No-arbitrage theorem for multi-factor uncertain stock model with floating interest rate

Transcription

1 Fuzzy Optim Decis Making : DOI 117/s No-arbitrage theorem for multi-factor uncertain stock model with floating interest rate Xiaoyu Ji 1 Hua Ke 2 Published online: 17 May 216 Springer Science+Business Media New York 216 Abstract In the stock models, the prices of the stocks are usually described via some differential equations So far, uncertain stock model with constant interest rate has been proposed, a sufficient necessary condition for it being no-arbitrage has also been derived This paper considers the multiple risks in the interest rate market stock market, proposes a multi-factor uncertain stock model with floating interest rate A no-arbitrage theorem is derived in the form of determinants, presenting a sufficient necessary condition for the new stock model being no-arbitrage In addition, a strategy for the arbitrage is provided when the condition is not satisfied Keywords Finance Stock model No-arbitrage principle Uncertain differential equation 1 Introduction In stochastic finance, the stock price is usually assumed to follow a stochastic differential equation In 19 s, Bachelier first modeled the stock price via a Brownian motion, studied the fluctuation of the stock market After that, Samuelson 1965 modeled the stock price via a geometric Brownian motion Considering the drift of the stock price the present value of the money, Black Scholes 1973 modified Samuelson s model, derived an option pricing formula Galai Masulis 1976 combined the capital asset pricing model the option pricing model, studied the systematic risk of the derivation of the stocks Recently, Guerard et al 215 B Hua Ke hke@tongjieducn 1 School of Business, Renmin University of China, Beijing 1872, China 2 School of Economics Management, Tongji University, Shanghai 292, China

2 222 X Ji, H Ke proposed a model to forecast the stock price by using the moments of the historical data, constructed an efficient portfolio Despite its widely application, the stochastic finance has been challenged in various aspects Wu 24 pointed out that the parameters such as the diffusion of the stock price behaves fuzzily owing to the fluctuation of financial market from time to time From this aspect, Wu 25, 27 derived the fuzzy Black-Scholes formula via the fuzzy set theory Thavaneswaran et al 213 studied the fuzzy option pricing problems by using binary tree model In addition, Yoshida 23 considered the romness fuzziness of the financial derivative instruments, modeled their prices via fuzzy stochastic processes Recently, Liu 213 pointed out that the Brownian motion is incompetent to describe the stock price because the lengths of its sample paths are infinite, which means the stock price changes at an infinite speed From this aspect, Liu 29 designed a canonical Liu process as a counterpart of the Brownian motion in the framework of uncertainty theory Liu 27, 21, modeled the stock price via a geometric Liu process Following that, Chen 211 derived the American option pricing formulas of Liu s stock model, Yao 215 derived a sufficient necessary condition for the uncertain stock market being no-arbitrage Inspired by the uncertain finance theory, Peng Yao 211 modeled the stock price via a mean-reverting uncertain process, Yu 212 modeled the stock price via a jump-diffusion uncertain process In addition, Chen et al 213 studied the uncertain stock model with dividends, Ji Zhou 215 studied the option pricing formulas of the uncertain stock model with jumps Except for the stock price, the interest rate in the open market also involves uncertain factors fluctuates over time Chen Gao 213 first modeled the interest rate via an uncertain differential equation Then the zero-coupon bond the interest rate ceiling instrument of the uncertain interest rate model were investigated by Jiao Yao 215, Zhu 215, Zhang et al 215, respectively Recently, Yao 215 assumed both the interest rate the stock price follow uncertain differential equations, derived some option pricing formulas In this paper, we generalize Yao s stock model to the case with multiple types of stocks, give a sufficient necessary condition for the generalized stock model being no-arbitrage In addition, a strategy for the arbitrage is provided when the condition is not satisfied The rest of this paper is organized as follows The next section introduces some concepts about uncertain variable uncertain differential equation Then Sect 3 proposes a stock model with multiple stocks, calculates the investor s wealth for a given portfolio Following that, a sufficient necessary condition is derived in the form of determinants for the stock model being no-arbitrage in Sect 4, the effectiveness of the condition is illustrated in Sect 5 via some examples Finally, some remarks are made in Sect 6 2 Preliminary In this section, we introduce some useful definitions about uncertain variables uncertain differential equations

3 No-arbitrage theorem for multi-factor uncertain stock model 223 Definition 1 Liu 27 LetL be a σ -algebra on a nonempty set Ɣ A set function M : L [, 1] is called an uncertain measure if it satisfies the following axioms: Axiom 1: Normality Axiom M{Ɣ} =1 for the universal set Ɣ Axiom 2: Duality Axiom M{ }+M{ c }=1for any event Axiom 3: Subadditivity Axiom For every countable sequence of events 1, 2,, we have { } M i M { i } Axiom 4: Product Axiom Let Ɣ k, L k, M k be uncertainty spaces for k = 1, 2, Then the product uncertain measure M is an uncertain measure satisfying { } M k = M k { k } k=1 where k are arbitrarily chosen events from L k for k = 1, 2,, respectively Uncertain variable is used to model a quantity with human uncertainty Essentially, it is a measurable function from an uncertainty space to the set of real numbers The uncertain variables ξ 1,ξ 2,,ξ n are said to be independent if { n } M ξ i B i = k=1 n M {ξ i B i } for any Borel set B 1, B 2,,B n of real numbers The uncertainty distribution of an uncertain variable ξ is defined by x = M{ξ x} for any x R An uncertain variable ξ is called normal if it has an uncertainty distribution x = πe x exp, x R 3σ denoted by Ne,σwhere e is the expected value σ 2 is the variance An uncertain process is a sequence of uncertain variables indexed by time As a special type of uncertain processes, canonical Liu processes are frequently used to model the dynamic continuous uncertain systems Definition 2 Liu 29 An uncertain process C t is called a canonical Liu process if i C = almost all sample paths are Lipschitz continuous, ii C t has stationary independent increments, iii every increment C s+t C s is a normally distributed uncertain variable with an uncertainty distribution t x = 1 + exp π x 1, 3t x R

4 224 X Ji, H Ke Definition 3 Liu 29 LetX t be an uncertain process 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 is an uncertain variable The Liu integral of an integrable function f s is a normal uncertain variable denoted by f sdc s N, f s ds Definition 4 Liu 28 Suppose C t is a canonical Liu process, f g are two given functions Then dx t = f t, X t dt + gt, X t dc t is called an uncertain differential equation The uncertain differential equation dx t = μx t dt + σ X t dc t has a solution X t = X exp μt + σ C t which is called a geometric Liu process 3 Multi-factor stock model with floating interest rate In the stock markets, there are multiple uncertain factors which influence the interest rate the stock price In this section, we assume these uncertain factors follow the geometric Liu processes, use the uncertain differential equations to describe both the interest rate the stock price Based on this assumption, a multi-factor uncertain stock model with floating interest rate could be formulated as below, dx t = ex t dt + n σ j X t dc jt dy it = e i Y it dt + n 1 σ ij Y it dc jt, i = 1, 2,,m Here, the uncertain process X t denotes the interest rate, e is the interest rate drift coefficient, σ j s are the interest rate diffusion coefficients, the uncertain processes

5 No-arbitrage theorem for multi-factor uncertain stock model 225 Y it s denote the stock prices, e i s are the stock drift coefficients, σ ij s are the stock diffusion coefficients, C jt s are independent canonical Liu processes, i = 1, 2,,m, j = 1, 2,,n There is one interest rate instrument m types of stocks in the stock model 1, so we have the choice of m + 1 different investment channels At each instant t, we may choose a portfolio β t,β 1t,,β mt ie, the investment fractions satisify β t + β 1t + +β mt = 1, then the increment of the wealth Z t could be represented by Z t = β t Z t X t According to the stock model 1, we have X t + β 1t Z t Y 1t + + β mt Z t Y mt Y 1t Y mt X t = ex t t + σ j X t C jt Y it = e i Y it t + σ ij Y it C jt, i = 1, 2,,m, so Z t = β t Z t ex t t + X t σ j X t C jt + = eβ t Z t + e i β it Z t t + β it Z t Y it e i Y it t + σ ij Y it C jt σ j β t Z t + σ ij β it Z t C jt Letting t, we get that the wealth Z t follows an uncertain differential equation dz t = eβ t Z t + e i β it Z t dt + whose solution is Z t = Z exp eβ s + e i β is ds + σ j β t Z t + σ ij β it Z t dc jt 2 σ j β s + σ ij β is dc js

6 226 X Ji, H Ke 4 No-arbitrage determinant theorem In the stock model 1, the uncertain process X t denotes the interest rate at the instant t, so the discount rate is X Xt 1 the present value of the total wealth Z t is X Xt 1 Z t Arbitrage means a possibility of a risk-free profit, in mathematical notation, there exists a portfolio β s,β 1s,,β ms such that for some time t, both M{X X 1 t Z t Z }=1 3 M{X X 1 t Z t > Z } > 4 hold Equality 3 implies that there is no chance that the investor will suffer a loss, Inequality 4 implies that there is some chance that the investor will earn a profit If there is not such a portfolio, then we say the stock model is no-arbitrage The following theorem provides a sufficient necessary condition for the stock model 1 being no-arbitrage Theorem 1 The stock model 1 is no-arbitrage if only if the system of linear equations has a solution Proof Since σ 11 σ 1 σ 12 σ 2 σ 1n σ n x 1 σ 21 σ 1 σ 22 σ 2 σ 2n σ n x 2 = σ m1 σ 1 σ m2 σ 2 σ mn σ n x n Z t = Z exp eβ s + e i β is ds + e 1 e e 2 e e m e σ j β s + σ ij β is dc js 5 X t = X exp et + σ j C jt, we have X X 1 t Z t = Z exp e i eβ is ds + σ ij σ j β is dc js

7 No-arbitrage theorem for multi-factor uncertain stock model 227 Then ln X Xt 1 Z t ln Z = e i eβ is ds + is a normal uncertain variable with an expected value e i eβ is ds σ ij σ j β is dc js 6 a variance 2 σ ij σ j β is ds Assume that the system 5 of linear equations has a solution The argument breaks into two cases Case 1: Given any time t, if there exists a portfolio β s,β 1s,,β ms such that t σ ij σ j β is ds =, then σ ij σ j β is =, j = 1, 2,,n, s, t Let x 1, x 2,,x n denote the solution of the system 5 of linear equations, ie, e i e = σ ij σ j x j, j = 1, 2,,m Then for any time s, t, wehave e i eβ is = σ ij σ j x j β is = m σ ij σ j β is x j = x j =

8 228 X Ji, H Ke which implies As a result, we have ie, e i eβ is ds = lnx X 1 t Z t = lnz, M{X X 1 t Z t > Z }= So the stock model 1 is no-arbitrage in the first case Case 2: Given any time t, for any portfolio β s,β 1s,,β ms, if t σ ij σ j β is ds =, then lnx Xt 1 Z t lnz is a normal uncertain variable with a positive variance As a result, we have ie, M{lnX X 1 t Z t lnz } < 1, M{X X 1 t Z t Z } < 1 So the stock model 1 is also no-arbitrage in the second case Conversely, assume that the system 5 of linear equations has no solution Then there exist real numbers β 1,,β m such that σ 11 σ 1 β 1 + σ 21 σ 1 β 2 + +σ m1 σ 1 β m = σ 1n σ n β 1 + σ 2n σ n β 2 + +σ mn σ n β m = e 1 eβ 1 + e 2 eβ 2 + +e n eβ n > according to the Farkas Lemma Take then we have β is = β i, i = 1, 2,,m, s, e i eβ is ds >

9 No-arbitrage theorem for multi-factor uncertain stock model 229 σ ij σ j β is ds = for any time t >, which imply in Eq 6, ie, As a result, both Equality 3 Inequality 4 lnx X 1 t Z t lnz > M{X X 1 t Z t > Z }=1 M{X X 1 t Z t Z }=1 M{X X 1 t Z t > Z } > hold Then β s,β 1s,,β ms = β,β 1,,β m is a portfolio for an arbitrage of the stock model 1, where β s = β = 1 β i, s The proof is complete Remark 1 If σ 1 = σ 2 = =σ n = in the stock model 1, ie, the interest rate is a constant, then Theorem 1 degenerates to the case that the stock market is no-arbitrage if only if the system of linear equations σ 11 σ 12 σ 1n σ 21 σ 22 σ 2n σ m1 σ m2 σ mn x 1 x 2 x n = has a solution This result was first presented in Yao 215 e 1 e e 2 e e m e

10 23 X Ji, H Ke Corollary 1 The stock model 1 is no-arbitrage if the m n m n matrix σ 11 σ 1 σ 12 σ 2 σ 1n σ n σ 21 σ 1 σ 22 σ 2 σ 2n σ n σ m1 σ 1 σ m2 σ 2 cdots σ mn σ n 7 has a rank m Proof Since the matrix 7 has a rank m, the system of linear equations 5 has a solution whatever values the vector e 1 e, e 2 e,,e m e T takes Then it follows from Theorem 1 that the stock model 1 is no-arbitrage Corollary 2 The stock model 1 is no-arbitrage if the drift coefficients of its interest rate stock prices satisfy e i = e, i = 1, 2,,m Proof Since e 1 e, e 2 e,,e m e T =,,, T, the system of linear equations 5 always has a solution x 1, x 2,,x n T =,,, T Then it follows from Theorem 1 that the stock model 1 is no-arbitrage Theorem 2 The stock model 1 has an arbitrage if only if the system σ 11 σ 1 β 1 + σ 21 σ 1 β 2 + +σ m1 σ 1 β m = σ 1n σ n β 1 + σ 2n σ n β 2 + +σ mn σ n β m = e 1 eβ 1 + e 2 eβ 2 + +e n eβ n > has a solution β 1,β 2,,β m Moreover, take β = 1 β i, then the portfolio β,β 1,,β m is a strategy for arbitrage Proof According to the Farkas Lemma, the system 8 has a solution if only if the system 5 has no solution Then it follows from Theorem 1 that the stock model 1 has an arbitrage if only if the system 8 has a solution In addition, substitute the equations inequality in the system 8 into Eq 6, we have ln X Xt 1 Z t ln Z = = e i eβ is ds + e i eβ is ds > 8 σ ij σ j β is dc js

11 No-arbitrage theorem for multi-factor uncertain stock model 231 Then M{X X 1 t Z t Z }=1 M{X X 1 t Z t > Z }=1 > Hence, the portfolio β,β 1,,β m is a strategy for arbitrage The proof is complete 5 Some examples In this section, we show the effectiveness of the above theorems via some examples Example 1 Consider the uncertain stock model dx t = X t dt + X t dc 1t + X 2t dc 2t dy 1t = 3Y 1t dt + Y 1t dc 1t + 2Y 1t dc 2t dy 2t = 2Y 2t dt + 2Y 2t dc 1t + 3Y 2t dc 2t Note that σ11 σ 1 σ 12 σ 2 = σ 21 σ 1 σ 22 σ 2 e1 e = e 2 e = = Since the system of linear equations 1 x1 2 = 12 x 2 1 has a solution x1 = x 2 3, 2 the uncertain stock model is no-arbitrage Example 2 Consider the uncertain stock model dx t = X t dt + X t dc 1t + 2X t dc 2t + X t dc 3t 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

12 232 X Ji, H Ke Note that σ 11 σ 1 σ 12 σ 2 σ 13 σ σ 21 σ 1 σ 22 σ 2 σ 23 σ 3 = = σ 31 σ 1 σ 32 σ 2 σ 33 σ e 1 e e 2 e = 3 1 = 2 e 3 e Since the system of linear equations x 1 x 2 x 3 1 = 2 3 has a solution x 1 x 2 x 3 1/2 = 1/2, the uncertain stock model is no-arbitrage Example 3 Consider the uncertain stock model dx t = X t dt + X t dc 1t + X t dc 2t + X t dc 3t dy 1t = 1Y 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 Note that σ 11 σ 1 σ 12 σ 2 σ 13 σ σ 21 σ 1 σ 22 σ 2 σ 23 σ 3 = = σ 31 σ 1 σ 32 σ 2 σ 33 σ e 1 e 1 1 e 2 e = 3 1 = 2 e 3 e 4 1 3

13 No-arbitrage theorem for multi-factor uncertain stock model 233 Since the system of linear equations x 1 x 2 x 3 = 2 3 has no solution, an arbitrage is available according to Theorem 1 Then it follows from Theorem 2 that the solution of the system β 1 + 1β 2 + 2β 3 = 1β 1 + 2β 2 + 3β 3 = 2β 1 + 3β 2 + 4β 3 = 9 β + β 1 + β 2 + β 3 = 1 β 1 + 2β 2 + 3β 3 >, provides a strategy for arbitrage Solving the system 9, we get β 1 β 1 β 2 = β 3 Theoretically, an arbitrage profit could be secured in this case by putting the wealth Z t into the bank, selling the first third stocks for Z t, respectively, buying the second stock with the 2Z t at each instant t 6 Conclusions This paper considered the multiple risks in the stock market the interest rate market, proposed a multi-factor uncertain stock model with floating interest rate Then it derived a sufficient necessary condition for the stock model being no-arbitrage in the form of determinants In addition, this paper provided a strategy for the arbitrage profit when it is available Acknowledgements This work was supported by the National Natural Science Foundation of China Grant Nos the Fundamental Research Funds for the Central Universities References Black, F, & Scholes, M 1973 The pricing of options corporate liabilities Journal of Political Economy, 81, Chen, X 211 American option pricing formula for uncertain financial market International Journal of Operations Research, 82, Chen, X, Liu, Y, & Ralescu, D A 213 Uncertain stock model with periodic dividends Fuzzy Optimization Decision Making, 121, 111 Chen, X, & Gao, J 213 Uncertain term structure model of interest rate Soft Computing, 174,

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