Bilateral Exchange of Contingent Claims

Transcription

1 Bilateral Exchange of Contingent Claims - an adaptive, behavioral, stochastic approach Sjur Didrik Flåm University of Bergen Workshop, Bonn, May 2013 Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

2 Abstract exchange of contingent claims only via voluntary, incentive compatible bilateral barters fully driven by di erences in gradients (or substitutions rates) no optimization, no coordination myopic & adaptive agents Still!: convergence to equilibrium Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

3 Framing the problem Consider an exchange economy with transferable utility and stochastic endowments stochastic endowment = random bundle = contingent claim state s 2 S 7! commodity vector e(s) 2 E = Euclidean What is meant by equilibrium? Can equilibrium be reached by bilateral barters? Chief example: A reinsurance market (say, Loyds of London). Insurance policy = contingent claim x() : event s 2 S 7! indemnity x(s) 2 R Everybody is * subject to feasibility constraints * operating with reduced non-smooth objectives. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

4 A preview of the story: 2 main stages 1) The ex ante stage: Agents exchange contingent claims in face of uncertainty. * They contend with repeated bilateral barters. * These are facilitated by side payments (NB. transferable utility). * Neither optimization nor coordination is needed. * Nobody needs full knowledge of vision of the data. 2) The ex post stage: Nature draws the contingency/ state s 2 S. * Contracts are executed. * No reneging, no default. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

5 Stochastics enter in 2 ways 1) The objects exchanged are random vectors 2) Who trades when with whom - and how - is random. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

6 Some background: 3 di erent institutions 1) A Walrasian auctioneer announces prices agents report their optimal net demands aggregate net demand > 0 () increase price. This story is pure ction! 2) Market makers announce ask & bid prices agents state demand & supply Used to be important, but recently less so. 3) Order markets: agents sumit limit or market orders there is computerized bilateral matching, random timing and adaptive design of orders. These platforms are increasingly popular. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

7 The setting There are several agents, each holding his stochastic endowment = random resource bundle = contingent claim. They proceed by repeated bilateral barters. Issues: Will equilibrium obtain? (Quit likely yes!) Is there room for "simple" agents? (Yes!) Is optimization really needed? (No!) Must prices be announced? (No!) Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

8 DATA agent i 2 I owns endowment e i 2 X = L 0 (S, F, E), (for computation S or F nite) faces constraint x i 2 X i closed convex X, and wants to maximize quasi-linear concave utility u i : X! R. But he lacks computational competence, perfect foresight, information, global vision,... There is no coordination, no auctioneer, no market maker,... Moreover: bargaining, matching, search is not made explicit Nonetheless: Maybe holdings converge to equilibrium? Inspiration from Pareto: "The economy is a great computing machine." Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

9 Equilibrium (x i ) an allocation i i x i = i e i =: e I feasible allocation i moreover, x i 2 X i for each i. De nition (Equilibrium) A feasible allocation (x i ) and a linear price p : X! R constitute and equilibrium i u i (x i ) + p(e i x i ) u i (χ) + p(e i χ) for each i 2 I and χ 2 X i. In equilibrium the agent s utility + his net value of sale is maximal! Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

10 Intermezzo: the nature of equilibrium Recall that a linear x : X! R is a supergradient of the proper function f : X! R[ f g at x, written x 2 f (x), i f (χ) f (x) + x (χ x) for all χ 2 X. Also recall: a linear n : X! R is a normal vector to a proper subset X X at x 2 X i n(χ x) 0 for all χ 2 X. Proposition (On equilibrium) A pro le (x i ) alongside a price p constitutes an equilibrium i p 2 u i (x i ) N i (x i ) for each i, and x i = e. i2i that is: there is a common price price "=" marginal utility (modulo normal components) Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

11 More on the nature of equilibrium: Cooperative aspects Suppose coalition C I uses e C := i2c e i to get u C (e C ) := sup ( ) u i (x i ) : x i = e C & x i 2 X i. i2c i2c Recall that a payment scheme (π i ) is in the core of this transferable-utility game i Pareto e cient: i2i π i = u I (e I ) stable against blocking: i2c π i u C (e C ) for each C I. Proposition (Equilibrium as core solution) For any equilibrium price p the payment scheme i 7! π i := sup fu i (χ) + p(e i χ) : χ 2 X i g is in the core. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

12 A rst simple guideline for exchange: A brave approach: When agent i meets agent j, they compare gradients: Let g i = u 0 i (x i ) and g j = u 0 j (x j ) Is g i 6= g 0 j? If yes, for suitable step-size σ > 0, transfer Similarly, j gets x i := σ [g i g j ] to i from j. x j := σ [g j g i ] = x i Note, no central coordination. Fully decentralized. The process stops when all gradients are equal. In equilibrium all gradients are equal = price p. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

13 Alas, it s not that simple: what about constraints? Constraints x i 2 X i? Two alternative strategies: 1) Use exact penalties say distance functions d i (χ) := min fkχ x i k : x i 2 X i g and modi ed objectives û i (x i ) = u i (x i ) c i d i (x i ), coe cient c i > 0 su ciently large. See Flåm, Godal, Soubeyran, Optimization (2012) 2) Here: Use projection to enforce feasibility throughout. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

14 Feasible exchange bewteen two agents Actual holdings: x i 2 X i and x j 2 X j. Updated holdings: x +1 i := x i + x 2 X i and x +1 j := x j x 2 X j When x 6= 0 with no loss of generality x = σd with positive stepsize σ > 0 and unit direction d 2 X, kdk = 1. Which stepsize? Which direction? Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

15 On maintaining feasibility throughout Cone of feasible directions of X i at x i 2 X i X Cone of common directions D i (x i ) := R + (X i x i ). D ij (x i, x j ) := D i (x i ) \ D j (x j ). Tangent cone T ij (x i, x j ) := cld ij (x i, x j ) Maximal slope of joint improvement S ij (x i, x j ) := max u 0 i (x i ; d) + uj 0 (x j ; d d) : d 2 T ij (x i, x j ) & kdk 1. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

16 More on the maximal slope S ij (x i, x j ) := max u 0 i (x i ; d) + uj 0 (x j ; d) : d 2 T ij (x i, x j ) & kdk 1. d P ij = orthogonal projection onto T ij (x i, x j ) : S ij (x i, x j ) = min fkp ij [g i g j ]k : g i 2 u i (x i ), g j 2 u j (x j )g. Also, with dist(c i, C j ) := inf kc i C j k, S ij (x i, x j ) = dist [ u i (x i ) N i (x i ), u j (x j ) N j (x j )]. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

17 Real transfers Fix ϕ ij 2 (0, 1) for each agent pair i, j. De nition: While holding x i 2 X i and x j 2 X j, agents i, j, make a real transfer x = σd, with σ > 0 and kdk = 1, if x i + σd 2 X i and x j σd 2 X j and satis es u ij := u i (x i + σd) + u j (x j σd) u i (x i ) u j (x j ) u ij σϕ ij S ij (x i, x j ). Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

18 Bilateral exchange "as algorithm": Start at any feasible allocation: i2i x i = e I, x i 2 X i. Pick two agents i, j. Actually these hold x i 2 X i and x j 2 X j. If S ij (x i, x j ) = 0, select two new agents. Otherwise, they make a real transfer. That is, they ppdate their holdings x i x i + σd 2 X i and x j x j σd 2 X j Continue to Pick two agents until Convergence. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

19 On side payments u ij > 0 =) 9 sidepayments r i and r j such that r i + r j = 0 and is solvable. u i (x +1 i ) + r i > u i (x i ) & u j (x +1 j ) + r j > u j (x j ) Money oils the transaction machinery. Deals and incentives are compatible. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

20 On common price and joint improvement Recall that in equilibrium p 2 \ i2i [ u i (x i ) N i (x i )]. We say agents i, j see a common price i [ u i (x i ) N i (x i )] \ [ u j (x j ) N j (x j )] 6=?. Proposition i, j see a common price i S ij (x i, x j ) = 0. Thus joint improvement is possible as long as S ij (x i, x j ) > 0. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

21 Complete trade (no more joint improvement) Trade is complete in the set C := ffeasible allocations (x i ) : each S ij (x i, x j ) = 0g Two questions: 1) Will (x k i ) "converge" to C? 2) Will each pro le (x i ) 2 C be an equilibrium? Standing hypotheses now: The set of feasible allocations is bounded, and T ij (x i, x j ) = cl [D i (x i ) \ D j (x j )]. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

22 Complete trade (no more joint improvement) Trade is complete in the set C := ffeasible allocations (x i ) : each S ij (x i, x j ) = 0g Two questions: 1) Will (x k i ) "converge" to C? 2) Will each pro le (x i ) 2 C be an equilibrium? Standing hypotheses now: The set of feasible allocations is bounded, and T ij (x i, x j ) = cl [D i (x i ) \ D j (x j )]. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

23 Convergence and equilibrium Proposition (On convergence) Suppose real transfers, and that * agents meet in quasi cyclical manner * or that periodically S ij (x i, x j ) is maximal. Then (x k i ) clusters to the set C := ffeasible allocations (x i ) : each S ij (x i, x j ) = 0g. Proposition (On equilibrium) Each pro le (x i ) 2 C is an equilibrium if 1) X is one-dimensional or 2) Some agent i has x i 2 intx i and u i di erentiable at x i. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

24 A simple example: 3 agents and 3 contingencies Agents i = 1, 2, 3, sample space S = f1, 2, 3g; X = R S, X i = R S +, u 1 (x 1 ) = 2x x x 13, e 1 = (1, 1, 0), u 2 (x 2 ) = 0x x x 23, e 2 = (0, 1, 1), u 3 (x 3 ) = 1x x x 33, e 3 = (1, 0, 1). Trade sequence: rst f1, 2g, second f1, 3g, third f2, 3g. Start from i x 0 i = [ei ], use direction d = P ij hui 0 uj 0 and step-size σ = 1, to get d = P 12 [u1 0 u2 0 ] = (0, 1, 0) ) xi 1 = [(1, 0, 0), (0, 2, 1), (1, 0, 1)], d = P 13 [u1 0 u3 0 ] = (1, 0, 0) ) xi 2 = [(2, 0, 0), (0, 2, 1), (0, 0, 1)], d = P 23 [u2 0 u3 0 ] = (0, 0, 1) ) xi 3 = [(2, 0, 0), (0, 2, 0), (0, 0, 2)]. x 3 i = the e cient allocation. The equilibrium price p = (2, 2, 2) becomes common by choosing normals [n i ] = [(0, 1, 0), (0, 0, 1), (1, 0, 2)] 2 Π i N i (x i ). Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

25 Example extended: Stochastic LP & exchange Linear production games. Let X = R S for some nite state space S, and Then by LP-duality x i u i (x i ) := sup fy i y j x i A i y i & y i 0g. 2 u i (x i ) () x i n o 2 arg min χi x i A T i χi yi & χi 0. The cone D i (x i ) is closed convex and easily computable. Let binding constraints S i (x i ) := fs 2 S j [x i A i y i ] s = 0 and y i is primal optimalg Then where D i (x i ) = Π s2si (x i )R + Π s /2Si (x i )R. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

26 Asymmetric information X = L 0 (S, F, π, E) with F nite. X i = L 0 (S, F i, π, E) with F i F. and d 2 T ij (x i, x j ) = X i \ X j =) x i 2 X i =) D i (x i ) = X i. d constant on A i [ A j when atoms A i 2 F i and A j 2 F j intersect. Projection = conditional expectation: Pr(x) s = s2a x s π s s2a π s for each s 2 atom A. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

27 Concluding remarks on equilibrium and dynamics? How can players arrive at equilibrium - if any? While underway, how much competence, coordination, and foresight is required? What are the roles of cognition and perception? Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

28 Lacunes in economic theory only concerned with equilibrium, modestly interested in computation, uniqueness, stability, or attainability most often out-of-equilibrium behavior gets no mention, cognition and perception are hardly in focus. Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

29 Here! emergence of market equilibrium requires little prices need not come from somewhere; they rather emerge price-taking or maximization is neither necessary nor quite realistic agents can do without public prices; no posting agents merely seek own improvements, avoiding set-backs everybody can contend with idiosyncratics, local information no coordination, central agency, or global konowledge is ever required Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

30 Bilateral barters as viewed here requires no coordination, experience, foresigth, or optimization,.. It s totally decentralized. Fully driven by low-complexity adaptive agents Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31

31 References Bauschke & Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review (1996) Feldman, Bilateral trading processes, pairwise optimality, and Pareto optimality, Rev Econ Studies (1973) Flåm &Koutsougeras, Private Information, Transferable Utility, and the Core, Economic Theory (2010) Flåm, On sharing of risks and resources, in Reich & Zaslavksi Optim Th. and Related Topics, Am Math Soc, Contemp Math (2012) Flåm, Exchanges and measures of risk, Math and Financial Econ (2012) Flåm et al, Gradient Di erences and Bilateral Barters, Optimization (2012) Flåm, Coupled Projects, Core Imputations, and the CAPM, J. Math Econ (2012) Flåm & Gramstad, Direct exchanges in linear economies, Int. J. Game Th (2013) Gode & Sunder, Allocative e ciency of markets with zero intelligence traders: markets as a partial substitute for individual rationality, J. Pol. Econ (1993) Shapley & Shubik, Trade using one commodity as a means of payment, J Pol Econ (1977) Sjur Didrik Flåm (University of Bergen) Bilateral Exchange of Contingent Claims Workshop, Bonn, May / 31