Computation of Arbitrage in a Financial Market with Various Types of Frictions

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1 Computation of Arbitrage in a Financial Market with Various Types of Frictions Mao-cheng Cai 1, Xiaotie Deng 2, and Zhongfei Li 3, 1 Institute of Systems Science, Chinese Academy of Sciences, Beijing , China caimc@iss.ac.cn 2 Department of Computer Science, City University of Hong Kong, Hong Kong csdeng@cityu.edu.hk 3 Lingnan (University) College, Sun Yat-Sen University, Guangzhou , China lnslzf@zsu.edu.cn Abstract. In this paper we study the computational problem of arbitrage in a frictional market with a finite number of bonds and finite and discrete times to maturity. Types of frictions under consideration include fixed and proportional transaction costs, bid-ask spreads, taxes, and upper bounds on the number of units for transaction. We obtain some negative result on computational difficulty in general for arbitrage under those frictions: It is NP-complete to identify whether there exists a cash-and-carry arbitrage transaction and it is NP-hard to find an optimal cash-and-carry arbitrage transaction. 1 Introduction No-arbitrage is a generally accepted condition in finance. In general, if there is any arbitrage opportunity, the market force would act as an invisible hand to drive the prices change and bring the market back to equilibria. An underlying assumption behind the general principle is the existence of active profit seeking agents in the financial market. Their restless effort in locating arbitrage possibilities is essential for the no-arbitrage condition to hold. For the above argument to work, it is essential that locating arbitrage possibilities is not a formidable task, computationally. For frictionless financial markets, the no-arbitrage condition is very well understood. See, for example, Ross (1978), Harrison and Kreps (1979), Green and Srivastava (1985), and Spremann (1986). In reality, however, financial markets are never short of friction. Investors are required to pay transaction costs, commissions and taxes. Selling and buying prices are differentiated with ask-bid spread. A security is available at a price only for up to a maximum amount. One may buy or sell a stock at an integer Correspondence author. N. Megiddo, Y. Xu, and B. Zhu (Eds.): AAIM 2005, LNCS 3521, pp , c Springer-Verlag Berlin Heidelberg 2005

2 Computation of Arbitrage in a Financial Market 271 number of shares (or an integer number of hundreds of shares). Friction is a de facto matter in financial markets. Study of arbitrage in frictional markets has attracted more and more attention in recent years. Garman and Ohlson (1981) extended the work of Ross (1978) to markets with proportional transaction costs. Later, Prisman (1986) studied the valuation of risky assets in arbitrage-free economies with taxation. Dermody and Prisman (1993) extended the results of Garman and Ohlson (1981) to markets with increasing marginal transaction costs. Jouini and Kallal (1995) investigated, by means of martingale method, the no-arbitrage problem under transaction costs. Ardalan (1999) showed that, in financial markets with transaction costs and heterogeneous information, the no-arbitrage imposes a constraint on the bid-ask spread. Deng, Li and Wang (2000, 2002) presented necessary and sufficient conditions for no-arbitrage in a finite-asset and finite-state financial market with proportional transaction costs. These results allows ones to use polynomial time algorithms to look for arbitrage opportunities by applying linear programming techniques. These works weer generalized to the case of multiperiod by Zhang, Xu and Deng (2002). Kabanov, Rásonyi and Stricker (2001) pointed out that, although the literature on models with friction is rapidly growing, arbitrage theory for markets with frictions still contains a number of questions with much less satisfactory answers than in the theory of frictionless markets and there are only a few papers dealing with necessary and sufficient conditions for the absence of arbitrage for markets with frictions. In addition, to the best of our knowledge, works on algorithmic study of arbitrage under friction are rare, although it is a central problem for discrete finite time models in finance. To capture the current price structure, to find out whether there is an arbitrage opportunity, and to price arbitrary cash stream, the study of algorithmic issues of arbitrage with realistic frictions is important, interesting and challenging. In the present paper we study computational issues of arbitrage with fixed and proportional transaction costs, bid-ask spreads, taxes, and upper bounds on transaction. The fixed transaction costs capture the situation in which an individual investor requests a broker to invest money on the securities exchange, paying a fixed sum for the service. The payment includes for example brokerage fees, fixed investment taxes to access to a market, operational and trade processing costs, information obtaining costs, or opportunity costs of looking at a market or of doing a specific trade, which are independent of the amount invested in each security. The proportional transaction costs are, as most usual, the fees that are proportional to the transaction size of each security. The bid-ask spreads are the difference between bid and ask prices of an individual security. The (income) taxes at every time to maturities are also set to be proportional to the transaction size of each security. 2 Notation and Definitions Consider a market of n fixed income securities (or bonds) i =1, 2,...,n.Let 0=t 0 <t 1 <t 2 <...<t m be all the payment dates (or the times to maturities)

3 272 M.-c. Cai, X. Deng, and Z. Li that can occur, which need not be equidistant. A cash stream is a vector w = (w 1,w 2,...,w m ) T, where T denotes the transposition of vector or matrix, and w j is the income received at time t j and may be positive, zero or negative. Assume that bond i pays the before-tax cash stream A i =(a 1i,a 2i,...,a mi ) T.Sowe have the m n payoff matrix A =(A 1,A 2,...,A n ). Bond i can be purchased at a current price p a i, the so-called ask price. There is also a bid price p b i at which bond i can be sold. The difference between these two prices, the so-called bid-ask spread, reflects a type of friction. This friction exists in most economic markets. We form the ask price vector p a =(p a 1,p a 2,...,p a n) T and the bid price vector p b =(p b 1,p b 2,...,p b n) T. The second type of friction considered in this paper is transaction costs including fixed and proportional. We assume that the fixed transaction cost is c i if bond i is traded and that no fixed transaction cost occurs if no trading of bond i. Thec i is a positive constant regardless of the amount of bond i traded. Denote c =(c 1,c 2,...,c n ) T the fixed transaction cost vector. Besides the fixed transaction cost, there is additional transaction cost that is proportional to the amount of the bond traded. Let λ a i and λ b i be such fees if one dollar of bond i is bought and sold respectively. Here 0 λ a i,λb i < 1,i =1, 2,...,n. Denote λ a =(λ a 1,λ a 2,...,λ a n) T and λ b =(λ b 1,λ b 2,...,λ b n) T. The third type of friction incorporated into our model is taxes. Here we concentrate only on a single investor as a member of just one tax class among many. For all investors in this class, the tax amount at time t j for holding one unit of bond i in long position is assumed to be t a ji, and the after-tax income at that time is then a ji t a ji ; whereas the tax amount for holding one unit of bond i in short position is t b ji as a credit against the obligation to pay a ji at time t j, and the net after-tax payment to be made is then a ji t b ji.letta be the m n matrix whose entries are t a ji,andtb the m n matrix whose entries are t b ji. Every investor in the fixed tax class under consideration will modify his or her position. Let the modification be x =(x 1,x 2,...,x n ) T R n, called also a portfolio, where x i is the number of units of bond i modified by the investor. If x i > 0, additional bond i is bought for the amount of x i ;andifx i < 0, additional bond i is sold for the amount of x i. Finally, the fourth type of friction considered in our model is bounds. An upper bound b + i > 0 (maximum amount of units that can be bought in bond i) and an upper bound b i > 0 (maximum amount of units that can be sold in bond i) are set on the modified amount x i for each bond i. Denote b + =(b + 1,b+ 2,...,b+ n ) T and b =(b 1,b 2,...,b n ) T.If b i x i b + i for i =1, 2,...,n, we call x a admissible portfolio. Now, the bond market considered in this paper can be described by the 10- tuple M = {p a,p b,λ a,λ b,b +,b,c,a,t a,t b }. For convenience, we use the vector notation x y to indicate that x i y i for all i. Denote, for i =1, 2,...,n and j =1, 2,...,m, τ i (x) = { { (1 + λ a i )pa i x if x>0, (a ji t a ji )x if x>0, (1 λ b i )pb i x if x 0, g ji (x) = (a ji t b ji )x if x 0,

4 Computation of Arbitrage in a Financial Market 273 and δ(x) =1ifx 0or0ifx = 0. If trading a portfolio x =(x 1,x 2,...,x n ) T, the investor pays the cost f(x) := n τ i(x i )+ n c iδ(x i ) in the present and receive the after-tax gain g j (x) := n g ji(x i ) at future time t j for j = 1, 2,...,m. The after-tax cash stream of gains generated by the portfolio x is then the vector G(x) :=(g 1 (x),g 2 (x),...,g m (x)) T. Definition 1. An after-tax cash stream w = (w 1,w 2,...,w m ) T is called no future obligations if k j=1 w j 0,k =1, 2,...,m, or, in matrix notation, if Bw 0, whereb is the lower-triangular m m-matrix whose diagonal and lower-triangular elements all are ones. Definition 2. A portfolio x is said to be a cash-and-carry arbitrage transaction if it is admissible (i.e., b x b + ) and if it has a negative payment (i.e., f(x) < 0) and generates an after-tax cash stream that implies no future obligations (i.e., BG(x) 0). Definition 3. The market M is said to exhibit weak no-arbitrage if there exists no cash-and-carry arbitrage transaction. 3 Characterizations of No-Arbitrage Theorem 1. The market M exhibits weak no-arbitrage if and only if the optimal value of the following nonlinear programming problem is zero: (P 1) { minimize f(x) subject to BG(x) 0, b x b +. Proof. Sufficiency. Assume that the optimal value of (P 1) is zero. Then, for any admissible portfolio x with BG(x) 0, x is feasible to (P 1) and hence f(x) 0. Thus, there exists no admissible portfolio x such that f(x) < 0andBG(x) 0. Therefore, the market M exhibits weak no-arbitrage. Necessity. Assume that the market M exhibits weak no-arbitrage. Then, for any x with BG(x) 0and b x b +, it must holds that f(x) 0 otherwise a cash-and-carry arbitrage transaction occurs. This means that the objective function of (P 1) is nonnegative at any feasible solution. On the other hand, it is clear that x = 0 is feasible to (P 1) and the objective function vanishes at x = 0. Hence, the optimal value of (P 1) is zero. Problem (P 1) states the lowest total cost or gain induced by trading a portfolio that generates a cash stream with no future obligation. Theorem 1 means that this lowest amount is zero if there exists a consistent term structure. Now we reformulate the model set up in the previous section. For any portfolio x =(x 0,x 1,...,x n ) T,letx + i = max{x i, 0} be the number of units of bond i bought and x i = min{x i, 0} the number of units of bond i sold. Denote x + =

5 274 M.-c. Cai, X. Deng, and Z. Li (x + 1,x+ 2,...,x+ n ) T, x =(x 1,x 2,...,x n ) T, p + = ((1 + λ a 1)p a 1,...,(1 + λ a n)p a n), and p = ((1 λ b 1)p b 1,...,(1 λ b n)p b n). Then, x i = x + i x i, x+ i x i =0, 0 x ± i b ± i, i =1, 2,...,n, f(x) = (1 + λ a i )p a i x + i (1 λ b i)p b ix i + c i δ(x + i x i ), g j (x) = Further we have (a ji t a ji)x + i f(x) =p + x + p x + (a ji t b ji)x i,j=1, 2,...,m. c i δ(x + i x i ) and G(x) =(A T a )x + (A T b )x. Hence, problem (P 1) can be equivalently formulated as the problem minimize p + x + p x + c i δ(x + i x i ) (P 2) subject to B[(A T a )x + (A T b )x ] 0 x + i x i =0, 0 x ± i b ± i,, 2,...,n. Theorem 2. The market M exhibits weak no-arbitrage if and only if the optimal value of problem (P 2) is zero. Thus, to identify whether the market exhibits weak no-arbitrage we need only to solve problem (P 2). Clearly, a cash-and-carry arbitrage transaction is a solution (x +,x )ofthe system p + x + p x + c i δ(x + i x i ) < 0 (S) B[(A T a )x + (A T b )x ] 0 x + i x i =0,i=0, 1,...,n 0 x ± b ±. The negative of optimal value of (P 2) can be interpreted as the maximal arbitrage profit. The optimal solutions of (P 2) with nonzero objective value are called optimal cash-and-carry arbitrage transactions. 4 Computational Complexity of Arbitrage In this section, we will discuss the computational complexity of finding an optimal cash-and-carry arbitrage transaction and of identifying whether there exists

6 Computation of Arbitrage in a Financial Market 275 a cash-and-carry arbitrage transaction. The technique which we use to reach this purpose is a polynomial time transformation of the EXACT COVER BY 3-SETS into an instance of the problem (P 2). The EXACT COVER BY 3-SETS (Garey and Johnson (1979)) is as follows: Given an arbitrary instance I of EXACT COVER BY 3-SETS with a ground set S = {s 1,,s 3h } and a collection C = {C 1,,C k } of 3-element subsets of S, doesc contain an exact cover for S, that is, a subcollection C C such that every element of S occurs in exactly one member of C? First we construct a digraph G =(V,E) from the instance I as follows: V = {w} {u 1,...,u 3h } {v 1,...,v k }, E = {(w, u i ),...,(w, u 3h )} k 3h ({(u i,v j ) s i C j } {(v 1,w),...,(v k,w)}. j=1 In this digraph, element s i corresponds to vertex u i, and subset C j corresponds to vertex v j. Further, there is an arc (u i,v j ) if and only if s i C j. Clearly, the indegrees d (u i )=1,d (v j )=3andd (w) =k; the outdegrees d + (u i )= {s i C j C}, d + (v j )=1andd + (w) =3h. The numbers of vertices and arcs of G are V =3h + k +1and E =3h +4k. Let D denote the incidence matrix of G, that is, the matrix with rows and columns indexed by V and E, respectively, where the entry in position (v, e) is 1, +1, or 0, if v is the head of e, the tail of e, or neither, respectively. Further, we assume that the first 3h columns of D is indexed by arcs (w, u 1 ),...,(w, u 3h ). To simplify expressions, we write B(A T a )=R + =(r + ji ), B(A T b )=R =(r ji ). Theorem 3. It is NP-hard to find an optimal cash-and-carry arbitrage transaction even if R ± are (0, ±1)-matrices, c 1 = = c n =1, and there is no constraint x + i x i =0, i =1, 2,...,n. Proof. Let us construct a reduction from instance I of EXACT COVER BY 3-SETS to the problem (P 2). For this purpose, set m =18h +10k +4 and n =3h +4k + 1. We compose m n-matrices R + and R as follows: R + = I 2, R = I 2 I 1 I D 0 2 D where D is the incidence matrix of G; I 1 and I 2 are the identity matrices of orders 3h and n; 0 1,0 3 and 0 4 are all-zero 3h 4k-, n n- and (12h +2k +2) n- matrices, respectively; 1 and 0 2 are the all-one and all-zero column vectors of dimensions 3h and 3h + k + 1, respectively.

7 276 M.-c. Cai, X. Deng, and Z. Li Further put c = p + =(1,, 1), p =(0,, 0, 7h + 2), and { b ± 3 if e =(vj,w), j=1,...,k, e = 1 otherwise. It is easy to see that the construction above can be accomplished in polynomial time. Then for the specified R ±,p ± and c, it is straightforward to check that problem (P 2) becomes ( ˆP 2): minimize δ(x i ) (7h +2)x n subject to x + = 0 (1) e δ + (v) x e x e x n =0 e δ + (w) (2) x e =0 v V (3) e δ (v) b x 0 (4) where δ + (v) ={(v, u) E} and δ (v) ={(u, v) E}. Clearly, (1) yields (x + ) T x =0and{x e : e E} is a circulation in G by (3) and (4). Further we have Claim. If x 0 satisfies (2) (4), then x n > 0, (5) δ(x i ) 7h +1. (6) Indeed, assume (5) to be false, then x e = 0 for all e δ + (w) by (2). It follows from (3) that x e = 0 for all e δ + (u i ),,...,3h, implying x e = 0 for all e δ + (v j ),j=1,...,k. Hence x = 0, a contradiction. To show (6), x e > 0 for all e δ + (w) by(2)and(5),thatis,x (w, u > 0 i) for i =1,...,3h. It follows from (3) that for each u i there is at least one arc e δ + (u i ) with x e > 0. As d (v j )=3,j =1,...,k, it derives from (3) that there are at least h vertices v j with x (v j,w) > 0. Therefore (6) holds. Claim. There is x 0 satisfying (2) (4) and δ(x i )=7h + 1 (7) if and only if the instance I of EXACT COVER BY 3-SETS has an exact cover C of S. First suppose x 0 satisfies (2) (4) and (7). Then it follows easily from the proof of (6) that

8 Computation of Arbitrage in a Financial Market 277 there is exactly one arc e δ + (u i ) with x e > 0foreachi =1,...,3h and there are exactly h vertices v jl, l =1,...,h, with x (v jl,w) > 0. For otherwise (7) cannot hold. Set C = {C jl C : x (v jl,w) > 0}. Then C is an exact cover of S. Indeed, each s i is in some C jl C as x e > 0 for some e δ + (u i )andc jp C jq = for all 1 p<q h since S =3h, C = h and {C jl C } = S. Conversely, suppose that there exists an exact cover C = {C j1,...,c jh } C of S. We need to find an n-vector x satisfying (2) (4) and (7). Now set x n =1, 1 if e =(w, u i ),,...,3h, x 1 if e =(u e = i,v jl )andu i C jl C, 3 if e =(v jl,w)andc jl C, 0 otherwise. It is straightforward to verify that the defined x satisfies (2) (4) and (7). Claim. The optimal value of Problem ( ˆP 2) is either 1 or0.moreover,the optimal value is 1 if and only if the instance I of EXACT COVER BY 3- SETS has an exact cover of S. Indeed, as x = 0 is a feasible solution to ( ˆP 2), the optimal value { n } minimize δ(x i ) (7h +2)x n 0. (8) If ( ˆP 2) has a optimal solution ˆx =(0, ˆx ) with ˆx 0, then by Claims 1 and 2, n δ(ˆx i )=7h + 1 if and only if the instance I has an exact cover of S, and n δ(ˆx i ) 7h + 2 otherwise as n δ(ˆx i ) is integer. Furthermore, ˆx n = 1 follows easily from the proof of Claim 2 and the optimality of ˆx. Therefore n δ(ˆx i ) (7h +2)ˆx n = 1 if and only if the instance I has an exact cover of S, and n δ(ˆx i ) (7h +2)ˆx n 0 otherwise, implying n δ(ˆx i ) (7h + 2)ˆx n = 0 by (8). So the claim is true. Now we come to the conclusion that the optimal value of Problem ( ˆP 2) is either 1 or 0 according to whether the instance I of EXACT COVER BY 3-SETS has an exact cover of S or not. To complete the proof, we have to show Claim. For the composed matrices R + and R, there exist matrices A, T a and T b satisfying B(A T a )=R + and B(A T b )=R. Clearly, the following m n linear systems { aji t a ji = m+ ji a ji t b ji =, i =1, 2,...,,n, j =1, 2,...,m m ji have feasible solutions, where a ji, t a ji and tb ji are variables, and ( m ± ji) = B 1 R ±. The proof is completed.

9 278 M.-c. Cai, X. Deng, and Z. Li Theorem 4. It is NP-complete to identify whether there exists a cash-and-carry arbitrage transaction in the market M. Proof. Equivalently we need only to show that it is NP-complete to determine feasibility of system (S). Clearly, the problem is in NP. We transform EXACT COVER BY 3-SETS to the identification problem by the same reduction used in the proof Theorem 3. To prove the theorem, it suffices to show Claim. There exists a cash-and-carry arbitrage transaction, that is, there is x satisfying (2) (4) with n δ(x i ) (7h +2)x n < 0, if and only if the instance I of EXACT COVER BY 3-SETS has an exact cover C of S. Clearly, the claim is a corollary of Claim 3. The theorem is proved. Note that m>nin the proofs of Theorems 3 and 4. Let us show the theorems to be still true for the case m n. Indeed, let R and R be m (n n)-matrices whose entries are non-negative, ˇp + and ˇp be the all-zero column vectors of dimension n,č be the all-one column n n {}}{ vector of dimension n,andˇb ± =(b ±, 0,...,0). Set ˇB(Ǎ Ť a )=(R +, R ), ˇB( Ǎ Ť b )=(R,R ), where R + and R are the matrices defined in the proof of Theorem 3. Consider the following programming: minimize ˇp +ˇx + ˇp ˇx n + č i δ(ˇx + i ˇx i ) ( ˇP 2) subject to ˇB( Ǎ Ť a )ˇx + ˇB(Ǎ Ť b )ˇx 0 ˇx + i ˇx i =0, ˇb ± i ˇx ± i 0,, 2,...,n. It is easy to see that the optimal values of ( ˇP 2) and (P 2) are equal. Furthermore, for any optimal solution (ˇx +, ˇx )of(ˇp 2), clearly ˇx + j =ˇx j =0forj = n + 1,...,n, and (x +,x )=(ˇx + 1, ˇx+ 2,...,ˇx+ n, ˇx 1, ˇx 2,...,ˇx n ) is an optimal solution of (P 2). Conversely, for any optimal solution (x +,x )of(p 2), then n n n n {}}{{}}{ (x +, 0,...,0,x, 0,...,0) is an optimal solution of ( ˇP 2). As (P 2) is NP-hard for m>n,sois(ˇp 2). Theorems 3 and 4 tell us that it is unlikely to find efficient optimal solution procedures and that one has to look for heuristic algorithms for problem (P 2).

10 Computation of Arbitrage in a Financial Market Conclusion In this paper, we have derived two necessary and sufficient conditions for the weak no-arbitrage in markets with fixed and proportional transaction costs, bidask spreads, taxes, and bounds for transaction. These characterizations extend some known results in discrete time frictionless security markets. With the help of the EXACT COVER BY 3-SETS, the computational complexity of the arbitrage problem is showed to be in NP. These motivate us to consider computational complexity in a more general setting of friction or/and time (period). Such extensions require more sophisticated tools and are worthy of being investigated further in future. Acknowledgements This work is partially supported by a grant from RGC of Hong Kong (CityU 1156/04E), a NSFC Major Research Program ( ), a Foundation for the Author of National Excellent Doctoral Dissertation of China (No ), and grants of the National Natural Science Foundation of China (Nos , , , ). References Ardalan, K.: The no-arbitrage condition and financial markets with transaction costs and heterogeneous information. Global Finance Journal 10 (1999) Deng, X. T., Li, Z. F., Wang S. Y.: On computation of arbitrage for markets with friction. In: Du, D. Z., et al. (eds.): Computing and Combinatorics. Lecture Notes in Computer Science, Vol Springer-Verlag, Berlin Heidelberg New York (2000) Deng, X. T., Li, Z. F., Wang, S. Y.: Computational complexity of arbitrage in frictional security market. International Journal of Foundations of Computer Science 3 (2002) Dermody, J. C., Prisman, E. Z.: No arbitrage and valuation in market with realistic transaction costs. Journal of Financial and Quantitative Analysis 28 (1993) Garey, M. R., Johnson, D. S.: Computers and Intractability: A Guide of the Theory of NP-Completeness. San Francisco, Freeman (1979) Garman, M. B., Ohlson, J. A.: Valuation of risky assets in arbitrage-free economies with transactions costs. Journal of Financial Economics 9 (1981) Green, R. C., Srivastava, S.: Risk aversion and arbitrage. The Journal of Finance 40 (1985) Harrison, J. M., Kreps, D. M.: Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20 (1979) Jouini, E., Kallal, H.: Martingales and arbitrage in securities markets with transaction costs. Journal of Economic Theory 66 (1995) Kabanov, Yu., Rásonyi, M., Stricker, Ch.: No-arbitrage criteria for financial markets with efficient friction. Université de Besançon, Preprint (2001)

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