz, Austria
Received August 25, 2011; revised October 1, 2011; accepted October 15, 2011
Abstract
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 set-
ting—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 Black-
Scholes 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 investiga-
tions initiated by an investment company carrying out certain subtle option trading strategies.
Keywords: Binomial Model, Black-Scholes Model, Options, Expectations, Mean Averting Strategies
1. Introduction
Often it is interesting for fonds-managers, asset managers,
or consultants to know which kind of investor is appro-
priate 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 amon-
gst 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 in-
vestors 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 in-
vestor 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 dif-
fering 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 com-
pared with a “non-average investor” who has the same
utility function U, but who follows investment strategies
differing from just buying the market portfolio. Espe-
cially 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 op-
tions 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 ex-
pectations.
It is the aim of this paper to fully discuss the above mod-
elling in a general binomial N-step model and subsequent
in a Black-Scholes model, and to give a complete answer to
the following questions:

L. DEL CHICCA, G. LARCHER 73
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 path-
independent case as in the path-dependent case?
arbitrary
2) Can we explicitly classify all the investment strate-
gies 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 ar-
bitrary 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 defini-
tion 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, re-
spectively for path-dependent contingent claims in full de-
tail. (Later, the general cases can be reduced to the 2-step
case to some extent.)
In Section 5 we provide the explicit computation tech-
nique 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 mean-
averting respectively mean-reverting investors in path-
independent 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 (path-
independent) 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 sum-
mary.
2. Lelands Approach. The Model
Leland [2] considers an average investor who has aver-
age expectations on market returns, average risk aversion
(i.e., a common average utility function) and therefore in-
vests 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
: thenumberofstepsN
: the initialvalue oftS
;
0he market portfolio
: the multiplicator for an up-move; u
;
:themultiplicator for a down-move,gidvenby= 1/
: the risk free interest rate in the model, assumed to be 0r
d u ;
These parameters determine the risk-neutral probabilities of
the model
d1
==
d1
r
u
e
Puu



and =.
1
d
u
Pu
These parameters are fixed for all the investors.
πu: is the probability of an up-move from the view of an
average investor (the consensus probability) and is time-
constant.
πd: the consensus probability of a down move
=1
du

 .
The risk aversion for the average investor is determined
by the utility function U. We assume that this average in-
vestor 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 aver-
age investor. From this assumption we can determine U as it
is done in [2] (see also [3]) and we obtain:

1
=1
x
Ux




with
log
=.
2log
u
d
u
u






Following Leland in [2], we are now interested in invest-
tors with the same risk aversion, i.e. with the same utility
function U, who, nevertheless, follow other trading strate-
gies than the average investor. Since, by assumption, these
investors also maximize their utility, they must have differ-
ent 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
iki
udSik kN



k
i



we will have one but sometimes even several such indi-
vidual 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 prob-
abilities. For each path


01 1
:=, ,,
k
Svv v
leading to we have exactly one probability
0
iki
udS






0110
;= ,,,;

k
pSkpv vvS k.
Here


,
i
vud stands for a vi-move in step 1i

. In
the path-independent case we just use the notation


0;
iki
pudS k
. 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-
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74 L. DEL CHICCA, G. LARCHER
pute these individual expectations from the individual trad-
ing strategies. Note that we carry out a certain “reverse en-
gineering”: 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 spe-
cific risk measure, and we determine the investor expecta-
tions.
In [3] Leland for example considers the following con-
crete example set of parameters
0
2
= 100;= 1.2;=,
3
u
Su 
and an European call-option buyer who is long 1.5 op-
tions 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 portfo-
lio is 100 like the initial value of the market portfolio. We
illustrate the situation and Leland’s results in Figures 1
and 2.
= 100K
The probabilities (path-independent) are the computed
individual probabilities for up-moves
e.g. .





000
,;2=, ;2=;2=0.643pudS pduS pudS
Figure 1. Market values in Leland’s example.
Figure 2. Implied Probabilities for call-option portfolio with
K = 100.
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 prob-
ability for an up-move in the step before. The analogous
property holds for down moves.
A further example in [2] with gives a simi-
lar 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 S0”) and two exam-
ples for path-dependent contingent claims in a 3-step model
(here he detected rather diffuse individual expectations).
= 110K
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 in-
vestor) 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






020 020
,,;1,,,;
kk
pv vSkpv vuSk


and








020 020
1,,;11,,,;
kk
pvvSkpvv dSk


for all


02
,,, ;=1,,1
k
vv udkN


.
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 strate-
gies resp. mean-reverting strategies.
For path-independent strategies the above properties
reduce to


11
00
;1 ;
ik iik i
pudS kpudS k
 


1


1
00
1;11
ik iiki
pudS kpudS k
 


;
for =0,, 1;=1,,1.ikkN


For our investigations we further need a suitable no-
tion for contingent claims (i.e. for trading strategies) in
our models. We denote trading strategies by






01
:=, ,;,
Ni
WWvvvud

where




01 1
,, ,
N
Wvv v

 denotes the payoff of the stra-
tegy if path


01
,,
N
vv

 happens. In the path-independent
case this reduces to
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L. DEL CHICCA, G. LARCHER 75




0
:=;= 0,,.
iNi
W WudSiN

We will also use the notation . We
restrict to “admissible” trading strategies, i.e. to strategies
with

0
:= iNi
i
WWudS




01 1
,, ,0
N
Wvv v
 always.
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.
3. Mean-Averting Investors in the Two-Step
Model: The Path-Independent Case
We start with our “reverse engineering” in the 2-step mo-
del by assuming an optimal strategy (portfolio) W, the av-
erage utility function U and by calculating from this the
investor expectations p. An arbitrary strategy in the 2-step
case is given by
 
= ,,,,,,,WWuuWudWduWdd.
The price at time zero of the strategy is
 


22
,, ,
uuddud
PW uuPPW udPPW duPW dd,.
Since we compare strategies W with the average strategy
of buying the market portfolio S we have the budget con-
straint








2
0=,,,, uuddud
SPWuuPPWudPPWdu PWdd
2
.
1
(1)
The trader following W is maximizing his utility
   
    
00
00
00
00
;0 ;1,
;0 1;1,
(1(;0))(;1)(( , ))
(1(;0))(1(;1))(( ,)).
pSpuSUWuu
pSpuS UWud
pSpdS UWdu
pSpdS UWdd


 
Hence by Lagrange we obtain the equations
   
    
 









2
00
00
00
2
00
,=;0;1
,= ;01;1
,=1 ;0;1
,=1;01 ;
u
ud
du
d
PU Wuup SpuS
PPUWudp SpuS
PPU Wdup Sp dS
PU Wddp Sp dS











(2)
where 1 =UU
 (for our special case

=UW W

). The
sum of the right hand sides is 1, so that


0
1
=EUWS 
 (3)
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


,= ,Wud Wdu

for simplicity we use the notation:






01 2
=,,=,=,,=,WWdd WWduWudWWuu
and




001 020
=;0,= ;1,= ;1ppSppuS ppdS

.
It is easily checked that (1),(2),(3) has a unique solution
p0, p1, p2, namely














20
1
2
10 10
2
20
2
1
21 10
21
0
2110
10
00
=
==
=
=
==
=
=
=
=
=
u
u
ud
u
u
ud
uud
uu ddu d
u
PEU WSudS
PW
pPWPWEUWSdS
PEUWSu S
PW
pPWPWEUWS uS
PPW PW
p
PPW PWPPW PW
PEU WSuS
EUWSS





 









 






(4)
(Si is the price of the market portfolio at time i).
In these relations W0, via the budget constraint (1), is
uniquely determined by W1 and W2. We only consider ad-
missible strategies, i.e. 012 So the definition
region for W is the triangle in Figure 3.
,, 0.WWW
Figure 3. Definition region for strategy W.
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L. DEL CHICCA, G. LARCHER
76
10
p
2
Strict mean-aversion now is determined by
and .
0
pp022 11ppp
Easy calculation based on (4) shows that the two con-
ditions are equivalent and reduce to the single vivid con-
dition
2
10
WWW (5)
(so in any case there are no investors who are neither mean-
averting nor mean-reverting).
It is worth noting that the condition is in no way de-
pendent on α or on the average expectations πu and πd. It
is dependent on Pu only via the dependence of W0 on W1,
W2 through the budget constraint: inserting
2
0
012
2
=2
uu
dd d
SPP
WWW
PP P




into (5) leads to

202
12
0=
u
d
WP SW
WW
P


 .
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 12
(See Figures 4 and 5). Buying the
market portfolio is mean-averting and mean-reverting,
hence it is located on
WW

. (Just insert
2
20
=,WuS
0
2
S
d
10 0
=, =WSW to check this once more).
Figure 4. Projection of 

to the W1, W2-plane.
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 Case
We just have to return to the system (2), now with


,Wdu and


,Wud not necessarily being equal. We
write


,du
1:=WW and , solve (2) and
obtain now

1:= ,WWud
















20
1
2
10 10
2
20
2
1
21 10
21
0
211 0
10
00
=
== =
=
== =
=
=
==
u
u
ud
u
u
ud
uu d
uu ddu d
u
PEU WSduS
PW
pPW PWEUWS dS
PEUWSuS
PW
pPW PWEUWS uS
PPW PW
pPPW PWPPW PW
PEU WSuS
EUWSS





 









 








(6)
The two mean-averting conditions
01 0
and 11ppp p
2

 (7)
now are not equivalent in general. Inserting (6) into (7)
leads to the following two mean-aversion conditions
2
112120duu d
PWPW WPWWPWW
2

  

 (8)
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L. DEL CHICCA, G. LARCHER 77
2
and 2
110010uddu
PWPW WPW WPW W

 

 (9)
For (i.e. path-independence) (8) and (9) are
equivalent, of course. If
1
 WW
1
11


WW
WW
then we easily check
that (8) implies (9). If then (9) implies (8).
11
Hence, an investor is strictly mean-averting if and only
if and (8) holds or and (9) holds.

1
WW
1 1
1
1
WW
Let () and () denote Equations (8) and (9) with
“greater-or-equal-sign”, i.e. (8) and () are the condi-
tions for strictly mean-reverting traders, then for
( 8) and () are equivalent. If
8

11
W
9

9 
 W9

1


WW

then
(9) implies (8) and if then (8) implies (9).
 
11
Hence, an investor is strictly mean-reverting if and
only if 11 and (9) does not hold or and
(8) does not hold.
W W
WW
1
WW
1
1
Finally, an investor is neither mean-averting nor mean-
reverting if and only if and (9) holds but (8) does
not hold or and (8) holds but (9) does not hold.
1
 WW
11
The conditions now (in the path-dependent case) de-
pend 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 W0 from the budget
constraint. The subsequent example should serve as an
illustration. Following Leland we choose the parameters
 WW
2
=1.2;=3
u
u, hence =2.4

and we set 1 = KW1 with K = 1 (path-independent
case, Figure 6), (Figure 7) and (Fig-
ure 8). We just show the W1, W2-plane (W3 again then is
uniquely determined by W1, W2 and K).

W
=0.8K=1.2K
5. Individual Investor Probabilities in a
Binomial N-Step Model
We now consider the N-step binomial model. First (like
Figure 6. Curve separating mean-averting (A) strategies
and mean-reverting (R) strategies for K = 1 (projection to
W1, W2-plane).
Figure 7. Curves separating mean-averting (A) strategies,
mean-reverting (R) strategies and other strategies (None)
for K = 0.8 (projection to W1, W2-plane).
Figure 8. Curves separating mean-averting (A) strategies,
mean-reverting (R) strategies and other strategies (None)
for K = 1.2 (projection to W1, W2-plane).
in Section 3) we again assume that a path-independent con-
tingent claim is the optimal choice for the investor. We
will give an explicit formula for the individual probabili-
ties 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


;
i
pSi are uniquely
determined by






1=
;=
ui
i
i
PEU WSuS
pSi EUWS





i
(10)
where 1 =UU

 (Note that this relation in fact holds for
any utility function U and not just for

1
=1
x
Ux




).
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).
=2N
pS
1kN
1i(;)
i
pS i
So it remains to prove the formula for . To
0
(;0)
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78 L. DEL CHICCA, G. LARCHER
this end we use the first of the Lagrange Equations (2) in
its N-step version










1
00 0
,, =;0;1;1
N
u
PUWuupSpuSpuS N 
N


with


0
1
=.
EUWS


Inserting for gives


1
00
;1 ,,;1
N
puSpuSN
 












2
20
0
010
0
1
10
=
,, =;0=
=
=










N
u
u
N
uN
N
N
PEUWSu S
PUWu upS
EUWS EUWS uS
PEU WSuS
EUWSu S
Since







0
== ,,


N
N
EUWS uSUWu u
, the re-
sult also follows for .

0;0pS 
We have stated and shown the result for the path-in-
dependent 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 adap-
tations). (Note that a path going through node N1(N2) (see
Figure 9) remains in R1(R2) so we again can use the in-
duction assumption).
Theorem 5.2. For a given path-dependent strategy W the
individual up- move probabilities in the N-step case


012 0
...; 1
k
pvvv Sk


are uniquely determined by








012 0
0120
10120
,, ,;1
=
=
=
k
ukk
kk
pvvv Sk
PEU WSvvvuS
EUWSvvvS








where


01 1
,, ,,
k
vv vud and (This again
holds for an arbitrary utility function U.)
=1,, .kN
6. The Martingale Property of the Individual
Expe
Figure 9. Regions R1, R2 where (10) holds by induction hy-
pothesis.
e-
cted Return of the Market Portfolio
land notes that the process of returns of the market portL
folio under the individual expectations is a martingale with
respect to the filtration of the binomial model, i.e. for
1
:=;= 0,,1


ii
i
SS
XiN
i
S
we have




=for
ji ii
EX SEXSij<. (E in this
es expectation with respect to th
es.) This fact easily can be obtaine
section denote individual
probabilitid for general
path-independent strategies from Theorem 5.1, respec-
tively from looking at the Lagrangean system (2) (in gen-
eral form).
The assertion (as also noted by Leland) is not at all
true for path-dependent strategies, as for example is im-
mediately seen from the geometric Asian future in Sec-
tion 10.
Theorem 6.1. In the path-independent case we have




=<.
ji ii
EX SEXSforij
Proof. Since the market portfolio has constant up-and-
down-move factors u, d it suffices to show that



Probup -move instep1=,
ii
jS pSi
(with respect to individual expectations). We show this
property for =1jN

and . The method easily =0i
extends to general i and j. We have


 
0
0
,, ,
02
030 20
Prob(up - moveinstep)NS
=
,,,,; 1
vv ud
N
NN
qv
qv vpvvSN







where






010
0
010
,,;if =
,, =
1,,;if
kk
k
kk
pvvSkv u
qv vpvv Skvd









 =
Using the Lagrangean Equations (2) the last sum equals








02
02
,, ,
,, ,
vvu N
N
vv ud
PPPWvvu





02
(*)
10
10
0
0
==
=
==
.
N
u
u
PE U WSuS
PEUWSuSp
EUWS











Note that the equality denoted by does not hold in
general for path-dependent strategies.
(*)
=
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L. DEL CHICCA, G. LARCHER 79
7.
e
Mean Averting Investors in a Binomial
N-Step Model: The Path Indpendent 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

=0, ,
=iiN WW (i
W is the strategy value at 0
iNi
udS
)

is strictly mean-averting if and only if
2
11iii
WWW


 )
for all =1,,1.
iN
(11
Wi is strictly mean-reverting if and only if
2WW
11ii
i
W


 (12)
fo
Remarks
we have connt individual probabilities if and
r all =1, ,1.iN
1) Sosta
only if

12
11
=
i
Wii
2) Trivially, for our special utility function U in-
equalitynt to 2
i
U

WW for all
(11) is equivale and (12) is
eq
from Theor
f both sides o11) and of (12)) the following
C
y mean-
=1, ,1.iN
11i i
UU


uivalent to 2
11
.
iii
UUU



We will refer to this later.
(

:=
kk
UUW
 )
We conclude em 7.1 (taking the logarithm
of (
orollary 7.2. If >0
i
W for all i, then the path- inde-
pendent strategy =0,, =( )
ii N
WW  is strictl
averting if and only if log,,log

0
N
WW is convex
and the path-indepe

=WW is nd
nd only if
ent strategy=0, ,
iiN
strictly mean-reverting if a

, log

0
log ,
N
WW
is concave.
Remarks
3) If 0foralland=0 for
i
WiW some k
k
version if and only if ,=0
i
iW
N that can be >0.
, then we
except for
W
for som
andnly if or
an
have mean-a
0 and W
4) If 0
i
W for all i and Wk = 0e k, then we
have mean-reversion if o1
d always when there are at least three successive

=0
k
W 1=0
k
W

12 1
,,>0,log,log,log
jj jjj
WW WWW
 is c
Proof of the Theorem: 7.1. Again we use induction on
is true for
2j
W
oncave.
N, and again we know that the assertion
=2 and assume that it is true for 1.KN Since
we have mean-aversion in regions R1 and R2 of Figure 9
if andly if (11) holds for 1,,
N
on
N
WW res
01
,,
N
WW

 we see that (11) is a necessary condition. Let
now (11) be satisfied. Furthell =0,,j
p. for
ar let for 3N


g condition hold true

PWPWPW PWPW
 

i.e. the condition (11) not only
the followin

1
(13)
holds for the terminal
wealth W at the time N but also for time
2
3212ujdjujdjujdj
PW

 

1N

for
values
the




1N
EUWS


 (see Figure 10).
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 R3 and hence e whole
strategy11) implies (13) an
ished. But this is just simple calculation (note that (11)
2
213
j
jj
WWW


 and
2
12123
j
jjj jjj
WWW WWWW

 
 ). Strictly mean-revert-
ing 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-in-
dependent case) who is neither mean-averting nor mean-
reverting 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 monoto-
nicity of expectations: just fix a certain point in time i.
Then calculate




i
EUW S
 for all possible values of i
S.
The resulting values (ordered from below) say 0,,
i
UU
determine in which nodes ,,NN (ordered from
01i
below) of step 1i

we h,
n (rever-
ave local mean-aversionor
local mean-reversion. We have mean-aversio
sion) in
001
,iff, ,NUUsatisfy the mean-aversion (reversion)
property in analogy to remark 2) in this section.
2
U
1
,i
i
N21 ff, ,
iii
UUU satisfy the mean-aversion (re-
version) property in analogy to remark 2) in this section
Figure 10. Region R3 where (11) holds by induction hy-
pothesis.
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80 L. DEL CHICCA, G. LARCHER
11
,0< <1,iff,,,

2
j
jjjj
Nji UUUU
satisfy the mean-
aversion (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).
8. Mean-Averting and Mean-Reverting
Investors in the Black-Scholes Model
Until now we have restricted ourselves to the binomial
N-step model. In this section we show how to extend the
results for the path-independent case of earlier sections to
the one-dimensional Black-Scholes model: here first we
need a suitable definition of a strictly mean-averting
(resp. mean-reverting) investor. Let χ be a simple con-
tingent claim with payoff-function Here we

=.
T
S
assume that  is left (or right) continuous and >0

.
Since geometric Brownian motion


d= dd0,
tt tt
SStSBtT  
can be approximated by a binomial model with parameters
=d
T
Nt

d
=t uN e

1d
=22
u
t 


(again we set =0r), implying for the utility exponent



logd logd
1
=,
22
tt
N
 

 

a given strategy

dt

 
:= i WNWN should-
=0, ,iN appro
ximate


T
S, i.e.



2
=iN
i
WNuN  
ial models with 0 NN
rem 7.1, is given if and onl



2
0=0,,
log iN
iN
uNS



0
S
larg
y
. It is tempt-
in all te enough.
This, if for all N
is convex.
Since
ing to define, that the investor is strictly mean-averting in
the Black-Scholes model, if he is strictly mean-averting
hese binom
by Theo
large enough


1 for N to infinity, and uNsince

is
one-sided continuous, this is


log x
satisfied if and only if the
function

:=
g
xe is convex on .
Conversely, a strategy, i.e. a contingent claim is strict-
ly mean-reverting if and only if g(x) is concave. Note,
that again the parameters

and

do not influenthe
mean-aversion property.
ep
ce
9. Example for the Path-Independent Case:
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
0
=
i
Wu S
2iN


0
log=2logloglinear in.
i
WiNuSi
Black-Scholes


=
x
x





= log=linear in.x
g
xexx
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:


2
0
=Max ,0
iN
i
WcuS K


The values logWi lie on a curve of the form in the Fig-
ur
So in any case we have neither convexity nor concavity if
there are at least three values for i with u2i–NS0 > K. If
there are two values for i with u2i–NS0 > K then acciden-
ta 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 paraeters but
then (see Figure 14) we do not have convexity
an
e 11.
lly we can have
=3N
m
=70K
y more.
Indeed the distribution of implied expectations in this
example is given as in Figure 15.
Figure 11. Logarithmic payoff of a call option portfolio.
C
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L. DEL CHICCA, G. LARCHER 81
Figure 12. Logarithmic convex payoff, Leland: Example 1,
K = 100.
Figure 13. Logarithmic convex payoff, Leland: Example 2,
K = 110.
Figure 14. Logarithmic non-convex payoff Leland’s exam-
ple with K = 70.
Figure 15. Implied expectations Leland’s example with K = 70.
Leland states that for larger N we recognize mean-
aversion (when buying a call option) for nodes near to
the initial wealth of the market portfolio. Our above dis-
cussion (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. 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
=10N
16 with a square in each node that is not mean-averting).
0. A Path-D1
01
,,

N vv
etric mean of
the payoff of this future is
given by the geom the path-values:
(we omit the substraction of S0 as it would be usual).

1/ 1
11
0011
N
NNN
N
Svvv



igure 16. Implied probabilities, Leland’s Example 1 with F
N = 10.
Figure 17. Mean-averting (•) and non-mean-averting (■)
nodes in Leland’s Example 1 with N = 10.
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L. DEL CHICCA, G. LARCHER
82
By Theorem 5.2 we now calculate the individual expectations










012 0
010
101 20
1
11
12
11
11
12 12
=
,,; ==
==
ukk
k
kk
Nk Nk
Nk NK
NN
uNk u
11
N
kNk
Nk NkNk Nk
NN
uNkd Nku
PEU WSvvvuS
pvv SkEUWSvvv S
PuE wwwPu
PuEw wwPdEw wwPuPd











 







NkNk
NN
d



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

,
i
wud.
independent of the path, and they even are independent
of the va

0=π



u
u
ud
pPuPd
by the definition of
Pu

and

1=
u
u
ud
P
pN P
PP
 .
In any case we do not have mean-aversion or mean-
reversion.
It is also easily checked (by using the definition of

g iff in Sectt pion 2) tha(k) is monotonically increasin
Based on the article [2] of Leland we have developed
some techniques to calculate and to interpret the indi-
vidual probabilities of an investor with average risk aver-
one investing strategies. It is
possible with the help of these techniques to completely
discuss path-independent strategies. Teby 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 fu-
e work should be possible: for example a classification
of mean-averting path-dependent strategies, the extension
to arbitrary discrete market models or to American op-
tions and the inclusion of transaction costs would be a
most interesting topic.
12. Acknowledgements
We thank Gunther Leobacher and Friedrich Pillichsham-
p-
entire time to maturity.
Based on the developed techniques a
tur
mer for technical support.
13. References
[1]
π
uu
and monotonically decreasing iff π
uu
P.
11. Conclusions
P
H. E. Leland, “Who Should Buy Portfolio Insurance?”
Journal of Finance, Vol. 35, No. 2, 1980, pp. 581-594.
doi:10.2307/2327419
[2] H. E. Leland, “Options and Expectations,” Institute of
Business and Economic Research, University of California,
Berkeley, 1996, pp. 43-51.
[3] M. Brennan, “The pr
sion but following n-averag
icing of Contingent Claims in Discrete
Time Models,” Journal of Finance, Vol. 34, No. 1, 1979, her
pp. 53-68. doi:10.2307/2 327143
rtly confirm assertions of Leland, partly we had to slightly