European Journal of Operational Research 201 (2010) 206–210

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Stochastics and Statistics

Computing option price for Levy process with fuzzy parameters

Piotr Nowak, Maciej Romaniuk *

Systems Research Institute Polish Academy of Sciences ul. Newelska 6, 01–447 Warszawa, Poland

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 April 2008Accepted 9 February 2009Available online 20 February 2009

Keywords:FinanceStochastic processesFuzzy setsSimulation

0377-2217/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.ejor.2009.02.009

* Corresponding author.E-mail addresses: pnowak@ibspan.waw.pl (P. Now

(M. Romaniuk).

In the following paper we propose the method for option pricing based on application of stochastic anal-ysis and theory of fuzzy numbers. The process of underlying asset trajectory belongs to a subclass of Levyprocesses with jumps. From practical point of view some parameters of such trajectory cannot be pre-cisely described. Therefore, some degree of the market uncertainly has to be reflected in the descriptionof the model itself. For example, the parameters (like interest rate, volatility) of financial market fluctuatefrom time to time and experts may have different opinions about such parameters. Using theory of fuzzynumbers and stochastic analysis enables us to take into account many sources of uncertainty, not onlythe probabilistic one. In our paper we apply the theory of Levy characteristics and present some numer-ical experiments based on Monte Carlo simulations. In detail, we present pricing formula for classicalexample of European call option.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The Black–Scholes model is one of the best known results in theoption pricing theory (see [2,8,9]). Using Black–Scholes formulaone could calculate the price of a derivative under the assumptionthat there is not a possibility of arbitrage in the market. In thismodel, the stochastic process which describes the price move-ments of an underlying financial instrument S is the geometricalBrownian motion. In such case the pricing formula is given in theanalytic form (see [2,8,9]). However, it is commonly known thatthe Black–Scholes formula does not describe the real behaviourof derivatives very well. For example, the returns of logarithmsof S in the real market have non-symmetric distribution (see [4]).One of our aims is to improve the pricing model. In the paper weuse the exponential function of a linear combination of Brownianmotion, drift and Poisson process. Such process belongs to naturalsubclass of Levy processes and its non-continuity may modeljumps of the underlying asset caused by external shocks. In our pa-per we would like to join two approaches. Firstly, we apply themartingale theory and Levy characteristics (see [12]). As neutralmartingale measure we use the minimal entropy measure(MEMM), which is closely related to exponential utility functionapproaches, known in e.g. economy and finance. Secondly, weuse the fuzzy sets theory (see [13,14]). Because of imprecise infor-mation and the fluctuations of financial market from time to time,some input parameters of our model cannot always be expected inthe precise sense. Therefore, this uncertainty has to be reflected in

ll rights reserved.

ak), mroman@ibspan.waw.pl

the construction of the model. Expert opinions or imprecise esti-mates of the market parameters may be expressed as fuzzy num-bers. Also statistical estimators may be transformed to fuzzynumbers (see [3]). Therefore, it is natural to consider fuzzy param-eters of the stochastic model. Such a treatment is particularly use-ful for financial analysts giving them the tool to pick option pricewith an acceptable belief degree for the latter use (see [13]). Ourconsiderations will be mainly focused on European options, how-ever they may be generalized for other types of derivatives.

The paper is organized as follows: Section 2 contains prelimi-naries. Section 3 is an exposition of basic definitions and facts con-cerning the minimal entropy martingale measure. In Section 4, wedescribe the model of underlying asset and prove the pricing for-mula. Section 5 is devoted to pricing the derivative with fuzzyparameters. In Section 6, we conduct some numerical experimentswhich illustrates our approach. The last section contains conclud-ing remarks.

2. Notation and definitions

2.1. Stochastic preliminaries

We recall some basic facts about stochastic analysis and martin-gale method.

Let ðX;F; ðFtÞt2½0;T�; PÞ be a probability space with filtration sat-isfying standard assumptions. Let T <1.

A stochastic process H ¼ ðHtÞt2½0;T� is cadlag, if its trajectories arefunctions, which are right-continuous with left limits.

H is ðFtÞ-adapted if Ht is ðFtÞ-measurable for each t 2 ½0; T�.A probability measure Q on ðX;FÞ is absolutely continuous

with respect to P (Q � P) if for all A 2F

P. Nowak, M. Romaniuk / European Journal of Operational Research 201 (2010) 206–210 207

PðAÞ ¼ 0) QðAÞ ¼ 0;

and it is equivalent to P if P and Q have the same sets with zeromeasure.

Let St be an ðFtÞ-adapted cadlag stochastic process describingthe underlying asset. Let r denote a constant risk-free interest rateand

Zt ¼ e�rtSt ð1Þ

be the discounted process of values of the underlying asset. Wehave to find the measure Q equivalent to P for which Zt is amartingale.

The next step is to find the form of the process St with respectto this new probability measure Q. The price of a derivative with apayment function f is given by formula:

Ct ¼ expð�rðT � tÞÞEQ ðf ðSÞjFtÞ; t 2 ½0; T�: ð2Þ

2.2. Fuzzy sets preliminaries

Now we remind some facts about fuzzy sets and numbers.Let X be a universal set and eA be a fuzzy subset of X. We denote by

leA its membership functionleA : X ! ½0;1�, and by eAa ¼fx : leA P agthe a-level set of eA, where eA0 is the closure of the set fx : leA – 0g. Inour paper we assume that X ¼ R.

Let ~a be a fuzzy number. Then, under our assumptions, the a-le-vel set ~aa is a closed interval, which can be denoted by ~aa ¼ ½~aL

a; ~aUa �

(see e.g. [14]).We can now introduce the arithmetics of any two fuzzy num-

bers. Let � be a binary operator �, �, � or between fuzzy num-bers ~a and ~b, where the binary operators correspond to :þ, �, or=, according to the ‘‘Extension Principle” in [14]. Then the member-ship function of ~a� ~b is defined by

l~a�~b zð Þ ¼ supðx;yÞ:xy¼z

minfl~aðxÞ;l~bðyÞg: ð3Þ

Let �int be a binary operator �int, �int, �int or int between twoclosed intervals ½a; b� and ½c;d�. Then

½a; b��int½c;d� ¼ fz 2 R : z ¼ x y; 8x 2 ½a; b�; 8y 2 ½c;d�g; ð4Þ

where is an usual operation þ;�; and =, if the interval ½c;d� doesnot contain zero in the last case.

Therefore, if ~a; ~b are fuzzy numbers, then ~a� ~b is also the fuzzynumber and its a-level set is given by

ð~a� ~bÞa ¼ ~aa�int~ba ¼ ½~aL

a þ ~bLa; ~a

Ua þ ~bU

a �;

ð~a� ~bÞa ¼ ~aa�int~ba ¼ ½~aL

a � ~bUa ; ~a

Ua � ~bL

a�;

ð~a� ~bÞa ¼ ~aa�int~ba

¼ ½minf~aLa~bL

a; ~aLa~bU

a ; ~aUa

~bLa; ~a

Ua

~bUag;

maxf~aLa~bLa; ~a

La~bUa ; ~a

Ua

~bLa; ~a

Ua

~bUag�;

ð~a ~bÞa ¼ ~aa int~ba

¼ ½minf~aLa=

~bLa; ~a

La=

~bUa ; ~a

Ua=

~bLa; ~a

Ua=

~bUag;

maxf~aLa=

~bLa; ~a

La=

~bUa ; ~a

Ua=

~bLa; ~a

Ua=

~bUag�

if a-level set ~ba does not contain zero for all a 2 ½0;1� in the case of.

Proposition 1. Let f : R! R be a function, FðRÞ be a set of all fuzzysubsets of R and eK 2FðRÞ. We assume that the membership functionleK of eK is upper semicontinuous and for all y the set fx : f ðxÞ ¼ yg iscompact. The function f ðxÞ can induce a fuzzy-valued function~f : FðRÞ !FðRÞ via the extension principle. Then the a-level set of~f ðeKÞ is ~f ðeKÞa ¼ ff ðxÞ : x 2 eKag.

For proof of this proposition see [13].Triangular fuzzy number ~a with membership function l~aðxÞ is

defined as

l~aðxÞ ¼

x�a1a2�a1

for a1 6 x 6 a2;

x�a3a2�a3

for a2 6 x 6 a3;

0 otherwise;

8><>: ð5Þ

where ½a1; a3� is the supporting interval and the membership func-tion has peak in a2. Triangular fuzzy number ~a is denoted as

~a ¼ ða1; a2; a3Þ:

Left–Right (or L–R) fuzzy numbers are special case of triangularfuzzy numbers (e.g. see [1,5]), where linear functions used in thedefinition are replaced by monotonic functions, i.e.

Definition 1. A fuzzy set eA on the set of real numbers is called L–Rnumber if the membership function may be calculated as

l~aðxÞ ¼

L a2�xa2�a1

� �for a1 6 x 6 a2;

R x�a2a3�a2

� �for a2 6 x 6 a3;

0 otherwise;

8>>><>>>: ð6Þ

where L and R are continuous strictly decreasing function definedon ½0;1� with values in ½0;1� satisfying the conditions

LðxÞ ¼ RðxÞ ¼ 1 if x ¼ 0; LðxÞ ¼ RðxÞ ¼ 0 if x ¼ 1:

The L–R fuzzy number ~a is denoted as

~a ¼ ða1; a2; a3ÞLR:

Next we turn to fuzzy estimation based on statistical approach(see [3]). Let X be a random variable with probability density func-tion fhð�Þ. Assume that parameter h is unknown and are to be esti-mated from a sample X1;X2; . . . ;Xn. Let h be a statistics based onX1;X2; . . . ;Xn which is used for such estimation. Then for the givenconfidence level 0 6 b 6 1 we have the b � 100% confidence inter-val ½hLðbÞ; hRðbÞ� for h which is established by the condition

Prfh ðhLðbÞ 6 h 6 hRðbÞÞ ¼ b: ð7Þ

If we suppose that ½hLð0Þ; hRð0Þ� ¼ ½h; h� then we could construct fuz-zy estimator ~h of ~h. We place the confidence intervals, one on top ofthe other, to produce a triangular shaped fuzzy ~h whose a-cuts arethe confidence intervals on b ¼ ð1� aÞ confidence levels. Therefore,we haveb~ha ¼ ½hLð1� aÞ; hRð1� aÞ� ð8Þ

for e.g. 0:01 6 a 6 1. In order to ‘‘finish” construction of the fuzzyestimator, we suppose thatb~ha ¼ ½hLð0:99Þ; hRð0:99Þ� ð9Þ

for 0 6 a < 0:01. It means that we drop the graph of b~h straight downto complete its a-cuts (see [3] for additional details).

3. MEMM for Levy processes

The cadlag stochastically continuous process Y ¼ ðYtÞt2½0;T�,Y0 ¼ 0 a.s., is called a Levy process if it satisfies the followingconditions.

1. Yt � Ys is independent of Fs for all 0 6 s 6 t 6 T:2. Yt � Ys and Yt�s have the same distributions for all

0 6 s 6 t 6 T:

We assume that a truncation function u is defined by the for-mula uðxÞ ¼ xIjxj61. We denote by MðRÞ the space of non-negative

208 P. Nowak, M. Romaniuk / European Journal of Operational Research 201 (2010) 206–210

measures on R. For Levy processes local characteristics, called Levycharacteristics are defined. They are functions of the followingform

Bt : ½0; T� ! R; Bt ¼ bt;

Ct : ½0; T� ! R; Ct ¼ ct;mt : ½0; T� !MðRÞ; mtðdxÞ ¼ mðdxÞt;

mðf0gÞ ¼ 0;Z

R

ðjxj2 ^ 1ÞmðdxÞ;

where b; c 2 R and m 2MðRÞ. Moreover, only constant b depends onthe form of u.

A stochastic process S ¼ ðStÞt2½0;T� is called the geometric Levyprocess, if it can be written in the following form

St ¼ S0 expðYtÞ; t 2 ½0; T�; ð10Þ

where Yt is a Levy process.Throughout this paper we assume that S is of the form (10),

F ¼FT and for t 2 ½0; T�

Ft ¼ rðSs; s 2 ½0; t�Þ ¼ rðYs; s 2 ½0; t�Þ:

The relative entropy IðQ ; PÞ of Q with respect to P is defined by

IðQ ; PÞ ¼ EPdQdP ln dQ

dP

� �� �if Q � P;

þ1 otherwise:

(If an equivalent martingale measure P0 satisfies the inequality

IðP0; PÞ 6 IðQ ; PÞ

for all equivalent martingale measures Q, P0 is called the minimalentropy martingale measure (MEMM).

Let

gðMEMMÞðhÞ ¼ bþ 12þ h

� �c þ

Zfjxj61g

ððex � 1Þehðex�1Þ � xÞmðdxÞ

þZfjxj>1g

ðex � 1Þehðex�1ÞmðdxÞ:

Let bY ¼ ðbY tÞt2½0;T� be the Levy process corresponding to the originalLevy process Y, such that the following equality holds

St ¼ S0 expðYtÞ ¼ S0EðbY tÞ; t 2 ½0; T�;

where EðbY tÞ is Doleans–Dade exponential of bY t .Let PðESSÞbY ½0;T�;h0

and PðESSÞbY T;h0

be the Esscher transformed measures of P

(see [10]).The following theorem from [10] will be useful in our paper.

Theorem 1. If the equation

gðMEMMÞðhÞ ¼ r ð11Þ

has a solution h0, then the MEMM of S, P0 exists and

P0 ¼ PðMEMMÞ ¼ PðESSÞbY ½0;T�;h0

¼ PðESSÞbY T;h0

:

The process Y is also a Levy process under P0 and the generating tripletof Y under P0, ðb0; c0; m0Þ is

b0 ¼ bþ h0c þZfjxj61g

xðeh0ðex�1Þ � 1ÞmðdxÞ;

c0 ¼ c;

m0ðdxÞ ¼ eh0ðex�1ÞmðdxÞ:

4. Pricing formula in the crisp case

Let ðX;F; ðFtÞt2½0;T�; PÞ be a probability space with filtration. LetT <1.

The price of the underlying asset St is the geometric Levy pro-cess given by (10), where

Yt ¼ lt þ rWt þ kNjt ; ð12Þ

Wt is Brownian motion, r > 0;l; k 2 R and Njt is Poisson process

with the intensity j > 0. We assume that Wt and Njt are

independent.

Theorem 2. The price of European call option with strike price K andpayment function f ðxÞ ¼ ðx� KÞþ at time 0 is given by formula

C0 ¼ e�j1TX1n¼0

ðj1TÞn

n!ðS0eðl1�rÞTþr2T

2 þknUðdnþÞ � e�rT KUðdn

�ÞÞ;

where U is cumulative distribution function of standard normal distri-bution and h0 is the solution of the equation

lþ 12þ h

� �r2 þ jðek � 1Þehðek�1Þ ¼ r;

l1 ¼ lþ h0r2; j1 ¼ jeh0ðek�1Þ;

dn� ¼

ln S0K þ l1T þ kn

rffiffiffiTp ; dn

þ ¼ln S0

K þ l1T þ r2T þ kn

rffiffiffiTp :

ð13Þ

Proof. We apply Theorem 1 to price the option. Eq. (11) has theform (13). Since limh!�1gðhÞ ¼ �1 and limh!1gðhÞ ¼ 1, the aboveequation has a solution. We denote it by h0. According to Theorem1, Y with respect to P0 has the form

Yt ¼ ðlþ h0r2Þt þ rW0t þ kNjeh0 ðe

k�1Þ

t ; t 2 ½0; T�; ð14Þ

where W0 is a Brownian motion and Njeh0ðek�1Þ

t is a Poisson processwith respect to P0 and the processes are independent. The price ofthe derivative is given by formula

C0 ¼ e�rTEP0 ST � Kð Þþ ¼ e�rTEP0 ST � Kð ÞI ST>Kf g

¼ e�rTEP0 S0el1TþrW0TþkN

j1T � K

� �I

el1TþrW0

TþkN

j1T > K

S0

n o¼ e�rTEP0

X1n¼0

I Nj1T¼nf g S0el1TþrW0

Tþkn � K� �

Il1TþrW0

Tþkn>ln KS0

n o¼ e�ðj1þrÞT

X1n¼0

ðj1TÞn

n!EP0 S0el1TþrW0

Tþkn � K� �

Il1TþrW0

Tþkn>ln KS0

n o:Since

ðl1 � rÞT þ rW0T þ kn � Nððl1 � rÞT þ kn;r

ffiffiffiTpÞ;

I ¼ e�rTEP0 ðS0el1TþrW0Tþkn � KÞIfl1TþrW0

Tþkn>ln KS0g

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pr2Tp

Z 1

ln KS0�rTðS0ex � e�rT KÞe�

½x�ððl1�rÞTþknÞ�2

2r2T dx:

Applying standard integral operations we obtain

I ¼ S0eðl1�rÞTþr2T2 þknUðdn

þÞ � e�rT KUðdn�Þ

and finally,

C0 ¼ e�j1TX1n¼0

ðj1TÞn

n!ðS0eðl1�rÞTþr2T

2 þknUðdnþÞ � e�rT KUðdn

�ÞÞ: �

It is easy to prove that the above series is convergent. From theTaylor expansion of the function exp, in our further considerationswe will replace the pricing formula byXN

n¼1

ðj1TÞn

n!ðS0eðl1�rÞTþr2T

2 þknUðdnþÞ � e�rT KUðdn

�ÞÞ ð15Þ

for sufficiently large N.

Fig. 1. Example of trajectory for the process (21).

P. Nowak, M. Romaniuk / European Journal of Operational Research 201 (2010) 206–210 209

5. Pricing formula in fuzzy case

In this section we will use the symbol ~ above model parame-ters to indicate they are fuzzy. Parameters without ~ are crisp.Methodology similar to the one presented below may be foundin [13].

We assume that drift l, volatility r, interest rate r, k and j arenot known precisely. Therefore we will treat them as L–R fuzzynumbers ~l; ~r;~r; ~k and ~j. We denote by l ;r ; r ; k and j theirdefuzzified versions. Then the pricing formula (15) has the form

eC 0 ¼ e�~j0�T � �N

n¼0ðð~j0 � TÞn n!Þ � ðS0 � e½ðð~l

0�~rÞ�TÞ�ð~r�~r�T 2Þ�ð~k�nÞ�

� ~Uð~dnþÞ � ðe�

~r�T � K � ~Uð~dn�ÞÞÞ; ð16Þ

where

~l0 ¼ ~l� ðh0 � ~r� ~rÞ; ~j0 ¼ ~j� eh0�ðe~k�1Þ; ð17Þ

~dn� ¼ ½lnðS0 KÞ � ð~l0 � TÞ � ð~k� nÞ� ð~r�

ffiffiffiTpÞ; ð18Þ

~dnþ ¼ ½lnðS0 KÞ � ð~l0 � TÞ � ð~r� ~r� TÞ � ð~k� nÞ� ð~r�

ffiffiffiTpÞ; ð19Þ

and h0 is the solution (with respect to h) of the equation

l þ ðr Þ2 12þ h

� �þ j ehðek �1Þðek � 1Þ ¼ r ; ð20Þ

where the parameters with were defuzzified using one of themaximum methods (e.g. Mean of Maximum Method) for L–R num-bers. The existence of solution for (20) follows from the same argu-ment as in the proof of Theorem 2.

Proposition 1 enables us to calculate the a-level set ofðeC0Þa ¼ ½eCL

0a;~CU

0a� using corresponding combinations of ~rLa or ~rU

a ,~lL

a or ~lUa , ~rL

a or ~rUa , ~kL

a or ~kUa and ~jL

a or ~jUa , respectively. From the

equality

leC 0ðcÞ ¼ sup

06a61aIðeC 0ÞaðcÞ;

we obtain the membership function of eC0. If c is the option priceand leC 0

ðcÞ ¼ a, then the value of the membership function maybe treated by a financial analyst as the belief degree of c.

Table 1Sets of fuzzy parameters.

~r ~j ~k

Set A [0.1,0.125,0.15] [0.05,0.075,0.1] [0.05,0.075,0.1]Set B [0.1,0.125,0.15] [3,4.5,6] [1,1.5,2]Set C [0.1,0.125,0.15] [3,4.5,6] [�2,�1.5,�1]Set D [0.2,0.25,0.3] [0.05,0.075,0.1] [0.05,0.075,0.1]

Table 2Prices of European call option.

Price Interval for fuzzy price

Set A 1.13969 [1.12787,1.1515]Set B 2.31048 [2.2793,2.34167]Set C 0.428761 [0.420751,0.43677]Set D 2.12243 [2.09533,2.14953]

6. Numerical experiments

It is easily seen, that even the pricing fuzzy formula for classicalEuropean call option (16) looks complicated, but for lower valuesof N may be numerically calculated.

However, for more complex derivatives it may be impossible toobtain even closed form for their price. Therefore, we have to usesome numerical methods for finding appropriate price.

For example, we may use Monte Carlo simulations (see [6]). Forthe given set of L–R fuzzy numbers ~l; ~r;~r; ~k; ~j and given a-level wefind their defuzzified versions using one of the maximum methods.These defuzzified values l ;r ; r ; k ;j are used in Eq. (20) to finda solution h0.

The necessary simulations may be derived from the iterativeversion of process (14), which is analogous to formula known asEuler scheme in Black–Scholes model (see [7,11]). In our case wehave

Stiþ1¼ Sti

expððlþ h r2Þdt þ rffiffiffiffiffidtp

�i þ kmiÞ; ð21Þ

where i ¼ 0;1; . . . ;m for ti0 ¼ 0 and tim ¼ T , dt ¼ tiþ1 � ti ¼ const,�0; . . . �m�1 are iid random variables from standard normal distribu-tion and m0; . . . mm�1 are iid realizations from Poisson process withintensity jdteh ðek�1Þ (compare with (14)). The value m is known asnumber of steps in each trajectory.

Because in order to conduct the simulations via the formula (21)we need crisp values of our parameters, firstly we find the a-level

sets ~la; ~ra;~ra;~ka; ~ja. Then we use simulation approach and we ran-

domly pick value of each parameter treating the appropriate a-levelsets as intervals for uniform distribution. We repeat this procedurefor each new trajectory SðjÞ0 ; S

ðjÞt1; . . . ; SðjÞtm

, where j ¼ 1;2; . . . ; n and n isthe number of trajectories. Hence, we can generate appropriatebunch of trajectories. The example of such trajectory may be foundin Fig. 1. These data could be used as approximation of the deriva-tive price, i.e. the present value of average of payments from theconsidered financial instrument.

Such an approach could be also used instead of the expansion in(16) for the case of European call option. In our example we use thefollowing triangular fuzzy numbers as constant parameters of themodel

~l ¼ ½0:01;0:03; 0:05�; ~r ¼ ½0:03;0:045;0:6�:

The other fuzzy parameters were gathered in a few sets(see Table 1).

For the considered European call option we have established

S0 ¼ 10; K ¼ 12; T ¼ 5; m ¼ 10; n ¼ 100000:

The results of the simulations may be found in Table 2 in thecolumn Price.

It is worth of noting, that from a practical point of view, the a-level set for fuzzy price like in (16) may be seen as the interval ofprices which has belief degree a for the financial analysts. There-fore, the analysts could find the interval of prices for their desireda, e.g. a ¼ 1 or a ¼ 0:95, which are comfortable for them. Suchconfidence interval based on simulation results and central limittheorem for a ¼ 0:95 may be found in Table 2 in the column Inter-val for fuzzy price.

210 P. Nowak, M. Romaniuk / European Journal of Operational Research 201 (2010) 206–210

7. Conclusions

In the paper we discussed some stochastic model from subclassof processes with stationary and independent increments. Afterapplication of the martingale method and fuzzy numbers theorywe obtained the pricing formula for classical example, i.e. Euro-pean call option. We also discussed generalization of the obtainedformulas for other kind of derivatives via simulations. Further de-tailed studies on more complex derivatives are possible.

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