# 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

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• Introduction

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• 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

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• 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)

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• 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)

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• 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 .

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• 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)

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• Geometry

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• 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?

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• 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 .

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• 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

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• 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

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• 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

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• 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(, )

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• 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.

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• Optimization

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• 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

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• 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.

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• 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.

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• 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

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• 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

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• Example

Wealth density for a family of constant-weighted portfolios:

1995

2000

2005

2010

2

1

0

1

2

0

1

2

3

4

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• 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)

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• 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().

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• Universal portfolio for FG

Wealth distribution of FG satisfies LDP:

FG

t

B

t(B) et infB I()

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• 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.

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• 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

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• 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)

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