Some Aspects of Universal Portfolio
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1 1 Some Aspects of Universal Portfolio Tomoyuki Ichiba (UC Santa Barbara) joint work with Marcel Brod (ETH Zurich) Conference on Stochastic Asymptotics & Applications Sixth Western Conference on Mathematical Finance September 2014
2 2 On (; F ; P) let us consider a model of equity market X := (X(t) := (X 1 (t); : : : ; X n (t)); 0 t < 1), n ( 2) dx i (t) = X i (t) i (t)dt + KX k =1 i ;k (t)dw k (t) ; for i = 1; : : : ; n ; 0 t < 1, where ( ) := ( 1 ( ); : : : ; n ( )), ( ) := ( i ;k ( )) i ;k progressively measurable w.r.t. a right-continuous filtration F := (F( )), (W 1 ; : : : ; W K ( )) K-dimensional BM with K n, adapted to augmented filtration G of F, we consider the model under integrability condition Z T where for every 1 i ; j n i ;j ( ) := KX k =1 0 j i (s)j + ii (s) ds < 1 i ;k ( ) j ;k ( ) = a:s: 1 X i ( ) X d j ( ) dt hx i ; X j i( ) :
3 3 Portfolios ( ) for a small investor Let us consider investments of a small investor who do not have power to affect market prices. Given the market model and some initial capital V (0) = v, the investor selects nonnegative, G -progressively measurable, portfolio weights ( ) := ( 1 ( ); : : : ; n ( )) 0 of the wealth process V ( ), that is, the resulting wealth process V ( ) V v ; ( ) satisfies dv v ; (t) V v ; (t) = i (t) dx i (t) X i (t) = 0 (t) (t) dt + (t) dw (t) with the no short selling constraints i (t) 2 n + for every 0 t < 1, where n + is the unit simplex in dimension n, i.e., n + := f(x 1; : : : ; x n ) 2 R n : x i 0 ; 1 i n ; x 1 + +x n = 1 g :
4 Market Portfolio The performance of the investor s selection can be evaluated by comparisons of the wealth process V v ; ( ) of portfolio ( ) with the the wealth process V v ; ( ) of the market portfolio ( ) := ( 1 ( ); : : : ; n ( )) with V v ; ( ) = v X ( ) X (0) ; i ( ) := X i ( ) X ( ) 1 i n ; where X ( ) := X 1 ( ) + + X n ( ) : The wealth process V v ; ( ) reflects the growth of the whole market and here we measure the comparative performance of V v ; ( ) relative to V v ; ( ).
5 5 Constant weighted portfolios i ( ) i, i = 1; : : : ; n for some 2 n +. By an application of Itô s formula, given a constant vector := ( 1 ; : : : ; n ) 2 n +, the wealth process V v ; ( ) of constant rebalanced portfolio i ( ) i can be written as ny h V v Xi ( ) i i Z ; ( ) = v exp X (s)ds ; i (0) where ( ) is the excess growth rate of the constant rebalanced portfolio defined by ( ) := 1 2 i i ;i ( ) i ;j =1 0 i i ;j ( ) j :
6 Universal portfolios u ( ) := (1 u ( ); : : : ; u n ( )), proposed by Cover ( 91), are portfolios based on the market performances defined by u i ( ) := h Z V v ; ( )d n + i 1h Z i V v ; (t) d n + with the resulting wealth process V v ;u ( ) given by V v ;u ( ) = h Z d n + i 1h Z n + V v ; ( ) d i = Z i (1) V v ; ( )m(d) ; n + where V v ; ( ) is the wealth process of the constant rebalanced portfolio 2 n +, the integration over n + means integration with respect to (n 1) variables ( 1 ; : : : ; n 1 ) with 0 n = 1 ( n 1 ) 1, and m( ) is the uniform probability measure on n +. What are comparative characteristics of Universal Portfolios?
7 Jamshidian ( 92) : under some weak regularity conditions the universal portfolio is asymptotically weakly optimal, and moreover, the universal portfolio outperforms constant rebalanced portfolios in the long-run. 7 More precisely, let us consider the target optimal portfolio (t) = arg max V v ; (t) with V v ; (t) := max V v ; (t) ; t 0 : 2 n + 2 n + under the weakly regular model that satisfies 1 T E log X i (T ) ; 1 T log X P i (T )! T!1 1 i := lim T!1 1 T hlog X P 1 i ; log X j i(t )! T!1 1 i ;j := lim T!1 where (α 1 ij ) 1i ;j n is assumed to be strictly positive definite. There exists asymptotically optimal constant rebalanced portfolio 1 such that 1 := arg max lim 2 n + T!1 1 T E log V v ; (T ) : T E hlog X i ; log X j i(t ) ;
8 Theorem [Jamshidian (1992)] Assume that the weakly regular market is asymptotically active, i.e., 1 i > 0, i = 1; : : : ; n. Then on a large time horizon the target optimal portfolio outperforms the universal portfolio u, however, the outperformance is only polynomial in time: jj j 1=2 T (n 1)=2 (n 1)! (2) (n 1)=2 V v u ; (T ) V v ; (T ) P! T!1 1 ; where jj j is the determinant of the positive-definite matrix J := ( 1 i ;j 1 i ;n 1 j ;n + 1 n ;n ) 1i ;j n 1. Moreover, for every 2 n + except there exists > 0 such that V v ; (T ) V v ;u (T e T P! 0 ) T!1 :
9 9 In the following we shall consider comparative characteristics of Universal Portfolio in a finite time. Let us define the cumulative excess growth process ' (; ) of constantly rebalanced portfolio 2 n + : ' (; ) := Z 0 (s)ds = 1 2 i Z 0 i ;i (s)ds i ;j =1 i j Z G -progressively measurable, continuous process of finite variation. 0 i ;j (s)ds ;
10 The universal portfolio is the moment Z i u ( ) = n + i (d; ) ; for i = 1; : : : ; n, where (d; ) is a measure-valued, stochastic process defined by (d; ) := h Z ny n + h Xi ( ) X i (0) i i (d; ) i 1 Y n h Xi ( ) i i (d; ) ; X i (0) with the exponentially tilted, measure-valued process (d; ) := h Z n + exp ' (; ) di 1 exp ' (; ) d ; 2 + n : Now we shall see the universal portfolio is generated by a function in the sense of Fernholz ( 02).
11 Let F : n + R m! R be a smooth function that is twice continuously differentiable in the first n variables in n +, continuously differentiable in the last m variables in R m. We denote by D i the derivative with respect to i -th variable and D i ;j := D i D j. We shall impose that x i D i log F (x ; y) is bounded in n + R m for i = 1; : : : ; n, that is, max 1in sup (x ;y)2 n + Rm j x i D i log F (x ; y) j < 1 : one can consider a portfolio F ; ( ) generated by F ( ), ( ) and some G -progressively measurable, continuous process ( ) := ( 1 ( ); : : : ; m ( )) of finite variation : F ; i (t) := D i log F ((t); (t))+1 for i = 1; : : : ; n, 0 t < 1. j =1 j (t)d j log F ((t); (t)) i (t)
12 Proposition 1 The wealth process V v ;F ; ( ) of the portfolio F ; ( ) generated by the above set (F ( ); ( )) of function and finite-variation process is compared with the market portfolio wealth process V v ; ( ) by V v ; F ; (t) d log V v ; (t) d (t) := ( 1) 2F ((t); (t)) = d logf ((t); (t)) + d (t) ; i ;j =1 D i ;j F ((t); (t)) i (t) j (t) i ;j (t) dt mx `=1 for 0 t < 1, where ( i ;j ( )) 1i ;j n = ( ) 0 ( ). D`+n log F ((t); (t)) d`(t) Proof. Follow the proof of Fernholz ( 02). See also Strong ( 12) and Brod ( 14).
13 13 Proposition 2 Let ( ) be a continuous, finite-variation process with values in R m and let us define a portfolio i (t) := f i ((t); (t)) ; i = 1; : : : ; n ; t 0 ; where f 1 ; : : : ; f n be C 1;1 (R n R m ; R) functions such that P n f i (x; y) = 1 for every i = 1; : : : ; n, x 2 n + and y 2 R m. Then the following are equivalent: the portfolio ( ) is functionally generated in the sense of Proposition 1. there exists a C 1;1 (R n R m ; R) function S such that fi (x; y) + S(x; y) dx i x i is an exact differential with respect to the first n arguments x := (x 1 ; : : : ; x n ).
14 Given the initial market weights (0) := ( 1 (0); : : : ; n (0)), let us apply Proposition 1 with function F (x ; y) = F (x 1 ; : : : ; x n ; y 1 ; : : : ; y m ) Z h ny xi i i h 1 := exp i (0) 2 n + j =1 j y c(j ;j ) and the G -progressively measurable process ( ) = ( 1 ( ); : : : ; m ( )) 0 with c(i ;j )( ) := Z 0 i ;j (s)ds i ;j =1 i j y c(i ;j ) of finite variation, where the indices c(i ; j ) are defined by c(i ; j ) := 8 >< >: i if i = j ; j + [ i(2n i 1) = 2 ] if i < j ; i + [ j (2n j 1) = 2 ] if i > j for 1 i ; j n with m := n(n + 1) = 2. Here c(1; 2) = n + 1,..., c(1; n) = 2n 1,..., c(n 1; n) = m. i d
15 Note that by direct calculations i (t)d i log F ((t); (t)) = 1 : Thus we observe the resulting portfolio F ; i (t) = D i log F ((t); (t)) + 1 = i (t)d i log F ((t); (t)) = F ((t); (t)) 1 h Z j =1 ny i n + j =1 D j log F ((t); (t)) i (t) i (t) j (0) j exp ' (; t) di and then F ; i (t) = Z n + i d (d; ) = u i (t) ; i = 1; : : : ; n ; t 0 ; where (d; ) is the measure-valued process.
16 Proposition 3 The universal portfolio u ( ) is functionally generated by (F ; ), i.e., u ( ) = F ; ( ). Its wealth process V v ;u ( ) is compared with the market portfolio wealth process V v ; ( ) by V v ;F ; (t) V v ; (t) = F ((t); (t)) F ((0); (0)) for t 0. Here there is no finite variation part in the comparison formula, i.e., ( ) 0 in the comparison given in Proposition 1 : V v ; F ; ( ) d log V v ; ( ) = d log F (( ); (t)) + d ( ) :
17 This answers a question by Fernholz & Karatzas ( 09) : Connection between portfolio generating functions and universal portfolios? Non trivial example of ( ) 0? Pal & Wong ( 14) : for portfolios generated by positive concave function ( ) of n variables ( 1 ( ); : : : ; n ( )), finite variation part of log(v v ; ( )=V v ; ( )) is zero (i.e., ( ) = 0 ), if and only if ( ) is affine. Here ( ) 0, although F ( ; (t)) is not affine.
18 The measure ( ; t) is a solution to the problem of maximizing the entropy ( ; t) = arg max 2S a(t) h Z n + d i log d m () (d) ; among ( ) 2 S, where m( ) is the uniform probability a(t) measure on n + and S is the family a(t) S a(t) := n( < m) : Z n + n i (t) i log i (0) o o +' (; t) (d) = a (t) ; of absolutely continuous probability measures ( ) with respect to m( ) for some G -progressively measurable process a ( ).
19 Corollary: Arbitrage Relative to Market Let us define the following performance measures of the market: S 1 i (t) (t) := log ; n i (0) T (t) := Z n + Then we have almost surely In particular, if P V v ;u (t) V v ; (t) S (T ) + T (T ) 0 ' (; t) m(d) ; t 0 : exp S (t) + T (t) ; t 0 : = 1 ; P(S (T ) + T (T ) > 0) > 0 for some T > 0, then the universal portfolio is an arbitrage opportunity relative to the market, i.e., P(V v ;u (T ) V v ; (T )) = 1 ; P(V v ;u (T ) > V v ; (T )) > 0 :
20 20 Corollary: Arbitrage Relative to Market Here is another look: let us define the following performance measures of the market : S(t) := h Z T (t) := log Z n + n + Then we have almost surely V v ;u (t) V v ; (t) i i i (t) (d; t) log ; i (0) e ' (;t) m(d) ; t 0 : exp S(t) + T (t) ; t 0 :
21 21 Example: Two stocks ( ) = 1 2 1;1( ) + (1 ) 2;2( ) 2 1;1( ) +2(1 ) 1;2( ) + (1 ) 2 2;2( ) ( ) = ( 2 ) ; 2 where ( ) := 1;1( ) R + 2;2( ) 2 1;2( ). Let us write ( ) := 0 (s)ds, f( ) := ( ( )) 1=2 and i ( ) := i ( ) = i (0) for i = 1; 2 for simplicity. Then ' (; ) = (1=2) ( )( 2 ) : The relative performance of universal portfolio u ( ) to the market portfolio ( ) is V v ;u ( ) V v ; ( ) = Z 1 h 1 ( ) i h 2 ( ) 0 1 (0) 2 (0) i 1 e ' (; ) d :
22 e (1=8)( ) V v ; u ( ) Z 1 V v ; ( ) = Z 1=2 = 0 Z 1=2 = 0 Z 1 + 1=2 1 ( ) 0 h 1 ( ) 1 (0) i h 2 ( ) 2 (0) i 1 e ' (; ) d e (1=8) ( ) n 1 ( ) ( ) 1 2 e 2o (1=2) ( )( (1=2)) d ( ) ( ) ( ) 2 e (1=2) ( )( 1=2) 2 d and then because of inequality of arithmetic and geometric means, we may have the lower bound V v ;u ( ) V v ; ( ) = e ( ) 8 f( ) Z f( ) 0 1 ( ) ( ) 1 2h 2 ( ) u 2 1 ( ) 2 1 ( ) 2 ( ) 1=2 e (f( )) 2 = 8 = h 1 ( ) i 2( ) 1=2 g(f( )) ; 1 (0) 2 (0) f( ) 1 ( ) i u f( ) + e 2 Z ( ) 1 f( )=2 e u 2 =2 du f( ) 0 u2 2 du
23 23 where g(c) := e c2 =8 2 c Z c=2 0 e u 2 =2 du ; c 0 is an increasing function in c 0 with g(0) = 1, and has its inverse function g 1 ( ) in R +. Thus sufficient volatility creates relative arbitrage to the market, i.e., for every " 0 " V v ;u ( " ) (1 + ")V ( " ) ; where " is the first time that cumulative volatility is large enough, i.e., n Z t 1=2 := inf t : ( 1;1(s) + 2;2(s) 2 1;2(s))ds 0 g 1 h 1 ( ) (1 + ") 1 (0) 2( ) 2 (0) i 1=2o :
24 In particular, if the market model is diverse (see section 2.2 of Fernholz (2002)), i.e., if there exists 2 (0; 1 =2) such that max( 1 ( ); 2 ( )) 1, and if the market is not degenerate in the sense of 1;1( ) + 2;2( ) 2 1;2( ) 2 for some > 0, then the universal portfolio is a strong arbitrage opportunity relative to the market : for every P(V v ;u (T ) (1 + ")V v ; (T )) = 1 T 1 hg i 2 (1 + ") : Thus the conclusion of the corollary holds in a stronger form. Diversity : Fernholz, Karatzas, Kardaras ( 05), Osterrieder & Rheinländer ( 06), Strong & Fouque ( 11),... Optimality : Fernholz & Karatzas ( 08),...
25 Case studies: comparisons under Atlas model Universal Portfolio versus Portfolio G ( ) generated by (G( ); ( )), G (t) := D i log G((t)) + 1 j =1 j (t)d j log G((t)) i (t) for i = 1; : : : ; n with wealth process V v ;G (t) 0 t < 1. Let us consider more specific dynamics of log stock price Y i ( ) := log(x i ( )) with reverse order statistics: Y (1) ( ) Y (2) ( ) Y (n) ( ) : dy i (t) = k 1 fyi (t) = Y(k)(t)g dt + dwi (t) k =1 for i = 1; : : : ; n, t 0 for simplicity. Atlas model (Banner, Fernholz & Karatzas ( 05), Pal & Pitman ( 08), Chatterjee & Pal ( 10), Dembo et al. ( 12), Jourdain & Reygner ( 13))
26 26 Let us assume the stability condition n kx `=1 p(`) < 0 ; k = 1; : : : ; n 1 = 0 and for every p of symmetric group of permutations of f1; : : : ; ng. Then the center 1 n Y i (t) = 1 n Y i (0) + 1 n W i (t) of log stock prices behaves like a Brownian motion. Moreover, the gaps (Y (1) ( ) Y (2) ( ); : : : ; Y (n 1) ( ) Y (n) ( )) have joint exponential distribution κ( ) as unique stationary distribution. In particular, it is weakly regular in the sense of Jamshidian (1992) (cf. Ichiba et al. (2011)).
27 Proposition 4 Under some additional assumptions by concentration of measure phenomenon we have V v u ; (t) P ν V e (r ν(eu))t v ;G (t) for every r ; t, where dκ dν L 2 (ν) r 2 c := max δ 2 (u) ; 4"(" + 2 ) := 2 min 1kn 1 G((0))F ((t); (t)) G((t))F ((0); (0)) exp ct ; s 1 + r 2 2"(" + 2 ) 2 kuk 2 1 X k ` 2 (1 cos( = n)) ; `=1 1 ;
28 28 2 := Z (R+) n 1 u 2 (y)ν(dy) δ(u) := supju(x ) u(y)j ; and eu( ) is chosen so that eu(y (1) (t); : : : ; Y (n) (t)) := Z u(y) := eu(y) ( 1) 2G((t)) (R+) n 1 u(y)ν(dy) X Z 2 ; (R+) n 1 eu(y)ν(dy) ; D i ;j G((t)) dhi ; j i dt i ;j =1 n (t) for every t 0 (cf. Ichiba, Shkolnikov & Pal ( 13)).
29 29 Large Equity Market Now let us assume that the drifts depends on the number of stocks in the market and consider the triangle array ( (n) ) n1 with 1 ; : : : ; n (n) and lim n!1 ((n) (n) i ) = 2 (0; 1=2) ; lim sup n!1 max 1in ((n) (n) i ) with (n) := ( (n) n (n) ) = n. What happens if the number n of the stocks goes to infinity? The stationary distribution of the ordered market weights ( (n) 1 (n) 2 : : :) converges weakly to one-parameter Poisson-Dirichlet distribution (2), as n! 1. Chatterjee & Pal ( 10), Pitman & Yor ( 97)
30 Furthermore, if p > 2, then under the invariant measure ν the ordered market weights ( (n) 1 (n) 2 : : :) satisfy lim n!1 E ν[ ( (n) i ) p ] = E[ 1X lim n!1 n c2 n 2 (0; 1) ; c p i ] ; where (c 1 ; c 2 ; : : :) is the random vector with Poisson-Dirichlet distribution (2). Combining these estimates with the previous corollary, we obtain h V v ; u lim n!1 E (t) i ν V v 1 ; t 0: ; (t)
31 Summary. Properties of universal portfolios Connection to portfolios generated by functions Concentration-of-measure inequality Large equity market model Thank you, and Happy Birthday, Professor Jean-Pierre Fouque! Research supported by NSF DMS
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