Geometry and Optimization of Relative Arbitrage

Transcription

1 Geometry and Optimization of Relative Arbitrage Ting-Kam Leonard Wong joint work with Soumik Pal Department of Mathematics, University of Washington Financial/Actuarial Mathematic Seminar, University of Michigan October 21, / 29

2 Introduction 2 / 29

3 3 / 29 Main themes Model-free and robust investment strategies Depend only on directly observable market quantities Stochastic portfolio theory (SPT) Connections with other approaches such as universal portfolio theory

4 4 / 29 Set up n stocks, discrete time Market weight µ i (t) = market value of stock i at time t total market value at time t (0, 1) µ(t) = (µ 1 (t),..., µ n (t)) n (open unit simplex) µ(t)

5 5 / 29 Portfolio For each t, pick weights Market portfolio: µ(t) π(t) = (π 1 (t),..., π n (t)) n Relative value w.r.t. market portfolio µ: V π (t) = V π (0) = 1 and value of portfolio π at time t value of market portfolio µ at time t V π (t + 1) V π (t) = n i=1 π i (t) µ i(t + 1) µ i (t)

6 6 / 29 Relative arbitrage Definition For t 0 > 0, a relative arbitrage over the horizon [0, t 0 ] (w.r.t. µ) is a portfolio π such that V π (t 0 ) > 1 for all possible realizations of {µ(t)} t=0. Interpretations: (i) (Probabilistic) If {µ(t)} t=0 is a stochastic process, we mean P (V π (t 0 ) > 1) = 1. (ii) (Pathwise) Let P be a path property. We require V π (t 0 ) > 1 for all sequences {µ(t)} t=0 n satisfying property P.

7 7 / 29 Stochastic portfolio theory Relative arbitrage exists (for t 0 sufficiently large) under realistic conditions: Stability of capital distribution; diversity µ(t) K K Sufficient volatility Functionally generated portfolios (Fernholz) e.g. π i (t) = µ i (t) log µ i (t) n j=1 µ j(t) log µ j (t) (entropy-weighted portfolio)

8 Geometry 8 / 29

9 9 / 29 Geometry of relative arbitrage Consider deterministic portfolios: π(t) = π(µ(t)), where π : n n is a portfolio map. Let K n be open and convex: K K Ask: If the portfolio map π : n n is a relative arbitrage given the market is volatile and satisfies the generalized diversity condition µ(t) K, what does π look like?

10 10 / 29 Relative arbitrage as FGP Theorem (Pal and W. (2014)) K n open convex, π : n n portfolio map. TFAE: (i) π is a pseudo-arbitrage over the market on K: there exists ɛ > 0 such that inf t 0 V π (t) ɛ for all sequences {µ(t)} t=0 K, and there exists a sequence {µ(t)} t=0 K along which lim t V π (t) =. (ii) π is functionally generated: there exists a non-affine, concave function Φ : n (0, ) such that log Φ is bounded below on K and n π i (p) q i Φ(q), for all p, q K. p i Φ(p) i=1 The ratios π i /µ i are given by supergradients of the exponential concave function log Φ.

11 11 / 29 Examples π i (µ) Φ(µ) Market µ i 1 Diversity-weighted µ λ i n j=1 µλ j ( n j=1 µλ j ) 1 λ Constant-weighted π i µ π 1 1 µπn n Entropy-weighted µ i log µ i n j=1 µ j log µ j n j=1 µ j log µ j R package RelValAnalysis on CRAN

12 12 / 29 FGP leads to relative arbitrage Recall V π (t + 1) V π (t) = n i=1 π i (µ(t)) µ i(t + 1) µ i (t) Φ(µ(t + 1)) Φ(µ(t)) Write log V π(t + 1) V π (t) = log Φ(µ(t + 1) Φ(µ(t)) + T (µ(t + 1) µ(t)) where is a measure of volatility T (µ(t + 1) µ(t)) 0

13 13 / 29 FGP leads to relative arbitrage log V π (t) A(t) osc K (log Φ) t 0 t log V π (t) = log Φ(µ(t)) Φ(µ(0)) } {{ } diversity t 1 + T (µ(s + 1) µ(s)) s=0 } {{ } =A(t) sufficient volatility

14 Relative arbitrage as optimal transport maps X, Y: Polish spaces c: cost function P: Borel probability measure on X Q: Borel probability measure on Y Coupling: R p.m. on X Y with marginals P and Q Monge-Kantorovich optimal transport problem: Intuition: π : µ π(µ) min R E R[c(X, Y)], (X, Y) R µ P pay c(µ, π) π Q market weights portfolio weights 14 / 29

15 Optimal transport and FGP Take X = n, Y = [, ) n Cost: Portfolio at µ given h: ( n ) c(µ, h) = log e h i µ i i=1 π i (µ) = µ i e h i n j=1 µ je h j (1) Theorem (Pal and W. (2014)) Given P on n and Q on Y \, suppose R solves the optimal transport problem and E R [c(µ, h)] is finite. Then the support of R defines via (1) a portfolio π generated by a concave function. Conversely, every FGP is given by an optimal solution for some P and Q. 15 / 29

16 Optimization 16 / 29

17 17 / 29 Point estimation of FGP Assume 1 M µ i(t+1) µ i (t) M (M is unknown to the investor) FG := { π : n n functionally generated } For π FG, log V π(t + 1) V π (t) Logarithmic growth rate: where = log ( n i=1 =: l π (µ(t), µ(t + 1)) 1 t log V π(t) = P t := 1 t is the empirical measure ) π i (µ(t)) µ i(t + 1) µ i (t) n n l π dp t t 1 δ (µ(s),µ(s+1)) s=0

18 18 / 29 Point estimation of FGP P: Borel probability measure on S := Given P, consider { (p, q) n n : Solution π is analogous to MLE sup l π dp π FG S 1 M q } i M p i Nonparametric MLE: given a random sample X 1,..., X N f 0 in R d, consider N log f (X k ) sup f F k=1 where F is a class of densities

19 19 / 29 Point estimation of FGP Theorem (W. 2015) Under suitable regularity conditions on P: (Solvability) The problem has an optimal solution which is a.e. unique. (Consistency) If π (N) is optimal for P N, P N P weakly, and π is optimal for P, then π (N) π a.e.

20 20 / 29 Point estimation of FGP Theorem (continued) (Finite-dimensional reduction) Suppose P is discrete: P = 1 N N δ (p(j),q(j)). j=1 Then the optimal solution ( π, Φ) can be chosen such that Φ is polyhedral over the data points. In particular, Φ is piecewise affine over a triangulation of the data points.

21 21 / 29 Universal portfolio for FG Aim: achieve the asymptotic growth rate of V (t) = sup V π (t) π FG Cover (1991): wealth-weighted average (for constant-weighted portfolios) We will construct a market portfolio π whose basic assets are the portfolios in FG

22 22 / 29 Wealth distribution Equip FG with topology of uniform convergence ν 0 : Borel probability measure on FG (initial distribution) At time 0, distribute ν 0 (dπ) wealth to π FG Total wealth at time t is V(t) := FG V π (t)dν 0 (π) Wealth distribution at time t: ν t (B) := 1 V π (t)dν 0 (π), V(t) B B FG Bayesian interpretation: ν 0 : prior, V π (t) : likelihood, ν t : posterior

23 Example Wealth density for a family of constant-weighted portfolios: / 29

24 24 / 29 Universal portfolio for FG In this context, define Cover s portfolio by the posterior mean π(t) := π(µ(t))dν t (π) Then V π (t) V(t) Suppose an asymptotically optimal portfolio π exists. Hope: Posterior ν t concentrates around π V(t) is close to V (t) FG

25 25 / 29 Universal portfolio for FG Theorem (W. (2015)) Let {µ(t)} t=0 be a sequence in n such that the empirical measure P t = 1 t 1 t s=0 δ (µ(s),µ(s+1)) converges weakly to an absolutely continuous Borel probability measure on S. Here, the asymptotic 1 growth rate W(π) := lim t t log V π(t) exists for all π FG. (i) For any initial distribution ν 0, the sequence {ν t } t=0 of wealth distributions satisfies the large deviation principle (LDP) with rate { W W(π) if π supp(ν 0 ), I(π) = otherwise, where W := sup π supp(ν0 ) W(π).

26 26 / 29 Universal portfolio for FG Wealth distribution of FG satisfies LDP: ν t ν t (B) e t inf θ B I(θ) B FG

27 27 / 29 Universal portfolio for FG Theorem (Continued) (ii) There exists an initial distribution ν 0 on FG such that W = sup π FG W(π) for any absolutely continuous P. For this initial distribution, Cover s portfolio satisfies the asymptotic universality property 1 V(t) lim log t t V (t) = 0.

28 28 / 29 Future research Connections between SPT, universal portfolio theory and online portfolio selection (statistical learning) Risk-adjusted universal portfolios Price impacts and transaction costs

29 29 / 29 References Available on arxiv: S. Pal and T.-K. L. Wong, The Geometry of Relative Arbitrage (2014) T.-K. L. Wong, Optimization of Relative Arbitrage (2015) T.-K. L. Wong, Universal Portfolios in Stochastic Portfolio Theory (2015)