Valuation in Incomplete Markets

 IV
market problem generalizing earlier results of Duffie and Richardson (1991)[DR91]. There 
have been multiple attempts to theoretically pick one for pricing purpose according to 
different optimal criteria, some of which are related to utility maximization. For instance, the 
Föllmer-Schweizer minimal measure by Föllmer and Schweizer (1991)[FS91]. As we can see 
the hedging of derivatives in incomplete financial markets is a frequently studied problem in 
mathematical finance. Several different approaches have been developed in literature, but no 
agreement on one uniformly superior method has emerged so far. 
The purpose of this thesis is to review two quadratic hedging approaches really interesting in 
the incomplete market literature: local risk-minimization and mean-variance hedging; and a 
minimal martingale measure approach. In a nutshell, the main difference between these two 
approaches is the following: one has either simple solution for hedging strategies (local risk-
minimization) or a control over total cost and risks (mean-variance hedging), but not both. 
The thesis is structured as follows. We first explain in Section 1 the general theoretical 
background of complete markets. Section 2 explains the importance and the difference 
between martingale, semimartingale, local semi-martingale and quadratic function, in order to 
proceed and better understand the different approaches. In Section 3 we give some 
preliminaries definitions and then in section 4, we explain the local risk-minimization theory, 
the minimal martingale measure approach and the mean-variance hedging. In Section 5 we 
study a particular case of incompleteness: due to information. In section 6 we give our 
conclusion on the argument. 
 
 
 1
1.  Complete Market 
 
 
In this section we describe in the discrete time case the model, we will use later. To do this we 
start with the definitions of the probability space and the financial market under the 
mathematical and statistical language. This will give to use the instrument to understand the 
martingale theory and the fundamental theorems in continuous time. 
 
1.1 The model 
 
We will work with a finite probability space     ,,P :F , with a finite number  :of points Ζ, 
each with positive probability:  ⊥    
0P  Ζ !. 
We specify a time horizon T, which is the terminal date for all economic activities considered. 
We use a filtration     
0
T
t
t  
  FF consisting of  ς-algebras 
01
...
T
      FF F: we take
 ⊥  
0
,    :F , the trivial  ς-field,    
T
P   :FF  (here     :P  is the power-set of  :, the 
class of all 2
 :
subset of   :). 
The financial market contains 1d    financial assets. The usual interpretation is to assume one 
risk-free asset (bond) labelled 0, and d risky assets (stocks) labelled 1 to d. the prices of the 
assets at time t are random variables,             
01
, , , ,..., ,
d
St St StΖ Ζ Ζsay, non-negative and 
ˆ
t
F -
measurable (adapted at time t, as price    
i
St). We write                 
01
, ,..., '
d
St S t S t S t  for 
the vector of prices at time t. The probability space     ,, :F P  is referred to be the set of 
trading dates, the prices process S and the information structure F , which is typically 
generated by the price process S, together as securities market model. 
It will be essential to assume that the price process of at least on asset follow a strictly 
positive process. 
 
Definition (1.1.1) A numeraire is a price process       
0
T
t
Xt
  
(a sequence of random 
variables), which is strictly positive for all  ⊥  0,1,...,tT . 
 
 2
A trading strategy (or dynamic portfolio)  Μ is a 
1d  
\  vector stochastic process 
                   
01
0
, , , ,..., , '
T
T
d
t
to
ttt Μ Μ Μ Ζ Μ Ζ Μ Ζ
  
  
 which is predictable (or previsible): each 
    
i
t Μ is 
1t  
F -measurable for 1t  τ. Here    
i
t Μ denotes the number of share assets i held in the 
portfolio at time t – to be determined on the basis of the information available before time t; 
the investor selects his time t portfolio after observing the prices     1St . However, the 
portfolio     t Μ  must be established before, and held until after, announcement of the 
prices     St. The components     
i
t Μ  may assume negative as well as positive values, reflecting 
the fact that we allow short sales and assume that the assets are perfectly divisible. 
 
Definition (1.1.2) The value of the portfolio at time t is the scalar product 
                    
0
:,
d
ii
i
Vt tSt tSt
 Μ
Μ Μ
  
 
ƒ
   1, 2,...,tT   and          010VS
 Μ
Μ . 
The process     ,Vt
 Μ
 Ζ is called the wealth or value process of the trading strategy Μ. 
 
The initial wealth    0V
 Μ
is called the initial investment or endowment of the investor. Now 
        1tSt Μ  reflects the market value of the portfolio just after it has been established at 
time 1t   , whereas        tSt Μ   is the value just after time t prices are observed, but before 
changes are made in the portfolio. Hence 
 
                  
1tStSt t StΜ Μ    ∋ 
 
Is the change in the time market value due to changes in security prices which occur between 
time 1t    and t. This motivates 
 
Definition (1.1.3) The gains process G
 Μ
 of trading strategy Μ is given by 
                    
11
:Gt t S S S
 Μ
Ω Ω
Μ Ω Ω Μ Ω Ω
 
    ∋
ƒ ƒ
,     1,2,...,tT  
 
 3
Definition (1.1.4). The strategy Μ is self-financing, Μ  ), if 
 
           1St t StΜ         1,2,..., 1tT  
 
When new prices 
    St
 are quoted at time t, the investor adjusts his portfolio from    t Μ  to 
    1t Μ  , without bringing in or consuming any wealth. 
 
Proposition (1.1.5) A trading strategy Μ belongs to  )if and only if 
 
          0Vt V Gt
Μ Μ  Μ

      0,1,2,...,tT  
 
We are allowed to borrow (so     
0
tΜ may be negative) and sell short (so     
i
tΜ may be negative 
for 1,2,...,id  ). So it is hardly surprising that if we decide what to do about the risky assets 
and fix an initial endowment, the numeraire will take care of itself, in the following sense. 
 
Proposition (1.1.6) If            
12
, ,..., '
d
tt tΜ Μ  Μis predictable and 
0
Vis 
0
F -measurable, 
there is a unique predictable process       
0
1
T
t
t Μ
  
such that     
01
, ,..., '
d
 Μ Μ Μ  Μ  is self-financing 
with initial value of the corresponding portfolio    
0
0VV
 Μ
  . 
Proof. If Μ is self-financing, then by (Definition 2.1.4), 
i
    
i
        
i
      
i
  
1
001
1
...
t
d
d
Vt VGt V S SΜ Μ
 Ω
 Μ Ω Ω Μ Ω Ω
  
 ∋    ∋
ƒ
. 
On the other hand, 
i
      
i
        
i
      
i
   1
01
...
d
d
Vt tSt t tSt tSt Μ Μ Μ Μ  Μ     . 
Equate these: 
        
i
      
i
 
   
i
      
i
   
 
11
00 1 1
1
... ...
t
dd
tV S S tSt tSt
 Ω
Μ Μ Ω Ω Μ Ω Ω Μ Μ
  
  ∋ ∋    
ƒ
,
which defines     
0
tΜ uniquely. The term in 
i
   iStare 
    
i
     
i
      
i
    1tS t tS t tS tΜ Μ  Μ  , 
 4
which is 
1t
F
  
-measurable. So 
        
i
      
i
  
           
i
 
 
1
1
00 1 11
1
... 1 ... 1
t
dd
tV S S tSt tSt
 Ω
Μ Μ Ω Ω Μ Ω Ω Μ  Μ
  
  
  ∋ ∋        
ƒ
, 
where as 
1
,...,
d
 Μ  Μare predictable, all terms on the right-hand side are 
1t
F
  
-measurable, so 
0
 Μ 
is predictable.          
            
  
             
This proposition has a further important consequence: for defining a gains process 
i
G Μ only 
the components             
12
, ,..., '
d
tt tΜ Μ  Μ are needed. If we require them to be predictable they 
correspond in a unique way (after fixing initial endowment) to a self-financing trading 
strategy. Thus for the discounted world predictable strategies and final cash-flows generated 
by them are all that matters. 
 
Definition (1.1.7) A contingent claim X with maturity date T is an arbitrary 
t
  FF-
measurable random variable (which is by the finites of the probability space bounded). We 
denote the class of all contingent claims by     
00
,,LL :F P . 
 
 5
1.2  Existence of Equivalent Martingale Measure 
 
The central and most important principle in any market model, is the no-arbitrage condition. 
Now we will define the mathematical part of this economic principle. 
 
Definition (1.2.1) Let 
i
 )   )be a set of self-financing strategies. A strategy 
i
 Μ  )is 
called an arbitrage opportunity or arbitrage strategy with respect to  )if      ⊥ 
001PV
 Μ
  , 
and the terminal wealth of Μ satisfies 
 
     ⊥ 
01PV T
 Μ
τ   and      ⊥  
00PV T
 Μ
! !. 
 
So an arbitrage opportunity is a self-financing strategy with zero initial value, which produces 
a non-negative final value with probability one and has a positive probability of a positive 
final value. Arbitrage opportunities are always defined with respect to a certain class of 
trading strategies. 
 
  Definition (1.2.2) We say that a security market M  is arbitrage-free if there are no 
arbitrage opportunities in the class  )of trading strategies. 
 
For example we can use this case. We observe a realization     ,St Ζof the price process    St. 
We want to know which sample point  Ζ  : we have. Information about  : is captured in 
the filtration  ⊥ 
t
  FF . In this setting we can switch to the unique sequence partitions 
 ⊥ 
t
P corresponding to the filtration ⊥ 
t
F . So at time t we know the set 
tt
A   Pwith
t
A Ζ  . 
Now recall the structure of the subsequent partitions. A set 
t
A  P is the disjoint union of sets 
12 1
, ,...,
kt
AA A
  
  P . Since     Suis 
u
F -measurable    St is constant on A and     1St  is 
constant on the
k
A , 1,2,..., K  . So we can think of A as the time 0 state in a single-period 
model and each 
k
A  corresponds to a state time 1 in the single-period model. We can therefore 
think of a multi-period market model as a collection of consecutive single-period markets. 
This is the effect of a “global” no-arbitrage condition on the single-period markets. 
 6
 
Lemma (1.2.3) If the market model contains no arbitrage opportunities, then for all 
 ⊥  0,1,..., 1tT   , for all self-financing trading strategies Μ  : and for any 
t
A  P , we have 
 ξ               
    
101 101Vt Vt A Vt Vt A
Μ Μ Μ Μ
  τ           
 
PP 
 ξ               
    
101 101Vt Vt A Vt Vt A
Μ Μ Μ Μ
  δ           
 
 
 
The conditions in the lemma are just the defining conditions of an arbitrage opportunity 
following the (2.2.1). They are formulated in a single-period model from t to t+1 with respect 
to the available information A Ζ  . The economic meaning: no arbitrage “globally” implies 
no arbitrage “locally”. 
The fundamental insight the single period example was the equivalence of the no-arbitrage 
condition and the existence of risk-neutral probabilities. For the multi-period case here is the 
explanation. 
Definition (1.2.4) A probability measure 
*
P on     ,
T
 :F equivalent to P  is called a 
martingale measure for S

if the process S

follows a 
*
P -martingale with respect to the 
filtration F . We denote by 
    
S

P the class of equivalent martingale measure. 
 
Proposition (1.2.5) Let
*
P be an equivalent martingale measure (     
*
S  P P ) and 
 Μ   ) any self-financing strategy. Then the wealth process 
i
   Vt Μ is a 
*
P -martingale with 
respect to the filtration F . 
 
 Proposition (1.2.6) If an equivalent martingale measure exists – that is, if 
i
    
S ζ  P - 
then the market M  is arbitrage-free 
 
Proof. Assume such a 
*
P exists. For any self-financing strategy Μ, we have as bifore 
i
            
i
   
1
0
t
Vt V S Μ
Μ
 Ω
Μ Ω Ω
  
 ∋
 ƒ
 
 7
Following Proposition 2.2.5, 
i
    St a (vector) 
*
P -martingale implies 
i
   Vt Μ  is a 
*
P -
martingale. So the initial and final 
*
P -expectations are the same, 
i
    
   
i
    
   
**
0EV T EVΜ Μ  
If the strategy is an arbitrage opportunity its initial value is zero. Therefore the left-hand side 
i
    
  
*
EV T Μ  is zero, but 
i
   0VT Μ τ (by definition). Also each  ⊥     
*
0Ζ !P  (by assumption, 
each  ⊥    
0 Ζ !P , so by equivalence each  ⊥     
*
0Ζ !P ). This and 
i
    0VT Μ τ force 
i
    0VT Μ  . So no arbitrage is possibile. 
            
  
 
Proposition (1.2.7) If the market M is arbitrage-free, then the class 
i
   
SP of 
equivalent martingale measures is non-empty. 
 
Theorem (1.2.8) (No-arbitrage Theorem). The market M  is arbitrage-free if and 
only if there exists a probability measure 
*
P equivalent to P  under which the discounted d-
dimensional asset price process
i
S is a 
*
P -martingale. 
 
 8
1.3 Risk-Neutral Pricing 
 
Before explaining which are the problems of finding a perfect pricing or hedging in an 
incomplete market, we would like to explain the complete market hypothesis and the no-
arbitrage condition. 
We say that a contingent claim is attainable if there exist a replicating strategy  Μ   )such 
that 
   VT X
 Μ
   
 
So the replicating strategy generates the same time T cash-flow as does X . 
Working with discounted values (using  Εas discount factor) we find 
            0TX V T V G T
Μ Μ
 Ε  

 
So the discounted value of contingent claim is given by the initial cost of setting up a 
replication strategy and the gains from trading. In a highly efficient security market we expect 
that the law of one price hold true, so that there exist only one price at any time instant. So the 
no-arbitrage condition implies that for an attainable contingent claim its time t  price must be 
given by the value (initial cost) on any replicating strategy (we the claim is uniquely 
replicated in this case). This is the basic idea of arbitrage pricing theory. 
 
Proposition (1.3.1) Suppose the market M is arbitrage-free. Then any attainable 
contingent claim X is uniquely replicated in M . 
 
Proof . Suppose there is an attainable contingent claim X and strategies Μ and ∴ such 
that: 
      VT VT X
Μ ∴
    , 
but there is a TΩ   such that 
        Vu Vu
Μ ∴
 for every u  Ω and       VV
Μ ∴
Ω Ω ζ . 
Define         ⊥  
::, ,AVV
Μ ∴
 Ζ Ω Ζ Ω Ζ  : !, then AF
 Ω
 and    0PA !. Define the F
 Ω
-
measurable random variable       :YV V
Μ ∴
Ω Ω  and consider the trading strategy  [defined by 
 9
    
      
             
,
1 1 ,0,...,0 ,
c
AA
uu
u
uu Y
Μ ∴
 [
Μ ∴  Ε Ω
↑
 °
→
 °
↓
    
u
uT
 Ω
 Ω
 δ
  δ
 
The idea here is to use  Μ and  ∴ to construct a self-financing strategy with zero initial 
investment (hence use their difference [) and put any gains at time  Ωin the savings account 
(i.e. invest them risk-free) up to time T. 
We need to show formally that  [satisfies the conditions of an arbitrage opportunity. By 
construction  [is predictable and the self-financing condition is clearly true for t  Ω ζ, and for 
t  Ω   we have using that ,Μ ∴   ) 
                            
,SSVV
Μ ∴
 [ Ω Ω Μ Ω ∴ Ω  Ω  Ω  Ω    
 
                          
0
1111 1
c
AA
YS [ Ω Ω Μ Ω ∴ Ω  Ω Ε Ω Ω    
                                                       
1
c
SVV
Μ ∴
 Μ Ω ∴ Ω Ω Ω Ω Ε Ω Ε Ω
  
  
                             .VV
Μ ∴
Ω Ω   
Comparing these two, [ is self-financing, and its initial value is zero. Also 
                      
1 1 ,0,...,0
c
A
A
VT T T ST Y ST
 [
Μ ∴ Ε Ω     
The first term is zero, as       VT VT
Μ ∴
 . The second term is 
      
0
10
A
YST Ε Ω τ 
As Y>0 on A, and indeed 
    ⊥    ⊥ 
00PV T P A
 [
!   !. 
Hence the market contains an arbitrage opportunity with respect to the class  ) of self-
financing strategies. But this contradicts the assumption that the market M is arbitrage-free. 
            
  
This uniqueness property allows us now to define the important concept of an arbitrage price 
process. 
 
Definition (1.3.2.) Suppose the market is arbitrage-free. Let X be any attainable 
contingent claim with time T maturity. Then the arbitrage price process     
X
t Σ , 0 tTδ δor 
simply arbitrage price of X is given by the value process of any replicating strategy Μfor X .