STOCHASTIC PORTFOLIO THEORY IOANNIS KARATZAS Mathematics and Statistics Departments Columbia University ik@math.columbia.edu Joint work with Dr. E. Robert FERNHOLZ, C.I.O. of INTECH Enhanced Investment Technologies, Princeton NJ Talk in Vienna, September 2007
CONTENTS 1. The Model 2. Portfolios 3. The Market Portfolio 4. Diversity 5. Diversity-Weighting and Arbitrage 6. Performance of Diversity-Weighted Portfolios 7. Strict Supermartingales 8. Completeness & Optimization without EMM 9. Concluding Remarks 1
SYNOPSIS The purpose of this talk is to offer an overview of Stochastic Portfolio Theory , a rich and flexible framework for analyzing portfolio behavior and eq- uity market structure. This theory is descriptive as opposed to normative, is consistent with observ- able characteristics of actual markets and portfo- lios, and provides a theoretical tool which is useful for practical applications. As a theoretical tool, this framework provides fresh insights into questions of market structure and ar- bitrage, and can be used to construct portfolios with controlled behavior. As a practical tool, Sto- chastic Portfolio Theory has been applied to the analysis and optimization of portfolio performance and has been the basis of successful investment strategies for close to 20 years. 2
REFERENCES Fernholz, E.R. (2002). Stochastic Portfolio The- ory. Springer-Verlag, New York. Fernholz, E.R. & Karatzas, I. (2005). Relative arbitrage in volatility-stabilized markets. Annals of Finance 1 , 149-177. Fernholz, E.R., Karatzas, I. & Kardaras, C. (2005). Diversity and arbitrage in equity markets. Finance & Stochastics 9 , 1-27. Karatzas, I. & Kardaras, C. (2007). The num´ eraire portfolio and arbitrage in semimartingale markets. Finance & Stochastics 11 , 447-493. Fernholz, E.R. & Karatzas, I. (2008) Stochastic Portfolio Theory: An Overview. To appear. 3
1. THE MODEL. Standard Model (Bachelier, Samuelson,...) for a financial market with n stocks and d ≥ n factors: for i = 1 , . . . , n , d � b i ( t ) dt + dX i ( t ) = X i ( t ) σ iν ( t ) dW ν ( t ) . ν =1 � �� � b ( · ) = ( b 1 ( · ) , . . . , b n ( · )) ′ . Vector of rates-of-return: Matrix of volatilities: σ ( · ) = ( σ iν ( · )) 1 ≤ i ≤ n, 1 ≤ ν ≤ d will be assumed bounded for simplicity. ♣ Assumption: for every T ∈ (0 , ∞ ) we have � T n � � � 2 dt < ∞ , � � � b i ( t ) a.s. � 0 i =1 All processes are adapted to a given flow of infor- mation (or “filtration”) F = {F ( t ) } 0 ≤ t< ∞ , which satisfies the usual conditions and may be strictly larger than the filtration generated by the driving Brownian motion W ( · ) = ( W 1 ( · ) , . . . , W d ( · )) ′ . No Markovian, or Gaussian, assumptions... 4
Suppose, for simplicity, that the variance/covariance matrix a ( · ) = σ ( · ) σ ′ ( · ) has all its eigenvalues bounded away from zero and infinity: that is, κ || ξ || 2 ≤ ξ ′ a ( t ) ξ ≤ K || ξ || 2 , ∀ ξ ∈ R d holds a.s. (for suitable constants 0 < κ < K < ∞ ). • Solution of the equation for stock-price X i ( · ): d � d (log X i ( t )) = γ i ( t ) dt + σ iν ( t ) dW ν ( t ) ν =1 � �� � with γ i ( t ) := b i ( t ) − 1 2 a ii ( t ) the growth-rate of the i th stock, in the sense � � � T 1 lim log X i ( t ) − 0 γ i ( t ) dt = 0 a.s. T t →∞ 5
2. PORTFOLIO. An adapted vector process π ( t ) = ( π 1 ( t ) , · · · , π n ( t )) ′ ; fully-invested, no short-sales, no risk-free asset: n � π i ( t ) ≥ 0 , π i ( t ) = 1 for all t ≥ 0 . i =1 Value Z π ( · ) of portfolio: n d dZ π ( t ) π i ( t ) dX i ( t ) � � X i ( t ) = b π ( t ) dt + σ π Z π ( t ) = ν ( t ) dW ν ( t ) i =1 ν =1 with Z π (0) = 1 . Here n n � � b π ( t ) := σ π π i ( t ) b i ( t ) , ν ( t ) := π i ( t ) σ iν ( t ) , i =1 i =1 are, respectively, the portfolio rate-of-return, and the portfolio volatilities. 6
¶ Solution of this equation: n � � � log Z π ( t ) = γ π ( t ) dt + σ π d ν ( t ) dW ν ( t ) . ν =1 � �� � Portfolio growth-rate n � γ π ( t ) := π i ( t ) γ i ( t ) + γ π ∗ ( t ) . i =1 Excess growth-rate n n n ∗ ( t ) := 1 � � � γ π π i ( t ) a ii ( t ) − π i ( t ) a ij ( t ) π j ( t ) . 2 i =1 i =1 j =1 � �� � Portfolio variance d n n � � � ν ( t )) 2 = a ππ ( t ) := ( σ π π i ( t ) a ij ( t ) π j ( t ) . ν =1 i =1 j =1 For a given portfolio π ( · ), let us introduce the ♠ “order statistics” notation, in decreasing order: π (1) := max 1 ≤ i ≤ n π i ≥ π (2) ≥ . . . ≥ π ( n ) := min 1 ≤ i ≤ n π i . 7
3. MARKET PORTFOLIO: Look at X i ( t ) as the capitalization of company i at time t . Then X ( t ) := X 1 ( t )+ . . . + X n ( t ) is the total capitalization of all stocks in the market, and µ i ( t ) := X i ( t ) X i ( t ) X ( t ) = X 1 ( t ) + . . . + X n ( t ) > 0 the relative capitalization of the i th company. Clearly � n i =1 µ i ( t ) = 1 for all t ≥ 0, so µ ( · ) is a portfolio process, called "market portfolio" . . Ownership of µ ( · ) is tantamount to ownership of the entire market, since Z µ ( · ) ≡ c .X ( · ). � � γ π ♣ ∗ ( · ) ≥ ( κ/ 2) · 1 − π (1) ( · ) . FACT 1: Moral: Excess-rate-of growth is non-negative, strictly positive unless the portfolio concentrates on a sin- gle stock: diversification helps not only to reduce variance, but also to “enhance growth”. 8
HOW? Consider, for instance, a fixed-proportion � n portfolio π i ( · ) ≡ p i ≥ 0 with i =1 p i = 1 and p (1) = 1 − η < 1 . Then � � � T n Z p ( T ) � ∗ ( t ) dt ≥ κη 0 γ p log p i log µ i ( T ) = − 2 T . Z µ ( T ) i =1 And if lim T →∞ 1 T log µ i ( T ) = 0 (no individual stock collapses very fast), then this gives almost surely � � Z p ( T ) 1 ≥ κη lim T →∞ T log 2 > 0 : Z µ ( T ) a significant outperforming of the market. Remark: Tom Cover’s “universal portfolio” � ∆ n p i Z p ( t ) d p Π i ( t ) := i = 1 , · · · , n , � ∆ n Z p ( t ) d p has value � ∆ n Z p ( t ) d p Z Π ( t ) = p ∈ ∆ n Z p ( t ) . ∼ max � ∆ n d p � � γ π ♣ ∗ ( · ) ≤ 2 K · 1 − π (1) ( · ) . FACT 2:
4. DIVERSITY. The market-model M is called • Diverse on [0 , T ], if there exists δ ∈ (0 , 1) such that we have a.s.: µ (1) ( t ) < 1 − δ � , ∀ 0 ≤ t ≤ T . � �� • Weakly Diverse on [0 , T ], if for some δ ∈ (0 , 1): � T 1 0 µ (1) ( t ) dt < 1 − δ , a.s. T � �� � If M is diverse, then γ µ ♣ ∗ ( · ) ≥ ζ for FACT 3: some ζ > 0 ; and vice-versa. ♣ FACT 4: If all stocks i = 1 , . . . , n in the market have the same growth-rate γ i ( · ) ≡ γ ( · ) , then � T 1 0 γ µ lim ∗ ( t ) dt = 0 , a.s. T T →∞ In particular, such a market cannot be diverse on long time-horizons : once in a while a single stock dominates such a market, then recedes; sooner or later another stock takes its place as absolutely dominant leader; and so on. 10
• Here is a quick argument: from γ i ( · ) ≡ γ ( · ) and X ( · ) = X 1 ( · ) + · · · + X n ( · ) we have � � � T 1 0 γ µ ( t ) dt lim log X ( T ) − = 0 , T T →∞ � � � T 1 lim log X i ( T ) − 0 γ ( t ) dt = 0 . T T →∞ for all 1 ≤ i ≤ n . But then � � � T 1 lim log X (1) ( T ) − 0 γ ( t ) dt = 0 , a.s. T T →∞ for the biggest stock X (1) ( · ) := max 1 ≤ i ≤ n X i ( · ) , and note X (1) ( · ) ≤ X ( · ) ≤ n X (1) ( · ) . Therefore, � � 1 lim log X (1) ( T ) − log X ( T ) = 0 , thus T T →∞ � T � � lim 1 γ µ ( t ) − γ ( t ) dt = 0 . T 0 But γ µ ( t ) = � n i =1 µ i ( t ) γ ( t ) + γ µ ∗ ( t ) = γ ( t ) + γ µ ∗ ( t ) , because all growth rates are equal. ✷ 11
♣ FACT 5: In a Weakly Diverse Market there exist potfolios π ( · ) that lead to arbitrage relative to the market-portfolio: with common initial capital Z π (0) = Z µ (0) = 1 , and some T ∈ (0 , ∞ ) , we have P [ Z π ( T ) ≥ Z µ ( T )] = 1 , P [ Z π ( T ) > Z µ ( T )] > 0 . And not only do such relative arbitrages exist; they can be described, even constructed, fairly explicitly. 5. DIVERSITY-WEIGHTING & ARBITRAGE For fixed p ∈ (0 , 1), set ( µ i ( t )) p π i ( t ) ≡ π ( p ) ( t ) := j =1 ( µ j ( t )) p , i = 1 , . . . , n . � n i Relative to the market portfolio µ ( · ), this π ( · ) de- creases slightly the weights of the largest stock(s), and increases slightly those of the smallest stock(s), while preserving the relative rankings of all stocks . 12
We shall show that, in a weakly-diverse market M , this portfolio ( µ i ( t )) p π i ( t ) ≡ π ( p ) ( t ) := , i = 1 , . . . , n , � n i j =1 ( µ j ( t )) p � �� � for some fixed p ∈ (0 , 1), satisfies 2 P [ Z π ( T ) > Z µ ( T )] = 1 , ∀ T ≥ T ∗ := pκδ · log n . In particular, π ( p ) ( · ) represents an arbitrage oppor- tunity relative to the market-portfolio µ ( · ). Suitable modifications of π ( p ) ( · ) can generate � such arbitrage over arbitrary time-horizons. The significance of such a result, for practical long- term portfolio management, cannot be overstated. Discussion and performance charts can be found in the monograph Fernholz (2002). 13
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