Exponential forward indi¤erence prices in
incomplete binomial models
M. Musiela, E. Sokolovayand T. Zariphopoulouz
November 18, 2015
Abstract
In this paper we initiate a study of indi¤erence prices under forward investment performance criteria in incomplete binomial models. The model
is non-reduced in that the nested market model is not complete. We focus
on time-monotone forward criteria and construct the associated prices. A
comparative study between the forward indi¤erence price with its traditional (backward) counterpart is carried out. Among others, we discuss
the pricing measures, compare the two prices as dynamic analogues of the
static certainty equivalent pricing rule and examine their dependence on
the trading horizon.
1
Introduction
Indi¤erence valuation o¤ers a way to price claims when perfect replication is not
possible. It produces the price through the optimal behavior of the writer (or
buyer) with and without the claim in consideration. There are two underlying
ingredients in this approach, namely, the trading horizon, [0; T ] ; during which
investment and pricing take place, and the individual’s utility, uT ; assigned at T .
A direct consequence of this construction is that once T and uT are chosen, the
writer (or buyer) cannot modify his risk preferences, for the latter are uniquely
identi…ed with the indirect utility (value function), Vt;T ; t 2 [0; T ] ; which is
consistent with the terminal datum VT;T = uT : In addition, because the value
function is essentially constructed backwards in time, as the Dynamic Programming Principle yields, modi…cation of upcoming market dynamics, learning, etc.
cannot be incorporated. Finally, the a priori horizon choice [0; T ] restricts the
applicability of the valuation method to the pricing of claims that mature on or
before T: As a result, the traditional indi¤erence valuation cannot accommodate
the pricing of claims that arise after the initial time and mature beyond T , or
of claims associated with rolling horizons, etc.
Oxford-Man Institute, University of Oxford; marek.musiela@oxford-man.ox.ac.uk.
SUEZ, Houston; katusha_s@yahoo.com.
z Departments of Mathematics and IROM, The University of Texas at Austin and the
Oxford-Man Institute, University of Oxford; zariphop@math.utexas.edu.
y GDF
1
To alleviate some of these shortcomings, two of the authors (see, among others, [10] and [13]) proposed an alternative approach in which risk preferences are
not chosen for a single investment horizon but, rather, can be modi…ed dynamically at all times. The associated criterion, the so-called forward investment
performance process is, then, de…ned for t 2 [0; 1) and provides an extension
of the traditional value function. The de…nition of this process is given in the
next section and we refer the reader to [13] for a detailed discussion.
The goal herein is to provide a complete study of the indi¤erence valuation
of claims under this new class of preferences in incomplete binomial models. We
are interested in the de…nition, construction and analysis of prices under forward performance criteria, as well as in their comparison with their traditional
counterparts.
As a …rst step, we concentrate on the family of time-monotone exponential
forward preferences. We choose to do this for a variety of reasons. Firstly,
exponential preferences have been primarily used in the existing indi¤erence
valuation literature. Secondly, albeit their simplicity, time-monotone forward
performances o¤er many valuable insights, as the analysis will show. In addition,
their structural properties facilitate the construction of the pricing algorithm.
For these reasons, we are able, not only, to explicitly compute the exponential
forward indi¤erence prices but, also, to make detailed and direct comparisons
with their classical analogues.
The paper contributes in two directions. On one hand, we work with a new
class of risk preferences and, on the other, we consider a more general binomial
model. Indeed, our model is more general than the binomial models studied, so
far, in the traditional exponential indi¤erence valuation literature because the
non-traded stochastic factor may a¤ect not only the claim’s payo¤ (as it is the
case, among others, in [1], [8], [22] and [23]) but, also, the transition probability
and the values of the traded stock. This extension is crucial in incorporating
models with stochastic investment opportunity sets.
In order to provide a meaningful comparison between forward and traditional
prices, we need to have for the latter class the analogous results for the extended
binomial model considered herein. This study was carried out by the authors in
[16], and for the reader’s convenience, we provide a summary of its main results
in section 4.1.
The …rst contribution herein is the construction of a pricing algorithm for
forward indi¤erence prices. The pricing scheme resembles the one found in [8]
(see, also, [22] and [23]) in that it is iterative and produces the price in two
sub-steps, locally in time. In the …rst sub-step, the forward price is calculated
in a non-linear way which re‡ects how the forward risk preferences distort the
iteratively produced intermediate payo¤. In the second sub-step, the remaining
risks are replicable and are, therefore, priced linearly.
Albeit this structural algorithmic similarity, there are fundamental di¤erences between the forward and traditional prices, which we study in detail.
This comparative study is our second contribution. Among others, we analyze
the representation of prices as dynamic certainty equivalents, compare the pricing measures and discuss the e¤ects of the horizon choice and the maturity of
2
the claim. A brief summary of our …ndings is given next.
The …rst, and in many ways quite striking, di¤erence between the two prices
is their connection with the static certainty equivalent. Speci…cally, we …nd that
the forward price yields the direct dynamic analogue of this static valuation rule.
In contrast, we show that such an intuitively pleasing representation fails for
the traditional price.
Another important di¤erence is that the pricing measures are not the same.
Indeed, when valuation is done under forward preferences, the emerging pricing
measure is the one that preserves the conditional distribution of unhedgeable
risks, given the hedgeable ones, in relation to its historical counterpart (see
(19)). On the other hand, in the classical setting, the relevant measure is the
minimal entropy one.
Forward and classical indi¤erence prices have very di¤erent behavior in terms
of the trading horizon and the maturity of the claim. In the traditional case,
the prices are a¤ected considerably by the horizon choice. Indirectly, they are
also a¤ected by the claim’s maturity, for the latter is constrained not to be
bigger than the horizon. However, such dependences dissipate in the forward
case. Indeed, because the forward preferences are de…ned for all times, there
is no constraint from a speci…c choice of a trading horizon. Moreover, neither
the pricing functionals or the associated pricing measure are a¤ected by the
trading horizon and the claim’s maturity. We note that this independence does
not imply that the price of a speci…c claim does not depend on its maturity,
an obviously wrong conclusion. Rather, it says that the exponential forward
pricing operator per se is independent on both the claim’s maturity and the
trading horizon, very much like the complete market pricing operator is.
We conclude with the comparison of forward and classical indi¤erence prices
in incomplete binomial models when market incompleteness comes only from
the presence of the non-traded factor in the claim’s payo¤. Such models are frequently called reduced or almost complete. In such settings, the nested model is
complete. We establish that, while the forward performance and the traditional
value function processes remain - by their de…nition - distinct, the associated
prices coincide. To show this, it …rst requires a rather detailed study of how internal market incompleteness is compiled and priced under the two kinds of risk
preferences. When the model becomes reduced, this component of incompleteness vanishes. We analyze how the remaining incompleteness, coming only from
the claim, is processed by the forward and traditional performance processes
and why the corresponding prices turn out to be identical.
The paper is organized as follows. In section 2, we introduce the model, its
forward investment performance process and the special case of time-monotone
exponential type, and the upcoming pricing measure. In section 3, we introduce the forward indi¤erence price, construct and study the associated pricing
algorithm. In section 4, we provide a summary of analogous results for the
traditional indi¤erence prices. In section 5, we carry out the comparative study
between the two prices and in section 6, we analyze the class of reduced binomial
models.
3
2
The model and its forward investment performance process
We start with the probabilistic setup of the incomplete multi-period binomial
model. There are two traded assets, a riskless bond and a stock. The bond is
assumed to o¤er zero interest rate. The values of the stock are denoted by St ;
t = 0; 1; ::: We assume St > 0 and de…ne the random variables
t+1
=
St+1
;
St
t+1
=
d
t+1 ;
u
t+1
with 0 <
d
t+1
<1<
u
t+1 :
(1)
Incompleteness comes from a non-traded stochastic factor. Its levels, denoted by Yt ; t = 0; 1; :::; satisfy Yt 6= 0. We introduce the variables
t+1
=
Yt+1
;
Yt
t+1
=
d
t+1 ;
u
t+1
with 0 <
d
t+1
<
u
t+1 :
(2)
We then view f(St ; Yt ) : t = 0; 1; :::g as a two-dimensional stochastic process
de…ned on the probability space ( ; F; (Ft ) ; P). The …ltration Ft is generated
by Si and Yi , or, equivalently, by i and i , for i = 0; 1; :::; t. We also consider the
…ltration FtS generated only by Si ; i = 0; 1; :::t. The real (historical) probability
measure on and F is denoted by P.
We assume that, for t = 0; 1; :::; the values dt+1 ; ut+1 ; dt+1 ; ut+1 of the
Ft+1 measurable random variables t+1 and t+1 satisfy
u
d
d
u
t+1 ; t+1 ; t+1 ; t+1
2 Ft :
(3)
An investor starts at s = 0; 1; :::; with initial endowment Xs = x; x 2 R; and
trades between the stock and the bond, following self-…nancing strategies. The
number of shares of stock held in his portfolio over the time interval [t 1; t);
t = s + 1; s + 2; :::; is denoted by t : It is throughout assumed that t 2 Ft 1 :
We denote the set of admissible policies, t ; t = 1; ::: by A. The investor’s
wealth is, then, given, for t = s + 1; s + 2; :::, by
Xt = x +
t
X
i=s+1
i
4 Si ;
(4)
where 4Si = Si Si 1 represents the stock price increment.
The performance of the various investment strategies is measured via a stochastic criterion, to so-called forward investment performance process, which
measures the output of any admissible portfolio and gives us a selection criterion. Speci…cally, a strategy is deemed optimal if it generates a wealth process
whose average performance is maintained over time. In other words, the average
performance of this strategy, at any future date, conditional on today’s information preserves the performance of this strategy up until today. Any strategy
that fails to maintain the average performance over time is, then, sub-optimal.
The aim, herein, is to construct such a performance criterion, analyze the
associated indi¤erence valuation problem and compare the emerging forward
4
indi¤erence prices with the ones generated by the classical expected utility approach.
We recall the de…nition of the forward investment performance process. We
denote the solution of (4) by Xt in a self-evident manner.
De…nition 1 An Ft adapted process Ut (x) is a forward investment performance process if:
i) for each t 0 and x 2 R; the mapping x ! Ut (x) is strictly increasing
and strictly concave,
ii) for each 2 A and t = 0; 1; :::;
Ut (Xt )
iii) there exists
and X
s = t; t + 1; :::
(5)
2 A for which
Ut Xt
with X
EP (Us (Xs ) jFt ) ;
= EP Us Xs
jFt ;
s = t; t + 1; :::;
(6)
given in (4).
Herein we focus on exponential forward performance processes which satisfy
the initial condition
U0 (x) =
e
x
; x 2 R and
> 0;
(7)
The concept of forward performance process was introduced by the authors
in [10]. Subsequently, the exponential case was analyzed in a multi-asset model
for prices following Ito-di¤usion processes in [12]. For a discussion on this concept, we refer the reader, among others, to [13], [15], [3] and [18] and references
therein.
Characterizing the entire family of forward performance processes remains an
open question and is being currently investigated by the authors and others. In
the case of Itô-di¤usion markets, two of the authors have proposed a stochastic
partial di¤erential equation that Ut (x) is expected to satisfy (see [14]). The
novel element therein is the forward performance volatility process, which is
an investor-speci…c input. Special classes of volatilities have proposed in [13]
which can be (re)interpreted as zero-volatility cases for alternative market setups under a di¤erent numeraire and/or di¤erent market views. For a complete
study of the zero-volatility case see [15]. More recent works include [19], [17]
and [21].
2.1
Time-monotone forward investment performance
We start with some auxiliary notions. Consider an arbitrary time interval, say
[0; t] ; and the set Qt of equivalent martingale measures de…ned on Ft . With a
slight abuse of notation, we denote the generic element of Qt by Q.
For s = 0; 1; :::; t; we de…ne the sets
As = f! :
s
(!) =
u
sg
and
5
Bs = f! :
s
(!) =
u
sg;
(8)
with s and s as in (1) and (2).
It is throughout assumed that 0 < P ( As+1 j Fs ) < 1; s = 0; 1; :::; t:
De…nition 2 Let t > 0: For s = 1; :::; t; let
risk neutral probabilities
d
1
s
qs = u
:
d
s
s
be as in (1) and consider the
(9)
s
The process hs ; s = 1; :::; t; is de…ned by
hs = qs ln
qs
P (As jFs
1)
+ (1
qs ) ln
1 qs
P (As jFs
1
1)
;
(10)
where the sets As are given in (8), P is the historical probability measure and
Fs is the …ltration generated by the random variables Si and Yi , i = 0; 1; :::; s.
Lemma 3 Let t > 0: Then, hs 2 Fs
hs = Q (As jFs
1 ) ln
Q (As jFs
P (As jFs
1)
1)
1;
s = 1; 2; ::; t, and, for all Q 2 Qt ;
+ (1
Q (As jFs
1 )) ln
1
1
Q (As jFs
P (As jFs
1)
1)
:
(11)
We are ready to present one of the main results.
Theorem 4 Let h as in (10). Then, for x 2 R; the process
8
Pt
< e x+ i=1 hi ; t = 1; :::;
Ut (x) =
:
e x;
t = 0;
(12)
is a time-monotone exponential forward investment performance.
Proof. We …rst show that the requirements in De…nition 1 are satis…ed. Lemma
3 yields that Ut (x) 2 Ft 1 : Equality (7) follows trivially. Next, we establish (5)
and (6). To this end, observe that in an arbitrary horizon, say [s; t) ; we have,
for initial endowment Xs = x and Xt given in (4),
sup
s+1 ;:::;
=
sup
s+1 ;:::;
EP e
t
(Xs +
t 1
i=s+1
1
t
i
EP (Ut (Xt ) jFs )
P
Si )+ ti=11 hi
sup EP
e
t
t
St +ht
jFt
1
Fs :
On the other hand, direct calculations show (see, for example, [2], [9]) that if
t > 0 and h as in (10), then, for s = 1; :::; t;
sup EP
e
s
Ss +hs
s
6
jFs
1
=
1;
(13)
with the maximum occurring at
s
1
=
Ss
1
u
s
d
s
ln
u
s
(
1) (P (As Bs jFs
d
s
1
1)
(P (Acs Bs jFs
+ P (As Bsc jFs
1)
1 ))
+ P (Acs Bsc jFs
: (14)
1 ))
We then deduce that
sup
s+1 ;:::;
t
EP (Ut (Xt ) jFs ) =
sup
s+1 ;:::;
EP
t
e
(Xs +
t 1
i=s+1
i
1
P
Si )+ ti=11 hi
Fs :
Proceeding iteratively, we obtain
sup
s+1 ;:::;
t
EP (Ut (Xt ) jFs ) = Us (Xs )
and we easily conclude. The time monotonicity follows easily (12) and that
hs > 0:
2.2
The pricing measure under forward preferences
We introduce a martingale measure that will be used in the construction of the
forward exponential indi¤erence prices. To this end, let T > 0 arbitrary and
Q 2 QT ; where QT is the set of martingale measures de…ned on FT :
For t = 0; 1; :::; T; consider the quantity
T
X
EP
ln
s=t+1
Q ( s;
P ( s;
jFs
jF
s
s
s
1)
1)
jFt
:
(15)
Each term can be written as
EP ln
Q ( s;
P ( s;
jFs
jF
s
s
s
1)
1)
jFt
= EP E P
ln
Q ( s;
P ( s;
jFs
jF
s
s
1)
s
1)
Fs
1
jFt
:
(16)
Moreover, with As ; Bs as in (8), we have
EP
=
= EP
ln
+EP
ln
ln
+ ln
Q (As Bs jFs
P (As Bs jFs
Q (Acs Bs jFs
P (Acs Bs jFs
Q (As Bs jFs
P (As Bs jFs
Q (Acs Bs jFs
P (Acs Bs jFs
1)
1)
1)
1)
ln
Q ( s;
P ( s;
1)
1)
1)
1)
jFs
s jFs
s
1As Bs + ln
1Acs Bs + ln
1)
1)
1
Q (As Bsc jFs
P (As Bsc jFs
Q (Acs Bsc jFs
P (Acs Bsc jFs
P (As Bs jFs
1)+
ln
P (Acs Bs jFs
1)+
ln
7
Fs
1)
1)
1)
1)
Q (As Bsc jFs
P (As Bsc jFs
Q (Acs Bsc jFs
P (Acs Bsc jFs
1As Bsc Fs
1
1Acs Bsc Fs
1
1)
1)
1)
1)
P (As Bsc jFs
1)
P (Acs Bsc jFs
1):
Observing that for ; > 0; arg maxx
ln x + ln q
that the above quantity is maximized for Q satisfying
Q ( As Bs j Fs
1)
=
P ( As Bs j Fs 1 )
qs , Q ( As Bsc j Fs
P ( As j Fs 1 )
Q ( Acs Bs j Fs
1)
=
P ( Acs Bs j Fs 1 )
q~s ; Q ( Acs Bsc j Fs
P ( Acs j Fs 1 )
with q~s = 1
x
=
+
q; we deduce
P ( As Bsc j Fs 1 )
qs
P ( As j Fs 1 )
(17)
P ( Acs Bsc j Fs 1 )
q~s ;
1) =
P ( Acs j Fs 1 )
(18)
1)
=
qs :
Using the above and that Q ( 1 ; :::;
T ; 1 ; :::; T )
=
T
Y
Q ( t;
t=1
t j Ft 1 ) ;
we
deduce that the martingale measure Q can be de…ned for any T > 0 and that
it minimizes (15) for each t = 0; 1; ::; T:
We can then use (17) and (18) to de…ne Q on F, for t = 0; 1; :::; or, equivalently,
Q Yt j Ft 1 _ FtS = P Yt j Ft 1 _ FtS :
(19)
3
Forward indi¤erence valuation
In this section, we recall the notion of forward indi¤erence price and provide an
iterative algorithm for its construction. Forward indi¤erence prices of European
claims were …rst introduced in [10]. Later, they were studied in di¤usion models
with stochastic volatility ([11]) and in the context of entropic risk measures in
[25]. More recently, forward indi¤erence prices of American-type claims were
examined in [17].
We consider a generic claim, written on both the traded stock and the nontraded factor, say at time t0 : For simplicity, we assume throughout that t0 = 0:
The claim matures at t > 0 yielding payo¤ Ct ; represented as an Ft -measurable
random variable. We de…ne and compute the claim’s forward indi¤erence price
at intermediate times s = 0; 1; :::; t; in reference to the exponential forward
criterion (12). For convenience, we eliminate the "exponential" terminology.
De…nition 5 Consider a claim, written at time t0 = 0 and yielding at t > 0
payo¤ Ct 2 Ft : Let Us ; s = 0; 1; :::; t; be the forward performance process given
in (12). For s = 0; 1; :::; t; the claim’s forward indi¤ erence price is de…ned as
the amount s (Ct ) 2 Fs for which
Us (x
s (Ct ))
=
sup
s+1 ;:::;
EP ( Ut (Xt
t
Ct )j Fs ) ;
for all initial wealth levels Xs = x; x 2 R and Xt given in (4) with
i = s + 1; :::; t:
(20)
i
2 A;
We note that the alignment of the expiry of the claim with the time at which
the forward process is calculated in the right hand side of the pricing condition
8
(20) is chosen for mere convenience. Indeed, the above de…nition can be directly
extended to times beyond the claim’s maturity, in that (20) can be replaced by
Us (x
s (Ct ))
=
sup
EP ( Ut0 (Xt0
s+1 ;:::;
t0
Ct )j Fs ) ;
t0
t + 1;
(21)
as it follows easily from the fact that Ct 2 Ft0 and the de…nition of the process
Ut ; for t = 0; 1; :::. Speci…cally,
sup
s+1 ;:::;
=
sup
s+1 ;:::;
EP e
(Xt Ct )+
EP (Ut0 (Xt0
t
i=s+1 hi
sup
s+1 ;:::;
with Xt = x +
tion of (13).
sup
t+1 ;:::;
t
=
t
i=s+1 i
Ct ) jFs )
t0
EP
e
EP
t0
i=t+1
i
t0
i=t+1 hi
Si +
t0
(Xt Ct )+
t
e
t
i=s+1 hi
Ft Fs
Fs ;
Si : The assertion then follows from iterative applica-
What the above tells us is that, because the forward investment process is
de…ned at all times t = 0; 1; :::; we have the ‡exibility to price, under the forward
criterion (12), any collection of claims of arbitrary maturities, say t1 ; t2 ; :::; tn ;
by applying the pricing condition (20) at the longest such maturity, tn . It is
worth noting that this cannot be done in the traditional indi¤erence framework.
Indeed, once the utility is pre-chosen for T; no claim with maturity beyond T
can be priced.
3.1
The forward indi¤erence pricing algorithm
(s;s+1)
The two fundamental pricing blocks are the non-linear functionals EQ
0
(s;s )
EQ ;
and
introduced below. In their form, both the forward process Ut ; t = 0; 1; :::;
( 1)
and its spatial inverse appear. The latter, denoted by Ut
is given by
8
t
P
>
1
>
ln ( x) + 1
hi ; t = 1; 2; :::
<
( 1)
i=1
Ut
(x) =
>
>
:
1
ln ( x) ; t = 0;
(22)
for x 2 R and h as in (10). We recall that Ft and FtS are the …ltrations
generated, respectively, by the random variables (Si ; Yi ) and Si , for i = 0; 1; :::; t:
Notice that the involved measure is Q , and not the minimal entropy measure1 which appears in the traditional exponential indi¤erence prices.
1 For similar observations in forward indi¤erence valuation in a di¤usion model with stochastic volatility, see [11].
9
De…nition 6 Let Z be a random variable on ( ; F, P) and Q as in (19)). Let
( 1)
Us (x) and Us
(x) ; s = 0; 1; :::; t; be the forward performance process and its
spatial inverse (cf. (12) and (22)).
We de…ne, for 0 s s0 t;
i) the single-step forward price functional
(s;s+1)
EQ
(Z) =
( 1)
EQ
Us+1
S
Us+1 ( Z) Fs _ Fs+1
EQ
Fs
(23)
and
ii) the multi-step forward price functional
(s;s0 )
EQ
(s;s+1)
(Z) = EQ
(s+1;s+2)
(EQ
(s0 1;s0 )
(:::EQ
(Z))):
(24)
We caution the reader that, in general, for s0 > s + 1;
(s;s0 )
EQ
(Z) 6=
Us0 1 EQ
EQ
Us0 ( Z) Fs _ FsS0
jFs :
(25)
The next result shows a remarkable simpli…cation of (23).
Proposition 7 Let t = 1; :::; and Z be a random variable in ( ; F; P). Let Q
(s;s+1)
and EQ
be, respectively, as in (19) and (23)). Then, for s = 0; 1; :::; t 1,
(s;s+1)
EQ
1
(Z) = EQ
ln EQ
e
Z
S
Fs _ Fs+1
jFs
:
(26)
Proof. Using (12) and Lemma 3, we have
EQ
S
Us+1 ( Z) Fs _ Fs+1
= EQ
=
s+1
i=1 hi
e
EQ
e
Therefore, (22) implies
( 1)
Us+1
=
1
ln e
EQ
s+1
i=1 hi
=
1
Z
e
Z+
s+1
i=1 hi
S
Fs _ Fs+1
S
Fs _ Fs+1
:
S
Us+1 ( Z) Fs _ Fs+1
EQ
ln EQ
e
e
Z
Z
S
Fs _ Fs+1
+
s+1
1X
hi
i=1
S
Fs _ Fs+1
;
and the assertion follows.
The reader familiar with existing indi¤erence pricing algorithms (see, among
others, [1], [8], [22], [23]), might …nd the form of (26) identical to the one
appearing in these references. This is not, however, the case. The results herein
are, not only, derived for entirely di¤erent risk preference criteria but, also,
for more general incomplete market environments. We will further explore this
issue in sections 4 and 5.
Next, we present an auxiliary result that will be used in the construction of
the forward pricing algorithm.
10
(s;s+1)
Lemma 8 Let t > 0; s = 0; 1; :::; t and EQ
sup EP
(Xs Z)
e
s
jFs
=
1
exp
be as in (23). For Z 2 ( ; F; P) ;
Xs
(s 1;s)
EQ
1
(Z)
hs ; (27)
with hs as in (10).
Proof. Let As ; s = 0; 1; :::; t; be given in (8). Then,
sup EP
(Xs Z)
e
s
=
Xs
e
+(1
1
P(As jFs
P(As jFs
1 )e
s Ss
1 ))e
jFs
s Ss
u
1( s
d
1( s
1)
1)
(28)
1
Z
EP e
EP e
Z
jFs
jFs
1
1
_ As
_ Acs
:
Di¤erentiating with respect to s yields that the optimum occurs at
0
1
EP e Z jFs 1 _ As P(As jFs 1 ) ( us 1)
1
A:
ln @
s =
d
d
Z jF
c ) (1
Ss 1 us
E
(e
_
A
P(A
jF
))
1
P
s 1
s s 1
s
s
s
From (19) we obtain
sup EP
e
(Xs Z)
s
=
exp
Xs
+ (1
Q (As jFs
P(As jFs 1 )
Q (As jFs 1 )
1
+ Q (As jFs
(s 1;s)
and using the de…nition of EQ
1)
1
1 ) ln EP
1 )) ln EP
Q (As jFs
jFs
e
Z
jFs
e
Z
1
_ Acs
jFs
1 P(As jFs 1 )
1 Q (As jFs 1 )
1
_ As
1 Q (As jFs
1)
;
(cf. (23)), (27) follows.
We are now ready to present the forward indi¤erence pricing algorithm.
Theorem 9 Consider a claim, introduced at time t0 = 0; yielding at time t
payo¤ Ct 2 Ft ; t = 0; 1; :::: Let s (Ct ) be de…ned in (20) with the forward
criterion, Us (x) ; s = 0; 1; :::; t as in (12). Let, also, Q be as in (19); and
(s;s+1)
(s;t)
EQ
and EQ
as in (23) and (24), respectively. The following statements
hold:
i) The forward indi¤ erence price, s (Ct ), is given, for s = 0; 1; :::; t; by the
iterative algorithm
t (Ct ) = Ct ;
s (Ct )
= EQ
1
ln EQ
(s;s+1)
= EQ
e
(
s+1 (Ct )
11
s+1 (Ct ))
S
Fs _ Fs+1
jFs
(29)
:
ii) The forward indi¤ erence price process s (Ct ) 2 Fs and satis…es, for
s = 0; 1; :::t;
(s;t)
(Ct ):
(30)
s (Ct ) = EQ
iii) The forward indi¤ erence price algorithm is consistent across time in that,
for 0 s s0 t, the semigroup property
(s;s0 )
s (Ct )
(s;s0 )
= EQ
= EQ
(
s0 (Ct ))
(s0 ;t)
(EQ
(Ct ))
(31)
(s0 ;t)
(Ct ))
s (EQ
=
holds.
Proof. We …rst show (29) for s = t
sup EP (Ut (Xt
Ct )jFt
1)
1. We have (cf. (12))
=e
t
i=1 hi
sup EP
t
(Xt Ct )
e
t
jFt
:
1
Using Lemma 8 for Z = Ct , we deduce
sup EP
e
(Xt Ct )
t
=
(t 1;t)
with EQ
exp
Xt
jFt
(t 1;t)
EQ
1
1
(Ct )
ht ;
(32)
as in (23). Using (12) once more, we have
sup EP (Ut (Xt
Ct )jFt
1)
= Ut
(t 1;t)
1 (Xt 1
EQ
t
(Ct ));
and (29) follows from (20).
We also show (29) for s = t 2; for there are subtleties that are not present
in the case s = t 1: To this end, (12) yields
sup EP (Ut (Xt
t
t 1
i=1 hi
=e
t 1
i=1 hi
=e
1
(Xt
e
2+
t 1
i=1 hi
sup EP
t
1
(Xt
t
St
1
(t
EQ
1
jFt
1;t)
1
jFt
2
(Ct )) ht
e
(Xt
2+
2+
t
t
1
St
1
t
1 (Ct ))
1
On the other hand, using (27) for s = t
e
(Xt Ct )
e
jFt
1
t
sup EP
2)
t
sup EP eht
t
Ct )jFt
t
sup EP eht sup EP
t
=e
1;
1
St
1
t
1 (Ct ))
1 and Z =
jFt
12
2
=
e
jFt
t 1 (Ct ),
Xt
2
:
we deduce
(t
2
2
EQ
2;t
1)
(
t
1 (Ct ))
ht
1
:
Combining the above, we obtain
sup EP (Ut (Xt
t
1;
Ct )jFt
2)
=
(t
Xt
e
2
EQ
2;t
1)
(
t
1 (Ct ))
t 2
i=1 hi
+
t
= Ut
(t 2;t 1)
Xt
2
EQ
2
(
t 1 (Ct ))
;
where we used (12). De…nition 5 then yields
t 2 (Ct )
(t 2;t 1)
= EQ
(
t 1 (Ct )) :
The proof of (29) for s < t 2 follows by similar arguments which are omitted.
Next, we establish (30). Since for s = t 1 it was already shown above, we
prove it for s = t 2: We need to show
sup EP (Ut (Xt
1;
t
(t 2;t)
with EQ
Ct )jFt
2)
= Ut
sup EP (Ut (Xt
t 2
i=1 hi
sup EP
t
=e
t 2
i=1 hi
e
1;
t
=
(Xt
2+
t
St
1
Ct )jFt
1 )+ht
1
(Ct ) ;
2)
sup EP
1
(
e
t
St Ct )+ht
t
sup EP
t
EQ
2
as in (24). We have
t
=e
(t 2;t)
Xt
2
t
exp
Xt
2
+
St
t 1
(t 1;t)
1
1
exp
Xt
(t 2;t 1)
EQ
2
(t 1;t)
EQ
EQ
(Ct )
(Ct ) + ht
+
t 2
i=1 hi
1
Ft
Ft
;
(t 2;t)
where we used Lemma 8 twice. Using the de…nition of EQ
, we conclude. To
show (30) for s < t 2 we work similarly.
Assertion (31) follows from straightforward, albeit tedious, arguments.
3.2
The case of multiple claims
We conclude this section by presenting the pricing algorithm when there is a
collection of claims to be priced. For simplicity, let us, for the moment, consider
two claims, written at t0 = 0 and maturing at points t and t0 , with t > t0 ;
yielding payo¤s Ct00 2 Ft0 and Ct 2 Ft . For Us , s = 0; 1; ::; t; as in (12) and
Xs = x, we deduce, using De…nition 5,
sup
s+1 ;:::;
=
sup
s+1 ;:::;
E
e
EP ( Ut (Xt
t
x+
t0
i=s+1
i
Si
t0
13
(Ct00 + Ct ))j Fs )
(
0
t0 (Ct )+Ct0
)
+
t0
i=s+1 hi
Fs ;
Ft
1
2
2
=
sup
s+1 ;:::;
with Mt0 = Ct00 +
t0
s
EP (Ut0 (Xt0
Mt0 ) jFs ) ;
t
(Ct ) : Applying De…nition 5 once more yields
(Ct00 + Ct ) =
s
(Mt0 ) =
s
(Ct00 +
t0
(Ct )) :
Generalizing the above argument, we obtain the following result. For convenience and ease of notation, we assume that in the interval [0; n] we price
a collection of n + 1 claims, C0 ; C1 ; :::; Cj ; :::Cn ; with each generic claim Cj
maturing at time j; j = 0; 1; :::; n:
Theorem 10 Consider a collection of n + 1 claims, written at t0 = 0; yielding
payo¤ s Cj 2 Fj ; with j = 0; 1; :::; n: Let Us ; s = 1; :::; n; be the forward criterion
(s;s+1)
given in (12) and EQ
; s = 0; 1; :::; n 1; as in (23).
The following statements hold:
i) The forward indi¤ erence price, s ( nj=s Cj ), is given, for s = 0; 1; :::; n 1;
by the iterative algorithm
n (Cn ) = Cn ;
s(
n
j=s Cj )
1
= Cs + EQ
(s;s+1)
= Cs + EQ
ln EQ
e (Cs+1 +
(Cs+1 +
s+1 (
n+1
j=s+2 Cj )
ii) The forward indi¤ erence price process s (
s = 0; 1; :::; n;
n
s ( j=s Cj )
(s;s+1)
= Cs + EQ
4
(s+1;s+2)
Cs+1 + EQ
s+1 (
n
j=s+2 Cj ))
) F _ FS
s
s+1 jFs
n
j=s Cj )
2 Fs and satis…es, for
(n 1;n)
Cs+2 + :::EQ
:
(Cn )
:
Indi¤erence prices in incomplete (non-reduced)
binomial models; review of results
In order to carry out a comparative study between the forward and the traditional indi¤erence prices, we need to have the analogous representation results
for the latter family. Such results were derived by the authors in [16] and extended previous works (see, for example, [4], [8], [22] and [23]), where the nested
model was complete, with the non-traded factor a¤ecting the claim’s payo¤ but
not the dynamics of the traded asset.
If the nested model is not complete, as the case herein, the internal market incompleteness is processed quite di¤erently by the forward and traditional
utilities. To our knowledge, the results in [16] were the …rst for exponential
indi¤erence prices in general incomplete binomial models. We call such models
non-reduced.
Indi¤erence prices for power utilities in incomplete markets were derived in
[7].
14
When internal market incompleteness is not present, substantial simpli…cation takes places which conceals the essential di¤erences between the forward
and the traditional approaches. We examine such models, which we call reduced,
in the last section.
In order to facilitate the exposition, we …rst provide a brief review of the main
results of [16] on the minimal entropy martingale measure, the value function
process and the traditional indi¤erence price.
4.1
Classical (backward) investment performance process
We start by recalling the familiar problem of maximizing the expected utility
from terminal wealth. A trading horizon, [0; T ]; is chosen at the end of which a
utility datum, uT ; is assigned. For the exponential case considered herein,
uT (x) =
for x 2 R and
e
x
;
(33)
> 0: The value function process is, then, de…ned as
Vt;T (x) =
sup
t+1 ;:::;
EP
e
T
XT
Ft ; Xt = x ;
(34)
for XT given by (4), with Xt = x; x 2 R: We introduce the T notation in order
to highlight the dependence of the maximal expected utility on the horizon
choice. Notice that this choice is made before any claim is introduced.
The process Vt;T (x) has been extensively studied (see, among others, [2],
[5] and [20]) for general market settings. Recalling the Dynamic Programming
Principle, we easily see that Vt;T (x) satis…es De…nition 1, for t = 0; 1; :::; T; with
the only di¤erence being that it is set at the end, and not at the beginning, of
the trading horizon.
To align Vt;T (x) with its forward counterpart, we will occasionally refer to
it as the backward investment performance process.
Explicit formulae for Vt;T (x) can be found, among others, in [2] and [20], for
general semi-martingale models. In [16] similar formulae were derived for the
speci…c binomial model at hand. They involve the aggregate entropy and iterative expressions for process de…ned in (10). To make the presentation concise
we present these results next and provide detailed description of the quantities
involved right after.
me
Proposition 11 For t = 0; 1; :::; T; let Ht;T
be the minimal aggregate entropy
(cf. (39)). Then, the value function process Vt;T (x) ; de…ned in (34), is given
by
me
Vt;T (x) = e x Ht;T
(35)
!!
T
X
(t;T )
= exp
x + JQ
hi
(36)
i=t+1
=
exp
x
(t;T )
JQme
T
15
T
X
i=t+1
hi
!!
;
(37)
(t;T )
with JQ
4.2
(t;T )
and JQme de…ned in (43) with Q = Q ; Qme
T ; respectively:
T
The minimal entropy measure
We recall the minimal entropy martingale measure. It is de…ned on FT and
denoted by Qme
T : It minimizes the relative entropy HT de…ned, for t = 0; 1; :::; T
and Q 2 QT ;
HT (Q ( jFt ) j P ( jFt ) ) = EQ
speci…cally,
ln
Q ( jFt )
Ft
P ( jFt )
HT (Qme
T ( jFt ) j P ( jFt ) ) = min HT (Q ( jFt ) j P ( jFt ) ) :
Q2QT
(38)
We refer the reader to [5] (see, also, [2]) for the properties and the role of
this measure in stochastic optimization with exponential criteria. We will be
using the condensed notation
me
Ht;T
= HT (Qme
T ( jFt ) j P ( jFt ) )
(39)
and referring to this process as the minimal aggregate entropy.
was derived for the speAn explicit representation for the density of Qme
T
ci…c model at hand in [24]. This construction, which is to the best of our
knowledge new, is based on an iterative procedure which yields the condiYt j Ft 1 _ FtS in terms of its historical counterpart
tional distribution Qme
T
S
P Yt j Ft 1 _ Ft and the (conditional on Ft 1 ) minimal aggregate entropy
me
Ht;T
: The latter is constructed through an independent iterative procedure
which involves the measure, Q (cf. (19)). To ease the presentation, the conme
is presented separately, in Proposition 13.
struction of Ht;T
Comparing (19) and (40), we see that in the non-reduced incomplete binodo not coincide. This fact will have
mial models the measures Q and Qme
T
important consequences on the distinct nature of the forward and backward
indi¤erence prices.
Proposition 12 The minimal entropy measure Qme
T satis…es, for t = 1; :::; T;
me;uu
Qme
P ( At Bt j Ft 1 ) e Ht;T
T ( At Bt j Ft 1 )
=
me;uu
Qme
T ( At j Ft 1 )
P ( At Bt j Ft 1 ) e Ht;T + P ( At Btc j Ft
c
Qme
T ( At Bt j Ft 1 )
me
QT ( At j Ft 1 )
=
1) e
Hme;ud
t;T
(40)
me;ud
P ( At Btc j Ft 1 ) e Ht;T
me;uu
me;ud
P ( At Bt j Ft 1 ) e Ht;T + P ( At Btc j Ft 1 ) e Ht;T
me;du
c
Qme
P ( Act Bt j Ft 1 ) e Ht;T
T ( At Bt j Ft 1 )
=
me;du
c
Qme
T ( At j Ft 1 )
P ( Act Bt j Ft 1 ) e Ht;T + P ( Act Btc j Ft
16
;
1) e
Hme;dd
t;T
;
and
me;dd
c c
Qme
P ( Act Btc j Ft 1 ) e Ht;T
T ( At Bt j Ft 1 )
=
me;du
c
Qme
T ( At j Ft 1 )
P ( Act Bt j Ft 1 ) e Ht;T + P ( Act Btc j Ft
Hme;dd
t;T
1) e
me;uu
me;ud
me;du
me;dd
where At ; Bt as in (8) and Ht;T
; Ht;T
; Ht;T
; Ht;T
are the values of
me
the Ft measurable random variable Ht;T , (cf. (39)), conditional on Ft 1 :
The above equalities can be expressed slightly di¤erently in order to provide
the direct analogues of (17) and (18). For example, (40) yields
P ( At Bt j Ft 1 ) Hme;uu
e t;T
Qme
(
A
B
j
F
)
P ( At j Ft 1 )
t t
t 1
T
=
P ( At Bt j Ft 1 ) Hme;uu
P ( At Btc j Ft 1 )
Qme
T ( At j Ft 1 )
e t;T +
e
P ( At j Ft 1 )
P ( At j Ft 1 )
Hme;ud
t;T
:
(41)
me;uu
me;ud
Notice that if Ht;T
= Ht;T
the above equality reduces to (17). This
observation will play a key role in the analysis of the reduced binomial model
in section 6.
Next, we provide two alternative iterative representations for the aggregate
minimal entropy involving Q and Qme
T ; respectively. To the best of our knowledge, these representations are new. Their proofs can be found in [24].
Proposition 13 Let Z be a random variable on ( ; F; P). For s = 0; 1; :::; T;
t = s + 1; :::; T; de…ne the single- and multi-step iterative functionals
(s;s+1)
JQ
and
(s;t)
JQ
(Z) = EQ
(s;s+1)
(Z) = JQ
S
jFs
eZ Fs _ Fs+1
ln EQ
(s+1;s+2)
JQ
(t 1;t)
:::JQ
(42)
(Z)
:
(43)
me
The minimal aggregate entropy Ht;T
; t = 0; 1; :::; T; is given by the following
iterative schemes:
i) Let h be as in (10). Then,
me
HT;T
=0
and, for t = 0; 1; :::; T
me
Ht;T
= ht+1
and
HTme 1;T = hT ;
(44)
2;
(t;t+1)
JQ
me
Ht+1;T
=
(t;T )
JQ
T
X
hi
i=t+1
!
:
(45)
ii) Let Qme
be the minimal entropy measure. Then, the minimal aggregate
T
me
entropy Ht;T
is given, for t = T; T 1; in (44) and, for t = 0; 1; :::; T 2; by
me
Ht;T
= ht+1 +
(t;t+1)
JQme
T
me
Ht+1;T
17
=
(t;T )
JQme
T
T
X
i=t+1
hi
!
:
(46)
4.3
Backward indi¤erence valuation and the backward pricing algorithm
We review the de…nition of backward indi¤erence prices, which is slightly reformulated so that we can draw more easily analogies between the two settings.
De…nition 14 Let T > 0 be …xed. Consider a claim, written at time t0 = 0
and yielding at t payo¤ Ct 2 Ft ; t = 0; 1; :::; T . Let Vt;T (x) ; t = 0; 1; :::; T; be
the exponential value function process given in (35).
For s = 0; 1; :::; t; the claim’s backward indi¤ erence price is de…ned as the
amount B
s (Ct ; T ) 2 Fs for which the equality
B
s
Vs;T Xs
(Ct ; T ) =
sup
s+1 ;:::;
EP ( Vt;T (Xt
Ct )j Fs )
t
(47)
is satis…ed for all initial wealth levels Xs = x; x 2 R and Xt given in (4) with
i 2 A; i = s + 1; :::; t:
In analogy to the analysis following the de…nition of the forward indi¤erence
prices, we comment that the above de…nition can be extended, without loss of
generality, for times beyond the claim’s maturity. Namely, for t0 = t + 1; :::; T ,
the pricing condition (47) may be replaced by
B
s
Vs;T Xs
(Ct ; T ) =
sup
s+1 ;:::;
EP ( Vt0 ;T (Xt0
t0
Ct )j Fs ) :
This follows easily from (47) and the Dynamic Programming Principle. Observe,
however, that this cannot be done for times exceeding T .
Next, we present the backward indi¤erence pricing algorithm, constructed
in [16] for non-reduced incomplete binomial models. The two fundamental pric(s;s+1)
(s;s0 )
ing blocks are the non-linear functionals EQme
and EQme introduced below.
T
T
Notice that their structural form is similar to their forward counterparts (see
(23) and (24)) but the involved measure is the minimal entropy measure and
not Q :
De…nition 15 Let t = 0; 1; :::; T , Z be a random variable on ( ; F; P) and Qme
T
be the minimal entropy measure de…ned on FT (cf. (38)). We de…ne
i) the single-step backward price functional
(s;s+1)
EQme
T
(Z) = EQme
T
1
ln EQme
e
T
Z
S
Fs _ Fs+1
(48)
and
ii) the multi-step backward price functional
(s;s0 )
(s;s+1)
EQme (Z) = EQme
T
T
across the time interval (s; s0 ); 0
(s+1;s+2)
(EQme
T
s
s0
18
(s0 1;s0 )
(:::EQme
T
T:
(Z)));
(49)
Theorem 16 Consider a claim, introduced at time t0 = 0; yielding at t payo¤
Ct 2 Ft ; t = 0; 1; :::; T: Let B
s (Ct ; T ) be de…ned in (47) with the value function
(s;s+1)
(s;t)
process Vt;T (x) given in (34). Let, also, EQme
and EQme be given in (48) and
T
T
(49). The following statements are true:
i) the backward indi¤ erence price, B
s (Ct ; T ) ; is given, for s = 0; 1; :::; t; by
the iterative algorithm
B
t (Ct ; T ) = Ct ;
B
s
(s;s+1)
(Ct ; T ) = EQme
T
(
B
s+1
(Ct ; T )):
(50)
ii) The backward indi¤ erence price process B
s (Ct ; T ) 2 Fs and satis…es, for
s = 0; 1; :::; t;
(s;t)
B
(51)
s (Ct ; T ) = EQme (Ct ):
T
iii) The backward indi¤ erence price algorithm is consistent across time in
that, for 0 s s0 t, the semigroup property
B
s (Ct ; T )
(s;s0 )
= EQme
T
(s;s0 )
= EQme
T
B
s0 (Ct ; T )
=
(s0 ;T )
EQme (CT )
T
B
s0
(52)
(s0 ;t)
EQme (Ct ) ; T
T
holds.
Remark 17 Comparing the single-step pricing iterations (29) and (50), we
might think that all it takes to "switch" from the forward to the backward formulation is to "switch" from the measure Q to Qme
T : This is not, however, the
case! As it will be discussed in the next section (see 5.2.1), the forward singlestep price functional (cf. (23)) carries a lot of intuition and provides a direct
stochastic extension of the classical static certainty equivalent.
5
5.1
Comparative study of the forward and the
backward indi¤erence prices
Similarities
Both forward and backward indi¤erence pries are constructed via the optimal
behavior of the investor with and without the claim in consideration. As such,
they incorporate and re‡ect the individual risk preferences. They are both
independent of the investor’s wealth.
5.1.1
Time consistency
Both indi¤erence pricing operators are time consistent, in that prices at any
intermediate time, say s; can be thought as the prices of a claim equal to the
corresponding indi¤erence prices at a future time s0 ; namely,
s (Ct )
=
s ( s0 (Ct ));
19
0
s
s0
and
B
s (Ct ; T )
=
B B
s ( s0 (Ct ; T ); T );
0
s0
s
T:
These equalities are readily deduced from (31) and (52). They are direct consequences of the so-called "self-generation" properties of the forward and the
backward investment performance processes,
Us (x) = sup E ( Us (Xs )j Fs ; Xs = x) ;
0
s
s0
Vs;T (x) = sup E ( Vs0 ;T (Xs )j Fs ; Xs = x) ;
0
s
s0
A
and
T;
AT
respectively. We refer the reader to [26] for a detailed discussion on the "selfgeneration" property.
5.1.2
A two-step iterative construction
Both indi¤erence prices are constructed via iterative pricing schemes which start
at the claim’s maturity and are applied backwards in time (see (29) and (50)).
The schemes have local and dynamic properties.
Dynamically, at each time interval, say (s; s + 1), the prices s (Ct ) and
B
s (Ct ; T ) are computed via the single-step forward and backward price func(s;s+1)
(s;s+1)
tionals, EQ
and EQme ; applied to the end of the period payo¤. The latter
T
payo¤s turn out to be the indi¤erence prices, s+1 (Ct ) and B
s+1 (Ct ; T ) , as
(s;s+1)
(s;s+1)
discussed earlier. Both functionals EQ
and EQme are independent of the
T
speci…c payo¤.
(s;s+1)
(s;s+1)
Locally, the pricing role of both these functionals, EQ
and EQme ; is
T
similar to their single-period counterpart, developed in [8], in that they are
non-linear and produce the price in two sub-steps.
Speci…cally, in the …rst sub-step, the end of the period payo¤s, s+1 (Ct ) and
B
s+1 (Ct ; T ); are distorted via a non-linear functional and produce two intermediate payo¤s. In the forward case, the new payo¤, say P (s;s+1) (Ct ); is given
by
1
S
P (s;s+1) (Ct ) = ln EQ
e s+1 (Ct ) Fs _ Fs+1
(53)
while in the backward case, the analogous payo¤, say P B;(s;s+1) (Ct ); is
P B;(s;s+1) (Ct ) =
1
ln EQme
T
e
B
s+1 (Ct ;T )
S
Fs _ Fs+1
:
These payo¤s are, in turn, priced by expectation yielding, the forward and
backward prices at the beginning of the time step,
s (Ct )
= EQ (P (s;s+1) (Ct ) jFs )
(54)
= EQme
(P B;(s;s+1) (Ct ) jFs ):
T
(55)
and
B
s (Ct ; T )
20
S
Notice that in the …rst step the conditioning is with regards to Fs _Fs+1
while,
in the second, it is only with respect to Fs :
5.1.3
Scaling, monotonicity, robustness and cash invariance properties
Using the explicit formulae appearing in the forward and backward valuation
algorithms, we easily deduce various properties of the associated indi¤erence
prices.
(s;s+1)
(s;s+1)
Because the single-step price functionals EQ
(Z) and EQme (Z), with
T
Z 2 Fs+1 representing s (Ct ) and B
s (Ct ; T ) ; respectively, only di¤er in the
pricing measure, which is independent of both the parameter and the input
Z, the technical steps needed to establish these properties are essentially the
same. The calculations are tedious and are, thus, omitted.
For the sake of the presentation, we only state the properties for the singlestep forward functional and not for the multi-step one, and also use the notations
(s;s+1)
(s;s+1)
EQ
(Z) and EQ
(Z; ); for Q = Q ; Qme
T :
(s;s+1)
i) The mapping
(s;s+1)
lim+ EQ
!0
! EQ
(Z; ) is increasing and continuous. Moreover,
(s;s+1)
(Z; ) = EQ (Z) and
lim EQ
!1
(Z; ) = EQ kZkL1
:
Q f jSt g
ii) We have
lim
!0
1
@ (s;s+1)
E
(Z; ) = EQ (V arQ (Z jSt )) ;
@ Q
2
and, thus,
(s;s+1)
EQ
iii) For
(s;s+1)
EQ
(Z; ) = EQ (Z) +
1
EQ (V arQ (Z jSt )) + o ( ) :
2
2 (0; 1) ;
1
( Zs+1
+ (1
2
)Zs+1
)
(s;s+1)
EQ
1
( Zs+1
) + (1
(s;s+1)
)EQ
Moreover,
(s;s+1)
EQ
(s;s+1)
( Z) 7 EQ
(Z) ; for
2 (0; 1) (resp. a
S
v) Let Z = Y + Y 0 such that Y 2 Fs+1 and Y 0 2 Fs+1
Then,
(s;s+1)
EQ
(s;t)
(Z) = EQ
(Yt ) + EQ ( Y 0 j Fs ) :
21
1):
2
(Zs+1
):
5.2
Di¤erences
While both the forward and backward indi¤erence prices are de…ned via similar
valuation conditions (cf. (20) and (47)), there are fundamental di¤erences between them. They stem from the very distinct nature of the associated forward
and backward investment performance processes, Ut (x) and Vt;T (x) which are
used to de…ne these prices. For the reader’s convenience, we recall the representations of these processes below.
The forward investment performance process is given by
Ut (x) =
P
x+ ti=1 hi
e
with hi de…ned in (10). ItPis Ft predictable and follows the market "path by
t
path" through the factor i=1 hi . It is set at t = 0 and de…ned for all future
times.
In contrast, the backward investment performance process is given by
Vt;T (x) =
e
x Hme
t;T
:
It is Ft adapted and a¤ected by market changes in a much
Pt coarser manner.
me
, in contrast to i=1 hi ; …rst aggreIndeed, the aggregate entropy process Ht;T
gates market information from current time t to terminal horizon T; and, then,
brings it back to today, as it can be seen in (42) and (43)
5.2.1
Analogies with the static certainty equivalent
The classical certainty equivalent is a static pricing rule which yields the price
of a generic claim, say Z, as
C (Z) =
u(
1)
(EP (u ( Z))) ;
(56)
for a concave and increasing utility function u (see, for example, [6]). Notice
that, in contrast to the indi¤erence prices, the above price is derived in the
absence of any trading activity. Notice, also, that the measure appearing above
is the historical probability measure and not a martingale one.
Given that both the forward and backward indi¤erence prices are constructed
taking into account the investor’s risk preferences, it is expected that they would
provide multi-period analogues of the static certainty equivalent rule. But, is
this really the case?
We explore this question next. We …nd that, indeed, the forward indi¤erence price o¤ers an intuitively pleasing direct multi-period analogue to (56).
However, the traditional indi¤erence price fails to do so.
In seeking multi-period analogues of (56), it is natural to assume that the
(1)
role of u and u( 1) will be played by the processes Ut (x) and Ut (x) ; and
( 1)
Vs;T (x) and Vs;T (x) ; respectively.
We start with the interpretation of the forward indi¤erence price as certainty
equivalents.
22
Proposition 18 Let t > 0 and consider a claim with payo¤ Ct 2 Ft : Let
be its forward indi¤ erence price at time s = 0; 1; :::; t:
For s = 0; 1; :::; t 1; and Z 2 Fs+1 de…ne
C (s;s+1) (Z) =
( 1)
S
EP Us+1 ( Z) Fs _ Fs+1
Us+1
Then, the forward indi¤ erence price
s
with
t
s
(Ct )
:
(Ct ) is given by
C (s;s+1) (
(Ct ) = EQ
s
s+1
(Ct )) Fs ;
(57)
(Ct ) = Ct :
Proof. From property (19) and the form of C (s;s+1) we deduce that
C (s;s+1) (Z) =
( 1)
Us+1
EQ
S
Us+1 ( Z) Fs _ Fs+1
:
In turn, Proposition 7 yields
C (s;s+1) (Z) =
1
ln EQ
e
Z
S
Fs _ Fs+1
and using (29) we conclude.
Next we examine whether the backward indi¤erence price provides a similar
analogue to (56). representation. For Z 2 Fs+1 ; let us de…ne the quantities
B;(s;s+1)
CQme
T
(Z) =
B;(s;s+1)
CP
(Z) =
( 1)
S
Vs+1;T EQme
Vs+1;T (Z)j Fs _ Fs+1
T
( 1)
( 1)
( 1)
(58)
S
Vs+1;T EP Vs+1;T (Z)j Fs _ Fs+1
where Vt;T (x) as in (34) and its spatial inverse Vt;T
0; 1; :::; T; by
Vt;T
;
(x) =
1
ln ( x)
1
me
Ht;T
;
(59)
(x) is given, for t =
x2R ;
me
with Ht;T
as in (39).
B;(s;s+1)
Proposition 19 Let CQme
T
di¤ erence price. Then,
B
s
B
s
be as in (58) and
B;(s;s+1)
(Ct ; T ) 6= EQme
CQme
T
T
B
s+1
(Ct ; T ) the backward in-
(Ct ; T ) Ft :
(s;s+1)
Proof. In [16] it is shown that for Z 2 Fs+1 ; and EQme
T
as in (48) and (58),
(s;s+1)
EQme
T
(Z; T ) = EQme
T
B;(s;s+1)
CQme
T
23
Z+
1
(60)
B;(s;s+1)
(Z; T ) and CQme
me
Hs+1;T
T
Fs
EQme
T
B;(s;s+1)
CQme
1
T
me
Hs+1;T
Fs ;
me
where Hs+1;T
is the minimal aggregate entropy (cf. (39)). Therefore, the iterative price equality (50) is equivalent, for s = 0; 1; :::; t 1; to
B
s
B;(s;s+1)
(Ct ; T ) = EQme
T
EQme
T
CQme
Z+
T
B;(s;s+1)
CQme
T
1
1
me
Hs+1;T
me
Hs+1;T
Fs
Fs :
(61)
and we easily conclude.
Similar calculations can be carried out to show that
B
s
B;(s;s+1)
with CP
5.2.2
B;(s;s+1)
(Ct ; T ) 6= EQme
CP
T
B
s+1
(Ct ; T ) Ft
as in (59). The calculations are rather tedious and are omitted.
The pricing measures
The pricing measure in the forward indi¤erence valuation is the measure Q
de…ned on F. Property (19) highlights its essential role, namely, Q preserves
the conditional distribution of unhedgeable risks, given the hedgeable ones, in
relation to its historical counterpart.
The pricing measure in the backward indi¤erence valuation is, as it is well
known, the minimal entropy measure, Qme
T , de…ned on FT : While the measure
Q has the above intuitively pleasing property, the minimal entropy measure
depends on the
does not have any analogous property. Notice also that Qme
T
choice of the horizon T:
5.2.3
Dependence on the terminal horizon
We start with the dependence of the backward indi¤erence prices on the horizon
choice T: This dependence is signi…cant and can be seen from the price func(s;s+1)
(s;t)
tionals EQme
and EQme appearing in the backward pricing algorithm: Indeed,
T
T
although the form of these functionals is horizon-independent, the involved measure, Qme
T ; is not. This is apparent from its density, (40), which involves the
me
horizon-dependent aggregate entropic values Ht;T
.
The next results quanti…es the relation between the backward indi¤erence
prices and their horizons (see [16]).
Proposition 20 Consider a claim written at t0 = 0 and yielding at t payo¤
Ct 2 Ft : Let T; T^ be two investment horizons with T < T^ and B
s (Ct ; T ) and
B
^
C
;
T
the
associated
backward
indi¤
erence
prices.
Then,
for
0 s t
t
s
T T^;
B
Ct ; T^
s
24
=
B
s
Ct
1
me
Ht;
T^
me
;T
Ht;T
1
+
me
Hs;
T^
me
:
Hs;T
(62)
We now revert our attention to the analogous dependence of the forward
indi¤erence prices. We recall that the forward performance process is de…ned
for all times. It is easy to see that forward indi¤erence prices are, also, de…ned
for all times and are independent of the origin. This can be seen from the form
(s;s+1)
(s;t)
of the forward pricing functionals, EQ
and EQ , and the density of the
forward pricing measure Q :
A question arises whether the results would change if the forward process is
de…ned for times t = s0 ; s0 + 1; :: with s0 > 0: To answer it, we would …rst need
to de…ne such a process. While this is, to some extent, an arti…cial situation,
we, nevertheless, explore it so that we can draw analogies with the backward
case. Imitating the proof of Theorem 4, we obtain that, for t = s0 ; s0 + 1; :::;
the process
t
(63)
Us0 ;t (x) = e x+ i=s0 +1 hi
is a time-monotone forward performance, with Us0 ;s0 (x) = e x :
Let us now consider a claim, written at t0 > s0 and maturing at time t:
We readily extend De…nition 5 to de…ne the forward price s (Ct ; s0 ), for s =
t0 ; t0 + 1; :::; t; by replacing the pricing condition (20) with
Us0 ;s (x
s (Ct ; s0 ))
=
sup
s+1 ;:::;
EP ( Us0 ;t (Xt
Ct )j Fs ) ;
t
(64)
where Us0 ;t (x) as above. Equivalently,
exp
=
(x
sup
s+1 ;:::;
EP
s (Ct ; s0 ))
e
s
i=s0 +1 hi
+
(Xt Ct )+
t
i=s0 +1 hi
t
Fs :
s0
Multiplying both sides by e i=1 hi and using the measurability properties of the
process h, we recover the original pricing condition (20). We easily conclude
that forward indi¤erence prices are not a¤ected by the choice of point s0 :
5.2.4
Dependence on the maturity of the claim
(s;s+1)
(s;t)
The forward pricing functionals EQ
and EQ are independent of the claim’s
maturity. Indeed, neither their form nor the involved measure depend on the
time t that the claim matures. This does not mean that the price is independent
of the claim’s maturity, an obvious wrong conclusion. Rather, it says that the
forward pricing operator per se does not depend on the speci…c maturity. This
setting is very much aligned with the one in complete markets where the pricing
operator, given by the conditional expectation of the (discounted) payo¤, is
independent of the claim’s maturity.
(s;s+1)
(s;t)
The backward pricing functionals EQme
and EQme ; however, depend indiT
T
rectly on the claim’s maturity because the latter cannot exceed T: While the
25
(s;s+1)
form of EQme
does not depend on T; the pricing measure does, because of its
T
me
dependence on the aggregate entropy Ht;T
, which itself depends on T:
5.2.5
Parity between backward and forward indi¤erence prices
Given that the forward and the backward indi¤erence prices are de…ned via
distinct investment performance processes, they are, in general, di¤erent in size.
The pricing formulae in Theorems 9 and 16 enable us to compute the price
di¤erential explicitly. It is easily seen that, in general, it is not possible to
deduce if one price dominates the other. It is worth noticing that the quantity
that di¤erentiates the prices is directly related to the local and the aggregate
entropies, which, as was discussed in detail earlier, are processed very di¤erently
by the forward and the backward performance processes.
Proposition 21 Let T > 0. Consider a claim written at t0 = 0 yielding at t
me
; s = 0; 1; :::; t be given by (10) and (39).
payo¤ Ct 2 Ft ; t T: Let hs and Hs;T
Then, the claim’s backward and forward indi¤ erence prices satisfy
B
s
(Ct ; T ) =
s
1
Ct
Mt;T
+
1
Ms;T ;
(65)
with Ms;T 2 Fs given by
Ms;T =
s
i=1 hi
me
+ Hs;T
:
(66)
Proof. From (47) we have
exp
=
B
s
x
sup
s+1 ;:::;
EP
me
Hs;T
=
(Ct ; T )
exp
s+1 ;:::;
1
Xt
Ct
Ut Xt
Ct
EP
sup
s+1 ;:::;
EP
t
i=1 hi
t
i=1 hi
t
= Us x
exp
1
x
s
s
1
Ct
Ct
1
t
i=1 hi
t
i=1 hi
e
(Xt Ct ) Hme
t;T
t
t
=
=
sup
me
+ Hs;T
+
me
+ Ht;T
Fs
t
i=1 hi
Fs
Fs
me
+ Ht;T
me
+ Ht;T
+
s
i=1 hi
;
where we used (12) and De…nition 5. We easily conclude.
6
Reduced incomplete binomial models
We focus on reduced incomplete (almost complete) binomial models, introduced
informally in section 1. Speci…cally, we assume that the values and the transition
26
probabilities of the stock price process St are not a¤ected by the non-traded
factor process Yt ; t = 0; 1; :::; (cf. (1) and (2), respectively).
Assumption: For t = 0; 1; :::; we have
u
t+1
2 FtS
d
t+1
and
2 FtS
(67)
and
P
t+1
=
u
t+1
jFt = P
t+1
=
u
t+1
FtS :
(68)
In essence, this is the case of a nested complete market model to which
an exogenous non-traded factor is added. In the incomplete binomial models
analyzed so far (see, among others, [1], [4], [8] and [22]) this factor a¤ects only
the claim’s payo¤.
As it is expected, the measures Q and Qme
must coincide since there is
T
now a unique nested martingale measure. An interesting fact is that the minimal aggregate entropy looses its non-linear character and reduces to a mere
conditional expectation.
Proposition 22 In the reduced binomial model,
Q = Qme
T
(69)
Moreover, the process, ht is FtS predictable while the minimal aggregate entropy
me
Ht;T
is given by
!
T
X
me
Ht;T
= EQ
hi FtS ;
(70)
i=t+1
with h as in (10) and Q = Q = Qme
T :
We are now ready to present the main result of this section.
Theorem 23 Let T > 0 and consider a claim written at t0 = 0 and maturing
at t, yielding payo¤ Ct 2 Ft ; t T: In the reduced binomial model, the claim’s
forward and backward indi¤ erence prices coincide,
s
for 0
s
t
(Ct ) =
B
s
(Ct ; T ) ;
(71)
T.
Proof. Let t be the expiry of the claim and consider its forward indi¤erence
price s (Ct ) de…ned in (20) and given in (30). Similarly, let B
s (Ct ; T ) be the
backward indi¤erence price, de…ned in (47) and given in (50). Using (69) and
the form of the forward and backward single-step price functionals (23) and
(48), we deduce that
B
t 1 (Ct ) = t 1 (Ct ; T ) :
Proceeding iteratively we easily conclude.
An immediate consequence of the above result is that the dependence of the
backward prices on the investment horizon vanishes.
27
The intuition behind equality (71) is that in the reduced binomial model,
there is, e¤ectively, no internal market incompleteness. Recall that the latter
is priced di¤erently by the backward and forward pricing operators, and this
a¤ects the tw prices. Once it is removed, however, the two dynamic utilities
process the incompleteness, which now comes exclusively from the stochastic
factor a¤ecting the claim, in the same manner. This, in turn, results in the two
prices becoming identical.
How to characterize other families of models and/or claims for which the
two prices coincide remains an open and, in our view, interesting problem.
Acknowledgment : The authors would like to thank B. Angoshtari for
fruitful comments.
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