Cover’s universal portfolio and stochastic portfolio theory Ting-Kam Leonard Wong University of Southern California joint with Christa Cuchiero and Walter Schachermayer WCMF 2017 1 / 20
Robust portfolio selection ■ Estimation error Monthly returns Optimized weights 0.10 stock1 0.00 6 −0.10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4 ● ● ● 0.10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● stock2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● stock 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● −0.10 ● ● ● ● ● ● ● ● ● ● ● ● 0.10 ● ● −2 stock 2 −3 −2 −1 0 1 2 3 stock3 −4 0.00 −6 −2 −1 0 1 2 3 −0.10 stock 1 0 10 20 30 40 50 60 ■ Changing parameters ■ Model uncertainty 2 / 20
Approaches we study model model independent specific universal functionally log-optimal/ portfolio generated numeraire portfolio portfolio (SPT) ■ Universal portfolio: Cover (1991), Cover and Ordentlich (1996) ■ Functionally generated portfolio: Fernholz (1998, 2002) ■ Log-optimal portfolio: Kelly (1956), Breiman (1962), Long (1990) 3 / 20
This work 1. Theoretical results that connect the three kinds of portfolios. 2. To work in the SPT set up, we use the market portfolio as the numeraire: portfolio value relative value of portfolio = market portfolio value 3. Portfolio performance is measured in terms of asymptotic growth rate (relative to the market portfolio). 4 / 20
Market weight X ( t ) We consider a stock market with n ✕ 2 assets: X i ( t ) = market cap of stock i ✖ ( t ) X i ( t ) ✖ i ( t ) = X 1 ( t ) + ✁ ✁ ✁ + X n ( t ) ∆ n The market weight vector ✖ ( t ) takes values in the simplex ∆ n . 5 / 20
Portfolio relative value ■ Having observed ❢ ✖ ( s ) ❣ 0 ✔ s ✔ t , the investor picks ✙ ( t ) ✷ ∆ n : ✖ ( t ) ✙ ( t ) ✖ ( t + 1 ) market weight portfolio weight ■ The portfolio is all-long, fully invested and self-financed. ■ In discrete time, portfolio relative value V ✙ ( t ) satisfies n V ✙ ( t + 1 ) ✙ i ( t ) ✖ i ( t + 1 ) � = ✿ V ✙ ( t ) ✖ i ( t ) i = 1 6 / 20
Cover’s universal portfolio The portfolio is constant-weighted, or constant-rebalanced, if ✙ ( t ) ✑ b for some fixed b ✷ ∆ n ✿ ■ Kelly (1956), Merton (1971), Cover (1991) ■ Volatility harvesting: Fernholz and Shay (1982), Luenberger (1997), ... Cover’s (1991) problem: ■ Find a portfolio strategy ❢ ✙ ( t ) ❣ such that 1 V ✙ ( t ) t log max b ✷ ∆ n V b ( t ) ✦ 0 for all sequences ❢ ✖ ( t ) ❣ ✶ t = 0 . ■ ✙ ( t ) = F t ( ❢ ✖ ( s ) ❣ 0 ✔ s ✔ t ) for some ‘universal’ functions F t . 7 / 20
Cover’s universal portfolio ■ Take a ‘prior’ probability distribution ✗ 0 on ∆ n and consider the ‘posterior’ distribution V b ( t ) ✗ t ( db ) := d ✗ 0 ( b ) ✿ � V ✁ ( t ) d ✗ 0 stock2 stock2 100 100 2 2 0 0 80 80 4 4 0 0 60 60 6 6 0 0 40 40 8 8 0 0 20 20 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 stock1 stock3 stock1 stock3 2 4 6 8 0 2 4 6 8 0 1 1 ■ Cover’s universal portfolio ˆ ✙ ( t ) is the posterior mean: � � ˆ ✙ ( t ) = ˆ bd ✗ t ( b ) ❀ V ( t ) = V b ( t ) d ✗ t ( b ) ✿ 8 / 20
Functionally generated portfolio ❋● ■ Portfolio map: ✙ ( t ) = ✙ ( ✖ ( t )) ❀ for some ✙ : ∆ n ✦ ∆ n ■ Functionally generated portfolio: ✙ ( ✁ ) is given in terms of the gradient of a generating function ✬ : ∆ n ✦ ❘ . ■ Relative arbitrage (i.e., beating the market portfolio w.p.1) under conditions on market stability and volatility ■ Lyapunov function: Karatzas and Ruf (2016) ■ Optimal transport and information geometry: Pal and Wong (2015, 2016) ■ More recent papers: Wong (2015), Vervuurt and Karatzas (2016), Vervuurt and Samo (2016), Pal (2016) ■ ❋● is convex and contains all constant-weighted portfolios. ■ Brod and Ichiba (2014): Cover’s portfolio is ‘functionally generated’ (answering a question in Fernholz and Karatzas (2009)) 9 / 20
UP , FGP and large deviation Theorem (W. (2016)) Under suitable conditions on ❢ ✖ ( t ) ❣ ✶ t = 0 ✚ ∆ n and ✗ 0 on ❋● : (i) The sequence ❢ ✗ t ❣ ✶ t = 0 of wealth distributions on ❋● satisfies a pathwise large deviation principle as t ✦ ✶ . (ii) Cover’s portfolio can be extended to ❋● : � ˆ ✙ ( t ) = ✙ ( ✖ ( t )) d ✗ t ( ✙ ) ❋● and the following universality property holds: ˆ 1 V ( t ) max ✙ ✷❋● V ✙ ( t ) = 0 ✿ lim t log t ✦✶ 10 / 20
Log-optimal/numeraire portfolio ■ A stochastic model for ❢ ✖ ( t ) ❣ is required. ■ In the SPT set-up (relative to the market): �� � � b ✁ ✖ ( t + 1 ) � � ✙ num ( t ) := arg max ❊ log � ❋ t � ✖ ( t ) b ✷ ∆ n ■ Al-Aradia and Jaimungal (2017): explicit solutions using stochastic control techniques ■ If ❢ ✖ ( t ) ❣ is a time homogeneous Markov chain, ✙ num can be realized by a portfolio map ✙ num : ∆ n ✦ ∆ n . ■ Györfi, Lugosi and Udina (2006): universal portfolios assuming stock returns are stationary and ergodic over time 11 / 20
UP for Lipschitz portfolio maps ■ For each M ❃ 0, let ▲ M be the family of M -Lipschitz portfolio maps (with some boundary conditions). ■ With topology of uniform convergence, ▲ M is compact. Let V ✄ ❀ M ( t ) := max ✙ ✷▲ M V ✙ ( t ) ✿ V M over ▲ M . ■ Consider Cover’s portfolio ˆ Theorem (Cuchiero, Schachermayer and W. (2016)) Assume ✗ 0 has full support on ▲ M . Then for every individual sequence ❢ ✖ ( t ) ❣ ✶ t = 0 in ∆ n we have 1 � � log V ✄ ❀ M ( t ) � log ˆ V M ( t ) lim = 0 ✿ t t ✦✶ 12 / 20
Approximating ✙ num by Lipschitz portfolio maps ■ Now let ❢ ✖ ( t ) ❣ ✶ t = 0 be a time homogeneous ergodic Markov chain on ∆ n with a unique variant measure ✚ , such that L := ❊ ✚ log V ✙ num ( 1 ) V ✙ num ( 0 ) ❁ ✶ ✿ M = 1 ▲ M by ■ We may construct Cover’s portfolio ˆ V ( t ) on � ✶ splitting the prior over each ▲ M . Theorem (Cuchiero, Schachermayer and W. (2016)) It holds almost surely that 1 1 1 t log V ✄ ❀ M ( t ) = lim t log ˆ M ✦✶ lim lim V ( t ) = lim t log V ✙ num ( t ) = L ✿ t ✦✶ t ✦✶ t ✦✶ 13 / 20
A continuous time analogue In continuous time, we let ❢ ✖ ( t ) ❣ t ✕ 0 be a continuous semimartingale in ∆ n . For a portfolio process ❢ ✙ ( t ) ❣ t ✕ 0 , n dV ✙ ( t ) ✙ i ( t ) d ✖ i ( t ) � V ✙ ( t ) = ✖ i ( t ) ✿ i = 1 ■ The previous theorem cannot be generalized directly because of stochastic integrals. ■ We will restrict to functionally generated portfolios where a pathwise decomposition for V ✙ ( t ) exists. ■ To compare with ✙ num , ❢ ✖ ( t ) ❣ t ✕ 0 needs to be a Markov diffusion process with a special structure. 14 / 20
Decomposition for functionally generated portfolio ■ Assume ✙ is generated by a positive C 2 concave function Φ = e ✬ on ∆ n . In fact n � ✿ ✙ i ( x ) = x i D i ✬ ( x ) + 1 � x j D j ✬ ( x ) j = 1 ■ Fernholz’s pathwise decomposition: V ✙ ( t ) = V ✙ ( 0 ) Φ( ✖ ( t )) Φ( ✖ ( 0 )) e Θ( t ) ❀ 1 where d Θ( t ) = � � i ❀ j D ij Φ( ✖ ( t )) d [ ✖ i ❀ ✖ j ] t . 2 Φ( ✖ ( t )) ■ The decomposition can be formulated using Föllmer’s Itô calculus (Schied, Speiser and Voloshehenko (2016)). 15 / 20
Analytical considerations ■ Analogous to ▲ M , we define a compact Hölder space ❋● M ❀☛ for M ❃ 0 and 0 ✔ ☛ ✔ 1. ■ Using the pathwise formulation in Schied et al (2016), we can define V ✄ ❀ M ❀☛ ( t ) := ✙ ✷❋● M ❀☛ V ✙ ( t ) max and prove the existence of a maximizer, for any continuous path ❢ ✖ ( t ) ❣ t ✕ 0 ✚ ∆ n whose quadratic variation exists. ■ Cover’s portfolio ˆ V M ❀☛ ( t ) can be generalized, in continuous time, to ❋● M ❀☛ , and asymptotic universality holds under suitable conditions. 16 / 20
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