The Pricing Method for Real Option with Fuzzy Multi-variable

The Pricing Method for Real Option with Fuzzy Multi-variable
ZHU Sheng
School of mathematics and information, Henan Polytechnic University, P.R.China, 454000
[email protected]
Abstract Traditional methods pricing for real option ignored uncertainty of underlying assets and
investment costs, when pricing the real option, which probably leads to a mistaken investment decision.
The pricing model of the real option is made research on, when the valuation of underlying assets and
investment costs are vague. It regards these uncertain elements as fuzzy number and fuzzy variant,
prices the real option using fuzzy set theory with binomial model and then obtains a new real option
pricing formula with fuzzy multi-variable.
Key word Finance, Real option, Fuzzy set theory, Risk investment
1. Introduction
In process of risk investment decision-making, underling assets and investment costs are uncertain.
The uncertainty influences the real option valuation by all appearances. However, Traditional methods
for pricing real option neglect the uncertainty, so we must provide a new pricing method for real option
in uncertain environment. As a result of the uncertainty, we have difficulty in obtaining an accurate real
option valuation, so decide to adopt a confidence interval for describing the real option valuation.
Otherwise, investors choose optimal investment tactic all the time. If they gain the most income when
inventing in the project of the moment, they have no hesitation to invent immediately [1]. If the project
brings non-market-oriented income, the claim is equivalent to American call option with dividend, so
early exercise may be taken into account. If not, the option can’t be exercised ahead [2].
2. Basic assumption
~
~
~
Suppose the value of underling assets and investment costs are S and X separately. We assume that S is
a normal fuzzy number, namely satisfies membership function as follows (Figure 1
x −U
）
µ (U , m)( x) = exp[−(
) 2 ], x ∈ R
m
~
We suppose that X is a triangle fuzzy number and its membership function (Figure 2) [4] is
（1）
b1 ≤ x ≤ b2
 ( x − b1 ) /(b2 − b1 )

µ X~ ( x ) = (b3 − x ) /(b3 − b2 ) b2 ≤ x ≤ b3
2

0
otherwise

~
which is denoted by X = (b1 , b2 , b3 ) .
~
~ L ~U
~
~
We shall use the notation S α = [ Sα , S α ] for α -lever set of S , and α -lever set of X is noted
~
~ L ~U
by X α = [ X α , X α ] , so that
（）
~
，
S
α
= (1 − α )b + αb ，
1
~
S αU = U + m ln
~
X αL
1
L
α
2
= U − m ln
1
α
~
X αU = (1 − α )b3 + αb2
For the sake of expositive facility, we provide the definition as follows [5]:
997
（3）
（ 4）
~
~
~
~
~
Aα is α -lever set of A , then max[ Aα ,0] = [max( AαL ,0), max( AαU ,0)]
µ1
.
µ
1
α
α
U − m ln
U
1
α
U + m ln
1
α
b2
b1
x
－ α )b + α b
(1
1
b3
x
(1 − α )b3 + αb2
2
Figure 2 investment cost is a triangle
fuzzy number
Figure 1 underlying asset valuation
is a normal fuzzy number
3. The pricing method for real option with fuzzy multi-variable
The maturity of investment project is time T , and we divide the life of the real option into n intervals.
~
We suppose that the risk-free interest rate is r and the real option valuation is C . We assume that the
investor considers the confidence lever as α . At the end of the first time step, the value of the
~
~
~
~
investment project S can either move up from S α to a new lever, uSα , or down from S α to a
~
new lever, dSα . The probability of an up movement is assumed to be p , so the variable 1 − p is the
probability of a down movement. If the price of investment assets can move up, the value of the real
~
~
option will become C u ,α . If move down, the value of the real option will become C d ,α (Figure 3) [6].
Now, we discuss the real option pricing model with n time steps.
~
C u 2 ,α
~
u Sα
2
~
Sα
~
uS α
~
Cα
~
udS α
~
Cu ,α
~
C ud ,α
~
C d ,α
~
dS α
~
C d 2 ,α
~
d 2 Sα
Figure 3 underling assets price and real option valuation
3.1 A condition without market-oriented income
We follow the method witch is adopted to deduce binomial trees[7], so the fuzzy option valuation can
be written
n
~
Cα = r − n {∑
j =0
~
n!
~
p j (1 − p) n− j max[0, u j d n − j S α − X α ]}
j!(n − j )!
where r = 1 + r .
998
（ 5）
From equation (5), we have
（）
n
~
CαL = r −n {∑
~
n!
~
p j (1 − p) n− j max[0, u j d n− j S αL − X αU ]}
6
j = 0 j!( n − j )!
n
n!
~
~
~
p i (1 − p ) n −i max[0, u i d n−i S αU − X αL ]}
CαU = r − n {∑
7
i = 0 i!( n − i )!
~
~
j n− j ~ L
S α − X αU < 0 ( ∀j ∈ [0, n], j ∈ N ), the low boundary will become CαL = 0 .
Obviously, if u d
~U
~L
i n −i ~ U
If u d S α − X α < 0 ( ∀i ∈ [0, n], i ∈ N ), the up boundary will become Cα = 0 . Let a and
b be
~
~
a = min{ j u j d n − j S αL − X αU > 0, j ∈ [0, n], j ∈ N}
~
~
b = min{i u i d n −i S αU − X αL > 0, i ∈ [0, n], i ∈ N }
8
（）
，
（）
From equation (6)
n
~
CαL = r − n {∑
j =a
n
~
= SαL [∑
j =a
~
n!
~
p j (1 − p ) n− j [u j d n− j SαL − X αU ]}
j!(n − j )!
u j d n− j
n!
n!
~ U −n n
p j (1 − p) n− j ]
−
X
[∑
)]
p j (1 − p ) n− j (
α r
n
j!(n − j )!
r
j = a j!( n − j )!
n
~ n
~
= S αL [∑ C nj p ′ j (1 − p ′) n− j ] − X αU r − n [∑ C nj p j (1 − p) n− j ]
j =a
j =a
which p ′ = (u / r ) p .
Substituting Φ[ x, n, y ] =
n
∑C
j=x
j
n
y j (1 − y ) n − j
， x ∈ [0, n], x ∈ N , y ∈ [0,1] , this becomes
~
~
~
CαL = S αL Φ[a, n, p ′] − X αU r − n Φ[a, n, p ]
~
For the same reason, the up boundary of C becomes
~
~
~
CαU = SαU Φ[b, n, p ′] − X αL r − n Φ[b, n, p ]
（9）
（10）
Substituting from equations (3) and (4) into equations (9) and (10), we get
1
~
CαL = (U − m ln )Φ[a, n, p ′] − r − n [(1 − α )b3 + αb2 ]Φ[a, n, p ]
（11）
1
~
CαU = (U + m ln )Φ[b, n, p ′] − r −n [(1 − α )b1 + αb2 ]Φ[b, n, p]
（12）
α
α
Therefore, if the investor considers that the confidence lever equal to α , we will obtain a fuzzy price of
~
~ L ~U
real option, Cα = [Cα , Cα ] .
3.2 A condition with market-oriented income
As already stated in our previous letter, if the project brings non-market-oriented income, the claim is
equivalent to American call option with dividend, so the option may be exercised ahead [8]. We denote
the non-market-oriented interest rate by δ . Actually, we mostly discuss the pricing method of real
option in uncertain environment, so we can’t obtain its accurate valuation.
999
L
U
We denote f i , j , f i , j and f i , j by
f i ,Lj = u i d j (1 − δ ) i + j S αL − X αU
f iU, j = u i d j (1 − δ ) i + j S αU − X αL
f i , j = u i d j (1 − δ )i + j S − X
where S is the accurate value of underlying assets and the value of investment costs is X .
L
U
We assume that f i , j is a random positive number in the interval [ f i , j , f i , j ] , and is considered as a
continuous random variable, noted by f i , j ( X ) . We assume that f i , j ( X ) accords with the distribution
witch density function is
1

δ ( f iU, j )
 U
f ( x ) =  f i , j − f i ,Lj

0
x ∈ [ w, f iU, j ]
otherwise
(13)
U
1 f i , j > 0
w = max( f ), δ ( f ) = 
U
where,
0 f i , j ≤ 0 .
L
Apparently, when f i , j > 0 , equation (13) means the density function of uniformity distributing. When
L
i, j
U
i, j
the real option is not advanced to exercise, f i , j = E[ f i , j ( X )] , where f i , j is the real option price in
the situation i of the step
j . If f i ,Lj > 0 , f i , j = ( f i ,Lj + f iU, j ) / 2 ; If f iU, j < 0 , f i , j = 0 ; If
f i ,Lj < 0 and f iU, j > 0 , f i , j = ( f iU, j ) 2 /( f iU, j − f i ,Lj ) .When the real option is exercised ahead,
f i , j = max{E[ f i , j ( X )], e − rδt [ pf i +1,1 + (1 − p ) f i , j +1 ]} .This means,
If f i , j > 0 , f i , j = max{( f i , j + f i , j ) / 2, e
L
L
If f i , j < 0 , f i , j = e
U
− rδt
U
− rδt
[ pf i +1,1 + (1 − p) f i , j +1 ]} ;
[ pf i +1,1 + (1 − p ) f i , j +1 ] ;
If f i , j < 0, f i , j > 0 , f i , j = max{( f i , j ) /( f i , j − f i , j ), e
L
U
U
2
U
L
− rδt
[ pf i +1,1 + (1 − p ) f i , j +1 ]} .
4. Example
A petroleum company has the opportunity to acquire a five-year licence on block. When developed,
the block is expected to yield 50 million barrels of oil. The current price of a barrel of oil from this field
is $10 and the present value of the development costs is $600 million [9]. Risk-free rate r is 5%. The
value of investment project can either increase by 30%, or reduce by 30%.
We now discuss whether the company should invest in the project from two follow methods.
First, we adopt the pricing model for real option which was put forward in reference [10].
NPV =50 million×$10 $600 million =$ -100 million
n
－
n!
) p j (1 − p) n− j max{u j d n − j S 0 − X ,0} = $ 139.03 million.
!
(
)!
j
n
j
−
j =0
Because of NPV + C = −100 + 139.03 = 39.03 > 0 , they consider that the project should be
C = r −n ∑ (
invested in.
Next, we make use of the method which was put forward in the paper, now. Using the notation
1000
~
introduced earlier, X = (55000,60000,65000) , T = 5 years
，
，
， U = 500 million ， u = 1.3 ，
r −d
= 0.58 so that p ′ = (u / r ) p = 0.73 . The life
u−d
of real option is divided into five time steps, each of which is one year, so n = 5 . Substituting above
d = 0.7 r = 5% m = 15000 and p =
parameters into equations (11) and (12), we obtain the results as follows (Table 1). The result from Table
1 is plotted in Figure 4.
α
0.60
0.62
0.64
0.66
0.68
0.70
0.72
0.74
0.76
0.78
0.80
0.82
0.84
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
When
~
C 0L,α
75.613
77.864
80.118
82.379
84.654
86.946
89.261
91.607
93.991
96.422
98.911
101.47
104.12
106.88
109.77
112.85
116.18
119.87
124.14
129.54
142.03
the
investor
Table 1 fuzzy valuation of real option and ENPV
~
C 0U,α
228.34
224.80
221.26
217.70
214.13
210.53
206.90
203.22
199.48
195.67
191.77
187.77
183.64
179.34
174.82
171.20
167.87
164.19
159.92
154.52
142.03
considers
~
Cα
~
NPV + C 0L,α
~
NPV + C 0U,α
-24.387
-22.136
-19.882
-17.621
-15.346
-13.054
-10.739
-8.393
-6.009
-3.578
-1.089
1.470
4.120
6.880
9.770
12.85
16.18
19.87
24.14
29.54
42.03
128.34
124.80
121.26
117.70
114.13
110.53
106.90
103.22
99.48
95.67
91.77
87.77
83.64
79.34
74.82
71.20
67.87
64.19
59.92
54.52
42.03
[75.613,228.34]
[77.864,224.80]
[80.118,221.26]
[82.379,217.70]
[84.654,214.13]
[86.946,210.53]
[89.261,206.90]
[91.607,203.22]
[93.991,199.48]
[96.422,195.67]
[98.911,191.77]
[101.47,187.77]
[104.12,183.64]
[106.88,179.34]
[109.77,174.82]
[112.85,171.20]
[116.18,167.87]
[119.87,164.19]
[124.14,159.92]
[129.54,154.52]
142.03
the
confidence
~U
lever
as
α = 0.82
~L
, NPV + Cα > 0
and NPV + Cα > 0 , therefore the project is worth to be invested in. According to the first method, we
get the same result. However, when α = 0.78 , the curve showing the low boundary of ENPV in Figure
~L
4 traverses x axis. This means NPV + Cα < 0 , so the investor should not invest in the project. The
conclusion is contrary to that we adopt the first method. Why are they different? The latter considers
underling assets and investment costs as crisp numbers, substitutes from them into binomial trees model,
and then obtains the result. In fact, underlying assets and investment costs are uncertain in uncertain
environment. However, the former has regard for fuzzy factors, so we obtain different conclusions.
1001
Figure 4
250.00
Low boundary
Of option price
200.00
Up boundary
Of option price
150.00
million
100.00
Low boundary
Of ENPV
50.00
0
0.6
-50.00
0.64
0.66
0.7
0.74
0.78
0.82
0.86
0.9
0.94
0.98
1
Up boundary
of NEPV
Membership
Figure 4 Relation between the fuzzy valuation of real option and confidence
5. Conclusion
Owing to the uncertainty of financial market, underling assets and investment costs cannot be expected
in a precise sense. Therefore, the fuzzy sets theory provides a useful tool for conquering this kind of
impreciseness. Under the considerations of fuzzy underling assets and investment costs, the real option
valuation turns into a fuzzy number. From two market conditions, corresponding fuzzy values of real
option are proposed in the paper. Finally, the illustrative example is introduced. Analysis shows the real
option pricing method proposed in the paper is prior to traditional pricing methods in uncertain
environment.
References
[1] GUO jing, CHEN Yin-wu, GUO Qin, LIAO dong-sheng. A Pricing Model of Real Option with
Ieakage in Value[J]. Systems Engineering, 2005(4):35-38.
[2] Song Feng-ming. Principles of Financial Engineering [M]. Beijing: Tsinghai University Press, 1999.
[3] Yuji, Yoshida. The valuation of European options in uncertain environment [J]. European Journal of
Operational Research [J].2003,145(1):221~229.
[4] Wu Hsien-Chung. Pricing European options based on the fuzzy pattern of Black-Scholes formula.
Computers & Operations Research [J]. 2004(7):1069~1081
[5] Wang Pei-zhang, Han li-xin. Applied Fuzzy Mathematic [M]. Beijing: Beijing College of Economics
Press, 1989.
[6] Cox J, Ross S, Rubinstein M. Option pricing, a simplified approach. Journal of Financial Economic
[J].1979( 7):229-263
[7] John C.Hull. Options, Futures, and Other Derivatives [M]. 4th edition, Prentice Hall, 2000.05
[8]Silvia Muzzioli, Costanza Torricelli. A multi-period binomial model for pricing options in a vague
world. Journal of Economic Dynamics & Control[J]. 2004 (28):861-887
[9] Christer Carlsson, Robert Fuller. A fuzzy approach to real option valuation [J]. Fuzzy Sets and
Systems [J]. 2003,139:221~229.
[10] LIU cheng-hua, HUANG Bei-xiao, LI Mei. Binomial Model in Process of Risk Investment.
Science and Technology Management Research [J], 2005(10):161-163.
1002