A new application of fuzzy set theory to the Black–Scholes
option pricing model
*
Cheng-Few Lee
a, Gwo-Hshiung Tzeng
b, Shin-Yun Wang
c,*
a
Department of Finance, Rutgers University, New Brunswick, NJ, USA;
Institure of Finance, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 3000, Taiwan
bDepartment of Business Administration, Kainan University, No. 1, Kai-Nan Rd., Luchu, Taoyuan 338, Taiwan;
Institute of Management of Technology, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300, Taiwan
c
Institute of Management of Technology, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300, Taiwan; Department of Finance, National Dong Hwa University, 1, Sec. 2, Da-Hsueh Rd., Shou-Feng, Hualier 974, Taiwan
Abstract
The Black–Scholes Option pricing model (OPM) developed in 1973 has always been taken as the cornerstone of option pricing model. The generic applications of such a model are always restricted by its nature of not being suitable for fuzzy environment since the decision-making problems occurring in the area of option pricing are always with a feature of uncertainty. When an investor faces an option-pricing problem, the outcomes of the primary variables depend on the investor’s estimation. It means that a person’s deduction and thinking process uses a non-binary logic with fuzziness. Unfortunately, the traditional probabilistic B–S model does not consider fuzziness to deal with the aforementioned problems. The purpose of this study is to adopt the fuzzy decision theory and Bayes’ rule as a base for measuring fuzziness in the practice of option analysis. This study also employs ‘Fuzzy Decision Space’ consisting of four dimensions, i.e. fuzzy state; fuzzy sample information, fuzzy action and evaluation function to describe the decision of investors, which is used to derive a fuzzy B–S OPM under fuzzy environment. Finally, this study finds that the over-estimation exists in the value of risk interest rate, the expected value of variation stock price, and in the value of the call price of in the money and at the money, but under-estimation exists in the value of the call price of out of the money without a consideration of the fuzziness.
q2005 Elsevier Ltd. All rights reserved.
Keywords: Black–Scholes; Option pricing model; Fuzzy set theory; Fuzzy decision space
1. Introduction
In this paper, we compare our results with the B–S results (Black & Scholes, 1973). The basic model of the B–S OPM is C Z SNðd1Þ K K eKRTNðd2Þ d1Z ½lnðS=KÞ C ðR C s 2=2ÞT=s ffiffiffiffi T p ; d2Z d1Ks ffiffiffiffi T p (1) where C call price;
S current stock price; K striking price; R riskless interest rate; T time until option expiration;
s standard deviation of return on the underlying security;
N(di) cumulative normal distribution function evaluated
at di.
In decision-making uncertainty is unknown. There are many factors that affect the decision-making, including human psychology state, external information input, which is usually difficult to be derived in terms of probabilistic or stochastic measurement (Cox & Ross, 1976). The well known B–S model, has a number of assumptions such as the riskless interest rate and the volatility are constant, which
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*
This project has been supported by NSC 93-2416-H-009-024. * Corresponding author. Tel.: C886 3571 2121x57505; fax: C886 3575 3926.
hardly catch human psychology state and external infor-mation input. Although, B–S model has been improved, for instant, Cox and Ross (1975) brought out the concept of Constant-Elasticity-of-Variance for volatility.MacBeth and
Merville (1979)pointed out that B–S model underprices
in-the-money options (SOK), and overprices out-of-the-money (S!K) options. Cox, Ross, and Rubinstein (1979)
used a simplified approach to estimate the volatility. Hull
and White (1987) released the assumption that the
distribution of price of underlying asset and volatility are constant. Wiggins (1987) and Scott (1987) let go the assumption that the volatility is constant and assumed the volatility follow Stochastic-Volatility. Amin (1993) con-sidered the Jump-Diffusion process of stock price and the volatility were random process. TheBakshi, Cao, and Chen
model (1997)derived call price when riskless interest rate
and volatility are uncertain. Kenneth (1996) and
Rabino-vitch (1989)have also used empirical data for verifying the
correctness of B–S model. They still did not adequately address the difficulties mentioned above. In this paper, we will use fuzzy concept to address the difficulties mentioned above. Its relevant decision-making is described with decision space BZ{S,D,P(Si),C(dl,Si)}, where SZ
{S1,S2,.,SI} stands for the state set of the environment is
element, Si, iZ1,2,.,I, stands for a possible state or an
actual condition of the state set; DZ{d1,d2,.,dJ} stands for
a decision action set; and di, lZ1,2,.,L, stands for an
action or alternative available for the investor. P(Si) is the
probability of Si, and C(dj,Si) stands for the premium which
is a function on D!S. In the B–S model, C(dl,Si) stands for
the call price. If the investors know for sure that (S,K,R,s,T) meet the requirements of a normal distribution, lognormal distribution, or other designated distribution with precise assessment of probabilities, then the optimal alternative ðdlÞ for the investors is such that:
CðdlÞ Z Min j XI iZ1 Cðdj; SiÞPðSiÞ ( ) (2)
However, the investor often encounters two difficulties when determining the optimal alternative ðdlÞ with the classical statistical decision model in a B–S model:
(i) An investor usually depends on an expert’s judgment to derive the probability distribution of primary variables in a B–S model. However, an investor often subjectively describes the uncertainty he/she faces with implicit fuzziness or imprecise-ness, which can be expressed as, for example, ‘there is a good chance for a riskless interest rate of 3% next year, the riskless interest rate is very unlikely to go below 1%, and it is most probable in the range of 1.5–2.5%.’ For another example, ‘In a booming economy, there is about a 60% probability that riskless interest rate will grow 10% next year.’ The phrases ‘booming economy’ and ‘about 60%’
mean implicitly that the probability for the event of ‘10% riskless interest rate’ could be 55, 58, 60, or 65%. In other words, an investor uses both random and fuzzy elements as a base to subjectively assess uncertainty. However, the precondition of the probabilistic and stochastic B–S model assumes that the probability used for the decision analysis is a ‘precise’ number. In addition, it is calculated and derived from repeated samples and the concept of relative frequency. Thus, it is different from the fuzzy probability calculated and derived in accord-ance with the ‘degree of belief’ by experts in the real world. Therefore, it is difficult to use the -traditional probabilistic B–S model under uncertainty for fuzzy decision-making (Bellman
and Zadeh, 1970). In this paper, the fuzzy decision
theory measures fuzziness and includes the con-clusion in the B–S OPM in order to determine an optimal decision ðdlÞ.
(ii) While assessing the distribution of a primary variable in a B–S model, an expert should evaluate the influence of sample information. This involves the fuzzy factor of the expert’s subjective judgment. That is, the fuzzy factor of the expert’s subjective judgment in the call price should not be overlooked. Otherwise, the evaluation will not accurately reflect the problem and will lead to inaccurate decision-making. However, the tra-ditional probability B–S model does not take into consideration on pricing the fact that investors face fuzzy (vague/imprecise/uncertain) factors in B–S analysis. In this paper, the posterior probability will be derived through sample information in accord-ance with Bayes’s rule. The fuzzy sample information will also be included in the B–S OPM to reflect more accurately the situation faced by the investor. An example is illustrated to demonstrate the fuzzy theory to the Black–Scholes call OPM. The results show that the fuzzy B–S OPM to determine an optimal pricing for option is superior to the traditional B–S model in explaining market prices in a fuzzy environment.
The remainder of this paper is organized as follows. The concepts of the probability of fuzzy events are introduced in Section 2. Section 3 describes the B–S model under fuzzy environment, which consist four dimensions: fuzzy state, fuzzy sample information, fuzzy action and evaluation function to describe the decision of investors. Section 4 describes the derivation of fuzzy B–S OPM. Section 5 compares three propositions that is superior to the traditional B–S OPM model in explaining market prices in a fuzzy environment. Section 6 assesses the accuracy of the approximation to the fuzzy B–S with an illustrative example, and conclusions are presented in Section 7.
2. Probability of fuzzy events
The concept of probability is employed in describing fuzzy events and in using sample information to make statistical inferences. An event is an experimental outcome that may or may not occur. Assume the probability of a fuzzy event that measures the chance, or likelihood, the degree of compatibility or degree of truth.
2.1. Prior probability of fuzzy events
In the B–S model under uncertainty, the distributions of the primary variables are assessed subjectively. Therefore, an investor faces the problem of implicit fuzziness. It is difficult to measure the impreciseness with the concept of probability (Zadeh, 1965) because probability is used to measure randomness. Randomness is relevant to the occurrence or non-occurrence of an event, while fuzziness is relevant to the degree of an event (Bellman and Zadeh, 1970). According to the definition given byZadeh (1965), a fuzzy set is used to describe the set of an event without clear boundaries. The membership function mA~ : X / ½0; 1
express the fuzzy set ~A in set X where if element x2X, then mA~ðxÞ 2½0; 1; mA~ðxÞ expresses the grade of membership
x (also, the degree of compatibility or degree of truth) of X in ~
A, which maps X to the membership space. The greater the value of mA~ðxÞ is, the higher the grade of membership of x
belong to ~A. According to the concept of a fuzzy set, when there is no extra sample information, the prior probability Pð ~AÞ of fuzzy event ~A can be defined as (Zadeh, 1968, 1972):
Pð ~AÞ ZX
m
xi
mA~ðxrÞPðxrÞ (3)
2.2. Posterior probability of fuzzy events
Let XZ{x1,x2,.,xm} be the sample information space.
In Si state, if the prior probability P(xrjSi) of acquiring
sample information xr is known, then the posterior
probability of acquiring sample information xris:
PðSijxrÞ Z
PðxrjSiÞPðSiÞ
PðxrÞ
(4) One could evaluate the sample information acquired through this method. Therefore, it still involves the subjective opinion of experts. For instance, provided that the prior probability Rtof riskless rate in t period is known,
and stdrops from 60 to 40% due to the change of pricing,
experts will then deduce the riskless interest rate as: ‘According to the new volatility of the company, the volatility currently drops from 60 to 40% with approxi-mately a 10% riskless rate growth.’
Therefore, the posterior probability of the sample information with fuzziness can be calculated as shown
in the following. Let sample information space be XZ {x1,x2,.,xm}, {xr}, rZ1,2,.,m be an independent event,
and let ~M Z f ~M1; ~M2; .; ~MJg be the concept of fuzzy sample
information. The posterior probability PðSij ~MjÞ is calculated
in accordance with Bayes’ rule after deriving ~Mj
PðSij ~MjÞ Z Pð ~MjjSiÞPðSiÞ Pð ~MjÞ where Pð ~MjjSiÞ Z Xm rZ1 PðxrjSiÞmM~ jðxrÞ Pð ~MjÞ Z Xm rZ1 PðxrÞmM~jðxrÞ Therefore PðSij ~MjÞ Z Pm rZ1PðxrjSiÞmM~ jðxrÞPðSiÞ Pm rZ1PðxrÞmM~jðxrÞ (5)
when fuzzy sample information exists, the occurrence probability of Si state can be described using the above
formula. Therefore, there is uncertainty in the future price of option; we want to bring in the concept of ‘fuzzy’ to describe the B–S model under fuzzy environment.
3. B–S model under fuzzy environment
When dealing with the actual B–S issues, an investor not only faces a fuzzy sample information space, but he/she also stays in a fuzzy state space. For example, industry forecasts its future riskless interest rate in accordance with the classification of ‘booming economy’, ‘fair economy’, or ‘depression’. The definitions of ‘booming economy’, ‘fair economy’, and ‘depression’, depend on the investor’s subjective opinion. Therefore, the state space encountered by the investor also involves implicit fuzziness. Besides, the actions that the investor plans to take will cause the price structure and change accordingly in the analysis. Let us take the pricing change for example, when st drops; Rt is
expected to go down. However, due to the investor’s environment, timing of the decision-making, and the inability to give it a trial, it is virtually impossible to wait a longtime for the collection of perfect information. Under these circumstances, the alternative adopted by the investor for B–S model under uncertainty contains fuzziness. In summary, the B–S model, which an investor actually deals with, is in a fuzzy state with fuzzy sample information and fuzzy action. As a result, the B–S model can be defined with fuzzy decision space ~BZ f ~F; ~A; Pð ~FÞ; Cð ~A; ~FÞg, in which ~F Zf ~F1; ~F2; .; ~FKg stands for fuzzy state set ~Fk, kZ1,2,.,K
stands for a fuzzy set in S; SZ{S1,S2,.,SI} stands for the
state set Si, iZ1,2,.,I stands for a state of the state set.
~
AZ f ~A1; ~A2; :::; ~ANg stands for fuzzy action set ~An, nZ
stands for action set dj, jZ1,2,.,J stands for an action or
alternative available for the investor. Pð ~FkÞ is the prior probability of ~Fk. Cð ~A; ~FÞ is the evaluation function of
~ A ! ~F.
3.1. The prior probability of fuzzy state ð ~FkÞ
The prior probability of fuzzy state Pð ~FkÞ is defined in
accordance with Eq. (3) as:
Pð ~FkÞ Z
XI iZ1
mF~kðSiÞPðSiÞ (6)
3.2. The posterior probability of fuzzy state ð ~FkÞ
Let ~M Z f ~M1; ~M2; .; ~MJg be the fuzzy sample
infor-mation space in X. ~Mj, jZ1,2,.,J, is the fuzzy sample
information; XZ{x1,x2,.,xm} is the sample information
space, and {xr}, rZ1,2,.,m is an independent event.
The posterior probability of fuzzy state ~Fk is defined in
accordance with the posterior probability of fuzzy events after the fuzzy sample information M~j is derived in
accordance with Eqs. (3) and (5)
Pð ~Fkj ~MjÞ Z XI iZ1 mF~ kðSiÞPðSij ~MjÞ Z XI iZ1 mF~ kðSiÞ Pm rZ1PðxrjSiÞmM~ jðxrÞPðSiÞ Pð ~MjÞ Z PI iZ1 Pm rZ1mF~kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ (7)
3.3. The expected call price of fuzzy action
An investor must consider the call price after he or she has realized the fuzzy state ~Fk and the fuzzy sample
information ~Mjof the industry in order to draft an optimal
decision and action ~An. Let the call price be the value
Cð ~An; ~FkÞ of evaluation function Cð ~A; ~FÞ, then the
expected call price Cð ~Anj ~MjÞ of ~An can be defined as:
Cð ~Anj ~MjÞ ZX
K
kZ1
Cð ~An; ~FkÞPð ~Fkj ~MjÞ (8) The optimal action ~An can be determined by Cð ~Anj ~MjÞ.
The call price Cð ~Anj ~MjÞ of the optimal fuzzy action ~An can be defined as
Cð ~Anj ~MjÞ Z o N
nZ1Cð ~Anj ~MjÞ (9)
where, oN
nZ1 is selecting the minimum value of N values.
The lower the call option prices the better for investor to reduce the loss.
4. The derivation of fuzzy B–S option pricing model The expected value S of Rt, st,and the fuzzy B–S call
option pricing are derived in the following.
4.1. Expected value of Rt
For instance, a company’s expert in its sales department has long been performed the riskless interest rate Rt. If this
sales expert always forecasts the company’s riskless interest rate of the next term in accordance with the economy’s condition, which might be classified as a ‘booming economy,’ a ‘fair economy,’ and a ‘depression’. Let the state set be SZ{S1,S2,.,SI}, where Si, iZ1,2,.,I stands for
the riskless interest rate, and the fuzzy stat set be ~
F Z f ~F1; ~F2; .; ~FKg, in which ~Fk, kZ1,2,.,K stands for
an economy condition. If this company prepares a new lower price plan to respond to the price competition in the market, the sales department expects riskless interest rate. Let the sample information space be XZ{x1,x2,.,xm}, where {xr},
rZ1,2,.,m stands for rate of riskless interest rates growth under the different price plans and {xr} is an independent
event. Also let the fuzzy sample information space ~
M Z f ~M1; ~M2; .; ~MJg, in which ~Mj, jZ1,2,.,J stands for
a riskless interest growth rate condition that might be classified as ‘high riskless interest growth rate’ or ‘fair riskless interest growth rate’. The expected value of Rtin ~Fk
state, Rtð ~FkÞ can be defined in accordance with Eq. (3) as:
Rtð ~FkÞ Z XI iZ1 Si 1 C Xm rZ1 xrPðxrjSiÞ ! " # mF~kðSiÞPðSiÞ (10) 4.2. Expected value of st
Taking a change of the price policy in fuzzy environ-ment, the company estimates its future st, which depends on
the investor’s subjective judgment. Therefore, the expected value of stin ~Fkstate, EA~ð ~stÞ can be defined in accordance
with Eq. (3) as:
EA~ð ~stÞ Z
Xv uZ1
stu$mA~ðstuÞ$PðstuÞ (11)
in which uZ1,2,.,v stands for different volatility in stu.
4.3. The Fuzzy B–S Option Pricing Model
Combining Eqs. (10) and (11), the Fuzzy B–S Option Pricing Model and the expected call price Cð ~Anj ~MjÞ can be
Cð ~Anj ~MjÞ Z XK kZ1 Cð ~An; ~FkÞPð ~Fkj ~MjÞ ZX K kZ1 Cð ~An; ~FkÞ XI iZ1 mF~kðSiÞPðSij ~MjÞ ZX K kZ1 Cð ~An; ~FkÞ ! PI iZ1PmrZ1mF~kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ (12) According to the expected call price Cð ~Anj ~MjÞ, the optimal
action ~A*n under the fuzzy B–S model can be defined as:
Cð ~Anj ~MjÞ Z o N
nZ1Cð ~Anj ~MjÞ (13)
5. General inference
According to the definition of the fuzzy set given byZadeh (1965), ~A is a fuzzy set of X and mA~ : X / ½0; 1. mA~ is the
membership function of A, that is, when x2X, then mA~ðxÞ 2½0; 1. When the range of the membership function
is improved to {0, 1}, then ~A will be transformed to a crisp set A. The mA~will be transformed into CA(characteristic function)
. This kind of transformation for a decision maker in the B–S model means that the estimates of the primary variable (S, K, T, R, s) are without fuzziness, that is, mA~ðxrÞZ 1 or 0.
Under these circumstances, mA~ðxÞZ CAðxÞZ 1 if x2A, or
mA~ðxrÞZ 0, if x;A, therefore, the probability for the
occurrence of fuzzy event ~A can be defined as:
Pð ~AÞ ZX m rZ1 mA~ðxrÞ$PðxrÞ Z Xm rZ1 CAðxrÞ$PðxrÞ Z PðAÞ (14)
According to the traditional probabilistic B–S model, the decision space faced by an investor is BZ{S,D,P(Si),C(dl,Si)},
in the B–S model, C Z SNðd1ÞK K eKRTNðd2Þ, the C value is
CðdljxrÞ. Therefore, under Sistate and xrsample information,
the expected call price CðdljxrÞ of the optimal alternative is
defined as: CðdljxrÞ Z Min j XI iZ1 Cðdj; SiÞPðSijxrÞ ( ) Z o J jZ1 XI iZ1 Cðdl; SiÞPðSijxrÞ (15)
The following results can be proved. To avoid distraction, the detailed mathematical proofs are put in the Appendices.
5.1. Proposition 1
Assume that EA~ð ~stÞ and E(st) stands for the expected
value of st of the fuzzy B–S model and the traditional
probabilistic B–S model, respectively. When ~Fkand ~Mjare
existentially, but its fuzziness has been neglected irration-ally, then EðstÞR EA~ð ~stÞ. This means that the expected
value of stwill be increased falsely and it will lead to false
decision-making.
Proof: please see Appendix A. 5.2. Proposition 2
Assume that Rtð ~FkÞ and Rtstands for the expected value
of riskless interest rate in the fuzzy B–S model and the traditional probabilistic B–S model, respectively. When ~Mj
and ~Fk are existentially, but its fuzziness has been
overlooked irrationally, then Rtð ~FkÞ% Rt. This means that
the expected value of riskless interest rate will be increased falsely and it will lead to a false decision-making.
Proof: please see Appendix B. 5.3. Proposition 3
(1) When ~Mj and ~Fk are existentially and under a fixed st and Rt condition, it is assumed that stuZst0 and
Rtð ~FkÞZ Rt0. Therefore, the option of in the money
(SOK) deriving CðdljxrÞR Cð ~A
nj ~MjÞ. This means that
the value of the expected call price of in the money will be overestimated.
Proof: please see Appendix C.
(2) When ~Mj and ~Fk are existentially and under a fixed
st and Rt condition, it is assumed that stuZst0 and
Rtð ~FkÞZ Rt0. Therefore, the option of at the money
(SZK) deriving CðdljxrÞR Cð ~A
nj ~MjÞ. This means that
the value of the expected call price of at the money will be overestimated.
Proof: please see Appendix D.
(3) When ~Mjand ~Fk are existentially and under a fixed st
and Rt condition, it is assumed that stuZst0 and
Rtð ~FkÞZ Rt0. Therefore, the option of in the money
(S!K) deriving CðdljxrÞ% Cð ~A
nj ~MjÞ. This means that
the value of the expected call price out of the money will be underestimated.
Proof: please see Appendix E.
From Proposition 3 we know that the option of in the money and at the money will be over-estimated, but the option of out of the money will be under-estimated. If the investor makes a decision in accordance with the estimated call price, then the optimal alternative might not be chosen, because of the target call price or requirement rate of return considerations without a consideration of the fuzziness.
6. Illustrative example for simulation
This paper takes the call option of stock Y, the target stock of Company Z, as an example to discuss the application of call prices derived by the investor using fuzzy OPM under uncertainty. Company Z’s fuzzy-decision space is described below in the following.
6.1. Fuzzy state
It is known to assume that the investor has acquired stock Y, the single target stock from Company Z, of which the estimation of risk interest rate (Rt) has long been carried
out by the sales specialists of the company, who have been projecting respective possible risk interest rates in the next term for different future outlooks of ‘booming economy’, ‘fair economy’, and ‘depression’. Suppose state set SZ {S1,S2,S3,S4,S5}, where Si denotes the risk interest rate of
call option, the set represents a collection of fuzzy states (Table 1).
6.2. Fuzzy sample information
In response to recent return fluctuations on the stock market, the investor has readjusted the magnitude of fluctuation for the rate of return on the stock (s). While S, K, T, and R can be derived directly from observation; s calculation requires the use of daily return data of the target stock over a past period of time. Based on historical statistics, s is revised downwards from its current level of 60–40%. It is expected that this change will cause the risk interest rate to drop. Suppose the sample message space XZ {x1,x2,x3,x4}, where xr, rZ1–4, denotes the growth of risk
interest rate and where (xr) is an independent event. The
investor produces estimations on two basic assumptions of ‘very high risk interest rate growth’ and ‘relatively flat risk
interest rate growth’. Hence, the fuzzy sample message space can be expressed as ~M Z f ~M1; ~M2g, where ~M1denotes
very high-risk interest rate growth and ~M2, a relatively flat
interest risk interest rate growth. It is also known that historically given Si, Company Z’s prior probability of the
occurrence of xr is P(xrjSi), as shown in Table 2, and the
prior probability of the membership function mM~jðxrÞ and xr
for fuzzy sample message ~Mj, jZ1, 2 is P(xr), as shown in
Table 3.
6.3. Fuzzy action
In response to the growth of risk interest rate in the next term and considering the market status, the investor has decided on his/her action set DZ{d1,d2}, where Solution 1
(d1) is to purchase large volumes of stock options under the
expectation of very high stock price fluctuation and Solution 2 (d2) is when the expected stock price fluctuation is low,
hence only small quantities of stock options will be purchased. The evaluation of the respective solutions shows the following results in the call option price:
Solution 1: When the investor purchases large quantities of stock options, the action will either fuel or dampen the market, causing s to rise creating a larger room for profit. Hence, the call option price will increase.
Solution 2: When the investor purchases only small quantities of stock options, the action has little effect on market fluctuation, while it will limit the level of rising in s and result in much smaller room for profit. Hence, the call option price will fall.
Suppose the fuzzy action set ~AZ f ~A1; ~A2g, where ~A1
denotes the fuzzy set for d1with s at around 60%, and ~A2
denotes the fuzzy set for d2with s at around 40%, we then
obtain mA~ nðstÞ, as shown inTable 4. Table 1 mF~kðSiÞ and P(Si) mF~kðSiÞ or P(Si) Si(Rt) 1% 2% 3% 4% 5% mF~1ðSiÞ 0 0 0.8 0.9 1.0 mF~2ðSiÞ 0 0.9 1.0 0.8 0 mF~3ðSiÞ 1.0 0.9 0.8 0.5 0 P(Si) 0.1 0.2 0.4 0.2 0.1 Table 2 P(xrjSi) Si Rt(%) xr 10% 15% 20% 25% S1 1 0.3 0.3 0.2 0.2 S2 2 0.5 0.4 0.1 0 S3 3 0.6 0.3 0.1 0 S4 4 0.8 0.2 0 0 S5 5 1.0 0 0 0 Table 3 mM~jðxrÞ and P(xr) mM~jðxrÞ or P(xr) xr 10% 15% 20% 25% mM~1ðxrÞ 0 0.2 0.8 0.8 mM~2ðxrÞ 0.2 0.8 0.6 0 P(xr) 0.2 0.3 0.3 0.2 Table 4 mA~nðstÞ and PðstuÞ mA~nðstÞ stu 20% 40% 60% 80% mA~1ðstÞ 0 0.6 1.0 0.8 mA~2ðstÞ 0.8 1.0 0.5 0 P(stu) 0.2 0.3 0.3 0.2 EA~ð ~stÞZ Pv uZ1stu$mA~ðstuÞ$PðstuÞ, where EA~ð ~s1ÞZ0:38, EA~ð ~s2ÞZ0:242.
6.4. Evaluation function
With ~Anand ~Fkgiven, the investor then introduces S, K,
and T into the B–S OPM to determine the call option price, which can be derived by calculating the value Cð ~An; ~FkÞ of
the pricing function Cð ~A; ~FÞ.
6.4.1. Expected value of risk interest rate under ~Fk
After taking into account the risk interest rate growth xr,
rZ1–4 of Company Z, and with Si, iZ1–5 given, the
expected value of the new risk interest rate Rt(Si) is:
RtðSiÞ Z Si 1 C X4 rZ1 xrPðxrjSiÞ ! According to Table 2, Rt(S1)Z0.012, Rt(S2)Z0.023, Rt(S3)Z0.034, Rt(S4)Z0.044, and Rt(S5)Z0.055, we can
obtain the expected values of risk interest rates Rtð ~FkÞ under
~ Fk, kZ1–3 as Pð ~FkÞ Z XI iZ1 mF~kðSiÞPðSiÞ; Rtð ~FkÞ Z Pð ~FkÞRtðSiÞ
Using the data provided inTable 1, we can obtain the values of Rtð ~FkÞ as shown inTable 5.
6.4.2. The value of evaluation function
Since the new stock price fluctuation of Company Z is set at 40%, we can bring the previously derived Rtð ~FkÞ values
into Eq. (1). We assume that under the current market state of Company Z, where SZ100 (NT $), KZ100 (NT $), TZ1 (year) and with Company Z having only one target stock Y, the call option price is then the value Cð ~An; ~FkÞ of Company
Z’s pricing function Cð ~A; ~FÞ. We derive the following: Cð ~A1; ~F1ÞZ15:925; Cð ~A2; ~F1ÞZ 10:552; Cð ~A1; ~F2ÞZ
16:128; Cð ~A2; ~F2ÞZ10:773; Cð ~A1; ~F3ÞZ 16:07;
Cð ~A2; ~F3ÞZ10:71.
6.5. Expected value of call option price at optimal actions To simplify or presentation that assume there are only two alternative under consideration to generalize from two to many alternative can be done similarly. With the investor’s derived Cð ~An; ~FkÞ and given the fuzzy sample
message ~Mj, jZ1, 2, the expected call option price value
Cð ~Anj ~MjÞ for ~Ancan be defined:
Cð ~Anj ~M1; ~M2Þ Z
XK kZ1
Cð ~An; ~FkÞPð ~Fkj ~M1; ~M2Þ
Cð ~A1j ~M1; ~M2Þ Z 8:385; Cð ~A2j ~M1; ~M2Þ Z 5:587
Therefore, the investor should adopt Action A2.
The investor’s expected call option price Cð ~Anj ~MjÞ at
optimal action can be as Cð ~Anj ~MjÞZo2nZ1Cð ~Anj ~M1; ~M2ÞZ
5:587.
6.6. Soundness analysis for fuzzy B–S option pricing model In pricing options, the fuzzy OPM argues that the investor’s estimation of the changes in both correlated variables R and s that contain hidden fuzzy factors. Therefore, unless the investor possesses complete infor-mation on correlated variables and has determined the values of the correlated variables under the constraints of the objective environment, the fuzzy factors cannot be completely excluded.
In the following section, the data from the case of Company Z discussed above will be used as a basis to compare the differences between CðdljxrÞ and Cð ~A
nj ~MjÞ, in
order to better understand the influence of fuzzy factors on the B–S OPM and to examine the soundness of a fuzzy OPM. 6.6.1. Expected value of risk interest rate
Let Rt(Si) be the expected value of risk interest rate
derived from the B–S OPM. According to Eq. (10) and Bayes’ theorem, Rt(Si) can be defined as:
RtðSiÞ Z XI iZ1 Si 1 CX m rZ1 xrPðxrjSiÞPðSiÞ " #
Using the data from the previous case of Company Z in
Tables 1 and 2, we obtain the expected value of risk interest
rate Rt(Si) as 0.034.
The figures are all higher than the expected values of risk interest rates derived for the same case of Company Z under fuzzy OPM. Rtð ~F1ÞZ 0:020, Rtð ~F2ÞZ 0:025, and
Rtð ~F3ÞZ 0:023.
This finding is consistent with the results from Proposition 1 of this study, suggesting that ignoring hidden fuzzy factors in the calculation of the expected values of risk interest rates will result in overestimations and thereby causing mistakes in investment decisions.
6.6.2. Expected value of stock price fluctuation
Let E(st) is the expected value of stock price fluctuation
derived from the B–S OPM. According to Eq. (12) and Bayes’ theorem, E(st) can be defined as:
EA~ð ~stÞ Z
Xv uZ1
stumA~ðstuÞPðstuÞ
Using the data from the previous case of Company Z in
Table 4, we obtain the expected value of stock price
fluctuation E(st) as 0.5.
The figures are all higher than the expected values of stock price fluctuations derived for the same case of
Table 5 Rtð ~FkÞ
~
Fk F~1 F~2 F~3
Company Z under fuzzy OPM. Here, EA~ð ~s1ÞZ 0:38 and
EA~ð ~s2ÞZ 0:242
This finding is consistent with the results from Proposition 2 of this study, suggesting that ignoring hidden fuzzy factors in the calculation of expected values of stock price fluctuations will result in overestimations and increased investors’ motivation for buying and selling of options, thereby fueling or dampening the target stock prices on the market. And this can easily lead to market volatility.
6.6.3. Expected value of call option price Let CðdljxrÞ and Cð ~A
nj ~MjÞ be the expected values of call
option prices of the two solutions derived from the B–S OPM
and the fuzzy OPM, respectively. Again using the previous case of Company Z with a stock return fluctuation below 40%, and with the values of Rtð ~FkÞ and EA~ð ~stÞ derived earlier,
we obtain CðdljxrÞ and Cð ~A
nj ~MjÞ for the solutions under the
two models as 21.09 and 5.587, respectively.
The result shows that the expected values of call option prices derived from the B–S OPM are higher than those obtained from the fuzzy OPM. This finding is again consistent with the results from Proposition 3 (SZK) of this study, suggesting that ignoring hidden fuzzy factors in the calculation of the expected values of call option prices will result in overestimations and thereby causing mistakes in relevant investment decisions.
7. Conclusions
The impact of implicit ‘Fuzziness’ is inevitable due to the subjective assessment made by investors in a B–S OPM. The fuzzy decision theory and Bayes’ rule are used to measure the effect of this fuzziness. It is included in the fuzzy B–S OPM to determine the optimal actions for B–S model under uncertainty. The thoughts and controlled behaviors of human involve both fuzziness and non-quantitative quality. Therefore, the fuzzy B–S model would result in a more realistic methodology for a B–S model. Further, corollaries have been made in this paper. It has been proved that if the fuzziness has been neglected irrationally, then the expected values of R, s and the value of the call price of in the money (SOK) and at the money (SZ K) will be over-estimated, but under-estimation exists in the value of the call price of out of the money (S!K) without
a consideration of the fuzziness. So, the expected call price will be inaccurately estimated and this will lead to inaccurate decision-making.
Appendix A
When ~Mjand ~Fkare existentially, E(st) can be defined in
accordance with Eq. (11):
EA~ð ~stÞ Z Xv uZ1 stumA~ðstuÞPðstuÞPð ~Fkj ~MjÞ Z PI iZ1 Pm rZ1 Pv uZ1stumA~ðstuÞPðstuÞmF~kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ (A1) Assume mA~ðstuÞZ1, mF~kðSiÞZ1, mM~jðxrÞZ1. It is
inputted into Eq. (A1) to derive E(st):
EðstÞ Z PI iZ1 Pm rZ1 Pv uZ1stuPðstuÞPðxrjSiÞPðSiÞ Pm rZ1PðxrÞ (A2) EðstÞ K EA~ð ~stÞ Z PI iZ1 Pm rZ1 Pv uZ1stuPðstuÞPðxrjSiÞPðSiÞ½1 K mA~ðstuÞmF~kðSiÞmM~jðxrÞ Pm rZ1mM~ jðxrÞPðxrÞ (A3) Due to stuO0, mA~ðstuÞ/ ½0; 1, mF~kðSiÞ/ ½0; 1, mM~jðxrÞ/ ½0; 1, 1KmA~ðstuÞ$mF~kðSiÞR 0. Therefore, EðstÞK EA~ð ~stÞR 0. Appendix B
When ~Mjand ~Fkare existentially, Rtð ~FkÞ can be defined
in accordance with Eq. (10) as:
Rtð ~FkÞ Z XI iZ1 RtðSiÞmF~kðSiÞPðSiÞPð ~Fkj ~MjÞ Z PI iZ1PmrZ1RtðSiÞðmF~kðSiÞÞ 2ðPðS iÞÞ2mM~jðxrÞPðxrjSiÞ Pm rZ1mM~ jðxrÞPðxrÞ (B1) Assume mF~ kðSiÞZ 1, mM~jðxrÞZ 1: It is inputted into
Eq. (B1) to derive the Rt:
RtZ PI iZ1PmrZ1RtðSiÞðPðSiÞÞ2PðxrjSiÞ Pm rZ1PðxrÞ (B2) RtKRtð ~FkÞ Z PI iZ1 Pm rZ1RtðSiÞðPðSiÞÞ2PðxrjSiÞ½1KðmF~ kðSiÞÞ 2m ~ MjðxrÞ Pm rZ1mM~jðxrÞPðxrÞ (B3) Due to RtðSiÞZSi 1C Pm rZ1xrPðxrjSiÞ , Rt(Si)O0, mF~kðSiÞ/½0;1, mM~jðxrÞ/½0;1, 1KðmF~kðSiÞÞ 2R0, there-fore, RtKRtð ~FkÞR0.
Appendix C
Assume option is in the money (SOK), let TZ1 then Cð ~Anj ~MjÞ can be defined in accordance with Eq. (12) as:
Cð ~Anj ~MjÞ Z SN lnðS=KÞ CPIiZ1mF~ kðSiÞPðSiÞRtðSiÞ C Pv uZ1stumA~ðstuÞPðstuÞ 2 =2 Pv uZ1stumA~ðstuÞPðstuÞ ( ) ! PI iZ1PmrZ1mF~kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~ jðxrÞPðxrÞ KK eKP I iZ1mFk~ ðSiÞPðSiÞRtðSiÞ !N lnðS=KÞ C PI iZ1mF~kðSiÞPðSiÞRtðSiÞ C Pv uZ1stumA~ðstuÞPðstuÞ 2 =2 Pv uZ1stumA~ðstuÞPðstuÞ K Xv uZ1 stumA~ðstuÞPðstuÞ ( ) ! PI iZ1PmrZ1mF~kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~ jðxrÞPðxrÞ ðC1Þ
Assume mA~nðstuÞZ mF~kðSiÞZ mM~jðxrÞZ1, when it is inputted into Eq. (C1), the Cð ~A
nj ~MjÞ will be transformed into CðdljxrÞ:
CðdljxrÞ Z SN lnðS=KÞ CPIiZ1PðSiÞRtðSiÞ C Pv uZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ ( ) PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ KK eKP I iZ1PðSiÞRtðSiÞ !N lnðS=KÞ C PI iZ1PðSiÞRtðSiÞ C PvuZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ K Xv uZ1 stuPðstuÞ ( ) PI iZ1PmrZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ ðC2Þ Let Nðd1Þ Z N lnðS=KÞ CPIiZ1PðSiÞRtðSiÞ C Pv uZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ ( ) Nðd2Þ Z N lnðS=KÞ CPIiZ1PðSiÞRtðSiÞ C PvuZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ K Xv uZ1 stuPðstuÞ ( ) Nð ~d1Þ Z N lnðS=KÞ CPIiZ1mF~kðSiÞPðSiÞRtðSiÞ C Pv uZ1stumA~ðstuÞPðstuÞ 2=2 Pv uZ1stumA~ðstuÞPðstuÞ ( ) Nð ~d2Þ Z N lnðS=KÞ CPIiZ1mF~kðSiÞPðSiÞRtðSiÞ C Pv uZ1stumA~ðstuÞPðstuÞ 2 =2 Pv uZ1stumA~ðstuÞPðstuÞ K Xv uZ1 stumA~ðstuÞPðstuÞ ( ) then CðdljxrÞ K Cð ~A nj ~MjÞ Z SNðd1Þ PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ KK eKRNðd2Þ PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ KSNð ~d1Þ PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ CK eK ~RNð ~d2Þ PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ Z PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ SNðd1Þ K K e KR Nðd2Þ K SNð ~d1Þ PI iZ1 Pm rZ1mF~kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ " CK eK ~RNð ~d2Þ PI iZ1 Pm rZ1mF~kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ # Z PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ ! SNðd1Þ K K e KR Nðd2Þ K PI iZ1 Pm rZ1mF~kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ ½SNð ~d1Þ K K e K ~R Nð ~d2Þ ( ) (C3)
where PI iZ1PmrZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ R0; due to, d1Od2and SOK.
So, SN(d1)KK e -R N(d2)R0, SNð ~d1ÞK K eK ~RNð ~d2ÞR 0, but PI iZ1 Pm rZ1mF~kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ %1 and SNðd1Þ K K eKRNðd2Þ K PI iZ1PmrZ1mF~kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ ½SNð ~d1Þ K K e K ~R Nð ~d2ÞR 0 Therefore, CðdljxrÞR Cð ~A nj ~MjÞ. Appendix D
Assume option is at the money (SZK), let TZ1 then Cð ~Anj ~MjÞ can be defined in accordance with Eq. (12) as:
Cð ~Anj ~MjÞ Z SN PI iZ1mF~kðSiÞPðSiÞRtðSiÞ C Pv uZ1stumA~ðstuÞPðstuÞ 2=2 Pv uZ1stumA~ðstuÞPðstuÞ ( ) PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ KK eK PI iZ1mFk~ ðSiÞPðSiÞRtðSiÞN PI iZ1mF~kðSiÞPðSiÞRtðSiÞ C Pv uZ1stumA~ðstuÞPðstuÞ 2=2 Pv uZ1stumA~ðstuÞPðstuÞ K Xv uZ1 stumA~ðstuÞPðstuÞ ( ) ! PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ (D1) Assume mA~
nðstuÞZ mF~kðSiÞZmM~jðxrÞZ1, when it is inputted into Eq. (D1), the Cð ~A
nj ~MjÞ will be transformed into CðdljxrÞ:
CðdljxrÞ Z SN PI iZ1PðSiÞRtðSiÞ C PvuZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ ( ) PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ KK eKP I iZ1PðSiÞRtðSiÞN PI iZ1PðSiÞRtðSiÞ C Pv uZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ K Xv uZ1 stuPðstuÞ ( ) PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ (D2) Let, Nðd1Þ Z N PI iZ1PðSiÞRtðSiÞ C PvuZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ ( ) Nðd2Þ Z N PI iZ1PðSiÞRtðSiÞ C Pv uZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ K Xv uZ1 stuPðstuÞ ( ) Nð ~d1Þ Z N PI iZ1mF~kðSiÞPðSiÞRtðSiÞ C Pv uZ1stumA~ðstuÞPðstuÞ 2 =2 Pv uZ1stumA~ðstuÞPðstuÞ ( ) Nð ~d2Þ Z N PI iZ1mF~kðSiÞPðSiÞRtðSiÞ C Pv uZ1stumA~ðstuÞPðstuÞ 2 =2 Pv uZ1stumA~ðstuÞPðstuÞ K Xv uZ1 stumA~ðstuÞPðstuÞ ( )
then CðdljxrÞKCð ~A nj ~MjÞZSNðd1Þ PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ KK eKRNðd2Þ PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ KSNð ~d1Þ PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ CK eK ~R Nð ~d2Þ PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ ZS PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ Nðd1ÞKe KR Nðd2ÞKNð ~d1Þ PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ " CeK ~R Nð ~d2Þ PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ # ZS PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ Nðd1ÞKe KR Nðd2ÞK PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ ½Nð ~d1ÞKe K ~R Nð ~d2Þ ( ) (D3) where PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ R0; due to d1Od2, So, N(d1)K e-RN(d2)R0, Nð ~d1ÞK eK ~RNð ~d2ÞR 0, but PI iZ1PmrZ1mF~kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ %1 and Nðd1Þ K e KR Nðd2Þ K PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ ½Nð ~d1Þ K e K ~R Nð ~d2ÞR 0 Therefore, CðdljxrÞR Cð ~A nj ~MjÞ. Appendix E
Assume option is out of the money (S!K), let TZ1 then, Cð ~Anj ~MjÞ can be defined in accordance with Eq. (12) as:
Cð ~Anj ~MjÞ Z SN lnðS=KÞ CPIiZ1mF~kðSiÞPðSiÞRtðSiÞ C Pv uZ1stumA~ðstuÞPðstuÞ 2 =2 Pv uZ1stumA~ðstuÞPðstuÞ ( ) PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ KK eKP I iZ1mFk~ ðSiÞPðSiÞRtðSiÞN lnðS=KÞ C PI iZ1mF~kðSiÞPðSiÞRtðSiÞ C Pv uZ1stumA~ðstuÞPðstuÞ 2 =2 Pv uZ1stumA~ðstuÞPðstuÞ ( K Xv uZ1 stumA~ðstuÞPðstuÞ ) PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ ðE1Þ Assume mA~
nðstuÞZmF~kðSiÞZ mM~jðxrÞZ 1, when it is inputted into Eq. (E1), the Cð ~A
nj ~MjÞ will be transformed into CðdljxrÞ:
CðdljxrÞ ZSN lnðS=KÞCPIiZ1PðSiÞRtðSiÞC PvuZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ ( ) PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ KK eKP I iZ1PðSiÞRtðSiÞN lnðS=KÞC PI iZ1PðSiÞRtðSiÞC Pv uZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ K Xv uZ1 stuPðstuÞ ( )PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ (E2)
Let Nðd1Þ ZN lnðS=KÞCPIiZ1PðSiÞRtðSiÞC PvuZ1stuPðstuÞ 2=2 Pv uZ1stuPðstuÞ ( ) Nðd2Þ ZN lnðS=KÞCPIiZ1PðSiÞRtðSiÞC PvuZ1stuPðstuÞ 2 =2 Pv uZ1stuPðstuÞ K Xv uZ1 stuPðstuÞ ( ) Nð ~d1ÞZN lnðS=KÞCPIiZ1mF~kðSiÞPðSiÞRtðSiÞC Pv uZ1stumA~ðstuÞPðstuÞ 2 =2 Pv uZ1stumA~ðstuÞPðstuÞ ( ) Nð ~d2ÞZN lnðS=KÞCPIiZ1mF~kðSiÞPðSiÞRtðSiÞC Pv uZ1stumA~ðstuÞPðstuÞ 2 =2 Pv uZ1stumA~ðstuÞPðstuÞ K Xv uZ1 stumA~ðstuÞPðstuÞ ( ) Then CðdljxrÞKCð ~A nj ~MjÞZSNðd1Þ PI iZ1PmrZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ KK eKRNðd2Þ PI iZ1PmrZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ KSNð ~d1Þ PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ CK eK ~R Nð ~d2Þ PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞPðxrjSiÞPðSiÞ Pm rZ1mM~jðxrÞPðxrÞ Z PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ SNðd1ÞKK e KR Nðd2ÞKSNð ~d1Þ PI iZ1 Pm rZ1mF~ kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ " CK eK ~R Nð ~d2Þ PI iZ1 Pm rZ1mF~kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ # Z PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ ! SNðd1ÞKK e KR Nðd2ÞK PI iZ1 Pm rZ1mF~kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ ½SNð ~d1ÞKK e K ~R Nð ~d2Þ ( ) (E3) where PI iZ1 Pm rZ1PðxrjSiÞPðSiÞ Pm rZ1PðxrÞ R0
although, d1!d2, but the degree of S!K is bigger than d1!d2. So, S$N(d1)KK eKRN(d2)%0, SNð ~d1ÞKK eK ~RNð ~d2Þ%0, but
PI iZ1 Pm rZ1mF~kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ %1 and SNðd1ÞKK eKRNðd2ÞK PI iZ1 Pm rZ1mF~kðSiÞmM~jðxrÞ Pm rZ1mM~jðxrÞ ½SNð ~d1ÞKK e K ~R Nð ~d2Þ%0 Therefore, CðdljxrÞ%Cð ~A nj ~MjÞ.
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