# Geometry and Optimization of Relative ?· Geometry and Optimization of Relative Arbitrage ... market…

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<ul><li><p>Geometry and Optimization ofRelative Arbitrage</p><p>Ting-Kam Leonard Wongjoint work with Soumik Pal</p><p>Department of Mathematics, University of Washington</p><p>Financial/Actuarial Mathematic Seminar, University of MichiganOctober 21, 2015</p><p>1 / 29</p></li><li><p>Introduction</p><p>2 / 29</p></li><li><p>Main themes</p><p>I Model-free and robust investment strategiesI Depend only on directly observable market quantitiesI Stochastic portfolio theory (SPT)I Connections with other approaches such as universal portfolio</p><p>theory</p><p>3 / 29</p></li><li><p>Set up</p><p>I n stocks, discrete timeI Market weight</p><p>i(t) =market value of stock i at time t</p><p>total market value at time t (0, 1)</p><p>I (t) = (1(t), . . . , n(t)) n (open unit simplex)</p><p>(t)</p><p>4 / 29</p></li><li><p>Portfolio</p><p>I For each t, pick weights</p><p>(t) = (1(t), . . . , n(t)) n</p><p>Market portfolio: (t)I Relative value w.r.t. market portfolio :</p><p>V(t) =value of portfolio at time t</p><p>value of market portfolio at time t</p><p>I V(0) = 1 and</p><p>V(t + 1)V(t)</p><p>=n</p><p>i=1</p><p>i(t)i(t + 1)i(t)</p><p>5 / 29</p></li><li><p>Relative arbitrage</p><p>DefinitionFor t0 &gt; 0, a relative arbitrage over the horizon [0, t0] (w.r.t. ) is aportfolio such that V(t0) &gt; 1 for all possible realizations of{(t)}t=0.</p><p>Interpretations:</p><p>(i) (Probabilistic) If {(t)}t=0 is a stochastic process, we mean</p><p>P (V(t0) &gt; 1) = 1.</p><p>(ii) (Pathwise) Let P be a path property. We require V(t0) &gt; 1 forall sequences {(t)}t=0 n satisfying property P .</p><p>6 / 29</p></li><li><p>Stochastic portfolio theory</p><p>I Relative arbitrage exists (for t0 sufficiently large) under realisticconditions:</p><p>I Stability of capital distribution; diversity (t) K</p><p>K</p><p>I Sufficient volatilityI Functionally generated portfolios (Fernholz)</p><p>e.g. i(t) =i(t) logi(t)nj=1j(t) logj(t)</p><p>(entropy-weighted portfolio)</p><p>7 / 29</p></li><li><p>Geometry</p><p>8 / 29</p></li><li><p>Geometry of relative arbitrage</p><p>I Consider deterministic portfolios:</p><p>(t) = ((t)),</p><p>where : n n is a portfolio map.I Let K n be open and convex:</p><p>K K</p><p>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?</p><p>9 / 29</p></li><li><p>Relative arbitrage as FGP</p><p>Theorem (Pal and W. (2014))K n open convex, : n n portfolio map. TFAE:</p><p>(i) is a pseudo-arbitrage over the market on K: there exists &gt; 0such that inft0 V(t) for all sequences {(t)}t=0 K, andthere exists a sequence {(t)}t=0 K along whichlimt V(t) =.</p><p>(ii) is functionally generated: there exists a non-affine, concavefunction : n (0,) such that log is bounded below onK and</p><p>ni=1</p><p>i(p)qipi (q)</p><p>(p), for all p, q K.</p><p>I The ratios i/i are given by supergradients of the exponentialconcave function log .</p><p>10 / 29</p></li><li><p>Examples</p><p>i() ()</p><p>Market i 1</p><p>Diversity-weighted in</p><p>j=1 j</p><p>(nj=1 </p><p>j</p><p>) 1</p><p>Constant-weighted i 11 nn</p><p>Entropy-weighted i loginj=1j logj</p><p>nj=1j logj</p><p>R package RelValAnalysis on CRAN</p><p>11 / 29</p></li><li><p>FGP leads to relative arbitrage</p><p>I Recall</p><p>V(t + 1)V(t)</p><p>=</p><p>ni=1</p><p>i((t))i(t + 1)i(t)</p><p> ((t + 1))((t))</p><p>I Write</p><p>logV(t + 1)</p><p>V(t)= log</p><p>((t + 1)((t))</p><p>+ T ((t + 1) | (t))</p><p>whereT ((t + 1) | (t)) 0</p><p>is a measure of volatility</p><p>12 / 29</p></li><li><p>FGP leads to relative arbitrage</p><p>t</p><p>log V(t)</p><p>t0</p><p>A(t)</p><p>oscK(log )</p><p>log V(t) = log((t))((0)) </p><p>diversity</p><p>+</p><p>t1s=0</p><p>T ((s + 1) | (s)) =A(t) sufficient volatility</p><p>13 / 29</p></li><li><p>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:</p><p>minR</p><p>ER[c(X,Y)], (X,Y) R</p><p>I Intuition: : 7 ()</p><p>market weights portfolio weights</p><p>P</p><p>Qpay c(, )</p><p>14 / 29</p></li><li><p>Optimal transport and FGPTake</p><p>I X = n, Y = [,)n</p><p>I Cost:</p><p>c(, h) = log</p><p>(n</p><p>i=1</p><p>ehii</p><p>)I Portfolio at given h:</p><p>i() =iehinj=1 je</p><p>hj(1)</p><p>Theorem (Pal and W. (2014))Given P on n and Q on Y \ , supposeR solves the optimaltransport problem and ER[c(, h)] is finite. Then the support ofRdefines via (1) a portfolio generated by a concave function.Conversely, every FGP is given by an optimal solution for some Pand Q.</p><p>15 / 29</p></li><li><p>Optimization</p><p>16 / 29</p></li><li><p>Point estimation of FGPI Assume 1M </p><p>i(t+1)i(t)</p><p> M (M is unknown to the investor)I FG :=</p><p>{ : n n | functionally generated</p><p>}I For FG,</p><p>logV(t + 1)</p><p>V(t)= log</p><p>(n</p><p>i=1</p><p>i((t))i(t + 1)i(t)</p><p>)=: `((t), (t + 1))</p><p>I Logarithmic growth rate:</p><p>1t</p><p>log V(t) =</p><p>nn`dPt</p><p>where</p><p>Pt :=1t</p><p>t1s=0</p><p>((s),(s+1))</p><p>is the empirical measure17 / 29</p></li><li><p>Point estimation of FGP</p><p>I P: Borel probability measure on</p><p>S :={</p><p>(p, q) n n :1M qi</p><p>pi M</p><p>}I Given P, consider</p><p>supFG</p><p>S`dP</p><p>Solution is analogous to MLEI Nonparametric MLE: given a random sample X1, ...,XN f0 in</p><p>Rd, consider</p><p>supfF</p><p>Nk=1</p><p>log f (Xk)</p><p>where F is a class of densities</p><p>18 / 29</p></li><li><p>Point estimation of FGP</p><p>Theorem (W. 2015)Under suitable regularity conditions on P:</p><p>I (Solvability) The problem has an optimal solution which isa.e. unique.</p><p>I (Consistency) If (N) is optimal for PN , PN P weakly, and isoptimal for P, then</p><p>(N) a.e.</p><p>19 / 29</p></li><li><p>Point estimation of FGP</p><p>Theorem (continued)</p><p>I (Finite-dimensional reduction) Suppose P is discrete:</p><p>P =1N</p><p>Nj=1</p><p>(p(j),q(j)).</p><p>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.</p><p>20 / 29</p></li><li><p>Universal portfolio for FG</p><p>I Aim: achieve the asymptotic growth rate of</p><p>V(t) = supFG</p><p>V(t)</p><p>I Cover (1991): wealth-weighted average (for constant-weightedportfolios)</p><p>I We will construct a market portfolio whose basic assets arethe portfolios in FG</p><p>21 / 29</p></li><li><p>Wealth distribution</p><p>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</p><p>V(t) :=FG</p><p>V(t)d0()</p><p>I Wealth distribution at time t:</p><p>t(B) :=1</p><p>V(t)</p><p>B</p><p>V(t)d0(), B FG</p><p>I Bayesian interpretation:</p><p>0 : prior, V(t) : likelihood, t : posterior</p><p>22 / 29</p></li><li><p>Example</p><p>Wealth density for a family of constant-weighted portfolios:</p><p>1995</p><p>2000</p><p>2005</p><p>2010</p><p>2</p><p>1</p><p>0</p><p>1</p><p>2</p><p>0</p><p>1</p><p>2</p><p>3</p><p>4</p><p>23 / 29</p></li><li><p>Universal portfolio for FG</p><p>I In this context, define Covers portfolio by the posterior mean</p><p>(t) :=FG</p><p>((t))dt()</p><p>I ThenV(t) V(t)</p><p>Suppose an asymptotically optimal portfolio exists. Hope:I Posterior t concentrates around </p><p>I V(t) is close to V(t)</p><p>24 / 29</p></li><li><p>Universal portfolio for FG</p><p>Theorem (W. (2015))Let {(t)}t=0 be a sequence in n such that the empirical measurePt = 1t</p><p>t1s=0 ((s),(s+1)) converges weakly to an absolutely</p><p>continuous Borel probability measure on S. Here, the asymptoticgrowth rate W() := limt 1t log V(t) exists for all FG.</p><p>(i) For any initial distribution 0, the sequence {t}t=0 of wealthdistributions satisfies the large deviation principle (LDP) withrate</p><p>I() =</p><p>{W W() if supp(0), otherwise,</p><p>where W := supsupp(0) W().</p><p>25 / 29</p></li><li><p>Universal portfolio for FG</p><p>Wealth distribution of FG satisfies LDP:</p><p>FG</p><p>t</p><p>B</p><p>t(B) et infB I()</p><p>26 / 29</p></li><li><p>Universal portfolio for FG</p><p>Theorem (Continued)</p><p>(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</p><p>limt</p><p>1t</p><p>logV(t)V(t)</p><p>= 0.</p><p>27 / 29</p></li><li><p>Future research</p><p>I Connections between SPT, universal portfolio theory and onlineportfolio selection (statistical learning)</p><p>I Risk-adjusted universal portfoliosI Price impacts and transaction costs</p><p>28 / 29</p></li><li><p>References</p><p>Available on arXiv:I S. Pal and T.-K. L. Wong, The Geometry of Relative Arbitrage</p><p>(2014)I T.-K. L. Wong, Optimization of Relative Arbitrage (2015)I T.-K. L. Wong, Universal Portfolios in Stochastic Portfolio</p><p>Theory (2015)</p><p>29 / 29</p></li></ul>