EQUITY MARKET STABILITY
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1 EQUITY MARKET STABILITY Adrian Banner INTECH Investment Technologies LLC, Princeton (Joint work with E. Robert Fernholz, Ioannis Karatzas, Vassilios Papathanakos and Phillip Whitman.) Talk at WCMF6 conference, Santa Barbara September 27, 2014 For Information Purposes Only 1 / 26
2 Model for an equity market Equity market framework with capitalizations modeled by continuous semimartingales; e.g., Bachelier, Samuelson... dx i (t) X i (t) = α i(t) dt + N ξ iν (t) dw ν (t), i = 1,..., N, ν=1 where W ( ) = (W 1 ( ),, W n ( )) is B.M. in dimension N N. Mean Return rates α( ) = (α 1 ( ),..., α N ( )) Volatility rates ξ( ) = (ξ iν ( )) 1 i N,1 ν N Covariance rates σ( ) = ξ( )ξ ( ), (N N) matrix given by σ ij (t) := N ξ iν (t)ξ jν (t) = ν=1 1 X i (t)x j (t) d dt X i, X j (t). Both α( ) and σ( ) are assumed to be locally integrable. 2 / 26
3 Logarithmic representation Define log capitalizations Y ( ) by so that the equation becomes Y i ( ) := log X i ( ), dx i (t) X i (t) = α i(t) dt + dy i (t) = γ i (t) dt + N ξ iν (t) dw ν (t) ν=1 N ξ iν (t) dw ν (t), ν=1 where the growth rate of the i th asset is γ i (t) := α i (t) 1 2 σ ii(t). 3 / 26
4 Market weights Define the total market capitalization X (t) := X 1 (t) + + X N (t). Consider now the relative weights of the various assets, in terms of capitalization: µ i (t) := X i(t) X (t) = X i (t), i = 1,..., N. X 1 (t) + + X N (t) These are strictly positive numbers, add up to 1, and are determined on the basis of the capitalizations of the market s various assets at any given time. They are perfectly observable, at all times. 4 / 26
5 Ranked log capitalizations Descending order statistics: max Y i( ) =: Y (1) ( ) Y (N) ( ) := min Y i( ). 1 i N 1 i N Ties are resolved lexicographically, by resorting to the lowest index (name) i. Similar ordering for the capitalizations X (k) ( ) = exp(y (k) ( )) and for the ordered market weights µ (k) ( ) = X (k) ( ) X 1 (t) X N (t), k = 1,, N. 5 / 26
6 Capital distribution, U.S. equity market % 1% 10 bps Market Weight 1 bp 0.1 bp 0.01 bp bp bp bp ,000 10,000 Stocks Ranked by Capitalization 6 / 26
7 Implications of market stability Many rules-based strategies ( smart beta ) are seen to outperform cap-weighted indexes over the long term. Equity market stability is critical to this outperformance. Example: a continuously rebalanced equal-weight portfolio E. (N.B. over $10B invested in equal-weight portfolios.) Its relative value process versus the market M is (Fernholz 2002): d log(v E (t)/v M (t)) = ds(t) + θ(t) dt where θ( ) is a.s. positive and S(t) := 1 N N log µ i (t). A stable capital distribution implies that S( ) is mean-reverting, hence the relative performance is positive over sufficiently long time periods. i=1 7 / 26
8 Collision local times The ordered processes Y (k) are themselves semimartingales (as we shall see), and for 1 k < l N, the gap ( ) ( ) G (k,l) X(k) ( ) µ(k)( ) ( ) := Y (k) ( ) Y (l) ( ) = log = log 0 X (l) ( ) µ (l) ( ) is a continuous, nonnegative semimartingale. We shall denote by Λ (k,l) ( ) := L G (k,l) ( ) its local time at the origin (collision local time of order l k + 1). For a nonnegative continuous semimartingale G( ), recall its local time process L G ( ) = {G(t)=0} dg(t) = lim ε 0 2ε 0 1 {0 G(t)<ε} d G (t). 8 / 26
9 Reasonable conditions for stability A: No single stock crashes relative to the entire market (coherence): lim T 1 T log µ i(t ) = 0, a.s., i = 1,..., N. B: Convergence in distribution: ( µ(1) (T ),..., µ (N) (T ) ) (M 1,, M N ), as T. C: Strong laws of large numbers for local times: lim T 1 2T Λ(k,k+1) (T ) =: λ k > 0, a.s., k = 1,, N. 9 / 26
10 A result of Fernholz Suppose that the eigenvalues of σ( ) are bounded from above and below a.s., and that γ i (t) = γ(t) for all i = 1,..., n (common growth rate). Then lim T 1 T µ T (1) (t) dt = 1. 0 This is a degenerate form of stability in which a single stock occupies virtually the entire market capitalization. This is not consistent with empirical observations. We seek a simple model with stability but not concentration into a subset of the stocks. 10 / 26
11 First-order models With given real numbers γ, γ 1,..., γ N and σ 1 > 0,..., σ N > 0, and independent Brownian motions W 1 ( ),..., W N ( ), we consider the system of SDEs with rank-based interactions dy i (t) = ( γ + ) N γ k 1 {Yi (t)=y (k) (t)} dt k=1 + N σ k 1 {Yi (t)=y (k) (t)}dw i (t). k=1 Bass & Pardoux (1987) show that this system has a weak solution, unique in distribution (well-posed martingale problem). 11 / 26
12 Pathwise uniqueness Ichiba, Karatzas & Shkolnikov (2013) show that there is pathwise uniqueness, thus also strength, for the above system up until the first time 3 particles collide simultaneously. For this never to happen, it is necessary and almost sufficient (IKS op. cit.) that the mapping {1,, N} k σ 2 k (0, ) be concave. Sarantsev has recently proved that concavity is indeed sufficient. However, the weaker property Λ (k,l) ( ) 0 for l k 2 is always satisfied. 12 / 26
13 In such a system, the ranked log-capitalizations are rather simple to describe: Y (k) (t) = Y (k) (0) + (γ + γ k )t + σ k B k (t) + 1 ( ) Λ (k,k+1) (t) Λ (k 1,k) (t). 2 In other words, they behave as independent Brownian motions B k ( ) := N i=1 0 1 {Yi (t)=y (k) (t)}dw i (t) (by the P. Lévy theorem) with drift, bouncing off each other with the help of the two-by-two collision local times. Alternative Picture: Brownian motion with drift, reflecting off the walls of the Weyl chamber W = {(y 1,..., y N ) R N : y 1 y 2 y N }. 13 / 26
14 Stochastic stability This system is stable if γ 1 < 0,..., γ γ N 1 < 0, γ γ N = 0. Most basic example ( Atlas Configuration): γ = g > 0, γ 1 = = γ N 1 = g, γ N = (N 1)g. Under this condition, we have the SLLN for local times lim T 1 2T Λ(k,k+1) (T ) = λ k := (γ γ k ) > 0, a.s. for k = 1,..., N / 26
15 Under this stability condition, the vector process of gaps ( ) G (k,k+1) µ(k) ( ) ( ) = Y (k) ( ) Y (k+1) ( ) = log, k = 1,..., N 1 µ (k+1) ( ) converges in distribution to a random vector (G 1,..., G N 1 ) with exponential marginals P (G k > u) = e ur k, u > 0 where r k := 4λ k σ 2 k + σ2 k+1 = 1 E(G k ). The components G 1,..., G N 1 of this vector are independent, if the variances are either all the same or grow linearly by rank: σ 2 2 σ 2 1 = = σ 2 N σ2 N / 26
16 Rank-based variances, U.S., (smoothed) Variance Rank 16 / 26
17 Estimated γ k, U.S. Equity market, g k Rank 17 / 26
18 Size-based approaches The study of rank-based models such as first-order models leads to local times. Can they be avoided? Consider a simplistic size-based model is of the form dy i (t) = G(Y i (t)) dt + S(Y i (t)) dw i (t) for suitable G, S. This is not realistic, since there may be no stability in (Y 1 ( ),..., Y N ( )). Refine by choosing G and S so that (Y 1 ( ),..., Y N ( )) has an invariant distribution, then set X i (t) = M(t) exp Y i (t) for some market growth factor M(t), independent of Y i (t). Alternatively, use the weight-based variant dy i (t) = G(µ i (t)) dt + S(µ i (t)) dw i (t). 18 / 26
19 Example: volatility-stabilized model With α 0, the weight-based model (Fernholz & Karatzas 2004) dy i (t) = α 2µ i (t) dt + 1 µi (t) dw i(t) is known as a volatility-stabilized model. The coordinates can be expressed as time-changed square Bessel processes of dimension 2(1 + α). Coherence holds for α > 0 but fails for α = 0. The invariant distribution of market weights is Dirichlet (Pal 2011). Many other properties are known (e.g. Goia 2009, Shkolnikov 2011). 19 / 26
20 Example: logarithmic potential Stabilization from the previous model follows from unbounded volatility (and growth rate, for α > 0). Consider instead the size-based model dy i (t) = ( g + σ2 2Y i (t) for g > 0, σ > 0, α 0. Coordinates are now independent. ) dt + σ exp( αy i (t)) dw i (t), With λ = 2g/σ 2, the stable density G( ) on (0, ) is { C exp ( 2αx λ G(x) = 2α e2αx + Ei(2αx) ), if α > 0 λ 2 xe λx, if α = 0. Very good agreement with observed data. 20 / 26
21 A jump model A potential barrier in the form of an unbounded growth rate may seem unrealistic (similar to Atlas stock). Consider instead a jump model: dy i (t) = γ dt + σ dw i (t) if Y i (t) > 0; when Y i (t) = 0, it is kicked back into (0, ) at random location η chosen (independently) according to the distribution F (x) = P(η x), 0 < x <. Interpretation: If the relative stock capitalization is too small, the stock exits the market (bankruptcy). It is immediately replaced by a new stock (IPO). This gives a constant number of stocks for convenience; a reinterpretation as a Poisson point process would allow a variable number of stocks. 21 / 26
22 Stability in the jump model As described in Gihman & Skorokhod (1972), the system has a stable distribution G( ) if ( x ) e 2γx e 2γy (1 F (y)) dy dx <. 0 Question: how can we ensure that F G? 0 Assume the kicks are absolutely continuous w.r.t. Lebesgue measure. Then 1 F (x) = for some pdf f ( ) on (0, ). x f (u) du 22 / 26
23 It develops that where f (x) = N f (x) 0 N f (u) du, x ( N f (x) := e 2γx e 2γy If f ( ) is C 2, it must then solve the ODE 0 y ) f (u) du dy. f (x) + 2γf (x) + f (0 + )f (x) = 0, x (0, ). A solution (not known to be unique!) is given by f (x) = ( γ σ 2 ) 2 xe γx/σ 2, leading to a Gamma distribution with parameter 2, bearing more than a passing resemblance to the analogous distribution for the logarithmic potential with constant variance (α = 0). 23 / 26
24 Summary Macroscopic models should be consistent with observations of stability and growth rates. All the models considered here have a stabilization mechanism at the bottom (Atlas stock, logarithmic potential, bankruptcy/ipo). Stabilization may also occur at the top, e.g. antitrust. Many open questions and generalizations, e.g.: Understand ranks and local time properties for non-rank-based models Study properties of canonical portfolios, e.g. equal-weight, universal, in these models Better linkage of common features between these models Thank you! 24 / 26
25 A few references A. Banner, E.R. Fernholz & I. Karatzas (2005) Atlas models of equity markets. Annals of Applied Probability 15, A. Banner & R. Ghomrasni (2008) Local times of ranked continuous semimartingales. Stochastic Processes & Applications 118, R. Bass & E. Pardoux (1987) Uniqueness for diffusions with piecewise constant coëfficients. Probability Theory & Related Fields 76, S. Chatterjee & S. Pal (2010) A phase transition behavior for Brownian motions interacting through their ranks. Probability Theory & Related Fields 147, / 26
26 E.R. Fernholz (2002) Stochastic Portfolio Theory. Springer-Verlag, New York. T. Ichiba, I. Karatzas & M. Shkolnikov (2013) Strong solutions of stochastic equations with rank-based coëfficients. Probability Theory & Related Fields 56, S. Pal & M. Shkolnikov (2013) Concentration of measure for Brownian particle systems interacting through their ranks. Probability Theory & Related Fields, to appear. M. Shkolnikov (2012) Large systems of diffusions interacting through their ranks. Stochastic Processes & Their Applications 122, / 26
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