Hybrid Atlas Model. of financial equity market. Tomoyuki Ichiba 1 Ioannis Karatzas 2,3 Adrian Banner 3 Vassilios Papathanakos 3 Robert Fernholz 3

Hybrid Atlas Model of financial equity market Tomoyuki Ichiba 1 Ioannis Karatzas 2,3 Adrian Banner 3 Vassilios Papathanakos 3 Robert Fernholz 3 1 University of California, Santa Barbara 2 Columbia University, New York 3 INTECH, Princeton November 2009

2 Outline Introduction Hybrid Atlas model Martingale Problem Stability Effective dimension Rankings Long-term growth relations Portfolio analysis Stochastic Portfolio Theory Target portfolio Universal portfolio Conclusion

3 Flow of Capital Figure: Capital Distribution Curves (Percentage) for the S&P 500 Index of 1997 (Solid Line) and 1999 (Broken Line).

Log-Log Capital Distribution Curves Figure: Capital distribution curves for 1929 (shortest curve) - 1999 (longest curve), every ten years. Source Fernholz( 02). What kind of models can describe this long-term stability?

A Model of Rankings [Hybrid Atlas model] Capital process X := {(X 1 (t),..., X n (t)), 0 t < }. Order Statistics: X (1) (t) X (n) (t) ; 0 t <. Log capital Y := log X : Y (1) (t) Y (n) (t) ; 0 t <. Dynamics of log capital: d Y (k) (t) = (γ + γ i + g k ) d t + σ k d W i (t) if Y (k) (t) = Y i (t) ; for 1 i, k n, 0 t <, where W ( ) is n dim. B. M. company name i kth ranked company Drift ( mean ) γ i g k Diffusion ( variance ) σ k > 0 Banner, Fernholz & Karatzas ( 05), Chatterjee & Pal ( 07, 09), Pal & Pitman ( 08). 5

6 Illustration (n = 3) of interactions through rank Time X 3 (t)>x 1 (t)>x 2 (t) X 1 (t)>x 3 (t)>x 2 (t) x 1 > x 3 > x 2 x 3 > x 1 > x 2 x 1 > x 2 > x 3 X 1 (t)>x 2 (t)>x 3 (t) x 3 > x 2 > x 1 x 2 > x 1 > x 3 X 3 X 2 X 1 x 2 > x 3 > x 1 Paths in R + Time. A path in different wedges of R n. Symmetric group Σ n of permutations of {1,..., n}. For n = 3, {(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)}.

Vector Representation d Y (t) = G(Y (t))d t + S(Y (t))dw (t) ; 0 t < Σ n : symmetric group of permutations of {1, 2,..., n}. For p Σ n define wedges (chambers) R p := {x R n : x p(1) x p(2) x p(n) }, R n = p Σn R p, (the inner points of R p and R p are disjoint for p p Σ n ), Q (i) k : = {x R n : x i is ranked kth among (x 1,..., x n )} = {p : p(k)=i} R p ; 1 i, k n, n j=1 Q(j) k = R n = n l=1 Q(i) l and R p = n k=1 Q(p(k)) k. G(y) = p Σ n (g p 1 (1) + γ 1 + γ,..., g p 1 (n) + γ n + γ) 1 Rp (y), S(y) = p Σ n diag(σ p 1 (1),..., σ p 1 (n) ) }{{} s p 1 Rp (y) ; y Rn. 7

8 Martingale Problem Theorem [Krylov( 71), Stroock & Varadhan( 79), Bass & Pardoux( 87)] Suppose that the coefficients G( ) and a( ) := SS ( ) are bounded and measurable, and that a( ) is uniformly positive-definite and piecewise constant in each wedge. For each y 0 R n there is a unique one probability measure P on C([0, ), R n ) such that P(Y 0 = y 0 ) = 1 and t f (Y t ) f (Y 0 ) L f (Y s ) d s ; 0 0 t < is a P local martingale for every f C 2 (R 2 ) where L f (x) = 1 n 2 i,j=1 a ij(x)d ij f (x) + n i=1 G i(x)d i f (x) ; x R n. This implies that the hybrid Atlas model is well-defined.

9 Model Market capitalization X follows Hybrid Atlas model: the log capitalization Y i = log X i of company i has drift γ + g k + γ i and volatility σ k, when company i is k th ranked, i.e., Y Q (i) k for 1 k, i n. d Y i (t) = ( γ + + n k=1 n k=1 g k 1 (i) Q (Y (t)) + γ i )d t k σ k 1 (i) Q (Y (t))d W i (t) ; 0 t <. k

10 Model assumptions Market capitalization X follows Hybrid Atlas model: the log capitalization Y i = log X i of company i has drift γ + g k + γ i and volatility σ k, when company i is k th ranked, i.e., Y Q (i) k for 1 k, i n. Assume σ k > 0, (g k, 1 k n), (γ i, 1 i n) and γ are real constants with stability conditions n n g k + γ i = 0, k=1 i=1 k (g l +γ p(l) ) < 0, k = 1,..., n 1, p Σ n. l=1 γ i = 0, 1 i n, g 1 = = g n 1 = g < 0, g n = (n 1)g > 0. γ i = 1 (2i)/(n + 1), 1 i n, g k = 1, k = 1,..., n 1, g n = n 1.

Model Summary The log-capitalization Y = log X follows d Y i (t) = ( γ + + n k=1 n k=1 g k 1 (i) Q (Y (t)) + γ i )d t k σ k 1 (i) Q (Y (t))d W i (t) ; 0 t < k where σ k > 0, (g k, 1 k n), (γ i, 1 i n) and γ are real constants with stability conditions n n g k + γ i = 0, k=1 i=1 k (g l +γ p(l) ) < 0, k = 1,..., n 1, p Σ n. l=1

Stochastic stability The average Y ( ) := n i=1 Y i( ) / n of log-capitalization: dy (t) = γ d t + 1 n n n k=1 σ k 1 (i) Q (Y (t))d W i (t) k i=1 }{{} d B k (t) is a Brownian motion with variance rate n k=1 σ2 k /n2 drift γ by the Dambis-Dubins-Schwartz Theorem. Proposition Under the assumptions the deviations Ỹ ( ) := (Y 1 ( ) Y ( ),..., Y n ( ) Y ( )) from the average are stable in distribution, i.e., there is a unique invariant probability measure µ( ) such that for every bounded, measurable function f we have the Strong Law of Large Numbers 1 T lim f (Ỹ T T (t)) d t = f (y)µ(d y), 0 Π where Π := {y R n : y 1 + + y n = 0}. a.s.

13 Average occupation times Especially taking f ( ) = 1 Rp ( ) or 1 (i) Q ( ), we define from µ k the average occupation time of X in R p or Q (i) k : 1 T θ p := µ(r p ) = lim 1 Rp (X(t))d t T T θ k,i := µ(q (i) k ) = lim T 1 T 0 T 0 1 (i) Q (X(t))d t, 1 k, i n, k since 1 Rp (Ỹ ( )) = 1 R p (X( )) and 1 (i) Q (X( )) = 1 (i) k Q (Ỹ ( )). By k definition 0 θ k,i = {p Σ n:p(k)=i} θ p 1 for 1 k, i n, n l=1 θ l,i = n j=1 θ k,j = 1 for 1 k, i n. What is the invariant distribution µ?

Attainability One-dimensional Brownian motion attains the origin infinitely often. Two-dimensional Brownian motion does not attain the origin. Does the process X( ) attain the origin? t t X(t) = X(0) + b(x(s)) d s + σ(x(s))d W (s) 0 0 where b and σ are bounded measurable functions. Friedman( 74), Bass & Pardoux( 87).

15 Effective Dimension Let us define effective dimension ED( ) by ED(x) = trace(a(x)) x 2 x A(x)x where A( ) = σ( )σ( ). ; x R n \ {0}, Proposition Suppose X(0) 0. If inf x R n \{0} ED(x) 2, then X( ) does not attain the origin. If sup x R n \{0} ED(x) < 2 and if there is no drift, i.e., b( ) 0, then X( ) attains the origin. Exterior Dirichlet Problem by Meyers and Serrin( 60). Removal of drift by Girsanov s theorem. If there is drift, take [trace(a(x))+x b(x)] x 2 x A(x)x.

16 Triple collision Now consider triple collision: { Xi (t) = X j (t) = X k (t) for some t > 0, 1 i < j < k n }. What is the probability of triple collision? Fix i = 1, j = 2, k = 3. Let us define the sum of squared distances: s 2 (x) := (x 1 x 2 ) 2 +(x 2 x 3 ) 2 +(x 3 x 1 ) 2 = x DD x ; x R n, where (n 3) matrix D is defined by D := (d 1, d 2, d 3 ) with d 1 := (1, 1, 0,..., 0), d 2 := (0, 1, 1, 0,..., 0), d 3 := ( 1, 0, 1,..., 0). Z := {x R n : s(x) = 0 }.

Define the local effective dimension: R(x) := trace(d A(x)D) x DD x x DD A(x)DD x ; x R n \ Z. Proposition Suppose s(x(0)) 0. If inf x R n \Z R(x) 2, then P(X 1 (t) = X 2 (t) = X 3 (t) for some t 0) = 0. If sup x R n \Z R(x) < 2 and if there is no drift, i.e., b( ) 0, then P(X 1 (t) = X 2 (t) = X 3 (t) for some t 0) = 1. R( ) 2 for n dim. BM, i.e., A( ) I. If there is drift, take [trace(d A(x)D)+x DD b(x)] x DD x x DD A(x)DD x. Idea of Proof: a comparison with Bessel process with dimension two. 17

18 Rankings Recall Y (1) ( ) Y (2) ( ) Y (n) ( ). Let us denote by Λ k,j (t) the local time accumulated at the origin by the nonnegative semimartingale Y (k) ( ) Y (j) ( ) up to time t for 1 k < j n. Theorem[Banner & Ghomrasni ( 07)] For a general class of semimartingale Y ( ), the rankings satisfy d Y (k) (t) = n i=1 1 (i) Q (Y (t))d Y i (t) k + (N k (t)) 1[ n l=k+1 k 1 ] d Λ k,l (t) d Λ l,k (t) l=1 where N k (t) is the cardinality {i : Y i (t) = Y (k) (t)}.

19 Rankings Recall Y (1) ( ) Y (2) ( ) Y (n) ( ). Let us denote by Λ k,j (t) the local time accumulated at the origin by the nonnegative semimartingale Y (k) ( ) Y (j) ( ) up to time t for 1 k < j n. Lemma Under the non-degeneracy condition σ k > 0 for k = 1,..., n, dy (k) (t) = ( γ + g k + + 1 2 n i=1 for k = 1,..., n, 0 t T. ) γ i 1 (i) Q (Y (t)) d t + σ k d B k (t) k ] [ d Λ k,k+1 (t) d Λ k 1,k (t) Idea of Proof: a comparison with a Bessel process with dimension one to show Λ k,l ( ) 0, k l 2..

20 Long-term growth relations Proposition Under the assumptions we obtain the following long-term growth relations: Y i (T ) log X i (T ) lim = lim = γ T T T T Thus the model is coherent: = lim T log n i=1 X i(t ) T 1 lim T T log µ i(t ) = 0 a.s.; i = 1,..., n a.s. where µ i ( ) = X i ( )/(X 1 ( ) + + X n ( )). Moreover, n g k θ k,i + γ i = 0 ; i = 1,..., n. k=1

n g k θ k,i + γ i = 0 ; i = 1,..., n. k=1 The log-capitalization Y grows with rate γ and follows d Y i (t) = ( γ + + n k=1 n k=1 g k 1 (i) Q (Y (t)) + γ i )d t k σ k 1 (i) Q (Y (t))d W i (t) ; 0 t < k for i = 1,..., n. The ranking (Y (1) ( ),..., Y (n) ( )) follows dy (k) (t) = ( γ + g k + + 1 2 n i=1 for k = 1,..., n, 0 t <. ) γ i 1 (i) Q (Y (t)) d t + σ k d B k (t) k ] [ d Λ k,k+1 (t) d Λ k 1,k (t).

Semimartingale reflected Brownian motions The adjacent differences (gaps) Ξ( ) := (Ξ 1 ( ),..., Ξ n ( )) where Ξ k ( ) := Y (k) ( ) Y (k+1) ( ) for k = 1,..., n 1 can be seen as a semimartingale reflected Brownian motion (SRBM): Ξ(t) = Ξ(0) + ζ(t) + (I n Q)Λ(t) where ζ( ) := (ζ 1 ( ),..., ζ n ( )), Λ( ) := (Λ 1,2 ( ),..., Λ n 1,n ( )), ζ k ( ) := n i=1 0 1 (i) Q (Y (s)) d Y (s) k n i=1 0 1 (i) Q (Y (s)) d Y (s) k+1 for k = 1,..., n 1, and Q is an (n 1) (n 1) matrix with elements

23 Q := 0 1/2 1/2 0 1/2 1/2......... 0 1/2 1/2 0 Thus the gaps Ξ k := Y (k) ( ) Y (k+1) ( ) follow. Ξ(t) = Ξ(0) + ζ(t) }{{} + (I n Q)Λ(t) }{{} semimartingale reflection part In order to study the invariant measure µ, we apply the theory of semimartingale reflected Brownian motions developed by M. Harrison, M. Reiman, R. Williams and others. In addition to the model assumptions, we assume linearly growing variances: σ 2 2 σ2 1 = σ2 3 σ2 2 = = σ2 n σ 2 n 1.

Invariant distribution of gaps and index Let us define the indicator map R n x p x Σ n such that x p x (1) x p x (2) x p(n), and the index process P t := p Y (t). Proposition Under the stability and the linearly growing variance conditions the invariant distribution ν( ) of (Ξ( ), P ) is ( ν(a B) = n 1 q Σ n k=1 ) 1 λ 1 q,k exp( λ p, z )d z p Σ A n for every measurable set A B where λ p := (λ p,1,..., λ p,n 1 ) is the vector of components λ p,k := 4( k l=1 g l + γ p(l) ) σk 2 + > 0 ; p Σ n, 1 k n 1. σ2 k+1 Proof: an extension from M. Harrison and R. Williams ( 87).

25 Average occupation time Corollary The average occupation times are ( θ p = n 1 q Σ n k=1 λ 1 q,k ) 1 n 1 j=1 λ 1 p,j and θ k,i = {p Σ n:p(k)=i} θ p for p Σ n and 1 k, i n. (1,10) (10,10) 0.15 If all γi = 0 and σ 2 1 = σ2 n, then θ k,i = n 1 for 1 k, i n. Heat map of θk,i when n = 10, σ 2 k = 1 + k, g k = 1 for k = 1,..., 9, g 10 = 9, and γ i = 1 (2i)/(n + 1) for i = 1,..., n. 0.05 0.1 (k,i) (1,1) 0.2 0.15 0.1 0.05 (10,1)

26 Market weights come from Pareto type Corollary The joint invariant distribution of market shares µ (i) ( ) := X (i) ( )/(X 1 ( ) + + X n ( )) ; i = 1,..., n has the density (m 1,..., m n 1 ) = λ p,1 λ p,n 1 θ p p Π n m λ p,1+1 1 m λ p,2 λ p,1 +1 2 m λ p,n 1 λ p,n 2 +1 n 1 0 < m n m n 1... m 1 < 1, m n = 1 m 1 m n 1. This is a distribution of ratios of Pareto type disribution. m λ p,n 1+1 n,

Expected capital distribution curves From the expected slopes E ν [ log µ (k) log µ (k 1) log(k+1) log k ] = Eν (Ξ k ) log(1+k 1 ) obtain expected capital distribution curves. e 1 e 1 we iv e 5 e 5 v Weight iii Weight e 10 i ii e 10 Rank 1 5 10 50 100 5001000 5000 Rank 1 5 10 50 100 5001000 5000 n = 5000, gn = c (2n 1), g k = 0, 1 k n 1, γ 1 = c, γ i = 2c, 2 i n, σ 2 k = 0.075 + 6k 10 5, 1 k n. (i) c = 0.02, (ii) c = 0.03, (iii) c = 0.04. (iv) c = 0.02, g 1 = 0.016, g k = 0, 2 k n 1, g n = (0.02)(2n 1) + 0.016, (v) g1 = = g 50 = 0.016, g k = 0, 51 k n 1, g n = (0.02)(2n 1) + 0.8.

28 Empirical data Variance Rate 0.2 0.3 0.4 0.5 0.6 0.7 Historical capital distribution curves. Data 1929-1999. Source: Fernholz ( 02) 0 1000 2000 3000 4000 5000 Rank Growing variances. 1990-1999.

29 Capital Stocks and Portfolio Rules Market X = ((X 1 (t),..., X n (t)), t 0) of n companies log X i(t ) T T n X i (0) = G i (t)dt + S i,ν (t)dw ν (t), 0 0 ν=1 with initial capital X i (0) = x i > 0, i = 1,..., n, 0 T <. Define a ij ( ) = n ν=1 S iν( )S jν ( ) and A ij ( ) = 0 a ij(t)d t. Long only Portfolio rule π and its wealth V π. Choose π n + := {x R n : x i = 1, x i 0 } invest π i V π of money to company i for i = 1,..., n, i.e., π i V π /X i share of company i : n dv π (t) = i=1 V π (0) = w. π i (t)v π (t) dx i (t), 0 t <, X i (t)

30 Portfolios and Relative Arbitrage Market portfolio: Take π(t) = m(t) = (m 1 (t),, m n (t)) n + where m i (t) = X i (t) X 1 + X n (t), i = 1,..., n, 0 t <. Diversity weighted portfolio: Given p [0, 1], take π i (t) = (m i (t)) P p n j=1 (m for i = 1,..., n, 0 t <. j (t)) p Functionally generated portfolio (Fernholz ( 02) & Karatzas ( 08)). A portfolio π represents an arbitrage opportunity relative to another portfolio ρ on [0, T ], if P(V π (T ) V ρ (T )) = 1, P(V π (T ) > V ρ (T )) > 0. Can we find an arbitrage opportunity π relative to m?

31 Constant-portfolio For a constant-proportion π( ) π, [ n { V π Aii (t) ( (t) = w exp π i 2 +log Xi (t) )} 1 X i (0) 2 for 0 t <. i=1 n ] π i A ij (t)π j i,j=1 Here A ij ( ) = 0 a ij(t)d t and a ij ( ) = n ν=1 S iν( )S jν ( ), d Y (t) = G(Y (t))d t + S(Y (t))dw (t) ; 0 t <, G(y) = p Σ n (g p 1 (1) + γ 1 + γ,..., g p 1 (n) + γ n + γ) 1 Rp (y), S(y) = p Σ n diag(σ p 1 (1),..., σ p 1 (n) ) }{{} s p 1 Rp (y) ; y Rn.

32 Target Portfolio(Cover( 91) & Jamshidian( 92)) [ n { V π Aii (t) ( ( ) = w exp π i 2 +log Xi ( ) )} 1 X i (0) 2 i=1 n ] π i A ij ( )π j i,j=1 Target Portfolio Π (t) maximizes the wealth V π (t) for t 0: V (t) := max V π (t), π n + Π (t) := arg max V π (t), π n + where by Lagrange method we obtain Π i (t) = (2A ii (t) n j=1 1 ) 1 [ 2 n 2 A jj (t) n j=1 + 1 2 + 1 ( A ii (t) log Xi (t) ) ; 0 t <. X i (0) 1 ( A jj (t) log Xj (t) )] X j (0)

Asymptotic Target Portfolio Under the hybrid Atlas model with the assumptions 1 v(π) := lim T T log V π (T ) = γ + 1 ( n n π i a ii π i a ) ii π j 2 i=1 i=1 }{{} γπ where (a ij ) 1 i n is the (i,i) element of a 1 := lim T T T 0 (a ij (t)) 1 i,j n d t = p Σ n θ p s p s p. Asymptotic target portfolio maximizes the excess growth γπ : ( n n π := arg max π i a π n ii π i a ) ii π j. + We obtain π i = 1 [ 1 n 2 ( n 2 a ii 1 a j=1 jj i=1 ) 1 ] 33 i=1 = lim Π i (t) ; i = 1,..., n. t

34 Universal Portfolio(Cover( 91) & Jamshidian( 92)) Universal portfolio is defined as π Π i ( ) := n i V π ( )dπ + V π ( )dπ, 1 i n, V bπ ( ) = n + V π ( )dπ n +. dπ n + Proposition Under the hybrid Atlas model with the model assumptions, 1 lim T T log V bπ (T ) V π (T ) = lim 1 T T log V bπ (T ) V (T ) = 0 P a.s.

35 Conclusion Ergodic properties of Hybrid Atlas model Diversity weighted portfolio, Target portfolio, Universal portfolio. Further topics: short term arbitrage, generalized portfolio generating function, large market (n ), numéraire portfolio, data implementation. References: 1. arxiv: 0909.0065 2. arxiv: 0810.2149 (to appear in Annals of Applied Probability) Tomoyuki Ichiba (UCSB) ichiba@pstat.ucsb.edu