Uncertain contour process and its application in stock model with ...
Fuzzy Optim Decis Making (2015) 14:399–424 DOI 10.1007/s10700-015-9211-y
Uncertain contour process and its application in stock model with floating interest rate Kai Yao1
Published online: 12 March 2015 © Springer Science+Business Media New York 2015
Abstract Uncertain process is an important tool to model dynamic uncertain systems. This paper proposes a special type of uncertain processes, named contour processes, whose sample paths can be classified by their inverse uncertainty distributions. It is shown that the set of contour processes is closed under the extreme value operator and the time integral operator as well as the monotone function. As an application, this paper considers an uncertain stock model with floating interest rate, in which both the interest rate and the stock price follow uncertain differential equations. By means of contour processes, some pricing formulas are derived for the European options, American options and Asian options of the stock model. Keywords Uncertain process · Stock model · Uncertain differential equation · Uncertain finance
1 Introduction Uncertainty theory, founded by Liu (2007), is used to model human uncertainty based on normality, duality, subadditivity and product axioms. In order to indicate the belief degree that an event is supposed to occur, an uncertain measure is defined as a set function on a σ -algebra. Then an uncertain variable is used to model a quantity with uncertainty, and an uncertainty distribution is used to describe an uncertain variable. In addition, some numerical characteristics of uncertain variables have also been studied such as expected value, variance and entropy.
B 1
Kai Yao [email protected] School of Management, University of Chinese Academy of Sciences, Beijing 100190, China
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In order to model the evolution of uncertain phenomena, a concept of uncertain process was proposed by Liu (2008) as a sequence of uncertain variables indexed by time. Uncertain independent increment process is an uncertain process with independent increments. A sufficient and necessary condition for a function being an inverse uncertainty distribution of an uncertain independent increment process was given by Liu (2014). Besides, the extreme values of an uncertain independent increment process was studied by Liu (2013b). Uncertain stationary independent increment process is an uncertain process with both stationary increments and independent increments. Liu (2010) proved that its expected value is a linear function of the time, and Chen (2011) proved that its standard deviation is proportional to the time. Canonical Liu process, designed by Liu (2009), is a special type of uncertain stationary independent increment processes whose increments are normal uncertain variables. Meanwhile, Liu (2009) founded an uncertain calculus theory to deal with the integral and differential of an uncertain process with respect to Liu process. Uncertain differential equation, a differential equation driven by the Liu process, was proposed by Liu (2008) to describe a dynamic uncertain system. Chen and Liu (2010) gave an analytic solution of a linear uncertain differential equation, and Liu (2012) and Yao (2013b) gave some methods to obtain the solutions of some special types of nonlinear uncertain differential equations. In addition, Yao and Chen (2013) verified some properties of the solution of an uncertain differential equation via the so-called α-path which is essentially the inverse uncertainty distribution of the solution. Based on these results, Yao (2013a) gave the uncertainty distributions of the extreme values and the time integral of the solution. Uncertain differential equation was introduced to finance in 2009. Liu (2009) assumed the stock price follows a geometric Liu process, and proposed a stock model which was named Liu’s stock model later. Chen (2011) derived its American option pricing formulas, and Yao (2015) gave a sufficient and necessary condition for the stock market being no-arbitrage. In addition, Peng and Yao (2011) proposed a stock model via mean-reverting uncertain differential equation to describe the stock price in a long term, and Chen et al. (2013) proposed a stock model with periodical dividends. Uncertain interest rate model was first proposed by Chen and Gao (2013), and was further studied by Jiao and Yao (2015). In addition, uncertain currency model was proposed by Liu et al. (2015) to model the currency exchange rate. For recent developments about uncertain finance, interested readers may refer to Liu (2013a) for details. Inspired by the solution of an uncertain differential equation, this paper will propose a concept of contour process, and will study an uncertain stock model with floating interest rate as an application. In Sect. 2, we review some basic concepts about uncertain variable, uncertain process and uncertain differential equation. Then in Sect. 3, we introduce the concept of contour process, and give its inverse uncertainty distribution and expected value. In Sects. 4 and 5, we study the extreme values and time integral of a contour process, respectively. Besides, we study the function of multiple contour processes in Sect. 6. After that, we derive some useful theorems about uncertain differential equations in Sect. 7. In Sect. 8, we introduce the new stock model with floating interest rate, and give its European option pricing formulas, American option pricing
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formulas, and Asian option pricing formulas. Finally, some conclusions are made in Sect. 9.
2 Preliminaries In this section, we introduce some concepts about uncertain variable, uncertain process and uncertain differential equation. 2.1 Uncertain variable Definition 1 (Liu 2007) Let L be a σ -algebra on a nonempty set . A set function M : L → [0, 1] is called an uncertain measure if it satisfies the following axioms: Axiom 1 (Normality Axiom) M{} = 1 for the universal set . Axiom 2 (Duality Axiom) M{} + M{c } = 1 for any event . Axiom 3 (Subadditivity Axiom) For every countable sequence of events 1 , 2 , . . . , we have ∞ ∞ i ≤ M {i } . M i=1
i=1
A product uncertain measure on multiple uncertainty spaces was defined by Liu (2009) to provide the operational law of uncertain variables. Axiom 4 (Product Axiom) Let (k , Lk , Mk ) be uncertainty spaces for k = 1, 2, . . . Then the product uncertain measure M is an uncertain measure satisfying ∞ ∞ M k = Mk {k } k=1
k=1
where k are arbitrarily chosen events from Lk for k = 1, 2, . . ., respectively. Definition 2 (Liu 2007) An uncertain variable is a measurable function ξ from an uncertainty space (, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {γ | ξ(γ ) ∈ B} is an event. Definition 3 (Liu 2007) The uncertainty distribution of an uncertain variable ξ is defined by (x) = M{ξ ≤ x} for any real number x.
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If the inverse function −1 exists and is unique for each α ∈ (0, 1), then it is called the inverse uncertainty distribution of ξ . In this case, the uncertainty distribution is said to be regular. Definition 4 (Liu 2007) Let ξ be an uncertain variable. Then its expected value E[ξ ] is defined by
+∞
E[ξ ] =
M{ξ ≥ r }dr −
0
0
−∞
M{ξ ≤ r }dr
provided that at least one of the two integrals is finite. Definition 5 (Liu 2009) The uncertain variables ξ1 , ξ2 , . . . , ξm are said to be independent if M
m k=1
(ξi ∈ Bi ) =
m
M{ξi ∈ Bi }
k=1
for any Borel sets B1 , B2 , . . . , Bm of real numbers. Let ξ1 , ξ2 , . . . , ξn be independent uncertain variables with regular uncertainty distributions 1 , 2 , . . . , n , respectively. Supppose f (x1 , x2 , . . . , xn ) is strictly increasing with respect to x1 , x2 , . . . , xm and strictly decreasing with respect to xm+1 , xm+2 , . . . , xn . Then Liu (2010) proved the uncertain variable ξ = f (ξ1 , ξ2 , . . . , ξn ) has an inverse uncertainty distribution
−1 −1 −1 −1 (r ) = f −1 (r ), . . . , (r ), (1 − r ), . . . , (1 − r ) , m n 1 m+1 and Liu and Ha (2010) proved ξ has an expected value
1
E[ξ ] = 0
−1 −1 −1 f −1 (r ), . . . , (r ), (1 − r ), . . . , (1 − r ) dα. m n 1 m+1
2.2 Uncertain process Definition 6 (Liu 2008) Let T be an index set, and (, L, M) be an uncertainty space. An uncertain process X t is a measurable function from T × (, L, M) to the set of real numbers, i.e., for each t ∈ T and any Borel set B of real numbers, the set {X t ∈ B} = {γ | X t (γ ) ∈ B} is an event. Definition 7 (Liu 2014) An uncertain process X t is said to have an uncertainty distribution t (x) if at each time t, the uncertain variable X t has the uncertainty distribution t (x).
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Definition 8 (Liu 2014) Uncertain processes X 1t , X 2t , . . . , X nt are said to be independent if for any positive integer k and any times t1 , t2 , . . . , tk , the uncertain vectors ξ i = (X it1 , X it2 , . . . , X itk ), i = 1, 2, . . . , n are independent, i.e., for any Borel sets B1 , B2 , . . . , Bn of k-dimensional real vectors, we have M
n
(ξ i ∈ Bi ) =
i=1
n
M{ξ i ∈ Bi }.
i=1
Definition 9 Let X t be an uncertain process on an uncertainty space (, L, M). Then for every γ ∈ , the function X t (γ ) is called a sample path of X t . For an uncertain process X t on an uncertainty space (, L, M), define uncertain processes Yt and Z t by Yt (γ ) = sup X s (γ ), ∀γ ∈ , 0≤s≤t
Z t (γ ) = inf X s (γ ), ∀γ ∈ . 0≤s≤t
Then Yt and Z t are called the supremum process and infimum process of X t , respectively. Definition 10 (Liu 2008) Let X t be an uncertain process. For any partition of a closed interval [a, b] with a = t1 < t2 < . . . < tk+1 = b, the mesh is written as = max |ti+1 − ti |. 1≤i≤k
Then the time integral of X t with respect to t is a
b
X t dt = lim
→0
k
X ti (ti+1 − ti )
i=1
provided that the limit exists almost surely and is finite. 2.3 Uncertain differential equation In order to found an uncertain calculus theory, a concept of canonical Liu process was defined by Liu (2009) as follows. Definition 11 (Liu 2009) An uncertain process Ct is said to be a canonical Liu process if
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(i) C0 = 0 and almost all sample paths are Lipschitz continuous, (ii) Ct has stationary and independent increments, (iii) every increment Cs+t − Cs is a normal uncertain variable with an uncertainty distribution
−1 πx t (x) = 1 + exp − √ , x ∈ . 3t Definition 12 (Liu 2009) Let X t be an uncertain process and Ct be a canonical Liu process. For any partition of closed interval [a, b] with a = t1 < t2 < . . . < tk+1 = b, the mesh is written as = max |ti+1 − ti |. 1≤i≤k
Then the Liu integral of X t is defined by
b
X t dCt = lim
→0
a
k
X ti · (Cti+1 − Cti )
i=1
provided that the limit exists almost surely and is finite. Definition 13 (Liu 2008) Suppose Ct is a canonical Liu process, and f and g are two given functions. Then (1) dX t = f (t, X t )dt + g(t, X t )dCt is called an uncertain differential equation. The uncertain differential Eq. (1) is equivalent to an integral equation Xt = X0 +
t
0
t
f (s, X s )ds +
g(s, X s )dCs ,
0
and a solution of (1) is just an uncertain process satisfying the integral equation. Yao and Chen (2013) defined a concept of α-path which is a function solving the ordinary differential equation dX tα = f (t, X tα )dt + |g(t, X tα )|−1 (α)dt where −1
√ α 3 ln (α) = π 1−α
is the inverse uncertainty distribution of Ct at t = 1, and showed M{X t ≤ X tα , ∀t} = α, M{X t > X tα , ∀t} = 1 − α.
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3 Contour process Inspired by the solution of an uncertain differential equation, we introduce a concept of contour process in this section, and study its inverse uncertainty distribution and expected value. Definition 14 Let X t be an uncertain process. If for each α ∈ (0, 1), there exists a real function X tα such that M{X t ≤ X tα , ∀t} = α, M{X t > X tα , ∀t} = 1 − α,
(2) (3)
then X t is called a contour process. In this case, X tα is called the α-path of the uncertain process X t . Example 1 A solution of an uncertain differential equation is a contour process. Please refer to Yao and Chen (2013) and Yao (2013a) for details. Example 2 Let ξ be a regular uncertain variable. Then the uncertain process Sn =
ξ, −ξ,
if n is odd if n is even
is not a contour process. Theorem 1 An uncertain process X t is a contour process if and only if for each α ∈ (0, 1), there exists a real function X tα such that M{X t < X tα , ∀t} = α, M{X t ≥
X tα , ∀t}
= 1 − α.
(4) (5)
Proof On the one hand, assume X t is a contour process with an α-path X tα . Note that for each α ∈ (0, 1) and for any given ε > 0, we have {X t ≤ X tα−ε , ∀t} ⊂ {X t < X tα , ∀t} ⊂ {X t ≤ X tα , ∀t}, {X t > X tα , ∀t} ⊂ {X t ≥ X tα , ∀t} ⊂ {X t > X tα−ε , ∀t}. By using the monotonicity of uncertain measure, we get that M{X t ≤ X tα−ε , ∀t} ≤ M{X t < X tα , ∀t} ≤ M{X t ≤ X tα , ∀t}, M{X t > X tα , ∀t} ≤ M{X t ≥ X tα , ∀t} ≤ M{X t > X tα−ε , ∀t}. According to Definition 14 of contour process, we have M{X t ≤ X tα−ε , ∀t} = α − ε, M{X t ≤ X tα , ∀t} = α, M{X t > X tα , ∀t} = 1 − α, M{X t > X tα−ε , ∀t} = 1 − α + ε.
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Then Eqs. (4) and (5) follow immediately. On the other hand, assume Eqs. (4) and (5) hold for each α ∈ (0, 1). Note that for each α ∈ (0, 1) and for any given ε > 0, we have {X t < X tα , ∀t} ⊂ {X t ≤ X tα , ∀t} ⊂ {X t < X tα+ε , ∀t}, {X t ≥ X tα+ε , ∀t} ⊂ {X t > X tα , ∀t} ⊂ {X t ≥ X tα , ∀t}. By using the monotonicity of uncertain measure, we get that M{X t < X tα , ∀t} ≤ M{X t ≤ X tα , ∀t} ≤ M{X t < X tα+ε , ∀t}, M{X t ≥ X tα+ε , ∀t} ≤ M{X t > X tα , ∀t} ≤ M{X t ≥ X tα , ∀t}. It follows from Eqs. (4) and (5) that M{X t < X tα , ∀t} = α, M{X t < X tα+ε , ∀t} = α + ε,
M{X t ≥ X tα+ε , ∀t} = 1 − α − ε, M{X t ≥ X tα , ∀t} = 1 − α. Then we have M{X t ≤ X tα , ∀t} = α, M{X t > X tα , ∀t} = 1 − α. According to Definition 14, the uncertain process X t is a contour process. Theorem 2 Let X t be a contour process with an α-path X tα . Then X t has an inverse uncertainty distribution α −1 t (α) = X t , ∀α ∈ (0, 1)
and an expected value
1
E[X t ] = 0
X tα dα.
Proof Given any time s, since {X s ≤ X sα } ⊃ {X t ≤ X tα , ∀t}, {X s > X sα } ⊃ {X t > X tα , ∀t}, we have M{X s ≤ X sα } ≥ M{X t ≤ X tα , ∀t} = α, M{X s > X sα } ≥ M{X t > X tα , ∀t} = 1 − α
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according to the monotonicity of uncertain measure. It follows from the duality of uncertain measure that M{X s ≤ X sα } + M{X s > X sα } = 1. Then we have M{X s ≤ X sα } = α, which means X t has an inverse uncertainty distribution X tα . As a result, we have
1
E[X t ] = 0
X tα dα.
The proof is completed.
Since the α-path X tα is just the inverse uncertainty distribution of a contour process X t , in the rest of this paper, we denote the inverse uncertainty distribution of X t by X tα for simplicity.
4 Extreme values In this section, we study the extreme values of a contour process. The results show that the set of contour processes is closed under the supremum operator and infimum operator. Theorem 3 Let X t be a contour process with an α-path X tα . Then its supremum process Yt = sup X s 0≤s≤t
is a contour process with an α-path Ytα = sup X sα . 0≤s≤t
Proof For a sample path X t (γ ) such that X t (γ ) ≤ X tα for any time t, we have sup X s (γ ) ≤ sup X sα , ∀t.
0≤s≤t
0≤s≤t
It implies
Yt ≤ sup X sα , ∀t 0≤s≤t
=
sup X s ≤ sup X sα , ∀t
0≤s≤t
0≤s≤t
⊃ X t ≤ X tα , ∀t .
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By the monotonicity of uncertain measure, we have α M Yt ≤ sup X s , ∀t ≥ M X t ≤ X tα , ∀t = α. 0≤s≤t
Similarly, we have
M Yt > sup 0≤s≤t
X sα , ∀t
≥ M X t > X tα , ∀t = 1 − α.
Since
M Yt ≤ sup 0≤s≤t
X sα , ∀t
+ M Yt > sup 0≤s≤t
X sα , ∀t
≤ 1,
we have
M Yt ≤ sup 0≤s≤t
X sα , ∀t
= α,
M Yt > sup X sα , ∀t
= 1 − α.
0≤s≤t
So Yt is a contour process with an α-path Ytα = sup X sα . 0≤s≤t
Theorem 4 Let X t be a contour process with an α-path X tα . Then its infimum process Z t = inf X s 0≤s≤t
is a contour process with an α-path Z tα = inf X sα . 0≤s≤t
Proof For a sample path X t (γ ) such that X t (γ ) ≤ X tα for any time t, we have inf X s (γ ) ≤ inf X sα , ∀t.
0≤s≤t
0≤s≤t
It implies Z t ≤ inf X sα , ∀t = inf X s ≤ inf X sα , ∀t ⊃ X t ≤ X tα , ∀t . 0≤s≤t
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0≤s≤t
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By the monotonicity of uncertain measure, we have α M Z t ≤ inf X s , ∀t ≥ M X t ≤ X tα , ∀t = α. 0≤s≤t
Similarly, we have M Z t > inf X sα , ∀t ≥ M X t > X tα , ∀t = 1 − α. 0≤s≤t
Since
M Z t ≤ inf X sα , ∀t + M Z t > inf X sα , ∀t < 1, 0≤s≤t
we have
0≤s≤t
M Z t ≤ inf X sα , ∀t = α, 0≤s≤t M Z t > inf X sα , ∀t = 1 − α. 0≤s≤t
So Z t is a contour process with an α-path Z tα = inf X sα . 0≤s≤t
5 Time integral In this section, we will study the time integral of a contour process. It is shown that the set of contour processes is closed under the time integral operator. Theorem 5 Let X t be a contour process with an α-path X tα . Then its time integral process
t
Yt =
X s ds 0
is a contour process with an α-path Ytα
t
=
X sα ds.
0
Proof For a sample path X t (γ ) such that X t (γ ) ≤ X tα for any time t, we have 0
t
X s (γ )ds ≤ 0
t
X sα ds, ∀t.
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It implies
t
Yt ≤ 0
X sα ds, ∀t =
t
X s ds ≤
0
0
t
X sα ds, ∀t ⊃ X t ≤ X tα , ∀t .
By the monotonicity of uncertain measure, we have M Yt ≤
t
0
X sα ds, ∀t ≥ M X t ≤ X tα , ∀t = α.
Similarly, we have
t
M Yt > 0
X sα ds, ∀t
≥ M X t > X tα , ∀t = 1 − α.
Since
M Yt ≤ 0
t
X sα ds, ∀t
t
+ M Yt > 0
X sα ds, ∀t
≤ 1,
we have
t
X sα ds, ∀t
= α, M Yt ≤ 0 t M Yt > X sα ds, ∀t = 1 − α. 0
So Yt is a contour process with α-path Ytα =
t 0
X sα ds.
6 Monotone function In this section, we will study the monotone function of multiple contour processes, and we will show the derived uncertain process is also a contour process. Theorem 6 Let X 1t , X 2t , . . . , X nt be independent contour processes with α-paths α , X α , . . . , X α , respectively. If f (x , x , . . . , x ) is strictly increasing with respect X 1t 1 2 n nt 2t to x1 , x2 , . . . , xm and strictly decreasing with respect to xm+1 , xm+2 , . . . , xn , then the uncertain process X t = f (X 1t , X 2t , . . . , X nt ) is a contour process with an α-path
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1−α α α 1−α . X tα = f X 1t , . . . , X mt , X m+1t , . . . , X nt Proof According to the monotonicity of the function f , we have
1−α α α 1−α , ∀t {X t ≤ X tα , ∀t} = f (X 1t , X 2t , . . . , X nt ) ≤ f X 1t , . . . , X mt , X m+1t , . . . , X nt ⊃
m
n X it ≤ X itα , ∀t ∩ X it ≥ X it1−α , ∀t
i=1
i=m+1
and {X t > X tα , ∀t} = ⊃
1−α α α 1−α , ∀t f (X 1t , X 2t , . . . , X nt ) > f X 1t , . . . , X mt , X m+1t , . . . , X nt
m
n X it > X itα , ∀t ∩ X it < X it1−α , ∀t .
i=1
i=m+1
By the independence of uncertain processes and the monotonicity of uncertain measure, we have M{X t ≤
X tα , ∀t}
≥M
m
X it ≤
X itα , ∀t
∩
i=1
=
m
n
X it ≥
X it1−α , ∀t
i=m+1
M X it ≤ X itα , ∀t ∧
i=1
n
M X it ≥ X it1−α , ∀t = α,
i=m+1
and M{X t
X itα , ∀t
∩
i=1
=
m i=1
M X it > X itα , ∀t ∧
n
X it
f L(X tα ), X tα , ∀t ⊃ {X t > X tα , ∀t}.
According to the monotonicity of uncertain measure, we have M f (L(X t ), X t ) ≤ f L(X tα ), X tα , ∀t ≥ M{X t ≤ X tα , ∀t} = α, M f (L(X t ), X t ) > f L(X tα ), X tα , ∀t ≥ M{X t > X tα , ∀t} = 1 − α. Since M f (L(X t ), X t ) ≤ f L(X tα ), X tα , ∀t + M f (L(X t ), X t ) > f L(X tα ), X tα , ∀t < 1, we have M f (L(X t ), X t ) ≤ f L(X tα ), X tα , ∀t = α, M f (L(X t ), X t ) > f L(X tα ), X tα , ∀t = 1 − α.
The proof is completed.
Example 3 Let X t be a contour process with an α-path X tα . Then the uncertain process Yt = sup X s + X t 0≤s≤t
is a contour process with an α-path Ytα = sup X sα + X tα , 0≤s≤t
and the uncertain process
t
Zt = 0
is a contour process with an α-path
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Z tα =
t
X sα ds + X tα .
0
7 Applications to uncertain differential equations In this section, we derive some theorems about uncertain differential equations. The applications of these theorems in uncertain finance will be given in the next section. Theorem 8 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t , respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have E exp −
T
X 1s ds (X 2T − K )
+
1
=
0
exp −
0
T 0
1−α X 1s ds
α (X 2T − K )+ dα,
(6) α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively. Proof Since the solution of an uncertain differential equation is a contour process, it follows from Theorems 5 and 6 that the uncertain process exp −
t
X 1s ds 0
is a contour process with an α-path exp −
t 0
1−α X 1s ds ,
α − K )+ . and the uncertain process (X 2t − K )+ is a contour process with an α-path (X 2t Then by Theorem 6, the uncertain process
exp −
t
X 1s ds (X 2t − K )+
0
is also a contour process with an α-path exp −
t 0
1−α X 1s ds
α (X 2t − K )+ .
According to Theorem 2, we have E exp −
T
X 1s ds (X 2T − K )+ =
0
The theorem is proved.
1 0
exp −
T 0
1−α α X 1s ds (X 2T − K )+ dα.
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Theorem 9 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t , respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have E exp −
T
X 1s ds (K − X 2T )+ =
0
1
exp −
0
T 0
α α + X 1s ds (K − X 2T ) dα,
(7) α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively. Proof Since the solution of an uncertain differential equation is a contour process, it follows from Theorems 5 and 6 that the uncertain process exp −
t
X 1s ds 0
is a contour process with an α-path exp −
t 0
1−α X 1s ds ,
1−α + ) . and the uncertain process (K −X 2t )+ is a contour process with an α-path (K −X 2t Then by Theorem 6, the uncertain process
exp −
t
X 1s ds (K − X 2t )+
0
is also a contour process with an α-path exp −
t 0
1−α X 1s ds
1−α + (K − X 2t ) .
According to Theorem 2, we have E exp −
T
X 1s ds (K − X 2T )
0
The theorem is proved.
+
= 0
1
exp −
T 0
α X 1s ds
α + (K − X 2T ) dα.
Theorem 10 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t , respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have
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sup exp −
E
0≤t≤T
1
=
415
t
X 1s ds (X 2t − K )
0
sup exp −
0 0≤t≤T
t 0
+
1−α α X 1s ds (X 2t − K )+ dα,
(8)
α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively.
Proof It follows from Theorem 8 that the uncertain process
t X 1s ds (X 2t − K )+ exp − 0
is a contour process with an α-path
t 1−α α X 1s ds (X 2t − K )+ . exp − 0
Then by Theorem 3, the uncertain process t
sup exp − X 1s ds (X 2t − K )+ 0
0≤t≤T
is a contour process with an α-path sup exp −
1−α α X 1s ds (X 2t − K )+ .
t 0
0≤t≤T
According to Theorem 2, we have
sup exp −
E
0≤t≤T
1
=
t
X 1s ds (X 2t − K )+
0
sup exp −
0 0≤t≤T
t
1−α X 1s ds
0
α (X 2t − K )+ dα.
The theorem is proved.
Theorem 11 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t , respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have E
sup exp −
0≤t≤T
=
1
t
X 1s ds (K − X 2t )
0
sup exp −
0 0≤t≤T
t 0
α X 1s ds
+
α + (K − X 2t ) dα,
(9)
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K. Yao
α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively.
Proof It follows from Theorem 9 that the uncertain process exp −
X 1s ds (K − X 2t )+
t 0
is a contour process with an α-path exp −
t 0
1−α 1−α + X 1s ds (K − X 2t ) .
Then by Theorem 3, the uncertain process sup exp −
t
X 1s ds (K − X 2t )+
0
0≤t≤T
is a contour process with an α-path sup exp −
t 0
0≤t≤T
1−α 1−α + X 1s ds (K − X 2t ) .
According to Theorem 2, we have E
sup exp −
0≤t≤T
= =
1
t
X 1s ds (K − X 2t )+
0
t
1−α sup exp − rs ds (K − X t1−α )+ dα
0 0≤t≤T 1
0
0 0≤t≤T
0
t
sup exp − rsα ds (K − X tα )+ dα.
The theorem is proved.
Theorem 12 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t , respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have
T
T
+ 1 E exp − X 1s ds X 2s ds − K T 0 0 T
T
+ 1 1 1−α α exp − X 1s ds · X 2s ds − K dα, = T 0 0 0
123
(10)
Uncertain contour process and its application...
417
α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively.
Proof Since the solution of an uncertain differential equation is a contour process, it follows from Theorems 5 and 6 that the uncertain process exp −
t
X 1s ds 0
is a contour process with an α-path exp −
t 0
1−α X 1s ds
,
and the uncertain process t
+ 1 X 2s ds − K t 0 is a contour process with an α-path t
+ 1 α X 2s ds − K . t 0 Then by Theorem 6, the uncertain process exp −
t 0
t
+ 1 X 1s ds X 2s ds − K t 0
is also a contour process with an α-path exp −
t 0
1−α X 1s ds
t
+ 1 α · X ds − K . t 0 2s
According to Theorem 2, we have
E exp −
T
X 1s ds 0
=
1
exp −
0
The theorem is proved.
T 0
1 T
1−α X 1s ds
T
+ X 2s ds − K
0
T
+ 1 α · X 2s ds − K dα. T 0
Theorem 13 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t ,
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K. Yao
respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have
+ 1 T X 1s ds K− X 2s ds T 0 0 T
+ 1 1 T α α = exp − X 1s ds K− X 2s ds dα, T 0 0 0
E exp −
T
(11)
α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively.
Proof Since the solution of an uncertain differential equation is a contour process, it follows from Theorems 5 and 6 that the uncertain process exp −
t
X 1s ds 0
is a contour process with an α-path exp −
1−α X 1s ds ,
t 0
and the uncertain process
1 t
K−
+
t
X 2s ds 0
is a contour process with an α-path
1 K− t
t 0
1−α X 2s ds
+
.
Then by Theorem 6, the uncertain process exp −
t
X 1s ds 0
1 K− t
+
t
X 2s ds 0
is also a contour process with an α-path exp −
t 0
1−α X 1s ds
According to Theorem 2, we have
123
+ 1 t 1−α · K− X ds . t 0 2s
Uncertain contour process and its application...
419
+ 1 T X 1s ds K− X 2s ds T 0 0 T
+ 1 1 T 1−α 1−α = exp − X 1s ds · K − X 2s ds dα T 0 0 0 T
+ 1 1 T α α exp − X 1s ds K− X 2s ds dα. = T 0 0 0
E exp −
T
The theorem is proved.
8 Stock model with floating interest rate Let rt denote the interest rate, and X t denote the stock price in an uncertain market. Liu (2009) proposed a stock model with fixed interest rate as follows,
drt = μ1rt dt dX t = μ2 X t dt + σ2 X t dC2t
where μ1 is the riskless interest rate, μ2 is the stock drift, σ2 is the stock diffusion, and C2t is a canonical Liu process. Liu (2009), Chen (2011), and Sun and Chen (2013) derived pricing formulas of the European options, American options and Asian options for Liu’s stock model, respectively. However, there are some uncertain factors influencing the interest rate in the financial market. So here we assume both the interest rate and the stock price follow uncertain differential equations, and propose a stock model with floating interest rate as follows, drt = μ1rt dt + σ1 rt dC1t (12) dX t = μ2 X t dt + σ2 X t dC2t , where μ1 and σ1 are the drift and the diffusion of the interest rate, respectively, μ2 and σ2 are the drift and the diffusion of the stock price, respectively, and C1t and C2t are independent canonical Liu processes. It is easy to verify that the α-path rtα of the interest rate rt is
√
rtα = r0 · exp μ1 t +
α 3σ1 t ln π 1−α
(13)
by solving the ordinary differential equation √ drtα
=
μ1rtα dt
+
α 3σ1rtα ln dt. π 1−α
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K. Yao
Similarly, the α-path X tα of the stock price X t is
X tα
√ 3σ2 t α = X 0 · exp μ2 t + . ln π 1−α
(14)
Consider a European call option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by f c = E exp −
T
rs ds (X T − K )
+
.
0
According to Theorem 8, we have
1
fc =
exp −
0
T
rs1−α ds
0
(X Tα − K )+ dα,
where exp −
T
0
rs1−α ds
⎞ √ 1 − exp μ1 T + 3σ1 T /π · (ln(1 − α) − ln α) ⎠ = exp ⎝r0 · √ μ1 + 3σ1 /π · (ln(1 − α) − ln α) ⎛
and (X Tα
+
− K) =
√
X 0 · exp μ2 T +
α 3σ2 T ln π 1−α
+ −K
.
Consider a European put option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by
f p = E exp −
T
rs ds (K − X T )
+
.
0
According to Theorem 9, we have fp =
1
0
exp −
T 0
rsα ds
(K − X Tα )+ dα,
where exp −
T 0
123
rsα ds
⎞ √ 1 − exp μ1 T + 3σ1 T /π · (ln α − ln(1 − α)) ⎠ = exp ⎝r0 · √ μ1 + 3σ1 /π · (ln α − ln(1 − α)) ⎛
Uncertain contour process and its application...
421
and (K −
X Tα )+
=
+ √ α 3σ2 T ln K − X 0 · exp μ2 T + . π 1−α
Consider an American call option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by
t
+ sup exp − rs ds (X t − K ) .
fc = E
0
0≤t≤T
According to Theorem 10, we have fc =
t
sup exp − rs1−α ds (X Tα − K )+ dα,
1
0 0≤t≤T
0
where
⎞ ⎛ √
t 1 − exp μ1 t + 3σ1 t/π · (ln(1 − α) − ln α) ⎠ rs1−α ds = exp ⎝r0 · exp − √ μ1 + 3σ1 /π · (ln(1 − α) − ln α) 0 and (X tα
+
− K) =
+ √ α 3σ2 t ln X 0 · exp μ2 t + . −K π 1−α
Consider an American put option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by fp = E
t
sup exp − rs ds (K − X t )+ . 0≤t≤T
0
According to Theorem 11, we have fp =
1
t
α sup exp − rs ds (K − X tα )+ dα,
0 0≤t≤T
0
where
⎞ ⎛ √
t 1 − exp μ1 t + 3σ1 t/π · (ln α − ln(1 − α)) ⎠ rsα ds = exp ⎝r0 · exp − √ μ1 + 3σ1 /π · (ln α − ln(1 − α)) 0
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K. Yao
and (K −
X tα )+
=
+ √ α 3σ2 t ln K − X 0 · exp μ2 t + . π 1−α
Consider an Asian call option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by
f c = E exp −
T
rs ds 0
1 T
T
+ X s ds − K
.
0
According to Theorem 12, we have
1
fc =
exp −
0
T 0
T
+ 1 rs1−α ds · X sα ds − K dα, T 0
where exp −
T 0
rs1−α ds
⎞
√ 1 − exp μ1 T + 3σ1 T /π · (ln(1 − α) − ln α) ⎠ = exp ⎝r0 · √ μ1 + 3σ1 /π · (ln(1 − α) − ln α) ⎛
and
⎞+ √ + ⎛ 1 − exp μ2 T + 3σ2 T /π · (ln α − ln(1 − α)) 1 T α X s ds − K = ⎝X0 · − K⎠ . √ T 0 μ2 T + 3σ2 T /π · (ln α − ln(1 − α))
Consider an Asian put option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by
f p = E exp −
T
rs ds 0
1 K− T
+
T
X s ds
.
0
According to Theorem 13, we have fp =
1
exp −
0
T 0
rsα ds
K−
1 T
0
T
X sα ds
+ dα,
where exp −
T 0
123
rsα ds
⎞ √ 1 − exp μ1 T + 3σ1 T /π · (ln α − ln(1 − α)) ⎠ = exp ⎝r0 · √ μ1 + 3σ1 /π · (ln α − ln(1 − α)) ⎛
Uncertain contour process and its application...
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and
⎞+ √ + ⎛ 1 − exp μ2 T + 3σ2 T /π · (ln α − ln(1 − α)) 1 T α ⎠ . X s ds = ⎝K − X0 · K− √ T 0 μ2 T + 3σ2 T /π · (ln α − ln(1 − α))
9 Conclusion This paper proposed a concept of contour process, and showed that the set of contour processes is closed under extreme value operator, time integral operator and monotone function. As a generalization of Liu’s stock model, this paper proposed an uncertain stock model with floating interest rate, and derived some pricing formulas of the European options, American options and Asian options of the stock model. Acknowledgments No. 61403360).
This work was supported by National Natural Science Foundation of China (Grant
References Chen, X. (2011). American option pricing formula for uncertain financial market. International Journal of Operations Research, 8(2), 32–37. Chen, X., & Gao, J. (2013). Uncertain term structure model of interest rate. Soft Computing, 17(4), 597–604. Chen, X., & Liu, B. (2010). Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization and Decision Making, 9(1), 69–81. Chen, X., Liu, Y., & Ralescu, D. A. (2013). Uncertain stock model with periodic dividends. Fuzzy Optimization and Decision Making, 12(1), 111–123. Jiao, D., & Yao, K. (2015). An interest rate model in uncertain environment. Soft Computing, 19(3), 775–780. Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer. Liu, B. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2(1), 3–16. Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10. Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer. Liu, B. (2013a). Toward uncertain finance theory. Journal of Uncertainty Analysis and Applications, 1. doi:10.1186/2195-5468-1-1 Liu, B. (2013b). Extreme value theorems of uncertain process with application to insurance risk model. Soft Computing, 17(4), 549–556. Liu, B. (2014). Uncertainty distribution and independence of uncertain processes. Fuzzy Optimization and Decision Making, 13(3), 259–271. Liu, Y. (2012). An analytic method for solving uncertain differential equations. Journal of Uncertain Systems, 6(4), 244–249. Liu, Y., & Ha, M. (2010). Expected value of function of uncertain variables. Journal of Uncertain Systems, 4(3), 181–186. Liu, Y., Chen, X., & Ralescu, D. A. (2015). Uncertain currency model and currency option pricing. International Journal of Intelligent Systems, 30(1), 40–51. Peng, J., & Yao, K. (2011). A new option pricing model for stocks in uncertainty markets. International Journal of Operations Research, 8(2), 18–26. Sun, J. J., & Chen, X. (2013). Asian option pricing formula for uncertain financial market. http://orsc.edu. cn/online/130511 Yao, K. (2013a). Extreme values and integral of solution of uncertain differential equation. Journal of Uncertainty Analysis and Applications, 1. doi:10.1186/2195-5468-1-2
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Yao, K. (2013b). A type of nonlinear uncertain differential equations with analytic solution. Journal of Uncertainty Analysis and Applications, 1. doi:10.1186/2195-5468-1-8. Yao, K. (2015). A no-arbitrage theorem for uncertain stock model. Fuzzy Optimization and Decision Making. doi:10.1007/s10700-014-9198-9. Yao, K., & Chen, X. (2013). A numerical method for solving uncertain differential equations. Journal of Intelligent and Fuzzy Systems, 25(3), 825–832.
123
Uncertain contour process and its application in stock model with floating interest rate Kai Yao1
Published online: 12 March 2015 © Springer Science+Business Media New York 2015
Abstract Uncertain process is an important tool to model dynamic uncertain systems. This paper proposes a special type of uncertain processes, named contour processes, whose sample paths can be classified by their inverse uncertainty distributions. It is shown that the set of contour processes is closed under the extreme value operator and the time integral operator as well as the monotone function. As an application, this paper considers an uncertain stock model with floating interest rate, in which both the interest rate and the stock price follow uncertain differential equations. By means of contour processes, some pricing formulas are derived for the European options, American options and Asian options of the stock model. Keywords Uncertain process · Stock model · Uncertain differential equation · Uncertain finance
1 Introduction Uncertainty theory, founded by Liu (2007), is used to model human uncertainty based on normality, duality, subadditivity and product axioms. In order to indicate the belief degree that an event is supposed to occur, an uncertain measure is defined as a set function on a σ -algebra. Then an uncertain variable is used to model a quantity with uncertainty, and an uncertainty distribution is used to describe an uncertain variable. In addition, some numerical characteristics of uncertain variables have also been studied such as expected value, variance and entropy.
B 1
Kai Yao [email protected] School of Management, University of Chinese Academy of Sciences, Beijing 100190, China
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K. Yao
In order to model the evolution of uncertain phenomena, a concept of uncertain process was proposed by Liu (2008) as a sequence of uncertain variables indexed by time. Uncertain independent increment process is an uncertain process with independent increments. A sufficient and necessary condition for a function being an inverse uncertainty distribution of an uncertain independent increment process was given by Liu (2014). Besides, the extreme values of an uncertain independent increment process was studied by Liu (2013b). Uncertain stationary independent increment process is an uncertain process with both stationary increments and independent increments. Liu (2010) proved that its expected value is a linear function of the time, and Chen (2011) proved that its standard deviation is proportional to the time. Canonical Liu process, designed by Liu (2009), is a special type of uncertain stationary independent increment processes whose increments are normal uncertain variables. Meanwhile, Liu (2009) founded an uncertain calculus theory to deal with the integral and differential of an uncertain process with respect to Liu process. Uncertain differential equation, a differential equation driven by the Liu process, was proposed by Liu (2008) to describe a dynamic uncertain system. Chen and Liu (2010) gave an analytic solution of a linear uncertain differential equation, and Liu (2012) and Yao (2013b) gave some methods to obtain the solutions of some special types of nonlinear uncertain differential equations. In addition, Yao and Chen (2013) verified some properties of the solution of an uncertain differential equation via the so-called α-path which is essentially the inverse uncertainty distribution of the solution. Based on these results, Yao (2013a) gave the uncertainty distributions of the extreme values and the time integral of the solution. Uncertain differential equation was introduced to finance in 2009. Liu (2009) assumed the stock price follows a geometric Liu process, and proposed a stock model which was named Liu’s stock model later. Chen (2011) derived its American option pricing formulas, and Yao (2015) gave a sufficient and necessary condition for the stock market being no-arbitrage. In addition, Peng and Yao (2011) proposed a stock model via mean-reverting uncertain differential equation to describe the stock price in a long term, and Chen et al. (2013) proposed a stock model with periodical dividends. Uncertain interest rate model was first proposed by Chen and Gao (2013), and was further studied by Jiao and Yao (2015). In addition, uncertain currency model was proposed by Liu et al. (2015) to model the currency exchange rate. For recent developments about uncertain finance, interested readers may refer to Liu (2013a) for details. Inspired by the solution of an uncertain differential equation, this paper will propose a concept of contour process, and will study an uncertain stock model with floating interest rate as an application. In Sect. 2, we review some basic concepts about uncertain variable, uncertain process and uncertain differential equation. Then in Sect. 3, we introduce the concept of contour process, and give its inverse uncertainty distribution and expected value. In Sects. 4 and 5, we study the extreme values and time integral of a contour process, respectively. Besides, we study the function of multiple contour processes in Sect. 6. After that, we derive some useful theorems about uncertain differential equations in Sect. 7. In Sect. 8, we introduce the new stock model with floating interest rate, and give its European option pricing formulas, American option pricing
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Uncertain contour process and its application...
401
formulas, and Asian option pricing formulas. Finally, some conclusions are made in Sect. 9.
2 Preliminaries In this section, we introduce some concepts about uncertain variable, uncertain process and uncertain differential equation. 2.1 Uncertain variable Definition 1 (Liu 2007) Let L be a σ -algebra on a nonempty set . A set function M : L → [0, 1] is called an uncertain measure if it satisfies the following axioms: Axiom 1 (Normality Axiom) M{} = 1 for the universal set . Axiom 2 (Duality Axiom) M{} + M{c } = 1 for any event . Axiom 3 (Subadditivity Axiom) For every countable sequence of events 1 , 2 , . . . , we have ∞ ∞ i ≤ M {i } . M i=1
i=1
A product uncertain measure on multiple uncertainty spaces was defined by Liu (2009) to provide the operational law of uncertain variables. Axiom 4 (Product Axiom) Let (k , Lk , Mk ) be uncertainty spaces for k = 1, 2, . . . Then the product uncertain measure M is an uncertain measure satisfying ∞ ∞ M k = Mk {k } k=1
k=1
where k are arbitrarily chosen events from Lk for k = 1, 2, . . ., respectively. Definition 2 (Liu 2007) An uncertain variable is a measurable function ξ from an uncertainty space (, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {γ | ξ(γ ) ∈ B} is an event. Definition 3 (Liu 2007) The uncertainty distribution of an uncertain variable ξ is defined by (x) = M{ξ ≤ x} for any real number x.
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K. Yao
If the inverse function −1 exists and is unique for each α ∈ (0, 1), then it is called the inverse uncertainty distribution of ξ . In this case, the uncertainty distribution is said to be regular. Definition 4 (Liu 2007) Let ξ be an uncertain variable. Then its expected value E[ξ ] is defined by
+∞
E[ξ ] =
M{ξ ≥ r }dr −
0
0
−∞
M{ξ ≤ r }dr
provided that at least one of the two integrals is finite. Definition 5 (Liu 2009) The uncertain variables ξ1 , ξ2 , . . . , ξm are said to be independent if M
m k=1
(ξi ∈ Bi ) =
m
M{ξi ∈ Bi }
k=1
for any Borel sets B1 , B2 , . . . , Bm of real numbers. Let ξ1 , ξ2 , . . . , ξn be independent uncertain variables with regular uncertainty distributions 1 , 2 , . . . , n , respectively. Supppose f (x1 , x2 , . . . , xn ) is strictly increasing with respect to x1 , x2 , . . . , xm and strictly decreasing with respect to xm+1 , xm+2 , . . . , xn . Then Liu (2010) proved the uncertain variable ξ = f (ξ1 , ξ2 , . . . , ξn ) has an inverse uncertainty distribution
−1 −1 −1 −1 (r ) = f −1 (r ), . . . , (r ), (1 − r ), . . . , (1 − r ) , m n 1 m+1 and Liu and Ha (2010) proved ξ has an expected value
1
E[ξ ] = 0
−1 −1 −1 f −1 (r ), . . . , (r ), (1 − r ), . . . , (1 − r ) dα. m n 1 m+1
2.2 Uncertain process Definition 6 (Liu 2008) Let T be an index set, and (, L, M) be an uncertainty space. An uncertain process X t is a measurable function from T × (, L, M) to the set of real numbers, i.e., for each t ∈ T and any Borel set B of real numbers, the set {X t ∈ B} = {γ | X t (γ ) ∈ B} is an event. Definition 7 (Liu 2014) An uncertain process X t is said to have an uncertainty distribution t (x) if at each time t, the uncertain variable X t has the uncertainty distribution t (x).
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Definition 8 (Liu 2014) Uncertain processes X 1t , X 2t , . . . , X nt are said to be independent if for any positive integer k and any times t1 , t2 , . . . , tk , the uncertain vectors ξ i = (X it1 , X it2 , . . . , X itk ), i = 1, 2, . . . , n are independent, i.e., for any Borel sets B1 , B2 , . . . , Bn of k-dimensional real vectors, we have M
n
(ξ i ∈ Bi ) =
i=1
n
M{ξ i ∈ Bi }.
i=1
Definition 9 Let X t be an uncertain process on an uncertainty space (, L, M). Then for every γ ∈ , the function X t (γ ) is called a sample path of X t . For an uncertain process X t on an uncertainty space (, L, M), define uncertain processes Yt and Z t by Yt (γ ) = sup X s (γ ), ∀γ ∈ , 0≤s≤t
Z t (γ ) = inf X s (γ ), ∀γ ∈ . 0≤s≤t
Then Yt and Z t are called the supremum process and infimum process of X t , respectively. Definition 10 (Liu 2008) Let X t be an uncertain process. For any partition of a closed interval [a, b] with a = t1 < t2 < . . . < tk+1 = b, the mesh is written as = max |ti+1 − ti |. 1≤i≤k
Then the time integral of X t with respect to t is a
b
X t dt = lim
→0
k
X ti (ti+1 − ti )
i=1
provided that the limit exists almost surely and is finite. 2.3 Uncertain differential equation In order to found an uncertain calculus theory, a concept of canonical Liu process was defined by Liu (2009) as follows. Definition 11 (Liu 2009) An uncertain process Ct is said to be a canonical Liu process if
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K. Yao
(i) C0 = 0 and almost all sample paths are Lipschitz continuous, (ii) Ct has stationary and independent increments, (iii) every increment Cs+t − Cs is a normal uncertain variable with an uncertainty distribution
−1 πx t (x) = 1 + exp − √ , x ∈ . 3t Definition 12 (Liu 2009) Let X t be an uncertain process and Ct be a canonical Liu process. For any partition of closed interval [a, b] with a = t1 < t2 < . . . < tk+1 = b, the mesh is written as = max |ti+1 − ti |. 1≤i≤k
Then the Liu integral of X t is defined by
b
X t dCt = lim
→0
a
k
X ti · (Cti+1 − Cti )
i=1
provided that the limit exists almost surely and is finite. Definition 13 (Liu 2008) Suppose Ct is a canonical Liu process, and f and g are two given functions. Then (1) dX t = f (t, X t )dt + g(t, X t )dCt is called an uncertain differential equation. The uncertain differential Eq. (1) is equivalent to an integral equation Xt = X0 +
t
0
t
f (s, X s )ds +
g(s, X s )dCs ,
0
and a solution of (1) is just an uncertain process satisfying the integral equation. Yao and Chen (2013) defined a concept of α-path which is a function solving the ordinary differential equation dX tα = f (t, X tα )dt + |g(t, X tα )|−1 (α)dt where −1
√ α 3 ln (α) = π 1−α
is the inverse uncertainty distribution of Ct at t = 1, and showed M{X t ≤ X tα , ∀t} = α, M{X t > X tα , ∀t} = 1 − α.
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3 Contour process Inspired by the solution of an uncertain differential equation, we introduce a concept of contour process in this section, and study its inverse uncertainty distribution and expected value. Definition 14 Let X t be an uncertain process. If for each α ∈ (0, 1), there exists a real function X tα such that M{X t ≤ X tα , ∀t} = α, M{X t > X tα , ∀t} = 1 − α,
(2) (3)
then X t is called a contour process. In this case, X tα is called the α-path of the uncertain process X t . Example 1 A solution of an uncertain differential equation is a contour process. Please refer to Yao and Chen (2013) and Yao (2013a) for details. Example 2 Let ξ be a regular uncertain variable. Then the uncertain process Sn =
ξ, −ξ,
if n is odd if n is even
is not a contour process. Theorem 1 An uncertain process X t is a contour process if and only if for each α ∈ (0, 1), there exists a real function X tα such that M{X t < X tα , ∀t} = α, M{X t ≥
X tα , ∀t}
= 1 − α.
(4) (5)
Proof On the one hand, assume X t is a contour process with an α-path X tα . Note that for each α ∈ (0, 1) and for any given ε > 0, we have {X t ≤ X tα−ε , ∀t} ⊂ {X t < X tα , ∀t} ⊂ {X t ≤ X tα , ∀t}, {X t > X tα , ∀t} ⊂ {X t ≥ X tα , ∀t} ⊂ {X t > X tα−ε , ∀t}. By using the monotonicity of uncertain measure, we get that M{X t ≤ X tα−ε , ∀t} ≤ M{X t < X tα , ∀t} ≤ M{X t ≤ X tα , ∀t}, M{X t > X tα , ∀t} ≤ M{X t ≥ X tα , ∀t} ≤ M{X t > X tα−ε , ∀t}. According to Definition 14 of contour process, we have M{X t ≤ X tα−ε , ∀t} = α − ε, M{X t ≤ X tα , ∀t} = α, M{X t > X tα , ∀t} = 1 − α, M{X t > X tα−ε , ∀t} = 1 − α + ε.
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Then Eqs. (4) and (5) follow immediately. On the other hand, assume Eqs. (4) and (5) hold for each α ∈ (0, 1). Note that for each α ∈ (0, 1) and for any given ε > 0, we have {X t < X tα , ∀t} ⊂ {X t ≤ X tα , ∀t} ⊂ {X t < X tα+ε , ∀t}, {X t ≥ X tα+ε , ∀t} ⊂ {X t > X tα , ∀t} ⊂ {X t ≥ X tα , ∀t}. By using the monotonicity of uncertain measure, we get that M{X t < X tα , ∀t} ≤ M{X t ≤ X tα , ∀t} ≤ M{X t < X tα+ε , ∀t}, M{X t ≥ X tα+ε , ∀t} ≤ M{X t > X tα , ∀t} ≤ M{X t ≥ X tα , ∀t}. It follows from Eqs. (4) and (5) that M{X t < X tα , ∀t} = α, M{X t < X tα+ε , ∀t} = α + ε,
M{X t ≥ X tα+ε , ∀t} = 1 − α − ε, M{X t ≥ X tα , ∀t} = 1 − α. Then we have M{X t ≤ X tα , ∀t} = α, M{X t > X tα , ∀t} = 1 − α. According to Definition 14, the uncertain process X t is a contour process. Theorem 2 Let X t be a contour process with an α-path X tα . Then X t has an inverse uncertainty distribution α −1 t (α) = X t , ∀α ∈ (0, 1)
and an expected value
1
E[X t ] = 0
X tα dα.
Proof Given any time s, since {X s ≤ X sα } ⊃ {X t ≤ X tα , ∀t}, {X s > X sα } ⊃ {X t > X tα , ∀t}, we have M{X s ≤ X sα } ≥ M{X t ≤ X tα , ∀t} = α, M{X s > X sα } ≥ M{X t > X tα , ∀t} = 1 − α
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Uncertain contour process and its application...
407
according to the monotonicity of uncertain measure. It follows from the duality of uncertain measure that M{X s ≤ X sα } + M{X s > X sα } = 1. Then we have M{X s ≤ X sα } = α, which means X t has an inverse uncertainty distribution X tα . As a result, we have
1
E[X t ] = 0
X tα dα.
The proof is completed.
Since the α-path X tα is just the inverse uncertainty distribution of a contour process X t , in the rest of this paper, we denote the inverse uncertainty distribution of X t by X tα for simplicity.
4 Extreme values In this section, we study the extreme values of a contour process. The results show that the set of contour processes is closed under the supremum operator and infimum operator. Theorem 3 Let X t be a contour process with an α-path X tα . Then its supremum process Yt = sup X s 0≤s≤t
is a contour process with an α-path Ytα = sup X sα . 0≤s≤t
Proof For a sample path X t (γ ) such that X t (γ ) ≤ X tα for any time t, we have sup X s (γ ) ≤ sup X sα , ∀t.
0≤s≤t
0≤s≤t
It implies
Yt ≤ sup X sα , ∀t 0≤s≤t
=
sup X s ≤ sup X sα , ∀t
0≤s≤t
0≤s≤t
⊃ X t ≤ X tα , ∀t .
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By the monotonicity of uncertain measure, we have α M Yt ≤ sup X s , ∀t ≥ M X t ≤ X tα , ∀t = α. 0≤s≤t
Similarly, we have
M Yt > sup 0≤s≤t
X sα , ∀t
≥ M X t > X tα , ∀t = 1 − α.
Since
M Yt ≤ sup 0≤s≤t
X sα , ∀t
+ M Yt > sup 0≤s≤t
X sα , ∀t
≤ 1,
we have
M Yt ≤ sup 0≤s≤t
X sα , ∀t
= α,
M Yt > sup X sα , ∀t
= 1 − α.
0≤s≤t
So Yt is a contour process with an α-path Ytα = sup X sα . 0≤s≤t
Theorem 4 Let X t be a contour process with an α-path X tα . Then its infimum process Z t = inf X s 0≤s≤t
is a contour process with an α-path Z tα = inf X sα . 0≤s≤t
Proof For a sample path X t (γ ) such that X t (γ ) ≤ X tα for any time t, we have inf X s (γ ) ≤ inf X sα , ∀t.
0≤s≤t
0≤s≤t
It implies Z t ≤ inf X sα , ∀t = inf X s ≤ inf X sα , ∀t ⊃ X t ≤ X tα , ∀t . 0≤s≤t
123
0≤s≤t
0≤s≤t
Uncertain contour process and its application...
409
By the monotonicity of uncertain measure, we have α M Z t ≤ inf X s , ∀t ≥ M X t ≤ X tα , ∀t = α. 0≤s≤t
Similarly, we have M Z t > inf X sα , ∀t ≥ M X t > X tα , ∀t = 1 − α. 0≤s≤t
Since
M Z t ≤ inf X sα , ∀t + M Z t > inf X sα , ∀t < 1, 0≤s≤t
we have
0≤s≤t
M Z t ≤ inf X sα , ∀t = α, 0≤s≤t M Z t > inf X sα , ∀t = 1 − α. 0≤s≤t
So Z t is a contour process with an α-path Z tα = inf X sα . 0≤s≤t
5 Time integral In this section, we will study the time integral of a contour process. It is shown that the set of contour processes is closed under the time integral operator. Theorem 5 Let X t be a contour process with an α-path X tα . Then its time integral process
t
Yt =
X s ds 0
is a contour process with an α-path Ytα
t
=
X sα ds.
0
Proof For a sample path X t (γ ) such that X t (γ ) ≤ X tα for any time t, we have 0
t
X s (γ )ds ≤ 0
t
X sα ds, ∀t.
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It implies
t
Yt ≤ 0
X sα ds, ∀t =
t
X s ds ≤
0
0
t
X sα ds, ∀t ⊃ X t ≤ X tα , ∀t .
By the monotonicity of uncertain measure, we have M Yt ≤
t
0
X sα ds, ∀t ≥ M X t ≤ X tα , ∀t = α.
Similarly, we have
t
M Yt > 0
X sα ds, ∀t
≥ M X t > X tα , ∀t = 1 − α.
Since
M Yt ≤ 0
t
X sα ds, ∀t
t
+ M Yt > 0
X sα ds, ∀t
≤ 1,
we have
t
X sα ds, ∀t
= α, M Yt ≤ 0 t M Yt > X sα ds, ∀t = 1 − α. 0
So Yt is a contour process with α-path Ytα =
t 0
X sα ds.
6 Monotone function In this section, we will study the monotone function of multiple contour processes, and we will show the derived uncertain process is also a contour process. Theorem 6 Let X 1t , X 2t , . . . , X nt be independent contour processes with α-paths α , X α , . . . , X α , respectively. If f (x , x , . . . , x ) is strictly increasing with respect X 1t 1 2 n nt 2t to x1 , x2 , . . . , xm and strictly decreasing with respect to xm+1 , xm+2 , . . . , xn , then the uncertain process X t = f (X 1t , X 2t , . . . , X nt ) is a contour process with an α-path
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Uncertain contour process and its application...
411
1−α α α 1−α . X tα = f X 1t , . . . , X mt , X m+1t , . . . , X nt Proof According to the monotonicity of the function f , we have
1−α α α 1−α , ∀t {X t ≤ X tα , ∀t} = f (X 1t , X 2t , . . . , X nt ) ≤ f X 1t , . . . , X mt , X m+1t , . . . , X nt ⊃
m
n X it ≤ X itα , ∀t ∩ X it ≥ X it1−α , ∀t
i=1
i=m+1
and {X t > X tα , ∀t} = ⊃
1−α α α 1−α , ∀t f (X 1t , X 2t , . . . , X nt ) > f X 1t , . . . , X mt , X m+1t , . . . , X nt
m
n X it > X itα , ∀t ∩ X it < X it1−α , ∀t .
i=1
i=m+1
By the independence of uncertain processes and the monotonicity of uncertain measure, we have M{X t ≤
X tα , ∀t}
≥M
m
X it ≤
X itα , ∀t
∩
i=1
=
m
n
X it ≥
X it1−α , ∀t
i=m+1
M X it ≤ X itα , ∀t ∧
i=1
n
M X it ≥ X it1−α , ∀t = α,
i=m+1
and M{X t
X itα , ∀t
∩
i=1
=
m i=1
M X it > X itα , ∀t ∧
n
X it
f L(X tα ), X tα , ∀t ⊃ {X t > X tα , ∀t}.
According to the monotonicity of uncertain measure, we have M f (L(X t ), X t ) ≤ f L(X tα ), X tα , ∀t ≥ M{X t ≤ X tα , ∀t} = α, M f (L(X t ), X t ) > f L(X tα ), X tα , ∀t ≥ M{X t > X tα , ∀t} = 1 − α. Since M f (L(X t ), X t ) ≤ f L(X tα ), X tα , ∀t + M f (L(X t ), X t ) > f L(X tα ), X tα , ∀t < 1, we have M f (L(X t ), X t ) ≤ f L(X tα ), X tα , ∀t = α, M f (L(X t ), X t ) > f L(X tα ), X tα , ∀t = 1 − α.
The proof is completed.
Example 3 Let X t be a contour process with an α-path X tα . Then the uncertain process Yt = sup X s + X t 0≤s≤t
is a contour process with an α-path Ytα = sup X sα + X tα , 0≤s≤t
and the uncertain process
t
Zt = 0
is a contour process with an α-path
123
X s ds + X t
Uncertain contour process and its application...
413
Z tα =
t
X sα ds + X tα .
0
7 Applications to uncertain differential equations In this section, we derive some theorems about uncertain differential equations. The applications of these theorems in uncertain finance will be given in the next section. Theorem 8 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t , respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have E exp −
T
X 1s ds (X 2T − K )
+
1
=
0
exp −
0
T 0
1−α X 1s ds
α (X 2T − K )+ dα,
(6) α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively. Proof Since the solution of an uncertain differential equation is a contour process, it follows from Theorems 5 and 6 that the uncertain process exp −
t
X 1s ds 0
is a contour process with an α-path exp −
t 0
1−α X 1s ds ,
α − K )+ . and the uncertain process (X 2t − K )+ is a contour process with an α-path (X 2t Then by Theorem 6, the uncertain process
exp −
t
X 1s ds (X 2t − K )+
0
is also a contour process with an α-path exp −
t 0
1−α X 1s ds
α (X 2t − K )+ .
According to Theorem 2, we have E exp −
T
X 1s ds (X 2T − K )+ =
0
The theorem is proved.
1 0
exp −
T 0
1−α α X 1s ds (X 2T − K )+ dα.
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K. Yao
Theorem 9 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t , respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have E exp −
T
X 1s ds (K − X 2T )+ =
0
1
exp −
0
T 0
α α + X 1s ds (K − X 2T ) dα,
(7) α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively. Proof Since the solution of an uncertain differential equation is a contour process, it follows from Theorems 5 and 6 that the uncertain process exp −
t
X 1s ds 0
is a contour process with an α-path exp −
t 0
1−α X 1s ds ,
1−α + ) . and the uncertain process (K −X 2t )+ is a contour process with an α-path (K −X 2t Then by Theorem 6, the uncertain process
exp −
t
X 1s ds (K − X 2t )+
0
is also a contour process with an α-path exp −
t 0
1−α X 1s ds
1−α + (K − X 2t ) .
According to Theorem 2, we have E exp −
T
X 1s ds (K − X 2T )
0
The theorem is proved.
+
= 0
1
exp −
T 0
α X 1s ds
α + (K − X 2T ) dα.
Theorem 10 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t , respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have
123
Uncertain contour process and its application...
sup exp −
E
0≤t≤T
1
=
415
t
X 1s ds (X 2t − K )
0
sup exp −
0 0≤t≤T
t 0
+
1−α α X 1s ds (X 2t − K )+ dα,
(8)
α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively.
Proof It follows from Theorem 8 that the uncertain process
t X 1s ds (X 2t − K )+ exp − 0
is a contour process with an α-path
t 1−α α X 1s ds (X 2t − K )+ . exp − 0
Then by Theorem 3, the uncertain process t
sup exp − X 1s ds (X 2t − K )+ 0
0≤t≤T
is a contour process with an α-path sup exp −
1−α α X 1s ds (X 2t − K )+ .
t 0
0≤t≤T
According to Theorem 2, we have
sup exp −
E
0≤t≤T
1
=
t
X 1s ds (X 2t − K )+
0
sup exp −
0 0≤t≤T
t
1−α X 1s ds
0
α (X 2t − K )+ dα.
The theorem is proved.
Theorem 11 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t , respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have E
sup exp −
0≤t≤T
=
1
t
X 1s ds (K − X 2t )
0
sup exp −
0 0≤t≤T
t 0
α X 1s ds
+
α + (K − X 2t ) dα,
(9)
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416
K. Yao
α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively.
Proof It follows from Theorem 9 that the uncertain process exp −
X 1s ds (K − X 2t )+
t 0
is a contour process with an α-path exp −
t 0
1−α 1−α + X 1s ds (K − X 2t ) .
Then by Theorem 3, the uncertain process sup exp −
t
X 1s ds (K − X 2t )+
0
0≤t≤T
is a contour process with an α-path sup exp −
t 0
0≤t≤T
1−α 1−α + X 1s ds (K − X 2t ) .
According to Theorem 2, we have E
sup exp −
0≤t≤T
= =
1
t
X 1s ds (K − X 2t )+
0
t
1−α sup exp − rs ds (K − X t1−α )+ dα
0 0≤t≤T 1
0
0 0≤t≤T
0
t
sup exp − rsα ds (K − X tα )+ dα.
The theorem is proved.
Theorem 12 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t , respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have
T
T
+ 1 E exp − X 1s ds X 2s ds − K T 0 0 T
T
+ 1 1 1−α α exp − X 1s ds · X 2s ds − K dα, = T 0 0 0
123
(10)
Uncertain contour process and its application...
417
α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively.
Proof Since the solution of an uncertain differential equation is a contour process, it follows from Theorems 5 and 6 that the uncertain process exp −
t
X 1s ds 0
is a contour process with an α-path exp −
t 0
1−α X 1s ds
,
and the uncertain process t
+ 1 X 2s ds − K t 0 is a contour process with an α-path t
+ 1 α X 2s ds − K . t 0 Then by Theorem 6, the uncertain process exp −
t 0
t
+ 1 X 1s ds X 2s ds − K t 0
is also a contour process with an α-path exp −
t 0
1−α X 1s ds
t
+ 1 α · X ds − K . t 0 2s
According to Theorem 2, we have
E exp −
T
X 1s ds 0
=
1
exp −
0
The theorem is proved.
T 0
1 T
1−α X 1s ds
T
+ X 2s ds − K
0
T
+ 1 α · X 2s ds − K dα. T 0
Theorem 13 Let X 1t and X 2t be the solutions of uncertain differential equations dX 1t = f 1 (t, X 1t )dt + g1 (t, X 1t )dC1t and dX 2t = f 2 (t, X 2t )dt + g2 (t, X 2t )dC2t ,
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K. Yao
respectively, where C1t and C2t are two independent canonical Liu processes. Then for any positive numbers T and K , we have
+ 1 T X 1s ds K− X 2s ds T 0 0 T
+ 1 1 T α α = exp − X 1s ds K− X 2s ds dα, T 0 0 0
E exp −
T
(11)
α and X α are the α-paths of the two uncertain differential equations, respecwhere X 1t 2t tively.
Proof Since the solution of an uncertain differential equation is a contour process, it follows from Theorems 5 and 6 that the uncertain process exp −
t
X 1s ds 0
is a contour process with an α-path exp −
1−α X 1s ds ,
t 0
and the uncertain process
1 t
K−
+
t
X 2s ds 0
is a contour process with an α-path
1 K− t
t 0
1−α X 2s ds
+
.
Then by Theorem 6, the uncertain process exp −
t
X 1s ds 0
1 K− t
+
t
X 2s ds 0
is also a contour process with an α-path exp −
t 0
1−α X 1s ds
According to Theorem 2, we have
123
+ 1 t 1−α · K− X ds . t 0 2s
Uncertain contour process and its application...
419
+ 1 T X 1s ds K− X 2s ds T 0 0 T
+ 1 1 T 1−α 1−α = exp − X 1s ds · K − X 2s ds dα T 0 0 0 T
+ 1 1 T α α exp − X 1s ds K− X 2s ds dα. = T 0 0 0
E exp −
T
The theorem is proved.
8 Stock model with floating interest rate Let rt denote the interest rate, and X t denote the stock price in an uncertain market. Liu (2009) proposed a stock model with fixed interest rate as follows,
drt = μ1rt dt dX t = μ2 X t dt + σ2 X t dC2t
where μ1 is the riskless interest rate, μ2 is the stock drift, σ2 is the stock diffusion, and C2t is a canonical Liu process. Liu (2009), Chen (2011), and Sun and Chen (2013) derived pricing formulas of the European options, American options and Asian options for Liu’s stock model, respectively. However, there are some uncertain factors influencing the interest rate in the financial market. So here we assume both the interest rate and the stock price follow uncertain differential equations, and propose a stock model with floating interest rate as follows, drt = μ1rt dt + σ1 rt dC1t (12) dX t = μ2 X t dt + σ2 X t dC2t , where μ1 and σ1 are the drift and the diffusion of the interest rate, respectively, μ2 and σ2 are the drift and the diffusion of the stock price, respectively, and C1t and C2t are independent canonical Liu processes. It is easy to verify that the α-path rtα of the interest rate rt is
√
rtα = r0 · exp μ1 t +
α 3σ1 t ln π 1−α
(13)
by solving the ordinary differential equation √ drtα
=
μ1rtα dt
+
α 3σ1rtα ln dt. π 1−α
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K. Yao
Similarly, the α-path X tα of the stock price X t is
X tα
√ 3σ2 t α = X 0 · exp μ2 t + . ln π 1−α
(14)
Consider a European call option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by f c = E exp −
T
rs ds (X T − K )
+
.
0
According to Theorem 8, we have
1
fc =
exp −
0
T
rs1−α ds
0
(X Tα − K )+ dα,
where exp −
T
0
rs1−α ds
⎞ √ 1 − exp μ1 T + 3σ1 T /π · (ln(1 − α) − ln α) ⎠ = exp ⎝r0 · √ μ1 + 3σ1 /π · (ln(1 − α) − ln α) ⎛
and (X Tα
+
− K) =
√
X 0 · exp μ2 T +
α 3σ2 T ln π 1−α
+ −K
.
Consider a European put option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by
f p = E exp −
T
rs ds (K − X T )
+
.
0
According to Theorem 9, we have fp =
1
0
exp −
T 0
rsα ds
(K − X Tα )+ dα,
where exp −
T 0
123
rsα ds
⎞ √ 1 − exp μ1 T + 3σ1 T /π · (ln α − ln(1 − α)) ⎠ = exp ⎝r0 · √ μ1 + 3σ1 /π · (ln α − ln(1 − α)) ⎛
Uncertain contour process and its application...
421
and (K −
X Tα )+
=
+ √ α 3σ2 T ln K − X 0 · exp μ2 T + . π 1−α
Consider an American call option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by
t
+ sup exp − rs ds (X t − K ) .
fc = E
0
0≤t≤T
According to Theorem 10, we have fc =
t
sup exp − rs1−α ds (X Tα − K )+ dα,
1
0 0≤t≤T
0
where
⎞ ⎛ √
t 1 − exp μ1 t + 3σ1 t/π · (ln(1 − α) − ln α) ⎠ rs1−α ds = exp ⎝r0 · exp − √ μ1 + 3σ1 /π · (ln(1 − α) − ln α) 0 and (X tα
+
− K) =
+ √ α 3σ2 t ln X 0 · exp μ2 t + . −K π 1−α
Consider an American put option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by fp = E
t
sup exp − rs ds (K − X t )+ . 0≤t≤T
0
According to Theorem 11, we have fp =
1
t
α sup exp − rs ds (K − X tα )+ dα,
0 0≤t≤T
0
where
⎞ ⎛ √
t 1 − exp μ1 t + 3σ1 t/π · (ln α − ln(1 − α)) ⎠ rsα ds = exp ⎝r0 · exp − √ μ1 + 3σ1 /π · (ln α − ln(1 − α)) 0
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K. Yao
and (K −
X tα )+
=
+ √ α 3σ2 t ln K − X 0 · exp μ2 t + . π 1−α
Consider an Asian call option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by
f c = E exp −
T
rs ds 0
1 T
T
+ X s ds − K
.
0
According to Theorem 12, we have
1
fc =
exp −
0
T 0
T
+ 1 rs1−α ds · X sα ds − K dα, T 0
where exp −
T 0
rs1−α ds
⎞
√ 1 − exp μ1 T + 3σ1 T /π · (ln(1 − α) − ln α) ⎠ = exp ⎝r0 · √ μ1 + 3σ1 /π · (ln(1 − α) − ln α) ⎛
and
⎞+ √ + ⎛ 1 − exp μ2 T + 3σ2 T /π · (ln α − ln(1 − α)) 1 T α X s ds − K = ⎝X0 · − K⎠ . √ T 0 μ2 T + 3σ2 T /π · (ln α − ln(1 − α))
Consider an Asian put option of the stock model (12) with a strike price K and an expiration date T , whose price is determined by
f p = E exp −
T
rs ds 0
1 K− T
+
T
X s ds
.
0
According to Theorem 13, we have fp =
1
exp −
0
T 0
rsα ds
K−
1 T
0
T
X sα ds
+ dα,
where exp −
T 0
123
rsα ds
⎞ √ 1 − exp μ1 T + 3σ1 T /π · (ln α − ln(1 − α)) ⎠ = exp ⎝r0 · √ μ1 + 3σ1 /π · (ln α − ln(1 − α)) ⎛
Uncertain contour process and its application...
423
and
⎞+ √ + ⎛ 1 − exp μ2 T + 3σ2 T /π · (ln α − ln(1 − α)) 1 T α ⎠ . X s ds = ⎝K − X0 · K− √ T 0 μ2 T + 3σ2 T /π · (ln α − ln(1 − α))
9 Conclusion This paper proposed a concept of contour process, and showed that the set of contour processes is closed under extreme value operator, time integral operator and monotone function. As a generalization of Liu’s stock model, this paper proposed an uncertain stock model with floating interest rate, and derived some pricing formulas of the European options, American options and Asian options of the stock model. Acknowledgments No. 61403360).
This work was supported by National Natural Science Foundation of China (Grant
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