In the paper the one-period (B, S) - market model with two securities is considered. In the introduction a replicating portfolio for the derivative security is obtained when the securities are primitive. In the second part we consider the situation where the stock price could have more than two values. We build an approximation of such a market using a fictive (B, S*) - market where the stock price could take only two values. In the third part the numerical illustration of the approximation is given

Lappo, P.

Zuev, N.

English

Электронная библиотека БГУ

1 
 
 
 
DERIVATIVES PRICING IN INCOMPLETE MARKETS 
Petr Lappo , Nickolay Zuev  
________________________________________________________________ 
Belarusian State University,  
Minsk, Belarus 
e-mail:lappopm@bsu.by 
60; Financial Risks (AFIR) 
      In the paper the one-period (B, S) - market model with two securities is considered. In 
the introduction a replicating portfolio for the derivative security is obtained when the 
securities are primitive. In the second part we consider the situation where the stock price 
could have more than two values. We build an approximation of such a market using a fictive 
),( *SB -market where the stock price could take only two values. In the third part the 
numerical illustration of the approximation is given. 
        Key words: incomplete market, derivative pricing, call option 
1. Introduction    
     We consider the one- period market model where there are two assets and two outcomes.  
Their time-0 prices are   0B , and  0S . At time 1 their prices are 01 )1()ω( BrB j +=  , and 
2,1,0),ω(1 =≥ jrS j . We note that the first security is riskless and the second security is 
risky. Such a model is a special case of ),( SB  -market (see, for example, [1,3]). We assume 
that all prices are strictly positive and all securities are infinitely divisible. There are no 
transaction costs. To ensure a complete market, the returns on these securities must be 
linearly independent. This implies that the payoff matrix  






)ω()ω(
)ω()ω(
2121
1111
SB
SB
 
is nonsingular so that its determinant is nonzero. 
      Suppose we have a derivative security whose payoff at time 1 is a function of )ω(1 jS , 
that is  
.2,1),ω(( 1 == jSfV jj  
2 
 
 
 
We can construct a portfolio of these two securities that has the same value as the derivative 
security in each state at time 1. Assume the portfolio consists of 1θ  units of 1B  and 2θ  units 
of 1S . We must have 
)ω(θ)1(θ
),ω(θ)1(θ
212012
112011
SBrV
SBrV
++=
++=
.
 
A unique solution of this system of two linear equations exists because of the complete 
market assumption and we have    
,)ω()1()ω()1(
)ω()ω(
θ
110210
112211
1 SBrSBr
SVSV
+−+
−
=                                                               (1)                                                       
.)ω()ω()ω()1()ω()1(
)1()1(
θ
1121
12
110210
0102
2 SS
VV
SBrSBr
BrVBrV
−
−
=
+−+
+−+
=                                    (2) 
If there is no an arbitrage, then the time-0 value of   the portfolio must be equal to the time -0 
price of the derivative. Hence the time-0 price value is 
                                   
02010 θθ SBV += ,                                                                              (3) 
where 1θ  and 2θ  are given by (1) and (2). It is convenient to simplify the notations. 
Let )1()ω( 011 dSS += , )1()ω( 021 uSS += . If there is no an arbitrage, then 
dru >> , and the equation (3) takes the form 
)))(1(
)))(1((
))(1(
)))(1(( 00
0 dur
druSf
dur
rudSfV
−+
−+
+
−+
−+
=
. 
Hence 
                          
20100 ))1((1
1))1((
1
1 quSf
r
qdSf
r
V ⋅+
+
+⋅+
+
=
                                   (4) 
where 
du
drq
du
ruq
−
−
=
−
−
= 21 ,
.
 
are  risk neutral probabilities. Denoting corresponding probability measure  
Q ,   we can rewrite (4) in the form 
3 
 
 
 




+
=
r
SfEV Q
1
)( 1
0 . 
This equation is well known (see, for example [2]). Further without loss of the generality we 
can assume 110 == BB  (see [1, 3]), and 0=r . 
2. Approximation of incomplete markets. 
      From the previous part we can conclude that if the stock price has only two values we can 
construct the replicating portfolio )θ,θ( 21 , and the derivative price is given by (1)-(3).  In [4] 
Takahashi   analyzed the situation where the stock price can take 3 values at time 1 and 
presented an approach of choosing the martingale measure which is the combination of the 
method of least squares and the embedded complete markets.  
       In this part we consider the situation where the stock price at time 1 can take more than 
two values. The equation (4) could not be applied because a risk neutral measure Q  is not 
unique. In this situation we introduce a fictive market ),( *SB  which is complete, and in 
some sense is close to our market.  In this fictive market the stock price takes only two 
values. To introduce the fictive market we need a measure of closeness. It is possible to use 
an expectation of some function .),( 21 yyR  of )(Sf  and the payoff of the   replicating 
portfolio in ),( *SB  market. 
      More formally, let )ρ1(01 += SS , )ρ1( *0*1 += SS , where ρ
 
and *ρ  are  random 
variables. We assume, that the random variable *ρ   values are u  and d,  
pdu ==−== )ρPr(1)ρPr( ** . The time-1 prices of the securities 1S  and *1S  are 
completely determined by ρ
,0S   , and     *ρ .  We can choose the distribution parameters of 
*ρ  to make the fictive market closer to the   (B, S) - market. So we have the problem: 
                                  
pdu
SrBSfER
,,
02011 min))ρ1(θ)1(θ),(( →+++   .                               (5)                                      
In the problem (5) the resulting portfolio depends on u  and d . In practice the problem could 
be solved numerically by using empirical distributions. 
Further we assume that  22121 )(),( yyyyR −=  . Then 
=−−−=
=+++
2*
02
*
11
02011
))ρρ(θ)()((
))ρ1(θ)1(θ),((
SSfSf
SrBSfR
 
4 
 
 
 
2
*00*
00 )ρρ(
))1(())1(()ρ1(()ρ1(( 





−
−
+−+
−+−+=
du
dSfuSfSfSf . 
We will look for the random variable *ρ  in the form 
                                         



<
≥
=
.ρ,
,ρ,
ρ
*
cifd
cifu
                                                                   (6)                                                                               
Thus the problem (5) in our case is to find such *ρ  of the type (6), which minimizes 
2
*00*
00 )ρρ(
))1(())1(()ρ1(()ρ1(( 





−
−
+−+
−+−+
du
dSfuSfSfSfE .                        (7)       
3. Numerical results   
     Having the observations   of the  ),,( 1100 nnnn SBSB  in the   discrete time  moments 
Nn ,...,2,1=
 and assuming that 1
1
1
11ρ
−
−
−
=
n
nn
n
S
SS
 are independent identically distributed 
random variables that have the same distribution as ρ  we can minimize empirical estimator 
of (7) numerically. 
The problem to be solved is 
cdu
nn
nn
n
n
n
n
N
n
du
dSfuSf
SfSf
N
,,
2
*00*
00
1
min
)ρρ())1(())1(()ρ1(()ρ1((1
→
→








−
−
+−+
−+−+∑
=
     (8)                                                                                                                             
where  



<
≥
=
.ρ,
,ρ,
ρ*
cifd
cifu
n
n
n
 
The no- arbitrage condition gives us the constrains 0<u  and 0>d . Remind, that we assume
NnBB nn ..,,2,1,1
10
=== . 
      Here we assume that a derivative security is a 1- period European call option. Then  
)0,max()( 11 KSSf −= , 
5 
 
 
 
where K -is  the strike price of the call. We have performed the calculations for two choices 
of
 K. For the first one  
NnSKK nn ,...,2,1),ρ1(01 =+== , 
∑
=
=
N
n
nρN 1
1
ρ , 
for the second :    
∑
=
=
N
n
nS
N
K
1
0
2 1
. 
From (1)-(2) for the first choice of   K   the replicating portfolio   approximation is given by 
the equalities 
,
)1)(0,)ρ1()1(max()1)(0,ρ)1()1(max(
θ
**
*
0
*
0
*
0
*
0
1 du
dSuSuSdS nnnnn
−
++−+−++−+
=  
.
)0,)ρmax()0,)ρmax(
θ
**
**
2 du
dun
−
−−−
=  
 Here  *u  and *d   are the solutions of (8).We can note that n2θ  does not depend on n. 
For the second choice of   K    
**
*2*
0
*2*
0
1
)1)(0,)1(max()1)(0,)1(max(
θ
du
dKuSuKdS nnn
−
+−+−+−+
= , 
**
2*
0
2*
0
2
)0,)1(max()0,)1(max(
θ
du
KdSKuS nnn
−
−+−−+
= . 
For the numerical illustration we consider the daily data of 3 Russian stocks SBER, GMKH, 
EERS since 01.04. 2003. We use N=50, and a derivative security is 1-month European call 
option. 
In Table 1 we can see the parameters of the −),( *SB  market with the strike price 1K
. In the 
sixth column Error is the square root of the minimum value of (8). This is the mean square  
error of the payoff function approximation. In  Figures 1-3 the values of nθ1  are shown 
6 
 
 
 
Table1 
Stock *d  *u  *c  2q  Error Coefficient 
of variation 
n
2θ  
SBER 
-0,004843 0,104167 0,02457 0,044424 
 
2.447484 
 
0,478456 
 
0.577214 
GMKH 
-0.032258 0.078838 0.071584 0.290361 
 
0.484335 
 
0.758302 
 
0.390904 
EERS  
-0.027388 
 
0.145867 
 
0.042231 
 
0.158087 
 
0.003529 
 
0.751422 
 
0.555813 
 
Figure 1 
 
Figure 2 
 
 
-12
-10
-8
-6
-4
-2
0
0 10 20 30 40 50 60
θ1 GMKH
θ1 GMKH
-0,1
-0,08
-0,06
-0,04
-0,02
0
0 20 40 60
θ1 EERS
θ1 EERS
7 
 
 
 
Figure 3 
 
 
In Table2 the parameters of −),( *SB  market for  2KK =  are given 
Table 2 
 
Stock 
 
*d  
 
*u  
 
*c  
 
2q  
 
Error 
Coefficient of 
variation 
 
SBER 
 
-0,000927 
 
 
0,120419 
 
0,006835 
 
0,007641 
 
 
1.34778 
 
 
0,147891 
 
 
GMKH 
 
-0,0082988 
 
 
0,075051 
 
 
0,071584 
 
 
0,099565 
 
 
0,226183 
 
 
0,29262 
 
 
EERS 
-0.07635 0.17636 -0.0735 0.302113 0.003561 
 
0.533157 
 
In Figure 4 nθ2  are shown for 
2KK =
 
 
-130
-125
-120
-115
-110
-105
0 10 20 30 40 50 60
θ1 SBER
θ1 SBER
8 
 
 
 
Figure 4 
 
Figures 5-7 illustrate the dependence nθ1  on n.   
Figure 5                                                                    
    
Figure 6                                                
                                    
  
0
0,2
0,4
0,6
0,8
1
1,2
0 10 20 30 40 50 60
θ2 EERS
θ2 GMKH
θ2 SBER
0
0,5
1
1,5
0 20 40 60
θ1 EERS
θ1 EERS
-30
-20
-10
0
0 20 40 60
θ1 GMKH
θ1GMKH
9 
 
 
 
Figure 7 
 
The values of the variation coefficients give a possibility to make outlines on the adequacy of 
the model under consideration. We see that the best result is for the SBER and GMKH stocks 
where 2KK =
.   
 
 
 References 
1. Shiriaev, A. N.  K teorii raschetov opcionov evropeyskogo i americanskogo tipov. 1. 
Diskretnoe vremia. / Shiriaev A. N. , Kabanov Y.M., Kramkov D.O., Melnikov A.V. // 
Teoriya veroyatn. i eyo primen. 1994.V.39.1.P.23-77.(in Russian). 
2. Panjer, H.H. Financial Economics: With applications to Investments, Insurance and Pensions/   
H. Panjer  et all//Schaumburg.:1998, 670 p. 
3. Shiriaev, A.N. Osnovy finansovoy I stohasticheskoy matematiki.Tom 2. /A.N,Shiriaev// 
Moskow: Fazis.1998. 544p. (in Russian). 
4. Takahashi H. A Note on Pricing Derivatives in an Incomplete Market./H.A.Nakahashi// 
Hitotsubashi University .2000.Discussion paper #2000-05.13p. 
-250
-200
-150
-100
-50
0
0 20 40 60
θ1 SBER
θ1 SBER


Derivatives Pricing in Incomplete Markets

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Derivatives Pricing in Incomplete Markets

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