Convex polytopes, interacting particles, spin glasses, and finance

Transcription

1 Convex polytopes, interacting particles, spin glasses, and finance Sourav Chatterjee (UCB) joint work with Soumik Pal (Cornell)

2 Interacting particles Systems of particles governed by joint stochastic differential equations. Drift of each particle depends on its relative position with respect to other particles. Continuous drifts: classical case (McKean-Vlasov). Completely different problem if drifts are prone to abrupt changes.

3 Example: The Atlas model X 1,..., X n are n diffusive particles, starting from zero. At each point of time, the lowest one gets a fixed upward drift. Formally, where δ i (t) = dx i (t) = δ i (t)dt + db i (t), { 1 if X i (t) = min j X j (t), 0 otherwise, and B 1,..., B n are independent Brownian motions. Occurs in finance (Banner, Fernholz & Karatzas 05). Uses reflecting Brownian motions and the Harrison-Williams machinery. More later.

4 Another example Again, X 1,..., X n are n Brownian particles starting from zero. At any point of time, the particle that is farthest from zero has a unit drift towards zero. Same definition in R d. Question: What is the stationary distribution of this system? Cannot use RBM tools in d 2.

5 A snapshot from the stationary law (n = 10, 000)

6 Explicit description for finite n Generate U 1,..., U n i.i.d. uniformly from unit ball in R d. Let V i = U i / max j U j, i = 1,..., n. Generate Γ Gamma(nd, 1) independently. Let Y i = 1 2 ΓV i, i = 1,..., n. Then (Y 1,..., Y n ) follows the stationary distribution of our system.

7 Solution in d = 1 We have dx i (t) = δ i (t)dt + db i (t), where 1 if X i (t) = max j X j (t), X i (t) < 0, δ i (t) = 1 if X i (t) = max j X j (t), X i (t) > 0, 0 otherwise. Alternatively, where k(x) = max i x i. dx(t) = k(x(t))dt + db(t),

8 Contd... Proposition Consider the s.d.e. dx(t) = k(x(t))dt + db(t), where k is any absolutely continuous function. Assume that exp( 2k(x)) is integrable and R n k(x) 2 e 2k(x) dx <. Then the probability distribution given by the un-normalized density exp( 2k(x)) provides a reversible, invariant probability distribution µ for the process X(t). Under some further conditions (...), the law of X(t) converges to µ in TV.

9 Contd... So in our problem, k is the Minkowski norm of the unit cube, and the stationary density is e 2 max i x i. Lemma If k is the Minkowski norm of a convex set C R n, then picking from exp( 2k(x)) is the same as: Picking U Unif (C) dividing by 2k(U) multiplying by independent Gamma(n, 1). Proof is simple, using polar coordinates induced by C. Picking from Unif (C) is can be handled theoretically if C can be easily triangulated, e.g. if C is a simplicial polytope.

10 Simplicial polytopes Each (n 1)-dimensional face is a simplex. 0 X = d j=1 X j I(Π = j) Unif (C). C = d j=1 C j ; simplices C j. Let X j Unif (C j ). P(Π = j) = Vol(C j )/Vol(C). Γ 2k(X) X = Γ 2k(X j ) X j, if Π = j. Γ 2k(X j ) X j - affine transformation of IID Exponentials.

11 Solving the Atlas model Recall: where δ i (t) = dx i (t) = δ i (t)dt + db i (t), { 1 if X i (t) = min j X j (t), 0 otherwise. Here k(x) = min i x i. This k is not the Minkowski norm of any convex set. Nor is exp( 2k(x)) integrable. The process does not converge to a stationary law.

12 Atlas model contd... Although k(x) = min i x i is not a Minkowski norm, it is so when restricted to i x i = 0. On this hyperplane, it is the norm induced by a regular simplex Figure: Level sets of k(x) and their intersection with {x : P x i = 0}

13 Atlas model contd... So, we project the process on to {x : x i = 0} and apply previous result. This gives the stationary law of (X 1 (t) X (t),..., X n (t) X (t)). Explicit description: Generate i.i.d. exponential r.v. s with mean n/2 and subtract off the mean. Obtained by Banner-Fernholz-Karatzas 05, and Pitman (tech. report). Both use Harrison-Williams RBM tools. General rank-dependent drifts: Stated as a major open problem in BFK 05. Solved by Pitman (tech. report). Easy solution by the following general result...

14 A general theorem Theorem (C. & Pal 06) Start with k : R n R. dx(t) = k(x(t))dt + db(t). Suppose there exists a subspace H such that k(x) = k 1 (y) + k 2 (z), y = P H (x), z = x y. k 1 0, cont., positively homogenous. {x H : k 1 (x) = 0} = {0}. Then Y (t) = P H X (t) = unique stationary distribution. Density exp( 2k 1 (y)). Can be generated using earlier trick. Exponentially fast. Reversible when stationary. Example: min x i = min(x i x) x, H = {x : x = 0}.

15 Some finance Equity market with n stocks. For the ith stock (company), define capital, Si = number of outstanding shares share price market weight, µ i = capital of ith stock total capital = S i P j S. j Capital distribution curve = log-log plot of µ versus rank. Ordered market wts: µ(1) µ (2)... µ (n). Plot log k versus log µ(n k+1). Economic theory (e.g. Simon 55) predicts that capital distribution curve should be a straight line. In reality, the curve is concave and remarkably stable in time.

16 Real data 1e!08 1e!06 1e!04 1e!02 log k versus log µ (n k+1). Dec 31, Includes all NYSE, AMEX, and NASDAQ Figure 1: Capital distribution curves: cesses represented by continuous semimartingales (see, e.g., Duffie (1992) or Karatzas and Shreve (1998)). The representation of market weights in terms of continuous semimartingales is straightforward, but in order to represent the ranked market weights, it is necessary to use semimartingale local times to capture the behavior when ranks change. The methodology for this analysis was developed in Fernholz (2001), and is outlined here in an Appendix. By using the representation of ranked market weights given in Fernholz (2001), we are able to determine the asymptotic behavior of the capital distribution. For a market with a stable capital distribution, this asymptotic behavior provides insight into the steady-state structure Sourav Chatterjee of the market. & Soumik Pal

17 Attempted explanations Jovanovic ( 82), Hopenhayn ( 92), Axtell ( 99), Hashemi ( 00), Kou & Kou ( 01). However, they all assume that the market converges rapidly to equilibrium, which is never true in reality! BFK s Atlas model tries to correct that, but recovers classical curve (straight line).

18 Stochastic modeling Log-capital: X i = log S i. Black-Scholes: Log-capital of a single company is modeled as a Brownian motion with drift. Natural extension for whole market: Log-capitals of all companies modeled as interacting particles on R. Empirical observation: Larger stocks have slower upward mobility than smaller stocks. One way to model this: The interacting particles (log-capitals) pull each other by a gravitational force.

19 We propose: The Gravity Model Let X i (t), i = 1,..., n be the log-capitals at time t. Gravitational force between i and j is proportional to sign(x i X j ). Resulting model: dx i (t) = α n n sign(x i (t) X j (t))dt + db i (t), i = 1,..., n. j=1 Represents a toy model of flow of capital from larger to smaller stocks. The parameter α determines the strength of the flow.

20 Capital distribution: Reality vs. Gravity model Real data (left) vs. simulations from gravity model with α = 1/2 (right). Simulations from Gravity model 1e!08 1e!06 1e!04 1e!02 log capital log rank Figure 1: Capital distribution curves: cesses represented by continuous semimartingales (see, e.g., Duffie (1992) or Karatzas and Shreve (1998)). The representation of market weights in terms of continuous semimartingales is straightforward, but in order to represent the ranked market weights, it is necessary to use semimartingale local times to capture the behavior when ranks change. The methodology for this analysis was developed in Fernholz (2001), and is outlined here in an Appendix. By using the representation of ranked market weights given in Fernholz (2001), we are able to determine the asymptotic behavior of the capital distribution. For a market with a stable capital distribution, this asymptotic behavior provides insight into the steady-state structure of the market. We shall assume that we operate in a continuously-traded, frictionless market in which the stock

21 Analysis Here k(x) = α n n x i x j. i,j=1 k(x) is not the Minkowski norm of any convex set in R n. Also, exp( 2k(x)) is not integrable. However, k(x) is the Minkowski norm of a polytope when restricted to H = {x : x i = 0}. The polytope is regular and simplicial. Uniform generation is easy to describe.

22 Stationary distribution X(t) does not converge in law. However, (X 1 (t) X (t),..., X n (t) X (t)) does converge to an equilibrium measure. Suppose (Y 1,..., Y n ) is drawn from this limiting distribution. Let Y (1) Y (n) denote the Y i s arranged in increasing order. Let i = Y (i+1) Y (i). Then 1,..., n 1 are independent, and ( ) 2αi(n i) i Exp. n Each possible ordering corresponds to one face of the polytope. Simplicial polytope exponentials.

23 Phase transition in the gravity model Recall: Market weight of ith company is µ i (t) = capital of ith stock total capital = ex i (t) j ex j (t). Market diversity, as defined by Fernholz ( 99): µ (n) (t) := max µ i (t) [0, 1]. i Under the gravity model, this has a limiting distribution as t.

24 Phase transition contd... Theorem Consider the gravity model with strength parameter α: dx i (t) = α n n sign(x i (t) X j (t))dt + db i (t), i = 1,..., n. j=1 Let µ (n) denote the diversity in equilibrium. Then as n, If α > 1/2, then µ (n) n (2α 1)/2α. Diversity exists. If α = 1/2 then µ (n) (log n) 1. Diversity exists to a lesser extent. If α < 1/2, then µ (n) 0. No diversity.

25 Sketch of proof µ (n) can be written as e ξ1 + e (ξ1+ξ2) + + e (ξ1+ +ξn 1), where ξ i Exp(2αi(n i)/n). When i n, Vague intuition: µ (n) E(ξ ξ i ) log i 2α /2α + 3 1/2α + n 1/2α n (2α 1)/2α if α > 1/2, (log n) 1 if α = 1/2, const. if α < 1/2. Rigorous proof via martingales and Poincaré inequalities.

26 Connection with spin glasses Let Σ be the set of possible configurations of a physical system. Let {Z σ, σ Σ} be a fixed collection of i.i.d. gaussian random variables. Derrida s Random Energy Model (REM) assigns a probability measure on Σ by putting mass exp(βz σ ) at each σ Σ. Exhibits phase transition as β varies. Mathematical reason is the same as for the gravity model phase transition. More complex models of spin glasses assume that the Z σ s have some correlation structure. Similar to more complex versions of the gravity model.

27 A generalized gravity model Gravity model on a graph G = (V, E): dx i (t) = α sign(x i (t) X j (t))dt + db i (t). j:(i,j) E As before, (X 1 (t) X (t),..., X n (t) X (t)) converges to a stationary law. Polytope is simplicial but not regular. Let (Y 1,..., Y n ) be a draw from the stationary distribution. Let Y (1) Y (n) be the ordered sample. Let Π i = the index j such that Y (i) = Y j. For each π S n and 1 i n 1, let f i (π) = #{(j, k) : j i < k, (π j, π k ) E}. Then P(Π = π) ( n 1 i=1 f i(π)) 1. Given Π = π, the increments Y (i+1) Y (i) are independent and Y (i+1) Y (i) Exp(2αf i (π)).