A No-Arbitrage Theorem for Uncertain Stock Model - Semantic Scholar
Fuzzy Optim Decis Making manuscript No. (will be inserted by the editor)
A No-Arbitrage Theorem for Uncertain Stock Model Kai Yao
Received: date / Accepted: date
Abstract Stock model is used to describe the evolution of stock price in the form of differential equations. In early years, the stock price was assumed to follow a stochastic differential equation driven by a Brownian motion, and some famous models such as Black-Scholes stock model and Black-Karasinski stock model were widely used. This paper assumes that the stock price follows an uncertain differential equation driven by Liu process rather than Brownian motion, and accepts Liu’s stock model to simulate the uncertain market. Then this paper proves a noarbitrage determinant theorem for Liu’s stock model and presents a sufficient and necessary condition for no-arbitrage. Finally, some examples are given to illustrate the usefulness of the no-arbitrage determinant theorem. Keywords finance · stock model · no-arbitrage · uncertainty theory · uncertain differential equation
1 Introduction Stock price in the financial market is usually described as a stochastic process. In 1900’s, Bachelier employed a Brownian motion to model the price fluctuation in Paris stock market. However, his model failed to evaluate both stock and option prices for many reasons, of which the major one was that the Brownian motion may become negative, contradicting the price of a stock. Samuelson (1965) assumed that the return rate of the stock followed a geometric Brownian motion and proposed a stock model, but he failed to consider the drift of the Brownian motion and the time value of money. Black and Scholes (1973) assumed that the stock price followed a geometric Brownian motion and proposed the famous Black-Scholes stock model. Merton (1973) analyzed the assumption of Black-Scholes stock model and performed a rigorous analysis. Nowadays, the Black-Scholes stock model has become an indispensable tool in the trading of options and other financial Kai Yao School of Management, University of Chinese Academy of Sciences, Beijing 100190, China E-mail: [email protected]
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Kai Yao
derivatives. Black and Karasinski (1991) proposed a stock model with OrnsteinUhlenbeck process, describing the stock price that diffuses around an average level in long run. Despite the fast development in stochastic finance, many critics argue that prices of some stocks may not behave like randomness. Thus some researchers such as Qin and Gao (2009) suggested to describe the financial markets by fuzzy theory. Unfortunately, this theory does not provide a satisfactory method for studying the real market. A new tool is the uncertainty theory that was invented by Liu (2007) and refined by Liu (2010) to model human’s belief degree. Based on an uncertain measure satisfying normality, duality, subadditivity and product axioms, a concept of uncertain variable was proposed to model a quantity with uncertainty. Then some concepts of uncertainty distribution, expected value, and variance were presented to describe an uncertain variable. In order to model an uncertain dynamic system, an uncertain process was defined by Liu (2008) as a sequence of uncertain variables driven by the time. Then Liu (2009) proposed a type of Liu process, that is a stationary independent increment process with normal uncertain increments. In addition, 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 was a type of differential equation driven by Liu process. Chen and Liu (2010) gave a sufficient condition for an uncertain differential equation having a unique solution. Then Yao et al. (2013) gave a sufficient condition for it being stable, which was first defined by Liu (2009). The analytic solution to a linear uncertain differential equation was obtained by Chen and Liu (2010), and some analytic methods for solving general uncertain differential equations were provided by Liu (2012) and Yao (2013b), respectively. In addition, some numerical methods were designed by Yao and Chen (2013), and Yao (2013a). By means of the uncertain differential equation, Liu (2009) proposed an uncertain stock model named Liu’s stock model. After that, Chen (2011) derived an American option pricing formula for Liu’s stock model. Peng and Yao (2011) proposed a mean-reverting stock model in an uncertain market. Besides, uncertain interest rate model was proposed by Chen and Gao (2013), and uncertain currency model was proposed by Liu et al. (2014) via uncertain differential equation. For recent developments of uncertain finance, please refer to Liu (2013). In this paper, we will derive a no-arbitrage determinant theorem for Liu’s stock model. The rest of this paper is structured as follows. The next section is intended to introduce some concepts of uncertain process and uncertain differential equation. Liu’s stock model is introduced in Section 3 and a sufficient and necessary condition for such a stock model being no-arbitrage is derived in Section 4. After that, we apply the theorem to some examples in Section 5. Finally, some remarks are made in Section 6. 2 Preliminary In this section, we will introduce some useful definitions about uncertain variable, uncertain process, uncertain calculus and uncertain differential equation. Definition 1 (Liu, 2007) Let L be a σ-algebra on a nonempty set Γ. A set function M : L → [0, 1] is called an uncertain measure if it satisfies the following
A No-Arbitrage Theorem for Uncertain Stock Model
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axioms: Axiom 1: (Normality Axiom) M{Γ } = 1 for the universal set Γ. Axiom 2: (Duality Axiom) M{Λ} + M{Λc } = 1 for any event Λ. Axiom 3: (Subadditivity Axiom) For every countable sequence of events Λ1 , Λ2 , · · · , we have (∞ ) ∞ [ X M Λi ≤ M {Λi } . i=1
i=1
Besides, the product uncertain measure on the product σ-algebre L is defined by Liu (2009) as follows, Axiom 4: (Product Axiom) Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, · · · Then the product uncertain measure M is an uncertain measure satisfying (∞ ) ∞ Y ^ M Λk = Mk {Λk } i=1
k=1
where Λk are arbitrarily chosen events from Lk for k = 1, 2, · · · , respectively. An uncertain variable is a measurable function from an uncertainty space to the set of real numbers. In order to describe an uncertain variable, a concept of uncertainty distribution is defined as follows. Definition 2 (Liu, 2007) The uncertainty distribution of an uncertain variable ξ is defined by Φ(x) = M{ξ ≤ x} for any x ∈ 0. In fact, the expected value of a normal uncertain variable N (e, σ) is e, and the variance is σ 2 . An uncertain process is essentially a sequence of uncertain variables driven by time or space. The formal definition of an uncertain process is given as follows. Definition 6 (Liu, 2008) Let T be an index set and (Γ, L, M) be an uncertainty space. An uncertain process is a measurable function from T × (Γ, L, M) to the set of real numbers, i.e., for each t ∈ T and any Borel set B of real numbers, the set {Xt ∈ B} = {γ | Xt (γ) ∈ B} is an event. Definition 7 (Liu, 2014) Uncertain processes X1t , X2t , · · · , Xnt are said to be independent if for any positive integer k and any times t1 , t2 , · · · , tk , the uncertain vectors ξ i = (Xit1 , Xit2 , · · · , Xitk ), i = 1, 2, · · · , n are independent, i.e., for any k-dimensional Borel sets B1 , B2 , · · · , Bn , we have ) ( n n ^ \ M {ξi ∈ Bi } . (ξ i ∈ Bi ) = M i=1
i=1
Definition 8 (Liu, 2009) An uncertain process Ct is said to be a canonical Liu process if (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 normally distributed uncertain variable with expected value 0 and variance t2 , whose uncertainty distribution is −1 πx Φ(x) = 1 + exp − √ , x ∈ Z0 } > 0 where Zt is determined by the uncertain differential equation (2) and represents the wealth at time t. The next theorem gives a sufficient and necessary condition for the stock model (1) being no-arbitrage. Meanwhile, the theorem gives a strategy for arbitrage when the condition is not satisfied. Theorem 2 (No-Arbitrage Determinant Theorem) The stock model (1) is noarbitrage if and only if (e1 − r, e2 − r, · · · , em − r) is a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ), i.e., the system of linear equations σ11 σ12 · · · σ1n x1 e1 − r σ21 σ22 · · · σ2n x2 e2 − r .. .. . . .. .. = .. . . . . . . σm1 σm2 · · · σmn xn em − r has a solution. Proof: The uncertain differential equation (2) has a solution Z t Z tX Z m m X n X Zt = Z0 exp r βs ds + ei βis ds + σij 0
0 i=1
Z = Z0 exp(rt) exp
m tX
0 i=1
(ei − r)βis ds +
i=1 j=1 n Z X j=1
m tX
0 i=1
t
βis dCjs
0
σij βis dCjs .
A No-Arbitrage Theorem for Uncertain Stock Model
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Thus we have m tX
Z ln (exp (−rt) Zt ) − ln (Z0 ) =
(ei − r)βis ds +
0 i=1
n Z X j=1
m tX
σij βis dCjs ,
0 i=1
which is a normal uncertain variable with expected value m tX
Z
(ei − r)βis ds
0 i=1
and variance
n Z X
t
j=1
0
2 m X σij βis ds . i=1
Assume that (e1 −r, e2 −r, · · · , em −r) is a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ). The argument breaks down into two cases. Case 1: Given any time t and portfolio (βs , β1s , · · · , βms ), if n Z t X m X σij βis ds = 0, 0 j=1
then
m X
σij βis = 0,
i=1
j = 1, 2, · · · , n, s ∈ (0, t).
i=1
Since (e1 − r, e2 − r, · · · , em − r) is a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ), we have m X
(ei − r)βis = 0,
s ∈ (0, t).
i=1
So we obtain Z
m tX
(ei − r)βis ds = 0
0 i=1
and ln(exp(−rt)Zt ) = ln(Z0 ). Thus
M{exp(−rt)Zt > Z0 } = 0 and the stock model (1) is no-arbitrage. Case 2: Given any time t and portfolio (βs , β1s , · · · , βms ), if m n Z t X X σij βis ds 6= 0, 0 j=1
i=1
then
M{ln(exp(−rt)Zt ) − ln(Z0 ) ≥ 0} < 1, i.e.,
M{exp(−rt)Zt ≥ Z0 } < 1.
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So the stock model (1) is also no-arbitrage. Conversely, assume that (e1 − r, e2 − r, · · · , em − r) cannot be represented by a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ). Then the system of linear equations
σ11 σ12 σ21 σ22 .. .. . . σm1 σm2
· · · σ1n x1 e1 − r · · · σ2n x2 e2 − r .. .. = .. .. . . . . · · · σmn xn em − r
has no solution, i.e., the system
σ11 σ21 .. . σm1
· · · σ1n −σ11 · · · σ2n −σ21 . .. .. . .. . · · · σmn −σm1
··· ··· .. . ···
y1 . −σ1n .. e1 − r e −r −σ2n 2 yn = , .. .. y . n+1 . −σmn ... em − r y2n
y1 .. . yn
yn+1 ≥ 0 . .. y2n has no solution. By Farkas Lemma, we obtain that the system σ11 .. . σ1n −σ11 . .. −σ1n
σ21 .. . σ2n −σ21 .. . −σ2n
··· .. . ··· ··· .. . ···
σm1 .. β 1 . β σmn 2 . ≤ 0, −σm1 .. .. . βm −σmn
β1 β2 (e1 − r, e2 − r, · · · , en − r) . > 0 .. βm has a solution. Thus there exist real numbers (β1 , · · · , βm ) such that σ11 β1 + σ21 β2 + · · · + σm1 βm = 0 ··· σ β + σ 1n 1 2n β2 + · · · + σmn βm = 0 (e1 − r)β1 + (e2 − r)β2 + · · · + (en − r)βn > 0.
A No-Arbitrage Theorem for Uncertain Stock Model
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Given some time t > 0, we have Z
m tX
(ei − r)βi ds > 0
0 i=1
which means ln(exp(−rt)Zt ) − ln(Z0 ) > 0. Set β0 = 1 −
m X
βi .
i=1
Then (β0 , β1 , · · · , βm ) is a portfolio such that
M{exp(−rt)Zt ≥ Z0 } = 1 and
M{exp(−rt)Zt > Z0 } > 0 for the given time t. Hence the stock model (1) has an arbitrage. The theorem is verified. Theorem 3 The stock model (1) has an arbitrage if and only if the system σ11 β1 + σ21 β2 + · · · + σm1 βm = 0 ··· σ β + σ 1n 1 2n β2 + · · · + σmn βm = 0 (e1 − r)β1 + (e2 − r)β2 + · · · + (en − r)βn > 0 has a solution (β1 , β2 , · · · , βm ). Moreover, the portfolio (β0 , β1 , · · · , βm ) is a strategy for arbitrage where β0 is a real number satisfying β0 = 1 −
m X
βi .
i=1
Proof: By Theorem 2, we just need to prove that the system σ11 β1 + σ21 β2 + · · · + σm1 βm = 0 ··· σ β + σ 1n 1 2n β2 + · · · + σmn βm = 0 (e1 − r)β1 + (e2 − r)β2 + · · · + (en − r)βn > 0 has a solution if and only if the system of linear equations
σ11 σ12 σ21 σ22 .. .. . . σm1 σm2 has no solution.
· · · σ1n x1 e1 − r · · · σ2n x2 e2 − r .. .. = .. .. . . . . · · · σmn xn em − r
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Kai Yao
Assume that the system of linear equations σ11 σ12 · · · σ1n x1 e1 − r σ21 σ22 · · · σ2n x2 e2 − r .. .. . . .. .. = .. . . . . . . σm1 σm2 · · · σmn xn em − r has no solution. Then the proof of Theorem (2) shows that the system σ11 β1 + σ21 β2 + · · · + σm1 βm = 0 ··· σ β + σ 1n 1 2n β2 + · · · + σmn βm = 0 (e1 − r)β1 + (e2 − r)β2 + · · · + (en − r)βn > 0 has a solution (β1 , β2 , · · · , βm ). Set β0 = 1 −
m X
βi .
i=1
Then the portfolio (β0 , β1 , · · · , βm ) is a strategy for arbitrage. Conversely, assume that the system σ11 β1 + σ21 β2 + · · · + σm1 βm = 0 ··· σ β + σ 1n 1 2n β2 + · · · + σmn βm = 0 (e1 − r)β1 + (e2 − r)β2 + · · · + (en − r)βn > 0 has a solution. Then the system σ11 σ21 .. .. . . σ1n σ2n −σ11 −σ21 . .. .. .
··· .. . ··· ··· .. . ···
σm1 .. β 1 . β 2 σmn . ≤ 0, −σm1 .. .. β m . −σ1n −σ2n −σmn β1 β2 (e1 − r, e2 − r, · · · , en − r) . > 0 .. βm
has a solution. By Farkas Lemma, the system
σ11 σ21 .. . σm1
· · · σ1n −σ11 · · · σ2n −σ21 . .. .. . .. . · · · σmn −σm1
y1 .. . yn
e −r · · · −σ1n 1 e −r · · · −σ2n 2 , .. .. .. yn+1 = . . . . · · · −σmn .. em − r y2n
A No-Arbitrage Theorem for Uncertain Stock Model
y1 .. . yn
11
yn+1 ≥ 0 . .. y2n has no solution, i.e., the system of linear equations
σ11 σ12 σ21 σ22 .. .. . . σm1 σm2
· · · σ1n x1 e1 − r · · · σ2n x2 e2 − r .. .. = .. .. . . . . · · · σmn xn em − r
has no solution. So the theorem is verified. Corollary 1 The stock model (1) is no-arbitrage if its diffusion matrix
σ11 σ12 σ21 σ22 .. .. . . σm1 σm2
· · · σ1n · · · σ2n . .. . .. · · · σmn
has rank m, i.e., the row vectors are linearly independent. Proof: Since the diffusion matrix has rank m, the vectors (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ) generate an m-dimension vector space. Thus (e1 −r, e2 −r, · · · , em −r) is a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ). So the stock model (1) is no-arbitrage by Theorem 2. Corollary 2 The stock model (1) is no-arbitrage if its stock drift coefficients ei = r,
i = 1, 2, · · · , m.
Proof: Since the stock drift coefficients ei = r, i = 1, 2, · · · , m, and (0, 0, · · · , 0) is a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ), the stock model (1) is no-arbitrage by Theorem 2.
5 Some Examples In this section, we will apply the above theorems to some numerical examples. Example 2 Consider a stock model given by dXt = Xt dt dY1t = 2Y1t dt + Y1t dC1t + 2Y1t dC2t dY2t = 3Y2t dt + 2Y2t dC1t + 3Y2t dC2t .
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Kai Yao
This uncertain market has one bond and two stocks. Thus we have three different investments. However, since the row vectors of the diffusion matrix 12 23 are independent, the stock model is no-arbitrage by Corollary 1 whatever portfolio we choose. Example 3 Consider a stock model given by dXt = Xt dt dY1t = 2Y1t dt + Y1t dC1t + 2Y1t dC2t + 3Y1t dC3t dY 2t = 3Y2t dt + 2Y2t dC1t + 3Y2t dC2t + 4Y2t dC3t dY3t = 4Y3t dt + 3Y3t dC1t + 4Y3t dC2t + 5Y3t dC3t . This uncertain market has one bond and three stocks, thus we have four different investments. Although the row vectors of the diffusion matrix 123 2 3 4 345 are not independent, the system of linear equations 123 x1 2−1 2 3 4 x2 = 3 − 1 345 x3 4−1 has a solution
x1 1 x2 = 0 . x3 0
So the stock model is no-arbitrage by Theorem 2 whatever portfolio we choose. Example 4 Consider a stock model given by dXt = Xt dt dY1t = 2Y1t dt + Y1t dC1t + 2Y1t dC2t + 3Y1t dC3t dY 2t = 3Y2t dt + 2Y2t dC1t + 3Y2t dC2t + 4Y2t dC3t dY3t = 5Y3t dt + 3Y3t dC1t + 4Y3t dC2t + 5Y3t dC3t . This uncertain market also has one bond and three stocks, thus we have four different investments too. Since the system of linear equations 123 x1 2−1 2 3 4 x2 = 3 − 1 345 x3 5−1 has no solution, an arbitrage is available by Theorem 2. We solve β1 + 2β2 + 3β3 = 0 2β1 + 3β2 + 4β3 = 0 3β1 + 4β2 + 5β3 = 0 β0 + β1 + β2 + β3 = 1 (2 − 1)β1 + (3 − 1)β2 + (5 − 1)β3 > 0,
A No-Arbitrage Theorem for Uncertain Stock Model
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and get a solution
β0 1 β1 1 = β2 −2 . β3 1 Thus we can obtain an arbitrage at some time t theoretically by putting the wealth Zs into the bank, selling the second stock for 2Zs , and buying the first and third stocks with Zs , respectively, at all instant s ∈ (0, t).
6 Conclusions In this paper, we derived a no-arbitrage determinant theorem for Liu’s stock model in uncertain markets. The theorem gives a sufficient and necessary condition for Liu’s stock model to be no-arbitrage. As two corollaries, the stock model is noarbitrage if its diffusion matrix has a full row rank or its stock drift coefficients are equal to the riskless interest rate. Acknowledgements This work was supported by National Natural Science Foundation of China (Grants No.61273044 and No.91224008).
References Black, F., & Karasinski, P. (1991). Bond and option pricing when short-term rates are lognormal. Financial Analysts Journal, 47(4), 52-59. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637-654. Chen, X., & Liu, B. (2010). Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization and Decision Making, 9(1), 69-81. 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. 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. (2013). Toward uncertain finance theory. Journal of Uncertainty Analysis and Applications, 1, Article 1. Liu, B. (2014). Uncertainty distribution and independence of uncertain processes. Fuzzy Optimization and Decision Making, to be published. Liu, Y.H. (2012). An analytic method for solving uncertain differential equations, Journal of Uncertain Systems, 6(4).
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Liu, Y.H., & Chen, X., & Ralescu, D.A. (2014). Uncertain currency model and currency option pricing. International Journal of Intelligent Systems, to be published. Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141-183. Peng, J., & Yao, K. (2011). A new option pricing model for stocks in uncertainty markets. International Journal of Operations Research, 8(2), 18-26. Qin, Z., & Gao, X. (2009). Fractional Liu process with application to finance. Mathematical and Computer Modelling, 50(9-10), 1538-1543. Samuelson, P. (1965). Rational theory of warrant pricing. Industrial Management Review, 6, 13-31. Yao, K., & Gao, J., & Gao, Y. (2013). Some stability theorems of uncertain differential equation. Fuzzy Optimization and Decision Making, 12(1), 3-13. Yao, K., & Chen, X. (2013). A numerical method for solving uncertain differential equations. Journal of Intelligent & Fuzzy Systems, 25(3), 825-832. Yao, K. (2013a). Extreme values and integral of solution of uncertain differential equation. Journal of Uncertainty Analysis and Applications, 1, Article 2. Yao, K. (2013b). A type of nonlinear uncertain differential equations with analytic solution. Journal of Uncertainty Analysis and Applications, 1, Article 8.
A No-Arbitrage Theorem for Uncertain Stock Model Kai Yao
Received: date / Accepted: date
Abstract Stock model is used to describe the evolution of stock price in the form of differential equations. In early years, the stock price was assumed to follow a stochastic differential equation driven by a Brownian motion, and some famous models such as Black-Scholes stock model and Black-Karasinski stock model were widely used. This paper assumes that the stock price follows an uncertain differential equation driven by Liu process rather than Brownian motion, and accepts Liu’s stock model to simulate the uncertain market. Then this paper proves a noarbitrage determinant theorem for Liu’s stock model and presents a sufficient and necessary condition for no-arbitrage. Finally, some examples are given to illustrate the usefulness of the no-arbitrage determinant theorem. Keywords finance · stock model · no-arbitrage · uncertainty theory · uncertain differential equation
1 Introduction Stock price in the financial market is usually described as a stochastic process. In 1900’s, Bachelier employed a Brownian motion to model the price fluctuation in Paris stock market. However, his model failed to evaluate both stock and option prices for many reasons, of which the major one was that the Brownian motion may become negative, contradicting the price of a stock. Samuelson (1965) assumed that the return rate of the stock followed a geometric Brownian motion and proposed a stock model, but he failed to consider the drift of the Brownian motion and the time value of money. Black and Scholes (1973) assumed that the stock price followed a geometric Brownian motion and proposed the famous Black-Scholes stock model. Merton (1973) analyzed the assumption of Black-Scholes stock model and performed a rigorous analysis. Nowadays, the Black-Scholes stock model has become an indispensable tool in the trading of options and other financial Kai Yao School of Management, University of Chinese Academy of Sciences, Beijing 100190, China E-mail: [email protected]
2
Kai Yao
derivatives. Black and Karasinski (1991) proposed a stock model with OrnsteinUhlenbeck process, describing the stock price that diffuses around an average level in long run. Despite the fast development in stochastic finance, many critics argue that prices of some stocks may not behave like randomness. Thus some researchers such as Qin and Gao (2009) suggested to describe the financial markets by fuzzy theory. Unfortunately, this theory does not provide a satisfactory method for studying the real market. A new tool is the uncertainty theory that was invented by Liu (2007) and refined by Liu (2010) to model human’s belief degree. Based on an uncertain measure satisfying normality, duality, subadditivity and product axioms, a concept of uncertain variable was proposed to model a quantity with uncertainty. Then some concepts of uncertainty distribution, expected value, and variance were presented to describe an uncertain variable. In order to model an uncertain dynamic system, an uncertain process was defined by Liu (2008) as a sequence of uncertain variables driven by the time. Then Liu (2009) proposed a type of Liu process, that is a stationary independent increment process with normal uncertain increments. In addition, 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 was a type of differential equation driven by Liu process. Chen and Liu (2010) gave a sufficient condition for an uncertain differential equation having a unique solution. Then Yao et al. (2013) gave a sufficient condition for it being stable, which was first defined by Liu (2009). The analytic solution to a linear uncertain differential equation was obtained by Chen and Liu (2010), and some analytic methods for solving general uncertain differential equations were provided by Liu (2012) and Yao (2013b), respectively. In addition, some numerical methods were designed by Yao and Chen (2013), and Yao (2013a). By means of the uncertain differential equation, Liu (2009) proposed an uncertain stock model named Liu’s stock model. After that, Chen (2011) derived an American option pricing formula for Liu’s stock model. Peng and Yao (2011) proposed a mean-reverting stock model in an uncertain market. Besides, uncertain interest rate model was proposed by Chen and Gao (2013), and uncertain currency model was proposed by Liu et al. (2014) via uncertain differential equation. For recent developments of uncertain finance, please refer to Liu (2013). In this paper, we will derive a no-arbitrage determinant theorem for Liu’s stock model. The rest of this paper is structured as follows. The next section is intended to introduce some concepts of uncertain process and uncertain differential equation. Liu’s stock model is introduced in Section 3 and a sufficient and necessary condition for such a stock model being no-arbitrage is derived in Section 4. After that, we apply the theorem to some examples in Section 5. Finally, some remarks are made in Section 6. 2 Preliminary In this section, we will introduce some useful definitions about uncertain variable, uncertain process, uncertain calculus and uncertain differential equation. Definition 1 (Liu, 2007) Let L be a σ-algebra on a nonempty set Γ. A set function M : L → [0, 1] is called an uncertain measure if it satisfies the following
A No-Arbitrage Theorem for Uncertain Stock Model
3
axioms: Axiom 1: (Normality Axiom) M{Γ } = 1 for the universal set Γ. Axiom 2: (Duality Axiom) M{Λ} + M{Λc } = 1 for any event Λ. Axiom 3: (Subadditivity Axiom) For every countable sequence of events Λ1 , Λ2 , · · · , we have (∞ ) ∞ [ X M Λi ≤ M {Λi } . i=1
i=1
Besides, the product uncertain measure on the product σ-algebre L is defined by Liu (2009) as follows, Axiom 4: (Product Axiom) Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, · · · Then the product uncertain measure M is an uncertain measure satisfying (∞ ) ∞ Y ^ M Λk = Mk {Λk } i=1
k=1
where Λk are arbitrarily chosen events from Lk for k = 1, 2, · · · , respectively. An uncertain variable is a measurable function from an uncertainty space to the set of real numbers. In order to describe an uncertain variable, a concept of uncertainty distribution is defined as follows. Definition 2 (Liu, 2007) The uncertainty distribution of an uncertain variable ξ is defined by Φ(x) = M{ξ ≤ x} for any x ∈ 0. In fact, the expected value of a normal uncertain variable N (e, σ) is e, and the variance is σ 2 . An uncertain process is essentially a sequence of uncertain variables driven by time or space. The formal definition of an uncertain process is given as follows. Definition 6 (Liu, 2008) Let T be an index set and (Γ, L, M) be an uncertainty space. An uncertain process is a measurable function from T × (Γ, L, M) to the set of real numbers, i.e., for each t ∈ T and any Borel set B of real numbers, the set {Xt ∈ B} = {γ | Xt (γ) ∈ B} is an event. Definition 7 (Liu, 2014) Uncertain processes X1t , X2t , · · · , Xnt are said to be independent if for any positive integer k and any times t1 , t2 , · · · , tk , the uncertain vectors ξ i = (Xit1 , Xit2 , · · · , Xitk ), i = 1, 2, · · · , n are independent, i.e., for any k-dimensional Borel sets B1 , B2 , · · · , Bn , we have ) ( n n ^ \ M {ξi ∈ Bi } . (ξ i ∈ Bi ) = M i=1
i=1
Definition 8 (Liu, 2009) An uncertain process Ct is said to be a canonical Liu process if (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 normally distributed uncertain variable with expected value 0 and variance t2 , whose uncertainty distribution is −1 πx Φ(x) = 1 + exp − √ , x ∈ Z0 } > 0 where Zt is determined by the uncertain differential equation (2) and represents the wealth at time t. The next theorem gives a sufficient and necessary condition for the stock model (1) being no-arbitrage. Meanwhile, the theorem gives a strategy for arbitrage when the condition is not satisfied. Theorem 2 (No-Arbitrage Determinant Theorem) The stock model (1) is noarbitrage if and only if (e1 − r, e2 − r, · · · , em − r) is a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ), i.e., the system of linear equations σ11 σ12 · · · σ1n x1 e1 − r σ21 σ22 · · · σ2n x2 e2 − r .. .. . . .. .. = .. . . . . . . σm1 σm2 · · · σmn xn em − r has a solution. Proof: The uncertain differential equation (2) has a solution Z t Z tX Z m m X n X Zt = Z0 exp r βs ds + ei βis ds + σij 0
0 i=1
Z = Z0 exp(rt) exp
m tX
0 i=1
(ei − r)βis ds +
i=1 j=1 n Z X j=1
m tX
0 i=1
t
βis dCjs
0
σij βis dCjs .
A No-Arbitrage Theorem for Uncertain Stock Model
7
Thus we have m tX
Z ln (exp (−rt) Zt ) − ln (Z0 ) =
(ei − r)βis ds +
0 i=1
n Z X j=1
m tX
σij βis dCjs ,
0 i=1
which is a normal uncertain variable with expected value m tX
Z
(ei − r)βis ds
0 i=1
and variance
n Z X
t
j=1
0
2 m X σij βis ds . i=1
Assume that (e1 −r, e2 −r, · · · , em −r) is a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ). The argument breaks down into two cases. Case 1: Given any time t and portfolio (βs , β1s , · · · , βms ), if n Z t X m X σij βis ds = 0, 0 j=1
then
m X
σij βis = 0,
i=1
j = 1, 2, · · · , n, s ∈ (0, t).
i=1
Since (e1 − r, e2 − r, · · · , em − r) is a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ), we have m X
(ei − r)βis = 0,
s ∈ (0, t).
i=1
So we obtain Z
m tX
(ei − r)βis ds = 0
0 i=1
and ln(exp(−rt)Zt ) = ln(Z0 ). Thus
M{exp(−rt)Zt > Z0 } = 0 and the stock model (1) is no-arbitrage. Case 2: Given any time t and portfolio (βs , β1s , · · · , βms ), if m n Z t X X σij βis ds 6= 0, 0 j=1
i=1
then
M{ln(exp(−rt)Zt ) − ln(Z0 ) ≥ 0} < 1, i.e.,
M{exp(−rt)Zt ≥ Z0 } < 1.
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Kai Yao
So the stock model (1) is also no-arbitrage. Conversely, assume that (e1 − r, e2 − r, · · · , em − r) cannot be represented by a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ). Then the system of linear equations
σ11 σ12 σ21 σ22 .. .. . . σm1 σm2
· · · σ1n x1 e1 − r · · · σ2n x2 e2 − r .. .. = .. .. . . . . · · · σmn xn em − r
has no solution, i.e., the system
σ11 σ21 .. . σm1
· · · σ1n −σ11 · · · σ2n −σ21 . .. .. . .. . · · · σmn −σm1
··· ··· .. . ···
y1 . −σ1n .. e1 − r e −r −σ2n 2 yn = , .. .. y . n+1 . −σmn ... em − r y2n
y1 .. . yn
yn+1 ≥ 0 . .. y2n has no solution. By Farkas Lemma, we obtain that the system σ11 .. . σ1n −σ11 . .. −σ1n
σ21 .. . σ2n −σ21 .. . −σ2n
··· .. . ··· ··· .. . ···
σm1 .. β 1 . β σmn 2 . ≤ 0, −σm1 .. .. . βm −σmn
β1 β2 (e1 − r, e2 − r, · · · , en − r) . > 0 .. βm has a solution. Thus there exist real numbers (β1 , · · · , βm ) such that σ11 β1 + σ21 β2 + · · · + σm1 βm = 0 ··· σ β + σ 1n 1 2n β2 + · · · + σmn βm = 0 (e1 − r)β1 + (e2 − r)β2 + · · · + (en − r)βn > 0.
A No-Arbitrage Theorem for Uncertain Stock Model
9
Given some time t > 0, we have Z
m tX
(ei − r)βi ds > 0
0 i=1
which means ln(exp(−rt)Zt ) − ln(Z0 ) > 0. Set β0 = 1 −
m X
βi .
i=1
Then (β0 , β1 , · · · , βm ) is a portfolio such that
M{exp(−rt)Zt ≥ Z0 } = 1 and
M{exp(−rt)Zt > Z0 } > 0 for the given time t. Hence the stock model (1) has an arbitrage. The theorem is verified. Theorem 3 The stock model (1) has an arbitrage if and only if the system σ11 β1 + σ21 β2 + · · · + σm1 βm = 0 ··· σ β + σ 1n 1 2n β2 + · · · + σmn βm = 0 (e1 − r)β1 + (e2 − r)β2 + · · · + (en − r)βn > 0 has a solution (β1 , β2 , · · · , βm ). Moreover, the portfolio (β0 , β1 , · · · , βm ) is a strategy for arbitrage where β0 is a real number satisfying β0 = 1 −
m X
βi .
i=1
Proof: By Theorem 2, we just need to prove that the system σ11 β1 + σ21 β2 + · · · + σm1 βm = 0 ··· σ β + σ 1n 1 2n β2 + · · · + σmn βm = 0 (e1 − r)β1 + (e2 − r)β2 + · · · + (en − r)βn > 0 has a solution if and only if the system of linear equations
σ11 σ12 σ21 σ22 .. .. . . σm1 σm2 has no solution.
· · · σ1n x1 e1 − r · · · σ2n x2 e2 − r .. .. = .. .. . . . . · · · σmn xn em − r
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Kai Yao
Assume that the system of linear equations σ11 σ12 · · · σ1n x1 e1 − r σ21 σ22 · · · σ2n x2 e2 − r .. .. . . .. .. = .. . . . . . . σm1 σm2 · · · σmn xn em − r has no solution. Then the proof of Theorem (2) shows that the system σ11 β1 + σ21 β2 + · · · + σm1 βm = 0 ··· σ β + σ 1n 1 2n β2 + · · · + σmn βm = 0 (e1 − r)β1 + (e2 − r)β2 + · · · + (en − r)βn > 0 has a solution (β1 , β2 , · · · , βm ). Set β0 = 1 −
m X
βi .
i=1
Then the portfolio (β0 , β1 , · · · , βm ) is a strategy for arbitrage. Conversely, assume that the system σ11 β1 + σ21 β2 + · · · + σm1 βm = 0 ··· σ β + σ 1n 1 2n β2 + · · · + σmn βm = 0 (e1 − r)β1 + (e2 − r)β2 + · · · + (en − r)βn > 0 has a solution. Then the system σ11 σ21 .. .. . . σ1n σ2n −σ11 −σ21 . .. .. .
··· .. . ··· ··· .. . ···
σm1 .. β 1 . β 2 σmn . ≤ 0, −σm1 .. .. β m . −σ1n −σ2n −σmn β1 β2 (e1 − r, e2 − r, · · · , en − r) . > 0 .. βm
has a solution. By Farkas Lemma, the system
σ11 σ21 .. . σm1
· · · σ1n −σ11 · · · σ2n −σ21 . .. .. . .. . · · · σmn −σm1
y1 .. . yn
e −r · · · −σ1n 1 e −r · · · −σ2n 2 , .. .. .. yn+1 = . . . . · · · −σmn .. em − r y2n
A No-Arbitrage Theorem for Uncertain Stock Model
y1 .. . yn
11
yn+1 ≥ 0 . .. y2n has no solution, i.e., the system of linear equations
σ11 σ12 σ21 σ22 .. .. . . σm1 σm2
· · · σ1n x1 e1 − r · · · σ2n x2 e2 − r .. .. = .. .. . . . . · · · σmn xn em − r
has no solution. So the theorem is verified. Corollary 1 The stock model (1) is no-arbitrage if its diffusion matrix
σ11 σ12 σ21 σ22 .. .. . . σm1 σm2
· · · σ1n · · · σ2n . .. . .. · · · σmn
has rank m, i.e., the row vectors are linearly independent. Proof: Since the diffusion matrix has rank m, the vectors (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ) generate an m-dimension vector space. Thus (e1 −r, e2 −r, · · · , em −r) is a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ). So the stock model (1) is no-arbitrage by Theorem 2. Corollary 2 The stock model (1) is no-arbitrage if its stock drift coefficients ei = r,
i = 1, 2, · · · , m.
Proof: Since the stock drift coefficients ei = r, i = 1, 2, · · · , m, and (0, 0, · · · , 0) is a linear combination of (σ11 , σ21 , · · · , σm1 ), (σ12 , σ22 , · · · , σm2 ), · · · , (σ1n , σ2n , · · · , σmn ), the stock model (1) is no-arbitrage by Theorem 2.
5 Some Examples In this section, we will apply the above theorems to some numerical examples. Example 2 Consider a stock model given by dXt = Xt dt dY1t = 2Y1t dt + Y1t dC1t + 2Y1t dC2t dY2t = 3Y2t dt + 2Y2t dC1t + 3Y2t dC2t .
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Kai Yao
This uncertain market has one bond and two stocks. Thus we have three different investments. However, since the row vectors of the diffusion matrix 12 23 are independent, the stock model is no-arbitrage by Corollary 1 whatever portfolio we choose. Example 3 Consider a stock model given by dXt = Xt dt dY1t = 2Y1t dt + Y1t dC1t + 2Y1t dC2t + 3Y1t dC3t dY 2t = 3Y2t dt + 2Y2t dC1t + 3Y2t dC2t + 4Y2t dC3t dY3t = 4Y3t dt + 3Y3t dC1t + 4Y3t dC2t + 5Y3t dC3t . This uncertain market has one bond and three stocks, thus we have four different investments. Although the row vectors of the diffusion matrix 123 2 3 4 345 are not independent, the system of linear equations 123 x1 2−1 2 3 4 x2 = 3 − 1 345 x3 4−1 has a solution
x1 1 x2 = 0 . x3 0
So the stock model is no-arbitrage by Theorem 2 whatever portfolio we choose. Example 4 Consider a stock model given by dXt = Xt dt dY1t = 2Y1t dt + Y1t dC1t + 2Y1t dC2t + 3Y1t dC3t dY 2t = 3Y2t dt + 2Y2t dC1t + 3Y2t dC2t + 4Y2t dC3t dY3t = 5Y3t dt + 3Y3t dC1t + 4Y3t dC2t + 5Y3t dC3t . This uncertain market also has one bond and three stocks, thus we have four different investments too. Since the system of linear equations 123 x1 2−1 2 3 4 x2 = 3 − 1 345 x3 5−1 has no solution, an arbitrage is available by Theorem 2. We solve β1 + 2β2 + 3β3 = 0 2β1 + 3β2 + 4β3 = 0 3β1 + 4β2 + 5β3 = 0 β0 + β1 + β2 + β3 = 1 (2 − 1)β1 + (3 − 1)β2 + (5 − 1)β3 > 0,
A No-Arbitrage Theorem for Uncertain Stock Model
13
and get a solution
β0 1 β1 1 = β2 −2 . β3 1 Thus we can obtain an arbitrage at some time t theoretically by putting the wealth Zs into the bank, selling the second stock for 2Zs , and buying the first and third stocks with Zs , respectively, at all instant s ∈ (0, t).
6 Conclusions In this paper, we derived a no-arbitrage determinant theorem for Liu’s stock model in uncertain markets. The theorem gives a sufficient and necessary condition for Liu’s stock model to be no-arbitrage. As two corollaries, the stock model is noarbitrage if its diffusion matrix has a full row rank or its stock drift coefficients are equal to the riskless interest rate. Acknowledgements This work was supported by National Natural Science Foundation of China (Grants No.61273044 and No.91224008).
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