Hybrid Atlas Model of financial equity market Tomoyuki Ichiba 1 - PowerPoint PPT Presentation

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 1

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 2

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

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? 4

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 k th 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

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 k th 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

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 k th 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

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 2 > x 3 > x 1 X 3 X 2 X 1 A path in different wedges of R n . Paths in R + × Time . 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 ) } . 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 2 > x 3 > x 1 X 3 X 2 X 1 A path in different wedges of R n . Paths in R + × Time . 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 ) } . 6

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 ∈ Σ n R p , (the inner points of R p and R p ′ are disjoint for p � = p ′ ∈ Σ n ), : = { x ∈ R n : x i is ranked k th among ( x 1 , . . . , x n ) } Q ( i ) k = ∪ { p : p ( k )= i } R p ; 1 ≤ i , k ≤ n , j = 1 Q ( j ) = R n = ∪ n ℓ = 1 Q ( i ) k = 1 Q ( p ( k )) ∪ n and R p = ∩ n . k ℓ k G ( y ) = � p ∈ Σ n ( g p − 1 ( 1 ) + γ 1 + γ, . . . , g p − 1 ( n ) + γ n + γ ) ′ · 1 R p ( y ) , S ( y ) = � y ∈ R n . p ∈ Σ n diag ( σ p − 1 ( 1 ) , . . . , σ p − 1 ( n ) ) · 1 R p ( y ) ; � �� � s p 7

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 ∈ Σ n R p , (the inner points of R p and R p ′ are disjoint for p � = p ′ ∈ Σ n ), : = { x ∈ R n : x i is ranked k th among ( x 1 , . . . , x n ) } Q ( i ) k = ∪ { p : p ( k )= i } R p ; 1 ≤ i , k ≤ n , j = 1 Q ( j ) = R n = ∪ n ℓ = 1 Q ( i ) k = 1 Q ( p ( k )) ∪ n and R p = ∩ n . k ℓ k G ( y ) = � p ∈ Σ n ( g p − 1 ( 1 ) + γ 1 + γ, . . . , g p − 1 ( n ) + γ n + γ ) ′ · 1 R p ( y ) , S ( y ) = � y ∈ R n . p ∈ Σ n diag ( σ p − 1 ( 1 ) , . . . , σ p − 1 ( n ) ) · 1 R p ( y ) ; � �� � s p 7

Martingale Problem Theorem [ Krylov(’71), Stroock & Varadhan(’79) , Bass & Pardoux(’87) ] Suppose that the coefficients G ( · ) and a ( · ) := SS ′ ( · ) are bounded and mea- surable, and that a ( · ) is uniformly positive-definite and piece- For each y 0 ∈ R n there is a wise constant in each wedge. 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 ≤ t < ∞ 0 is a P local martingale for every f ∈ C 2 ( R 2 ) where � n i , j = 1 a ij ( x ) D ij f ( x ) + � n L f ( x ) = 1 i = 1 G i ( x ) D i f ( x ) ; x ∈ R n . 2 This implies that the hybrid Atlas model is well-defined. 8

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 ) for 1 ≤ k , i ≤ n . k � � n � d Y i ( t ) = γ + g k 1 Q ( i ) k ( Y ( t )) + γ i d t k = 1 n � + σ k 1 Q ( i ) k ( Y ( t )) d W i ( t ) ; 0 ≤ t < ∞ . k = 1 9

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 ) for 1 ≤ k , i ≤ n . k Assume σ k > 0 , ( g k , 1 ≤ k ≤ n ) , ( γ i , 1 ≤ i ≤ n ) and γ are real constants with stability conditions n n k � � � g k + γ i = 0 , ( g ℓ + γ p ( ℓ ) ) < 0 , k = 1 , . . . , n − 1 , p ∈ Σ n . k = 1 i = 1 ℓ = 1 ◮ γ i = 0 , 1 ≤ i ≤ n , g 1 = · · · = g n − 1 = − g < 0, g n = ( n − 1 ) g > 0 . ◮ γ i = 1 − ( 2 i ) / ( n + 1 ) , 1 ≤ i ≤ n , g k = − 1 , k = 1 , . . . , n − 1 , g n = n − 1 . 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 ) for 1 ≤ k , i ≤ n . k Assume σ k > 0 , ( g k , 1 ≤ k ≤ n ) , ( γ i , 1 ≤ i ≤ n ) and γ are real constants with stability conditions n n k � � � g k + γ i = 0 , ( g ℓ + γ p ( ℓ ) ) < 0 , k = 1 , . . . , n − 1 , p ∈ Σ n . k = 1 i = 1 ℓ = 1 ◮ γ i = 0 , 1 ≤ i ≤ n , g 1 = · · · = g n − 1 = − g < 0, g n = ( n − 1 ) g > 0 . ◮ γ i = 1 − ( 2 i ) / ( n + 1 ) , 1 ≤ i ≤ n , g k = − 1 , k = 1 , . . . , n − 1 , g n = n − 1 . 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 ) for 1 ≤ k , i ≤ n . k Assume σ k > 0 , ( g k , 1 ≤ k ≤ n ) , ( γ i , 1 ≤ i ≤ n ) and γ are real constants with stability conditions n n k � � � g k + γ i = 0 , ( g ℓ + γ p ( ℓ ) ) < 0 , k = 1 , . . . , n − 1 , p ∈ Σ n . k = 1 i = 1 ℓ = 1 ◮ γ i = 0 , 1 ≤ i ≤ n , g 1 = · · · = g n − 1 = − g < 0, g n = ( n − 1 ) g > 0 . ◮ γ i = 1 − ( 2 i ) / ( n + 1 ) , 1 ≤ i ≤ n , g k = − 1 , k = 1 , . . . , n − 1 , g n = n − 1 . 10

Model Summary The log-capitalization Y = log X follows � � n � d Y i ( t ) = γ + g k 1 Q ( i ) k ( Y ( t )) + γ i d t k = 1 n � + σ k 1 Q ( i ) k ( Y ( t )) d W i ( t ) ; 0 ≤ t < ∞ k = 1 where σ k > 0 , ( g k , 1 ≤ k ≤ n ) , ( γ i , 1 ≤ i ≤ n ) and γ are real constants with stability conditions n n k � � � g k + γ i = 0 , ( g ℓ + γ p ( ℓ ) ) < 0 , k = 1 , . . . , n − 1 , p ∈ Σ n . k = 1 i = 1 ℓ = 1 11