A No-Arbitrage Theorem for Uncertain Stock Model

Transcription

1 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 19 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 119, China yaokai@ucasaccn

2 2 Kai Yao derivatives Black and Karasinski (1991) proposed a stock model with Ornstein- Uhlenbeck 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 (29) 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 (27) and refined by Liu (21) 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 (28) as a sequence of uncertain variables driven by the time Then Liu (29) proposed a type of Liu process, that is a stationary independent increment process with normal uncertain increments In addition, Liu (29) 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 (21) gave a sufficient condition for an uncertain differential equation having a unique solution Then Yao et al (213) gave a sufficient condition for it being stable, which was first defined by Liu (29) The analytic solution to a linear uncertain differential equation was obtained by Chen and Liu (21), and some analytic methods for solving general uncertain differential equations were provided by Liu (212) and Yao (213b), respectively In addition, some numerical methods were designed by Yao and Chen (213), and Yao (213a) By means of the uncertain differential equation, Liu (29) proposed an uncertain stock model named Liu s stock model After that, Chen (211) derived an American option pricing formula for Liu s stock model Peng and Yao (211) proposed a mean-reverting stock model in an uncertain market Besides, uncertain interest rate model was proposed by Chen and Gao (213), and uncertain currency model was proposed by Liu et al (214) via uncertain differential equation For recent developments of uncertain finance, please refer to Liu (213) 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, 27) Let L be a σ-algebra on a nonempty set Γ A set function M : L [, 1] is called an uncertain measure if it satisfies the following

3 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{Λ c } = 1 for any event Λ Axiom 3: (Subadditivity Axiom) For every countable sequence of events Λ 1, Λ 2,, we have { } M Λ i M {Λ i } Besides, the product uncertain measure on the product σ-algebre L is defined by Liu (29) as follows, Axiom 4: (Product Axiom) Let (Γ k, L k, M k ) be uncertainty spaces for k = 1, 2, Then the product uncertain measure M is an uncertain measure satisfying { } M Λ k = M k {Λ k } where Λ k are arbitrarily chosen events from L k 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, 27) The uncertainty distribution of an uncertain variable ξ is defined by Φ(x) = M{ξ x} for any x R The inverse function Φ 1 is called the inverse uncertainty distribution of ξ if it exists and is unique for each α (, 1) Inverse uncertainty distribution plays a crucial role in operations of independent uncertain variables k=1 Definition 3 (Liu, 29) The uncertain variables ξ 1, ξ 2,, ξ n independent if { n } n M (ξ i B i ) = M {ξ i B i } for any Borel set B 1, B 2,, B n of real numbers are said to be Theorem 1 (Liu, 21) Let ξ 1, ξ 2,, ξ n be independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively If f(x 1, x 2,, x n ) is strictly increasing with respect to x 1, x 2,, x m and strictly decreasing with respect to x m+1, x m+2,, x n, then ξ = f(ξ 1, ξ 2,, ξ n ) is an uncertain variable with an inverse uncertainty distribution ( ) Φ 1 (r) = f Φ 1 1 (r),, Φ 1 m (r), Φ 1 m+1(1 r),, Φ 1 n (1 r) Definition 4 (Liu, 27) The expected value of an uncertain variable ξ is defined by + E[ξ] = M{ξ x}dx M{ξ x}dx provided that at least one of the two integrals exists

4 4 Kai Yao Definition 5 (Liu, 27) Let ξ be an uncertain variable with a finite expected value e Then the variance of ξ is defined by V [ξ] = E[(ξ e) 2 ] Let ξ be an uncertain variable with an uncertainty distribution Φ If E[ξ] exists, then and E[ξ] = + (1 Φ(x))dx V [ξ] = + Φ(x)dx = (x E[ξ]) 2 dφ(x) + xdφ(x), An uncertain variable ξ is called normal if it has a normal uncertainty distribution ( ( )) 1 π(e x) Φ(x) = 1 + exp, x R 3σ denoted by N(e, σ) where e and σ are real numbers with σ > 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, 28) 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, ie, for each t T and any Borel set B of real numbers, the set is an event {X t B} = {γ X t (γ) B} Definition 7 (Liu, 214) Uncertain processes X 1t, X 2t,, X nt are said to be independent if for any positive integer k and any times t 1, t 2,, t k, the uncertain vectors ξ i = (X it1, X it2,, X itk ), i = 1, 2,, n are independent, ie, for any k-dimensional Borel sets B 1, B 2,, B n, we have { n } n M (ξ i B i ) = M {ξ i B i } Definition 8 (Liu, 29) An uncertain process C t is said to be a canonical Liu process if (i) C = and almost all sample paths are Lipschitz continuous, (ii) C t has stationary and independent increments, (iii) every increment C s+t C s is a normally distributed uncertain variable with expected value and variance t 2, whose uncertainty distribution is ( ( Φ(x) = 1 + exp πx )) 1, x R 3t If C t is a canonical Liu process, then the uncertain process G t = exp(et + σc t ) is called a geometric Liu process

5 A No-Arbitrage Theorem for Uncertain Stock Model 5 Definition 9 (Liu, 29) Let X t be an uncertain process and C t be a canonical Liu process For any partition of closed interval [a, b] with a = t 1 < t 2 < < t k+1 = b, the mesh is written as = max 1 i k t i+1 t i Then the Liu integral of X t with respect to C t is b a X t dc t = lim k X ti (C ti+1 C ti ) provided that the limit exists almost surely and is an uncertain variable Liu (21) proved that, given a deterministic and integrable function f(s), its Liu integral f(s)dc s is a normal uncertain variable at each time t, ie, ( f(s)dc s N, ) f(s) ds Definition 1 (Liu, 29) Suppose C t is a canonical Liu process, and f and g are some given functions Then dx t = f(t, X t )dt + g(t, X t )dc t is called an uncertain differential equation A solution is an uncertain process that satisfies the uncertain differential equation identically in t Example 1 Let C t be a canonical Liu process Then the uncertain differential equation dx t = α t X t dt + β t X t dc t has a solution ( X t = X exp α s ds + which is just a geometric Liu process β s dc s ) 3 Liu s Stock Model Assume that there are multiple stocks whose prices are determined by multiple Liu processes Liu (29) proposed a stock model in which the bond price X t and the stock price Y it are determined by dx t = rx t dt dy it = e i Y it dt + n σ ij Y it dc jt, i = 1, 2,, m (1) j=1

6 6 Kai Yao where r is the riskless interest rate, e i are the stock drift coefficients, σ ij are the stock diffusion coefficients, and C jt are independent canonical Liu processes, i = 1, 2,, m, j = 1, 2,, n For the stock model (1), we have the choice of m + 1 different investments At each instant t, we may choose a portfolio (β t, β 1t,, β mt ) (ie, the investment fractions meeting β t + β 1t + + β mt = 1) Then the wealth Z t at time t should follow the uncertain differential equation n dz t = rβ t Z t dt + e i β it Z t dt + σ ij β it Z t dc jt (2) That is Z t = Z exp r j=1 n β s ds + e i β is ds + σ ij β is dc js j=1 4 No-Arbitrage Determinant Theorem The stock model (1) is said to be no-arbitrage if there is no portfolio (β t, β 1t,, β mt ) such that for some time s, and M{exp( rs)z s Z } = 1 M{exp( rs)z s > Z } > where Z t 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 (e 1 r, e 2 r,, e m r) is a linear combination of (σ 11, σ 21,, σ m1 ), (σ 12, σ 22,, σ m2 ),, (σ 1n, σ 2n,, σ mn ), ie, the system of linear equations has a solution σ 11 σ 12 σ 1n σ 21 σ 22 σ 2n σ m1 σ m2 σ mn x 1 x 2 x n = e 1 r e 2 r e m r Proof: The uncertain differential equation (2) has a solution n Z t = Z exp r β s ds + e i β is ds + σ ij β is dc js = Z exp(rt) exp (e i r)β is ds + j=1 n j=1 σ ij β is dc js

7 A No-Arbitrage Theorem for Uncertain Stock Model 7 Thus we have ln (exp ( rt) Z t ) ln (Z ) = (e i r)β is ds + n j=1 which is a normal uncertain variable with expected value σ ij β is dc js, and variance n (e i r)β is ds j=1 2 σ ij β is ds Assume that (e 1 r, e 2 r,, e m r) is a linear combination of (σ 11, σ 21,, σ m1 ), (σ 12, σ 22,, σ m2 ),, (σ 1n, σ 2n,, σ mn ) The argument breaks down into two cases Case 1: Given any time t and portfolio (β s, β 1s,, β ms ), if n t σ ij β is ds =, then j=1 σ ij β is =, j = 1, 2,, n, s (, t) Since (e 1 r, e 2 r,, e m r) is a linear combination of (σ 11, σ 21,, σ m1 ), (σ 12, σ 22,, σ m2 ),, (σ 1n, σ 2n,, σ mn ), we have So we obtain and Thus (e i r)β is =, s (, t) (e i r)β is ds = ln(exp( rt)z t ) = ln(z ) M{exp( rt)z t > Z } = and the stock model (1) is no-arbitrage Case 2: Given any time t and portfolio (β s, β 1s,, β ms ), if n t σ ij β is ds, then ie, j=1 M{ln(exp( rt)z t ) ln(z ) } < 1, M{exp( rt)z t Z } < 1

8 8 Kai Yao So the stock model (1) is also no-arbitrage Conversely, assume that (e 1 r, e 2 r,, e m r) cannot be represented by a linear combination of (σ 11, σ 21,, σ m1 ), (σ 12, σ 22,, σ m2 ),, (σ 1n, σ 2n,, σ mn ) Then the system of linear equations has no solution, ie, the system σ 11 σ 12 σ 1n x 1 σ 21 σ 22 σ 2n x 2 = σ m1 σ m2 σ mn x n σ 11 σ 1n σ 11 σ 1n σ 21 σ 2n σ 21 σ 2n σ m1 σ mn σ m1 σ mn y 1 y n y n+1 y 2n y 1 y n y n+1 y 2n e 1 r e 2 r e m r = has no solution By Farkas Lemma, we obtain that the system σ 11 σ 21 σ m1 β 1 σ 1n σ 2n σ mn β 2 σ 11 σ 21 σ m1, β m σ 1n σ 2n σ mn β 1 β 2 (e 1 r, e 2 r,, e n r) > β m e 1 r e 2 r e m r has a solution Thus there exist real numbers (β 1,, β m ) such that σ 11 β 1 + σ 21 β σ m1 β m = σ 1n β 1 + σ 2n β σ mn β m = (e 1 r)β 1 + (e 2 r)β (e n r)β n >,

9 A No-Arbitrage Theorem for Uncertain Stock Model 9 Given some time t >, we have which means Set (e i r)β i ds > ln(exp( rt)z t ) ln(z ) > β = 1 β i Then (β, β 1,, β m ) is a portfolio such that and M{exp( rt)z t Z } = 1 M{exp( rt)z t > Z } > 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 σ 11 β 1 + σ 21 β σ m1 β m = σ 1n β 1 + σ 2n β σ mn β m = (e 1 r)β 1 + (e 2 r)β (e n r)β n > has a solution (β 1, β 2,, β m ) Moreover, the portfolio (β, β 1,, β m ) is a strategy for arbitrage where β is a real number satisfying β = 1 β i Proof: By Theorem 2, we just need to prove that the system σ 11 β 1 + σ 21 β σ m1 β m = σ 1n β 1 + σ 2n β σ mn β m = (e 1 r)β 1 + (e 2 r)β (e n r)β n > has a solution if and only if the system of linear equations has no solution σ 11 σ 12 σ 1n x 1 σ 21 σ 22 σ 2n x 2 = σ m1 σ m2 σ mn x n e 1 r e 2 r e m r

10 1 Kai Yao Assume that the system of linear equations σ 11 σ 12 σ 1n x 1 σ 21 σ 22 σ 2n x 2 = σ m1 σ m2 σ mn x n e 1 r e 2 r e m r has no solution Then the proof of Theorem (2) shows that the system σ 11 β 1 + σ 21 β σ m1 β m = σ 1n β 1 + σ 2n β σ mn β m = (e 1 r)β 1 + (e 2 r)β (e n r)β n > has a solution (β 1, β 2,, β m ) Set β = 1 β i Then the portfolio (β, β 1,, β m ) is a strategy for arbitrage Conversely, assume that the system σ 11 β 1 + σ 21 β σ m1 β m = σ 1n β 1 + σ 2n β σ mn β m = (e 1 r)β 1 + (e 2 r)β (e n r)β n > has a solution Then the system σ 11 σ 21 σ m1 β 1 σ 1n σ 2n σ mn β 2 σ 11 σ 21 σ m1, β m σ 1n σ 2n σ mn β 1 β 2 (e 1 r, e 2 r,, e n r) > β m has a solution By Farkas Lemma, the system σ 11 σ 1n σ 11 σ 1n σ 21 σ 2n σ 21 σ 2n σ m1 σ mn σ m1 σ mn y 1 y n y n+1 y 2n = e 1 r e 2 r e m r,

11 A No-Arbitrage Theorem for Uncertain Stock Model 11 y 1 y n y n+1 y 2n has no solution, ie, the system of linear equations σ 11 σ 12 σ 1n x 1 σ 21 σ 22 σ 2n x 2 = σ m1 σ m2 σ mn x n has no solution So the theorem is verified e 1 r e 2 r e m r Corollary 1 The stock model (1) is no-arbitrage if its diffusion matrix σ 11 σ 12 σ 1n σ 21 σ 22 σ 2n σ m1 σ m2 σ mn has rank m, ie, the row vectors are linearly independent Proof: Since the diffusion matrix has rank m, the vectors (σ 11, σ 21,, σ m1 ), (σ 12, σ 22,, σ m2 ),, (σ 1n, σ 2n,, σ mn ) generate an m-dimension vector s- pace Thus (e 1 r, e 2 r,, e m r) is a linear combination of (σ 11, σ 21,, σ m1 ), (σ 12, σ 22,, σ m2 ),, (σ 1n, σ 2n,, σ mn ) 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 e i = r, i = 1, 2,, m Proof: Since the stock drift coefficients e i = r, i = 1, 2,, m, and (,,, ) is a linear combination of (σ 11, σ 21,, σ m1 ), (σ 12, σ 22,, σ m2 ),, (σ 1n, σ 2n,, σ mn ), 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 dx t = X t dt dy 1t = 2Y 1t dt + Y 1t dc 1t + 2Y 1t dc 2t dy 2t = 3Y 2t dt + 2Y 2t dc 1t + 3Y 2t dc 2t

12 12 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 ( ) are independent, the stock model is no-arbitrage by Corollary 1 whatever portfolio we choose Example 3 Consider a stock model given by dx t = X t dt dy 1t = 2Y 1t dt + Y 1t dc 1t + 2Y 1t dc 2t + 3Y 1t dc 3t dy 2t = 3Y 2t dt + 2Y 2t dc 1t + 3Y 2t dc 2t + 4Y 2t dc 3t dy 3t = 4Y 3t dt + 3Y 3t dc 1t + 4Y 3t dc 2t + 5Y 3t dc 3t This uncertain market has one bond and three stocks, thus we have four different investments Although the row vectors of the diffusion matrix are not independent, the system of linear equations x x 2 = x has a solution x 1 1 x 2 = x 3 So the stock model is no-arbitrage by Theorem 2 whatever portfolio we choose Example 4 Consider a stock model given by dx t = X t dt dy 1t = 2Y 1t dt + Y 1t dc 1t + 2Y 1t dc 2t + 3Y 1t dc 3t dy 2t = 3Y 2t dt + 2Y 2t dc 1t + 3Y 2t dc 2t + 4Y 2t dc 3t dy 3t = 5Y 3t dt + 3Y 3t dc 1t + 4Y 3t dc 2t + 5Y 3t dc 3t This uncertain market also has one bond and three stocks, thus we have four different investments too Since the system of linear equations x x 2 = x has no solution, an arbitrage is available by Theorem 2 We solve β 1 + 2β 2 + 3β 3 = 2β 1 + 3β 2 + 4β 3 = 3β 1 + 4β 2 + 5β 3 = β + β 1 + β 2 + β 3 = 1 (2 1)β 1 + (3 1)β 2 + (5 1)β 3 >,

13 A No-Arbitrage Theorem for Uncertain Stock Model 13 and get a solution β β 1 β 2 = β Thus we can obtain an arbitrage at some time t theoretically by putting the wealth Z s into the bank, selling the second stock for 2Z s, and buying the first and third stocks with Z s, respectively, at all instant s (, 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 and No912248) References Black, F, & Karasinski, P (1991) Bond and option pricing when short-term rates are lognormal Financial Analysts Journal, 47(4), Black, F, & Scholes, M (1973) The pricing of options and corporate liabilities Journal of Political Economy, 81, Chen, X, & Liu, B (21) Existence and uniqueness theorem for uncertain differential equations Fuzzy Optimization and Decision Making, 9(1), Chen, X (211) American option pricing formula for uncertain financial market International Journal of Operations Research, 8(2), Chen, X, & Gao, J (213) Uncertain term structure model of interest rate Soft Computing, 17(4), Liu, B (27) Uncertainty Theory (2nd ed) Berlin: Springer Liu, B (28) Fuzzy process, hybrid process and uncertain process Journal of Uncertain Systems, 2(1), 3-16 Liu, B (29) Some research problems in uncertainty theory Journal of Uncertain Systems, 3(1), 3-1 Liu, B (21) Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty Berlin: Springer Liu, B (213) Toward uncertain finance theory Journal of Uncertainty Analysis and Applications, 1, Article 1 Liu, B (214) Uncertainty distribution and independence of uncertain processes Fuzzy Optimization and Decision Making, to be published Liu, YH (212) An analytic method for solving uncertain differential equations, Journal of Uncertain Systems, 6(4)

14 14 Kai Yao Liu, YH, & Chen, X, & Ralescu, DA (214) 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), Peng, J, & Yao, K (211) A new option pricing model for stocks in uncertainty markets International Journal of Operations Research, 8(2), Qin, Z, & Gao, X (29) Fractional Liu process with application to finance Mathematical and Computer Modelling, 5(9-1), Samuelson, P (1965) Rational theory of warrant pricing Industrial Management Review, 6, Yao, K, & Gao, J, & Gao, Y (213) Some stability theorems of uncertain differential equation Fuzzy Optimization and Decision Making, 12(1), 3-13 Yao, K, & Chen, X (213) A numerical method for solving uncertain differential equations Journal of Intelligent & Fuzzy Systems, 25(3), Yao, K (213a) Extreme values and integral of solution of uncertain differential equation Journal of Uncertainty Analysis and Applications, 1, Article 2 Yao, K (213b) A type of nonlinear uncertain differential equations with analytic solution Journal of Uncertainty Analysis and Applications, 1, Article 8