Geometry and Optimization of Relative Arbitrage ?· Geometry and Optimization of Relative Arbitrage…

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  • Geometry and Optimization ofRelative Arbitrage

    Ting-Kam Leonard Wongjoint work with Soumik Pal

    Department of Mathematics, University of Washington

    Math Finance Colloquium, University of Southern CaliforniaNovember 30, 2015

    1 / 29

  • Introduction

    2 / 29

  • Main themes

    I Model-free and robust investment strategiesI Depend only on directly observable market quantitiesI Stochastic portfolio theory (SPT), volatility pumpingI Connections with universal portfolio theory and other approaches

    3 / 29

  • Set up

    I n stocks, discrete timeI Market weight

    i(t) =market value of stock i at time t

    total market value at time t (0, 1)

    I (t) = (1(t), . . . , n(t)) n (open unit simplex)

    (t)

    4 / 29

  • Portfolio

    I For each t, pick weights

    (t) = (1(t), . . . , n(t)) n

    Market portfolio: (t) = (1(t), . . . , n(t))I Relative value w.r.t. market portfolio :

    V(t) =growth of $1 of portfolio at time t

    growth of $1 of market portfolio at time t

    I V(0) = 1 and

    V(t + 1)V(t)

    =

    ni=1

    i(t)i(t + 1)i(t)

    5 / 29

  • Relative arbitrage

    DefinitionFor t0 > 0, a relative arbitrage over the horizon [0, t0] (w.r.t. ) is aportfolio such that V(t0) > 1 for all possible realizations of{(t)}t=0.

    Interpretations:

    (i) (Probabilistic) If {(t)}t=0 is a stochastic process, we mean

    P (V(t0) > 1) = 1.

    (ii) (Pathwise) Let P be a path property. We require V(t0) > 1 forall sequences {(t)}t=0 n satisfying property P .

    6 / 29

  • Stochastic portfolio theory

    I Relative arbitrage exists (for t0 sufficiently large) under realisticconditions:

    I Stability of capital distribution; diversity (t) K

    K

    I Sufficient volatilityI Functionally generated portfolios (Fernholz)

    e.g. i(t) =i(t) logi(t)nj=1j(t) logj(t)

    (entropy-weighted portfolio)

    7 / 29

  • Geometry

    8 / 29

  • Geometry of relative arbitrage

    I Consider deterministic portfolios:

    (t) = ((t)),

    where : n n is a portfolio map.I Let K n be open and convex:

    K K

    I Ask: If the portfolio map : n n is a relative arbitragegiven the market is volatile and satisfies the generalized diversitycondition (t) K, what does look like?

    9 / 29

  • Relative arbitrage as FGP

    Theorem (Pal and W. (2014))K n open convex, : n n portfolio map. TFAE:

    (i) is a pseudo-arbitrage over the market on K: there exists > 0such that inft0 V(t) for all sequences {(t)}t=0 K, andthere exists a sequence {(t)}t=0 K along whichlimt V(t) =.

    (ii) is functionally generated: there exists a non-affine, concavefunction : n (0,) such that log is bounded below onK and

    ni=1

    i(p)qipi (q)

    (p), for all p, q K.

    I The ratios i/i are given by supergradients of the exponentialconcave function log .

    10 / 29

  • Examples

    i() ()

    Market i 1

    Diversity-weighted in

    j=1 j

    (nj=1

    j

    ) 1

    Constant-weighted i 11 nn

    Entropy-weighted i loginj=1j logj

    nj=1j logj

    R package RelValAnalysis on CRAN

    11 / 29

  • FGP leads to relative arbitrage

    I Recall

    V(t + 1)V(t)

    =

    ni=1

    i((t))i(t + 1)i(t)

    ((t + 1))((t))

    I Write

    logV(t + 1)

    V(t)= log

    ((t + 1)((t))

    + T ((t + 1) | (t))

    whereT ((t + 1) | (t)) 0

    is a measure of volatility

    12 / 29

  • FGP leads to relative arbitrage

    t

    log V(t)

    t0

    A(t)

    oscK(log )

    log V(t) = log((t))((0))

    diversity

    +

    t1s=0

    T ((s + 1) | (s)) =A(t) sufficient volatility

    13 / 29

  • Relative arbitrage as optimal transport mapsI X , Y: Polish spacesI c: cost functionI P: Borel probability measure on XI Q: Borel probability measure on YI Coupling: R p.m. on X Y with marginals P and QI Monge-Kantorovich optimal transport problem:

    minR

    ER[c(X,Y)], (X,Y) R

    I Intuition: : 7 ()

    market weights portfolio weights

    P

    Qpay c(, )

    14 / 29

  • Optimal transport and FGPTake

    I X = n, Y = [,)n

    I Cost:

    c(, h) = log

    (n

    i=1

    ehii

    )I Portfolio at given h:

    i() =iehinj=1 je

    hj(1)

    Theorem (Pal and W. (2014))Let P be any Borel probability measure on n, and let be an FGP.Define h = h() via (1), and let Q be the law of h when P . Thenthe coupling (, h) minimizes ER[c(, h)] whereR has marginals Pand Q.

    15 / 29

  • Optimization

    16 / 29

  • Point estimation of FGPI Assume 1M

    i(t+1)i(t)

    M (M is unknown to the investor)I FG :=

    { : n n | functionally generated

    }I For FG,

    logV(t + 1)

    V(t)= log

    (n

    i=1

    i((t))i(t + 1)i(t)

    )=: `((t), (t + 1))

    I Logarithmic growth rate:

    1t

    log V(t) =

    nn`dPt

    where

    Pt :=1t

    t1s=0

    ((s),(s+1))

    is the empirical measure17 / 29

  • Point estimation of FGP

    I P: Borel probability measure on

    S :={

    (p, q) n n :1M qi

    pi M

    }I Given P, consider

    supFG

    S`dP

    Solution is analogous to MLEI Nonparametric MLE: given a random sample X1, ...,XN f0 in

    Rd, consider

    supfF

    Nk=1

    log f (Xk)

    where F is a class of densities

    18 / 29

  • Point estimation of FGP

    Theorem (W. 2015)Under suitable regularity conditions on P:

    I (Solvability) The problem has an optimal solution which isa.e. unique.

    I (Consistency) If (N) is optimal for PN , PN P weakly, and isoptimal for P, then

    (N) a.e.

    19 / 29

  • Point estimation of FGP

    Theorem (continued)

    I (Finite-dimensional reduction) Suppose P is discrete:

    P =1N

    Nj=1

    (p(j),q(j)).

    Then the optimal solution (, ) can be chosen such that ispolyhedral over the data points. In particular, is piecewiseaffine over a triangulation of the data points.

    20 / 29

  • Universal portfolio for FG

    I Aim: achieve the asymptotic growth rate of

    V(t) = supFG

    V(t)

    I Cover (1991): wealth-weighted average (for constant-weightedportfolios)

    I We will construct a market portfolio whose basic assets arethe portfolios in FG

    21 / 29

  • Wealth distribution

    I Equip FG with topology of uniform convergenceI 0: Borel probability measure on FG (initial distribution)I At time 0, distribute 0(d) wealth to FGI Total wealth at time t is

    V(t) :=FG

    V(t)d0()

    I Wealth distribution at time t:

    t(B) :=1

    V(t)

    B

    V(t)d0(), B FG

    I Bayesian interpretation:

    0 : prior, V(t) : likelihood, t : posterior

    22 / 29

  • Example

    Wealth density for a family of constant-weighted portfolios:

    1995

    2000

    2005

    2010

    2

    1

    0

    1

    2

    0

    1

    2

    3

    4

    23 / 29

  • Universal portfolio for FG

    I In this context, define Covers portfolio by the posterior mean

    (t) :=FG

    ((t))dt()

    I ThenV(t) V(t)

    Suppose an asymptotically optimal portfolio exists. Hope:I Posterior t concentrates around

    I V(t) is close to V(t)

    24 / 29

  • Universal portfolio for FG

    Theorem (W. (2015))Let {(t)}t=0 be a sequence in n such that the empirical measurePt = 1t

    t1s=0 ((s),(s+1)) converges weakly to an absolutely

    continuous Borel probability measure on S. Here, the asymptoticgrowth rate W() := limt 1t log V(t) exists for all FG.

    (i) For any initial distribution 0, the sequence {t}t=0 of wealthdistributions satisfies the large deviation principle (LDP) withrate

    I() =

    {W W() if supp(0), otherwise,

    where W := supsupp(0) W().

    25 / 29

  • Universal portfolio for FG

    Wealth distribution of FG satisfies LDP:

    FG

    t

    B

    t(B) et infB I()

    26 / 29

  • Universal portfolio for FG

    Theorem (Continued)

    (ii) There exists an initial distribution 0 on FG such thatW = supFG W() for any absolutely continuous P. For thisinitial distribution, Covers portfolio satisfies the asymptoticuniversality property

    limt

    1t

    logV(t)V(t)

    = 0.

    27 / 29

  • Future research

    I Connections between SPT, universal portfolio theory and onlineportfolio selection (statistical learning)

    I Risk-adjusted universal portfoliosI Relative arbitrage under price impacts and transaction costsI Economic models inspired by SPT

    28 / 29

  • References

    I The Geometry of Relative ArbitrageS. Pal and T.-K. L. Wong. MAFE (2015)

    I Optimization of Relative ArbitrageAnnals of Finance (2015)

    I Universal Portfolios in Stochastic Portfolio TheoryPreprint (2015)

    29 / 29

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