








































DYNAMIC ECONOMETRIC MODELS


© 2012 Nicolaus Copernicus University Press. All rights reserved.  
http://www.dem.umk.pl/dem 

DYNAMIC ECONOMETRIC MODELS 
Vol. 12 (2012) 53−71 

Submitted October 29, 2011  ISSN 
Accepted July 1, 2012 1234-3862 

Maciej Kostrzewski* 

Bayesian Pricing of the Optimal-Replication Strategy 
for European Option in the JD(M)J Model† 

A b s t r a c t. In incomplete markets replication strategies may not exist and pricing of derivatives 
is not an easy task. This paper presents an application of Bertsimas, Kogan and Lo’s algorithm of 
determining an optimal-replication strategy. In the Merton model the likelihood function is 
a product of a mixture of infinite number of components. In the paper this number is assumed to 
be equal to a fixed value M+1. To determine the optimal strategy, we should estimate unknown 
parameters. To this end we resort to Bayesian estimation techniques. The presented methodology 
is exemplified by an empirical research. 

K e y w o r d s: incomplete markets, Bayesian inference, jump-diffusion process, pricing of deriv-
atives. 

J E L Classification: C11, C15, C58, C61, C63, G17 

Introduction 
 The Black-Scholes model assumes a continuous path of underlying instru-
ment values. Pricing European options under the Black-Scholes model is an 
easy task (Black, Scholes, 1973). Unfortunately, if we additionally include 
a component responsible for jumps, we obtain a model of a risky instrument in 
an incomplete market. Then, the pricing of options is more of a challenge. 
In general, replication strategies do not exist (Lamberton, Lapeyre, 2000; 
Shreve, 2004). Bertsimas, Kogan and Lo (2001) propose an algorithm of deter-
mining an optimal-replication strategy. The optimality is understood in a mean-
squared sense. Apart from the major drawbacks of the strategy – it is self-
                                                 
* Correspondence to:  1. Maciej Kostrzewski, Department of Econometrics and Operational Re-
search, Cracow University of Economics, ul. Rakowicka 27, 31-510 Krakow, Poland or 1. Maciej 
Kostrzewski, Faculty of Applied Mathematics, AGH University of Science and Technology, al. 
Mickiewicza 30, 30-059 Krakow, Poland, e-mail: maciej.kostrzewski@uek.krakow.pl 

† The research was supported by the Polish Ministry of Science and Higher Education. Re-
search Project 2010-2012; No. N N111 429139. 



Maciej Kostrzewski 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

54 

financing but it is not an admissible strategy, and its calculation may be time-
consuming – it plays a significant role in hedging and pricing derivatives. 
 To determine the optimal-replication strategy, the unknown parameters of 
the model need to be estimated. In this paper we resort to the Bayesian estima-
tion techniques. Accounting for the parameter uncertainty inherent to the esti-
mated parameters, for which the Bayesian methodology is widely appreciated, 
is relevant not only to the estimation itself, but extends also to the pricing of 
optimal strategy, providing the researcher with a full (posterior) distribution of 
the strategy cost (instead of a single value). 

1. The Jump-Diffusion Model with M-jumps 
 Let ( )P,,ℑΩ  denote a probability space. We consider a standard Wiener 
process ( ) 0≥= ttWW , a Poisson process ( ) 0≥= ttNN  with intensity 0>λ , and 
a family of independent random variables { },...2,1: == jQQ j . The variables 

jQ ’s have Gaussian distributions: ( )2,~ QQj NQ σµ . It is also assumed that  
σ –algebras generated by W, N and Q are independent. 
 In the Merton model (Merton, 1976) the price of a risky instrument is gov-
erned by a jump-diffusion process ( ) 0≥= ttPP which is the solution of the equa-
tion: 

( ) .1 tt
Q

tttt dNPedWPdtPdP −++= σµ  

The first two elements on the right-hand side define a pure diffusion process. 
The last element corresponds to jumps. ( ) 0≥ttP  is an adapted and right-
continuous process. It can be shown that: 

.
2
1exp

1

2
0













++





 −= ∑

=
j

N

j
tt QWtPP

t

σσµ  

The price between consecutive jumps is governed by a geometric Brownian 
motion. The process P  has a finite number of jumps on each interval [ ]t,0 . 
The logarithm of the price is the solution of the equation: 

.
2
1ln 2

ttt QdNdWdtPd ++





 −= σσµ  

Hence, for a given time step 0>∆  we obtain: 

( ) ( ) ( ) .
2
1lnln

1

2
j

N

Nj
tttt QWWPP

t

t

∑
∆+

+=
∆+∆+ +−+∆






 −+= σσµ  

It follows that the probability density function of logarithmic rates of return 
is given by (Hanson, Westman, 2002): 



Bayesian Pricing of the Optimal-Replication Strategy for the European Option… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

55 

( )( ) ( ) ( ) ( )( )kkxxp QQkk

k

tP
tP

222
2
1

!0ln
,;exp σσµσµφλ λ +∆+∆−∆−∑= ∆

∞

=
∆+ ,   (1) 

where ( )2,; sm⋅φ  denotes the density of a normal distribution with mean m  and 
variance 2s . Therefore, the likelihood function is given by the product of an 
infinite mixture of normal densities, which highly complicates the statistical 
inference for the model. 
 In order to define a jump-diffusion model with M jumps, let us consider 
a finite approximation of series (1): 

( ) ( )








+∆+∆






 −

∆
∆−∑

∞

=

kkx
k QQ

k

k

222

0
,

2
1;

!
exp σσµσµφλλ , 

( ) ( ) ,,
2
1;

!
exp 222

0








+∆+∆






 −

∆
∆−≈ ∑

=

kkx
k QQ

kM

k
σσµσµφλλ  (2) 

where { }...,2,1,0∈M  is a fixed constant. In the Black-Scholes framework, the 
process of logarithmic returns of risky instrument is governed by an arithmetic 
Brownian motion which is a pure diffusion process. Under 0=M  the above 
approximation collapses to the density of this arithmetic Brownian motion with 
time step ∆ (Kloeden, Platen, 1992). In the general case the integral of sum (2) 
may not equal one. Therefore, to obtain a probability density function, let us 
normalize the approximation given by (2): 

( ) 







+∆+∆






 −= ∑

=

kkxwxp QQik

M

k
i

222

0
,

2
1; σσµσµφθ , (3) 

with ,
!
)(

!
)(

1

0

−

= 






 ∆∆
= ∑M

j

jk

k jk
w λλ  Mk ...,,0= , being the normalizing weights1.  

In the paper the logarithmic rates of return ( ),, 21 xxx = are assumed to follow 
the distribution defined by (3), and the resulting model is termed the jump-
diffusion model with M jumps, or JD(M)J, in short. The construction of the 
process restricts the number of jumps over any time interval ∆  to M, with the 
magnitude of each jump model with normal distribution ( )2, QQN σµ . Finally, let 
us note that the jump-diffusion specification under study is some approximation 
to the Merton model. Therefore, and on a more statistical note, estimators of 
JD(M)J model parameters could be treated as approximations of the Merton 
model parameters.  

                                                 
1 See Frühwirth-Schnatter (2006) for a thorough exposition on mixture modeling. 



Maciej Kostrzewski 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

56 

2. Bayesian Inference for the JD(M)J Model 
 In the JD(M)J model there are five unknown parameters 
( ) Θ∈22 ,,,, QQ σµλσµ , where ( ) ( ) ( ) 5,0,0,0 RRR ⊂∞××∞×∞×=Θ . The 
likelihood function is given by: 

( ) .,
2
1; 222

01








+∆+∆






 −= ∑∏

==

kkxwxp QQik

M

k

n

i
σσµσµφθ  (4) 

 If we observe a path of some JD(M)J process we do not  know whether the 
observations or which of them have resulted from jumps. Moreover, we are not 
able to (directly) identify an impact of the pure diffusion process and the jumps. 
In other words, we do not know which component of sum (3) is “responsible” 
for each observation. To solve this problem we introduce latent variables 

( )nZZZ ...,,1=  such that { }MZi ...,,1,0∈  and ( ) ji wjZP == θ , where 
{ }ni ...,,1∈  and { }Mj ...,,1,0∈ . By means of iZ ’s, we can assess the contribu-

tion made by the jumps (as compared with the pure diffusion compound) to 
explain each of n observations. Increments of the Poisson process N  are inde-
pendent, variables nZZ ,...,1  are also independent. Consequently, 

( ) 







+∆+∆






 −= iQiQiii ZZxZxp 222 ,

2
1;, σσµσµφθ  

and  

( ) .,
2
1;, 222

1








+∆+∆






 −=∏

=
iQiQi

n

i
ZZxZxp σσµσµφθ  

It is advisable to consider the following reparametrization of the model parame-
ters:  

∆= λL , 2
1
σσ =h , 2

1
Q

Qh
σ

= , 

under which vector of all the n+15 unknown quantities is given by: 
( ) ( ).,,,,,...,,, 1 QQn hLhZZZ µµθ σ=  

The Bayesian model is defined by the joint distribution of the observables x, the 
hidden variables Z and the parameters θ : 

( ) ( ) ( ).,,,, θθθ ZpZxpZxp =  

Under a common assumption of the mutual prior independence among the co-
variates of θ , the joint prior density is formed: 

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),,
1

θµµθθθ σ i

n

i
QQ ZphppLphpppZpZp

=
∏==  



Bayesian Pricing of the Optimal-Replication Strategy for the European Option… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

57 

where: 

( ) ( ) ( ) ,,,0for 
!!

 ,
1

0
Mk

jk
wwjZP

jM

j

k

kji =










 ∆∆
===

−

=
∑ λλθ  

( ) ( ),,; 2
µµµφµ smp =  

( ) ( ) ( ),2/exp2/2
σ

ν
σσ

σ Ahhhp −∝ −  

( ) ( ) ( ),2/exp2/2 LLLp L −∝ −ν  

( ) ( ),,; 2
QQQQ smp µφµ =  

( ) ( ) ( ).2/exp2/2
QQQ Bhhhp Q −∝ −ν  

Such a choice of the prior structure (normal distributions for µ  and Qµ , the 

gamma distributions 







2
,

2
AGamma σν  and 








2

,
2

BGamma Qν  for σh  and Qh , 

respectively, and the 2

Lνχ  distribution for L ) densities guarantees that the poste-
rior density is a bounded function even though the likelihood function is unbo-
unded (Lin, Huang, 2002). The prior structure is determined under 1== BA , 

∆= 10Lν , 01.0=µm , 12 =µs , 5=σν , 01.0=Qm , 12 =Qs , 5=Qν . 
 Posterior characteristics of the unknown quantities are calculated via the 
Markov Chain Monte Carlo (MCMC) methods (Gamerman, Lopes, 2006), 
combining the Gibbs sampler, the independence and the sequential Metropolis-
Hastings algorithms, as well as the acceptance-rejection sampling. For more 
details on the technicalities we refer to Kostrzewski (2011). 

3. The Optimal-Replication Strategy 
 Pricing and hedging derivatives are among investors’ fundamental prob-
lems. Investors employ replication strategies to hedge derivatives. Unfortunate-
ly, in the case of incomplete markets such a strategy may not exist. Some idea is 
to create a self-financing strategy, the value of which is “close” (at maturity) to 
the one of the derivative’s payoff function. We apply the results of Bertsimas, 
Kogan and Lo (2001) to define and calculate the optimal-replication strategy for 
portfolios comprising a risky asset and riskless bonds. The approach involves 
buying, selling, borrowing and lending the portfolio constituents. Let tP  and tB  
denote price of the risky instrument and value of the riskless investment at 

Tt ≤≤0  respectively. The payoff of an European option at maturity T is deno-
ted by ( )TPF . Finally, let tθ  be the amount of stocks in the portfolio at time t. 
Then tttt BPV +=θ  is the value of the portfolio at t. Bertsimas, Kogan and Lo 
(2001) consider a mean-squared-error criterion to define the optimal-replication 



Maciej Kostrzewski 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

58 

strategy *
tθ , under which *

tV  is the value of the optimal portfolio. It follows that 
*

tV and *
tθ  minimize: 

( )[ ]( )0
2 VPFVE TT −  

over { }tV θ,0 . Moreover, 

{ }
( )[ ]( )0

2

,0

min VPFVE TTV t

−=∗

θ
ε

 
 

constitutes the minimum replication error, that is an error of fitting the strategy 
into the payoff F at T. If the replication strategy exists, then 0=∗ε  and 

( )TT PFV =* . The error ∗ε  is construed as a relative measure of the market in-
completeness, with its relativity justified by ∗ε  corresponding only to a given 
derivative and a given model. To evaluate the optimal-replication strategy Bert-
simas, Kogan and Lo (2001) make some additional assumptions: 

1. There are no taxes and transaction costs. 
2. Purchasing, selling, borrowing and shortsale are possible without any re-

strictions. 
3. The borrowing and lending interest rate r is constant and equal zero. 
4. P  is a Markov process. 
5. Trading takes place at known and fixed times { }Ni ttt ,...,0∈ , where 

Ttt N =<= 00 . 

To simplify the notation let iti ≡ . The aim is to calculate strategy ( )ii PVi ,,∗θ , 

the initial value ∗
0V  of the optimal portfolio and the error ∗ε . Let ix  be a loga-

rithmic rate of return such that ( ).explnexp 1
1 ii

i

i
ii xP

P
PPP =


















= +

+  Bellman’s 

principle of optimality (Bertsekas, 1995) yields the following theorem: 

Theorem 1 (Bertsimas, Kogan, Lo, 2001) 

If ( )
( )

( )( )( )iiNN

Nki
PVkiii PVPFVEPVJ

kk

,min, 2

1
,,

−=
−≤≤

θ
, then: 

( ) ( )( )
( )

( )
( )( ),,,min,

,,

111,,

2

iiiiiPViiii

NNNNN

PVPVJEPVJ
PFVPVJ

ii
+++=

−=

θ

 

for 1,...,0 −= Ni . 



Bayesian Pricing of the Optimal-Replication Strategy for the European Option… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

59 

 Theorem 1 suggests that the strategy is set recursively. The problem of op-
timal replication is solved via stochastic dynamic programming. The main re-
sults are formulated in the theorem below. 

Theorem 2 (Bertsimas, Kogan, Lo, 2001) 
Under the above assumptions: 

1. There are functions: ( )ii Pa , ( )ii Pb  and ( )iPc , such that: 

( ) ( ) ( )[ ] ( )iiiiiiiiii PcPbVPaPVJ +−− 2, , .,...,0 Ni =  

2. ( ) ( ) ( )iiiiiii PqVPpPVi −=∗ ,,θ , where functions ia , ib , ic , ip  and iq  are 
evaluated recursively. Starting with 1)( =NN Pa , )()( NNN PFPb = , 

0)( =NN Pc , the calculations for 0,...,1−= Ni  proceed as follows: 

( ) ( ) ( ) ( )[ ]
( ) ( )[ ] ,2

111

11111

iiiii

iiiiiii

PPPPaE

PPPPbPaE
ii Pp

−⋅

−⋅⋅

+++

+++++=  

( ) ( ) ( )[ ]
( ) ( )[ ] ,2

111

111

iiiii

iiiii

PPPPaE

PPPPaE
ii Pq

−⋅

−⋅

+++

+++=  

( ) ( ) ( ) ( )( ) ],1[ 2
111 iiiiiiiii PPPPqPaEPa −⋅−⋅= +++  

( ) ( ) ( ) ( ) ( ) ( )( )⋅−⋅−⋅⋅= +++++ iiiiiiiiPaii PPPpPbPaEPb
ii 11111

1 [

( ) ( )( ) ],1 1 iiiii PPPPq −⋅−⋅ +  
( ) ( ) ( ) ( ) ( )( ) +−⋅−⋅= +++++ ][ 2

11111 iiiiiiiiiii PPPPpPbPaEPc

( ) ( ) ( ) .][][ 2
11 iiiiiii PbPaPPcE ⋅−+ ++  

3. ( ) 0>ii Pa , ( ) 0≥ii Pc  for 0,...,1−= Ni . 

4. Under the optimal-replication strategy ( )ii PVi ,,∗θ  we obtain: 
( ) ( ) ( )[ ] ( ),, 00

2
00000000 PcPbVPaPVJ +−=  ( )000 PbV =∗  and ( ).00 Pc=∗ε  

Properties (Bertsimas, Kogan, Lo, 2001) 

a) The error of replication ( )00 Pc=∗ε  is the same for put and call options. 

b) If prices tP  follow a geometric Brownian motion and ∞→N , then the 

cost ∗
0V  of the optimal-replication strategy converges to the Black-Scholes 

price. 
c) ( )00 Pb  meets the put-call parity.  

d) ( )ii PVi ,,∗θ  is the self-financing strategy which does not guarantee 0≥∗
iV . 

e) The value of the optimal-replication strategy could be lower or higher at  
maturity T than the value of the payoff function. 



Maciej Kostrzewski 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

60 

 The optimal-replication strategy could be less or more attractive than other 
strategies, e.g. the delta-hedging strategy. It is because the optimal-replication 
strategy is optimal only in the mean-squared-error sense. In general, calculation 
of expectations defined in Theorem 2 may not be straightforward. In the case of 
the JD(M)J models numerical techniques should be employed to approximate 
their values. 

 To calculate the cost of the optimal-replication strategy *
0V  and the relative 

measure of market incompleteness ∗ε  the conditional expectations defined in 
Theorem 2 need to be evaluated. Obviously, these are given by relevant inte-
grals, as for instance: 

( ) ( )[ ]=−⋅ +++ iiiii PPPPaE 111  
( )( ) ( )( ) ( ) =⋅−⋅∫= +

∞
∞− dxPxpPxPxPa iiiii expexp1  

( )( ) ( )( )
( ) ,2

1

exp
2exp2exp

0

dy
y

PymPymPaw
M

k

ikkikkiik −+⋅+∞
∞−

=

+∫= ∑ σσ
π

 

where ( ) ( )[ ] 1

!0!

−
∆

=
∆ ∑= j

M
jkk

jk

w λλ , ( ) km Qk µσµ +∆−= 2
2
1  and kQk

222 σσσ +∆= . 
Analytical calculations of such formulae are difficult or positively impossible, 
which is why numerical approximations, such as the piecewise cubic Hermite 
interpolation and Gauss-Hermit quadrature, are utilized. All numerical calcula-
tions are carried out in R using the pracma and glmmML packages. 

4. Empirical Studies 
 In this section we present the results of Bayesian estimation, model compar-
ison and pricing of the optimal-replication strategy. The calculations are per-
formed for two stock market indices WIG20 and S&P100. 
 The WIG20 is a stock market index comprising 20 biggest and most liquid 
companies on the Warsaw Stock Exchange (WSE)2. The considered time series 
x  consists of 946 daily logarithmic rates of return on the WIG20 index closing 
quotations from June 5, 2007 to March 11, 20113.  
 The S&P100 index includes 100 leading US stocks recorded by Standard & 
Poor’s. The considered data x  contains 1,077 daily log-returns on the index 
over a period from April 2, 2007 to July 8, 20114.  
 Daily closing quotations of the WIG20 and S&P100 indices are presented in 
Figure 1, whereas Figure 2 plots the logarithmic rates of return.  

                                                 
2 www.gpw.pl. 
3 The data were downloaded from www.gpwinfostrefa.pl. 
4 The data were downloaded from http://finance.yahoo.com/. 



Bayesian Pricing of the Optimal-Replication Strategy for the European Option… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

61 

WIG20 

 

S&P100 

 
Figure 1. Daily closing quotations of the WIG20 and S&P100 indices 

 The horizontal lines present bands of plus/minus two or three standard devi-
ations (dashed and dotted lines, respectively).  

WIG20 

 

S&P100 

 
Figure 2. Daily logarithmic rates of return on the WIG20 and S&P100 closing quota-

tions 

As evidenced in Figure 2 the outlying log-returns on the S&P100 index are 
more prominent than the ones featured by the WIG20 series, which may hint at 
the jump component playing a more crucial role in modeling the former. 

4.1. WIG20 

 It is assumed that time interval between consecutive observations equals 
=∆ 1/252. For the WIG20 series we restrict the analysis to two model specifi-

cations: JD(0)J (i.e. a pure diffusion process) and JD(1)J (i.e. the one allowing 
for a single jump over a given time interval ∆).  



Maciej Kostrzewski 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

62 

4.1.1. General Results 

Table 1 presents posterior means and standard deviations (in parentheses) of 
the parameters. The results are based on 600,000 and 1,000,000 draws of poste-
rior distributions, preceded by 10,000 and 300,000 burn-in passes for M=0 and 
M=1, respectively. The results of the MCMC sampler are robust to the choice 
of the starting points. Convergence of the chains is confirmed by the CUMSUM 
statistics (Yu, Mykland, 1998), as well as the ergodic means and standard devia-
tions plots. 
 Noteworthy, posterior characteristics of the pure diffusion parameters, i.e. 
µ  and 2σ , are close to their JD(1)J counterparts, which may hint at there being 
no need for jumps to be accounted for. The conclusion is also supported by the 
close to zero posterior mean of the jump intensity λ, accompanied with relative-
ly large posterior dispersion of the parameters 

Table 1. Posterior means and standard deviations (in parentheses) of the parameters for 
the WIG20 index 

Parameters JD(0)J JD(1)J 

λ –  0.0557 (0.3053) 

µ -0.0364 (0.1556) -0.0346 (0.1545) 

µQ –  0.0085 (0.9770) 

σ2 0.0917 (0.0042) 0.0919 (0.0044) 

σ2
Q –  0.3520 (1.3100) 

 We now focus on the choice of the appropriate value of M. The model with 
the highest posterior probability is referred to as the best one. The best model 
points the value of M. We have to compare: 

( ) ( ) ( )
( ) ( ) ( ) ( )JJDxPJJDPJJDxPJJDP

JJDxPJJDP
xJJDP

)1()1()0()0(
)0()0(

)0(
+

=  

and 

( ) ( ).)0(1)1( xJJDPxJJDP −=  

 The Newton-Raftery estimators (Newton, Raftery, 1994; Raftery, Newton, 
Satagopan, Krivitsky, 2007) are employed to assess the posterior probabilities 
( )JJDxP )0(  and ( )JJDxP )1( . These estimators are consistent, but their 

asymptotic variances do not exist. In practice, the values of the estimator may 
not stable. A longer Monte Carlo chain (of 1,000,000 draws) was generated to 
increase the credibility of the estimator. 



Bayesian Pricing of the Optimal-Replication Strategy for the European Option… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

63 

 Under equal prior probabilities of each model, i.e. ( ) ( )JJDPJJDP )1()0( = , 
we obtain ( ) ( ).|)1(|)0( xJJDPxJJDP ≈  However, invoking Occam’s razor 
that promotes parsimony (and thereby models with lower number of parame-
ters) we set ( ) 22)0( −∝JJDP  and ( ) 52)1( −∝JJDP , which results in 
( ) 9.0|)0( ≈xJJDP . The JD(0)J model is more likely a posteriori than the 

JD(1)J specification. In other words, jumps are non-essential in modeling dy-
namics of daily (closing) quotations of the WIG20 index5. 

4.1.2. Calculating the Optimal-Replication Strategy Cost 

 Under market completeness of the Black-Scholes model replication strate-
gies do exist. Continuous trading opportunity is one of the model’s underlying 
assumptions. However, in practice this assumption is quite unrealistic. If we 
limit trading opportunities to discrete times we get an incomplete model (Bert-
simas, Kogan, Lo, 2001). In the JD(0)J framework and under the assumption of 
the fixed time Δ between consecutive trading times, the replication strategy may 
not exist. Further, we calculate the costs and errors of the optimal strategies for 
some European options. 
 Let us consider two European call options. The date of pricing the optimal 
strategy is March 14, 2011, and the maturity date T is March 18, 2011. Strike 
prices are equal 2700=K  and 2800=K . The closing quotation value of the 
WIG20 index on March 14, 2011 equals 2757.76. The first option is in the mo-
ney and the second one is out of the money. The riskless interest rate is arbitra-
rily set at 0362.0=r  (r equals an arithmetic mean of WIBID ON and WIBOR 
ON on March 14, 2011). The theory of optimal-replication strategy was origi-
nally presented under the restriction of 0=r , but, fortunately, it could be gene-
ralized so as to incorporate any constant riskless rate 0>r . The pillar of the 
extension is normalization of all prices with the price of a zero-coupon bond 
(Bertsimas, Kogan, Lo, 2001). 
 Figures 3 and 4 display posterior distributions of the optimal-replication 
strategy cost *

0V  and the relative measure of the market incompleteness ∗ε  for 
each strike price. The histograms are calculated on the basis of 1,000 states of 
Markov chains. 

The maturity T is specified as 252/4 , which may appear a short period of 
time, but is long enough to judge the convergence of the optimal-replication 
strategy to the replication strategy (the strategy exists in Black-Scholes frame-
work). For the time being let us assume that the unknown parameters equal the 
assessed posterior means, i.e. 03644597.0−=µ  and 09171522.02 =σ . Let l 
denote the number of times the portfolio changes over the duration of the 
                                                 

5 The Barndorff-Nielsen and Shephard’s nonparametric test also rejects jumps in the consid-
ered time series (Barndorff-Nielsen, Shephard, 2006). 



Maciej Kostrzewski 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

64 

 

V0
* for K=2700 

75 76 77 78 79 80 81

0.
0

0.
1

0.
2

0.
3

0.
4

 

ε* for K=2700 

 

Figure 3. Histograms of the posterior distributions of *
0V  and ∗ε  for 2700=K  

V0
* for K=2800 

23 24 25 26 27 28

0.0
0.1

0.2
0.3

0.4

 

ε* for K=2800 

14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0

0.0
0.1

0.2
0.3

0.4
0.5

0.6
0.7

 

Figure 4. Histograms of the posterior distributions of *
0V  and ∗ε  for 2800=K  

options (a strategy is a sequence of portfolios). Tables 2 and 3 present the cost 
of the optimal-replication strategy *

0V  and the relative measure of the market 
incompleteness ∗ε  for each strike price K. The prices of options calculated 
under the Black-Scholes assumptions are presented in the last row of Table 2. 
Recall that the market completeness of the Black-Scholes model warrants a zero 
replication error, i.e. 0=∗ε . 
 We note that as the number l of times the portfolio changes over the option 
duration increases the optimal-replication strategy cost converges to the Black-
Scholes price. The relation is accompanied by a systematic decrease in the rep-
lication error (see Table 3), indicating that the market is “nearing” complete-
ness. 
 The prices of the options on March 14, 2011 equaled 52 and 5, for strike 
prices K=2700 and K=2800, respectively. Posterior histograms and expected 
values of *

0V  suggest that hedging of the options by the optimal-replication is  
 

13 14 15 16 17 18 

0.0 
0.2 
0.4 
0.6 
0.8 



Bayesian Pricing of the Optimal-Replication Strategy for the European Option… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

65 

Table 2. Values of *
0V  calculated under 03644597.0−=µ  and 09171522.02 =σ  for 

increasing values of l, along with the Black-Scholes (BS) prices 

l  
*

0V  
K=2700 K=2800 

1 77.7705 26.1447 
4 77.7768 24.7248 

10 77.7477 25.0945 
30 77.7488 25.0621 
BS 77.7427 25.0372 

Table 3. Values of ∗ε  calculated under 03644597.0−=µ  and 09171522.02 =σ  for 
increasing values of l 

l  ∗ε  
K=2700 K=2800 

1 27.8907 28.6173 
4 15.1117 16.6562 

10 9.8949 10.5713 
30 5.8611 6.2656 

100 3.2616 3.4959 
200 2.3230 2.4952 
500 1.4778 1.5844 
BS 0 0 

  
strategy expensive in comparison with the prices of the options. Note that the 
above results depend on estimation of the model’s parameters and the choice of 
the observation set. If the estimation is based on a shorter series, avoiding the 
period of time with more volatile changes of the index, the estimation and pri-
cing results are affected. We additionally consider a dataset from May 5, 2010 
to March 11, 2011. Then the values of the relative measure of market incomple-
teness ∗ε  are smaller than in the case of the full sample, and so is the posterior 
mean of the volatility parameter σ , with its value declining from 0.03 in the 
case of the full sample model to 0.02 for the trimmed series. Figure 5 presents 
the costs of the optimal-replication strategy *

0V  for a strike price K=2700 and 
two sets of observations. However, the cost of the new strategy is still high 
(or the price of the option is low). 

4.2. S&P100 
 Let us consider the S&P100 index and three model specifications: JD(0)J, 
JD(1)J and JD(10)J. It is assumed that the time interval between consecutive 
observations equals 252/1=∆ . 
 



Maciej Kostrzewski 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

66 

 
V0

* (05-Jun-2007 to 11-Mar-2011) 

75 76 77 78 79 80 81

0.0
0.1

0.2
0.3

0.4

 

V0
* (05-May-2010 to 11-Mar-2011) 

65 66 67 68 69

0.0
0.1

0.2
0.3

0.4
0.5

 

Figure 5. Costs of the optimal-replication strategy *
0V  for the strike price K=2700 and 

two sets of observations: 05-Jun-2007 to 11-Mar-2011 and 05-May-2010 to 
11-Mar-2011 

4.2.1. General Results 
 Table 4 contains results of Bayesian estimation - posterior means and stand-
ard deviations (in parentheses). The outcomes are based on 1,000,000 draws of 
the Markov chain and 25,000 burn-in passes. The results of the MCMC sampler 
are robust to the choice of the starting points. Convergence of the chains is con-
firmed by the CUMSUM statistics (Yu, Mykland, 1998), as well as the ergodic 
means and standard deviations plots. 
 Posterior means of the pure diffusion parameters µ  and 2σ  calculated in 
the JD(1)J and JD(10)J models – though almost identical across the two specifi-
cations – differ quite substantially from their counterparts in the JD(0)J model 
(i.e. the one that precludes any jumps). Particularly, note that 
E( 2σ |y, JD(0)J) = 0.0677 as opposed to E( 2σ |y, JD(M)J) = 0.047 for M=1 and 
M=10. The difference is justified by the jump component “absorbing” some of 
the log-returns’ volatility, whereas exclusion of jumps in the JD(0)J specifica-
tion is compensated with a higher value of the volatility parameter’s posterior 
mean.  
 Noteworthy, the posterior results for the JD(M)J specifications featuring 
M>0 are very close, which may be indicative of there being no empirical need 
for allowing for more than a single jump per ∆. Particularly, posterior means of 
the jump intensity parameter λ in both the JD(1)J and the JD(10)J model con-
sistently imply that on average there are 4 jumps per year. 



Bayesian Pricing of the Optimal-Replication Strategy for the European Option… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

67 

Table 4. Posterior means and standard deviations (in parentheses) of the JD(M)J  
models’ parameters for the S&P100 index 

Parameters JD(0)J JD(1)J JD(10)J 

λ –  4.4234 (1.3363) 4.3507 (1.3030) 

µ 0.0145 (0.1256) 0.0426 (0.1089) 0.0430 (0.1085) 

µQ –  -0.0090 (0.1850) -0.0087 (0.1845) 

σ2 0.0677 (0.0029) 0.0470 (0.0028) 0.0470 (0.0028) 

σ2
Q –  0.5440 (1.1183) 0.5451 (1.3625) 

 Turning to the formal pair-wise model comparison, we calculate decimal 
logarithms of Bayes factors (Bernardo, Smith, 2002): 

( )
( ) ,17

|JJD(0)
|JJD(1)log)(log 100,110 ≈








=

xP
xPB   

( ) ( )
( ) .7.1

|JJD(10)
|JJD(1)loglog 1010,110 ≈








=

xP
xPB  

It appears that the JD(1)J specification beats the competition, being as much as 
ca. 17 orders of magnitude more likely a posteriori than the simplest model 
structure (JD(0)J)6. Although only marginally, the former is also favored against 
the other jump-diffusion specification, i.e. JD(10)J, which seems to be penali-
zed for its excessively large number M=10 of jumps allowed per ∆.  Admittedly, 
the result fits in well with the overall pursuit for parsimony. 

4.2.2. Calculating the Optimal-replication Strategy Cost 
 We confine our further considerations to the JD(1)J model. It is known that 
markets are incomplete when sources of randomness outnumber the underlying 
traded risky instruments (Björk, 2004). In the JD(1)J specification there are 
three sources of randomness – the Wiener process W, the Poisson process N, 
and random variables Q . In our setting we consider a market with only one 
risky underlying instrument (a stock market index) accompanied by as much as 
three sources of randomness, so the market is incomplete. Therefore, we resort 
to the optimal-replication strategies for selected options. 
 Let us consider two European call options. A date of pricing of the optimal 
strategy is July 11, 2011, and the maturity date T of the options is July 15, 2011. 
Strike prices equal 590=K  and 610=K . The closing quotation of the 

                                                 
6 The Barndorff-Nielsen and Shephard’s nonparametric test reject the pure diffusion at signif-

icance level 0.05 (p-value equals 0.011). 



Maciej Kostrzewski 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

68 

S&P100 index on July 11, 2011 equals 588.15. Both of the options are out of 
the money. The riskless interest rate is set at 0075.0=r  and it equals the Fed 
Funds Discount Rate at the considered option duration. 
 Table 5 presents posterior means and standard deviations (in parentheses) of 
the optimal-replication strategy (initial) cost *

0V  and the relative measure of the 
market incompleteness ∗ε  for each strike price. On the day of the pricing, ac-
cording to our knowledge, there were no transactions of selling the considered 
options. 

Table 5. Posterior means (and standard deviations) of *
0V  and ∗ε  calculated for the 

S&P100 index as an underlying instrument 
Quantity K=590 K=610 

*
0V  12.12 (2.8927) 6.619 (3.0739) 
*ε  35.68 (11.6726) 35.47 (12.8357) 

 Table 6 contains quantiles of the posterior distributions of *
0V  and ∗ε . In 

general, the call option with a lower strike price is more expensive than the 
option with a higher strike price. Note that the cost of the optimal-replication 
strategy *

0V  is higher for the more attractive option.  

Table 6. Quantiles of the posterior distributions of *
0V  and ∗ε  calculated for the 

S&P100 index as an underlying instrument 
 

Orders of the 
quantiles 

K=590 K=610 
*

0V  *ε  *
0V  *ε  

5% 8.4071 20.2538 2.2598 16.2821 
25% 9.9695 27.0364 4.6371 26.7550 
50% 11.4782 32.7958 6.2447 33.4317 
75% 13.5252 42.2325 8.1295 43.2222 
95% 17.8427 57.2116 12.6826 58.7495 

 Figures 6 and 7 display histograms of the posterior distributions of the op-
timal-replication strategy cost *

0V  and the relative measure of the market in-
completeness ∗ε . These histograms are based on (only) 150 (randomly chosen) 
states of the Markov chains. The reason behind such a small sample is time-
consuming calculations of the optimal-replication strategy for each parameter 
vector. These calculations took about twenty hours on a standard PC. The appli-
cation of parallel calculations reduced that time to seven hours. 
 A fairly large dispersion of the posterior distributions of *

0V  and ∗ε  may 
stem from a relatively large parameter uncertainty (as evidenced by the posteri-
or standard deviations). 



Bayesian Pricing of the Optimal-Replication Strategy for the European Option… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

69 

V0
* for K=590 

5 10 15 20 25

0.
00

0.
05

0.
10

0.
15

 

∗ε  for K=590 

10 20 30 40 50 60 70 80

0.
00

0
0.

00
5

0.
01

0
0.

01
5

0.
02

0
0.

02
5

0.
03

0

 

Figure 6. Histograms of the posterior distributions of *
0V  and ∗ε  for 590=K  

V0
* for K=610 

0 5 10 15

0.
00

0.
05

0.
10

0.
15

 

∗ε  for K=610 

0 20 40 60 80

0.
00

0
0.

00
5

0.
01

0
0.

01
5

0.
02

0
0.

02
5

0.
03

0

 

Figure 7. Histograms of the posterior distributions of *
0V  and ∗ε  for 610=K  

5. Conclusions 
 This paper concerns the issue of option hedging in incomplete market mod-
els using stochastic dynamic programming and Bayesian statistics. 
 Familiar models of option pricing are complete. Unfortunately, the assump-
tions these structures usually rest upon are quite unrealistic. For instance, the 
Black-Scholes model is hinged upon continuous trading and continuous paths of 
a risky underlying instrument. Relaxing these assumptions leads to incomplete 
market models, such as the JD(M)J structures considered in the present study. 
 It is shown that incorporation of jumps in modeling financial time series 
may improve the model fit (as compared with a pure diffusion process). Unfor-



Maciej Kostrzewski 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

70 

tunately, the market incompleteness in the models featuring jumps (JD(M)J 
with M>0) renders the task of pricing and hedging derivatives more demanding. 
In general, as the replication strategy does not exist, the investor needs to resort 
to some optimal strategy. In the study we succeeded in employing the optimal-
replication strategy algorithm, derived by Bertsimas, Kogan and Lo (2001), in 
the JD(M)J framework. 
 Contrary to what seems a common practice in the financial mathematics 
works, where the model’s parameters are set arbitrarily, we estimate the param-
eters using Bayesian methodology, taking advantage of its accounting for the 
parameters uncertainty. Moreover, the results are further employed to infer up-
on the degree of market incompleteness as well as to price the optimal-
replication strategy. Specifically, posterior densities (rather than point values 
solely) of both the optimal strategy costs along and its relative error are calcu-
lated (using the MCMC techniques), providing us with some insight into their 
uncertainty.  

Acknowledgements 

Useful comments and remarks by two anonymous referees are highly appreciat-
ed. The author would also like to thank Łukasz Kwiatkowski for language veri-
fication of the manuscript. 

References 
Barndorff-Nielsen, O.E., Shephard, N. (2006), Econometrics of Testing for Jumps in Financial 

Economics Using Bipower Variation, Journal of Financial Econometrics, 4, 1, 1–30. 
Bernardo, J. M., Smith, A. F. M. (2002), Bayesian theory, Wiley Series in Probability and Statis-

tics. 
Bertsekas, D. (1995), Dynamic Programming and Optimal Control, Vol. I, Athena Scientific, 

Belmont, MA. 
Bertsimas, D., Kogan L., Lo, A.W. (2001), Hedging Derivative Securities and Incomplete Mar-

kets: an ε-Arbitrage Approach, Operations Research, 49, 3, 372–397. 
Björk, T. (2004), Arbitrage Theory in Continuous Time, Oxford University Press. 
Black, F., Scholes, M. (1973), The Pricing of Options and Corporate Liabilities, Journal of Politi-

cal Economy, 81, 637–654. 
Frühwirth-Schnatter, S. (2006), Finite Mixture and Markov Switching Models, Springer Science + 

Business Media, LCC. 
Gamerman, D., Lopes, H.F. (2006), Markov Chain Monte Carlo. Stochastic Simulation for Bayes-

ian Inference, Chapman & Hall/CRC. 
Hanson, F.B., Westman, J.J. (2002), Stochastic Analysis of Jump-Diffusions for Financial Log-

Return Processes, Stochastic Theory and Control, 280/2002, 169–183. 
Kloeden, P.E., Platen, E. (1992), Numerical Solution of Stochastic Differential Equations, Spring-

er-Verlag, Berlin-Heidelberg-New York. 
Kostrzewski, M. (2011), Bayesian inference for the jump-diffusion model with M jumps, Work-

ing paper: http://home.agh.edu.pl/ kostrzew/BayesianInferenceJDMJ.pdf. 
Lamberton, D., Lapeyre, B. (2000), Introduction to Stochastic Calculus Applied to Finance, 

CHAPMAN & HALL/CRC. 



Bayesian Pricing of the Optimal-Replication Strategy for the European Option… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 53–71 

71 

Lin, S-J., Huang, M-T. (2002), Estimating Jump-Diffusion Models Using the MCMC Simulation, 
National Tsing Hua University Department of Economics NTHU Working Paper Series, 
Working paper 0215E, October 2002. 

Merton, R.C. (1976), Option pricing when underlying stock return rates are discontinuous,  
Journal of Financial Economics, 3, 141–183. 

Newton, M.A., Raftery, A.E. (1994), Approximate Bayesian inference by the weighted likelihood 
bootstrap (with discussion), Journal of the Royal Statistical Society B, 56 (1), 3–48. 

Raftery, A. E., Newton, M. A., Satagopan, J. M., Krivitsky, P. N.  (2007), Estimating the Inte-
grated Likelihood via Posterior Simulation Using the Harmonic Mean Identity, Bayesian 
Statistics, 8, 1–45. 

Schweizer, M. (1992), Variance-Optimal Hedging in Discrete Time, Mathematics of Operations 
Research, 20, 1–31. 

Shreve, S.E. (2004), Stochastic Calculus for Finance II. Continuous-Time Models, Springer Sci-
ence+Business Media, Inc.  

Yu, B., Mykland, P. (1998), Looking at Markov Samplers Through CUMSUM Path Plots:  
A Simple Diagnostic Idea, Statistics and Computing, 8, 275–286. 

Bayesowska wycena kosztu optymalnej strategii replikującej europejską 
opcję w modelu JD(M)J 

Z a r y s  t r e ś c i. Wycena opcji w modelu niezupełnym jest nietrywialnym zagadnieniem. Przy-
kładem modelu niezupełnego jest wprowadzony przez Mertona model dyfuzji ze skokami. Gę-
stość logarytmu procesu dyfuzji ze skokami jest nieskończoną mieszanką rozkładów normalnych. 
W badaniu przyjęto, że liczba mieszanek jest skończona. Otrzymany w ten sposób model nazwa-
no modelem JD(M)J. W praktyce parametry modelu są nieznane i wymagają estymacji. W bada-
niu zastosowano wnioskowanie bayesowskie. JD(M)J jest modelem niezupełnym dla którego, w 
ogólnym przypadku, nie można wskazać strategii replikujących instrumenty pochodne. W bada-
niu zaprezentowano algorytm wyznaczający optymalną w sensie średniokwadratowym strategię 
replikującą europejski instrument pochodny. Do zilustrowania omówionej teorii wykorzystano 
indeksy WIG20 i S&P100. Przedstawiona metodologia jest użyteczna dla inwestorów, którzy 
chcą uwzględnić w wycenie instrumentów pochodnych oraz analizach szacowania ryzyka 
,,niepewność” estymacji parametrów modelu. 

S ł o w a  k l u c z o w e: rynki niezupełne, wnioskowanie bayesowskie, procesy dyfuzji  
ze skokami, wycena instrumentów pochodnych. 




	Introduction
	1. The Jump-Diffusion Model with M-jumps
	2. Bayesian Inference for the JD(M)J Model
	3. The Optimal-Replication Strategy
	4. Empirical Studies
	5. Conclusions
	References

