PROBABILISTIC ASPECTS OF ARBITRAGE IOANNIS KARATZAS INTECH Investment Management LLC, Princeton, and Department of Mathematics, Columbia University, New York Joint work with D. FERNHOLZ, Austin, Texas Talk at Walter-Schachermayer-Fest, Vienna, July 2010
“ YOU CANNOT BEAT THE MARKET ” In mathematical terms: absence/existence of Arbitrage. 30’s: DeFinetti’s “Theory of Coherence” 70’s: Ross, Harrison, Kreps, Pliska,... 80’s: Dalang, Morton, Willinger,.... 90’s: Delbaen, Schachermayer, Levental, Skorohod,... More often than not: in the context of one risky asset (“stock”) and one riskless (“money market”).
A possible approach: Try to find conditions under which you cannot, and conditions under which you might (be able to outperform an equity mar- ket)... and then, show how. For instance: examples from the early 1990’s involving Bessel processes, due to Delbaen, Shirakawa, Schachermayer, Skoro- hod. And do it in the context of a large equity market – not for a single risky asset.
1. PRELIMINARIES Canonical filtered probability space (Ω , F , P ) , F = {F ( t ) } 0 ≤ t< ∞ . Vector X ( · ) = ( X 1 ( · ) , · · · , X n ( · )) ′ of strictly positive semimartin- gales; they represent the capitalizations of various assets in an equity market, say n = 500 or n = 7 , 000 . Then X ( · ) := X 1 ( · ) + · · · + X n ( · ) is the total capitalization, and Z 1 ( · ) := X 1 ( · ) · · · , Z n ( · ) := X n ( · ) X ( · ) , X ( · ) , the corresponding relative market weights.
The vector Z ( · ) = ( Z 1 ( · ) , · · · , Z n ( · )) ′ of these weights is a semi- martingale with values in the interior ∆ o of the simplex n � � � ( z 1 , · · · , z n ) ′ = z ∈ [0 , 1] n : ∆ := z i = 1 ; i =1 Γ := ∆ \ ∆ o will be the boundary of ∆ . π ( · ) = ( π 1 ( · ) , · · · , π n ( · )) ′ is an F − predictable 2. PORTFOLIO process with � n i =1 π 1 ( · ) ≡ 1 . Collection Π . Here π i ( t ) stands for the proportion of wealth V v,π ( t ) that gets invested at time t > 0 in the i th asset, for each i = 1 , · · · , n .
Dynamics of wealth corresponding to portfolio π ( · ) is n d V v,π ( t ) � π i ( t ) dX i ( t ) V v,π (0) = v . = X i ( t ) , V v,π ( t ) i =1 Portfolios with values in ∆ , that is with π 1 ( · ) ≥ 0 , · · · , π n ( · ) ≥ 0 , will be called “long-only”. The most conspicuous long-only port- folio is the Market Portfolio Z ( · ) = ( Z 1 ( · ) , · · · , Z n ( · )) ′ itself. This takes values in ∆ o and generates wealth proportional to the total market capitalization at all times: V v, Z ( · ) = vX ( · ) /X (0) .
3. RELATIVE ARBITRAGE Given a horizon T ∈ (0 , ∞ ) and any two portfolios π ( · ) and ρ ( · ) , we say that π ( · ) is an arbitrage relative to ρ ( · ) over [0 , T ], if we have � � � � V 1 ,π ( T ) ≥ V 1 ,ρ ( T ) V 1 ,π ( T ) > V 1 ,ρ ( T ) = 1 and > 0 . P P • When in fact � � V 1 ,π ( T ) > V 1 ,ρ ( T ) P = 1 , we call such relative arbitrage strong .
We shall be interested in performance relative to the mar- ket , and consider for any given portfolio π ( · ) ∈ Π its relative performance Y q,π ( · ) := V qX (0) ,π ( · ) = V qX (0) ,π ( · ) V X (0) , Z ( · ) , 0 < q < ∞ ; X ( · ) the scalar parameter q > 0 measures initial wealth v = qX (0) as a proportion of total market capitalization at time t = 0 . The dynamics of this relative performance process are n d Y q,π ( t ) � π i ( t ) dZ i ( t ) Y q,π (0) = q . = (1) Z i ( t ) , Y q,π ( t ) i =1
It is convenient to describe π ( · ) in terms of its scaled relative weights ψ ( · ) = ( ψ 1 ( · ) , · · · , ψ n ( · )) ′ , with ψ i ( · ) := π i ( · ) / Z i ( · ) . Then the relative performance dynamics (1) become n d Y q,π ( t ) � ψ i ( t ) dZ i ( t ) = ψ ′ ( t ) d Z ( t ) , Y q,π (0) = q . = Y q,π ( t ) i =1 � �� � Since � n i =1 Z i ( t ) ≡ 1 , the vector ψ ( t ) of scaled portfolio weights need be specified only modulo a scalar factor; then we can re- cover from ψ 1 ( t ) , · · · , ψ n ( t ) the ordinary portfolio weights � � n � π i ( t ) = Z i ( t ) ψ i ( t ) + 1 − Z j ( t ) ψ j ( t ) , i = 1 , · · · , n . j =1 (2)
4. RELATIVE ARBITRAGE FUNCTION The smallest amount of relative initial wealth � � � � Y q, � π ( T ) ≥ 1 U ( T, z ) := inf q > 0 : ∃ � π ( · ) ∈ Π s.t. = 1 P required at t = 0 , in order to attain at time t = T relative wealth of (at least) 1 with respect to the market, P − a.s. Equivalently, 1 /U ( T, z ) gives the maximal relative amount by which the market portfolio can be outperformed over [0 , T ] . We have 0 < U ( T, z ) ≤ 1 . We shall try to obtain several charac- terizations of this function, look at it through many lenses; and describe a (super-hedging) portfolio � π ( · ) as well.
(The inequality U ( T, z ) > 0 is a consequence of conditions to be imposed: these amount to NoBPBR. ) If U ( T, z ) = 1 , it is not possible to outperform (“beat”) the • market over [0 , T ] . • If U ( T, z ) < 1 , then for every q ∈ ( U ( T, z ) , 1) – and even for q = U ( T, z ) when the infimum is attained – there exists a portfolio π q ( · ) ∈ Π such that Y q,π q ( T ) ≥ 1 ; equivalently, V 1 ,π q ( T ) ≥ 1 q > 1 , holds P − a.s. V 1 , Z ( T ) That is, strong arbitrage relative to the market portfolio Z ( · ) exists then over the time-horizon [0 , T ] . ¶ In order to be able to say something about this function U ( · , · ) , we need a Model.
5. MODEL: MARKET WEIGHTS DIRECTLY o process model for the ∆ o − valued relative Hybrid Markovian-Itˆ � � ′ process, of the form market weight Z ( · ) = Z 1 ( · ) , · · · , Z n ( · ) � � � � Z (0) = z ∈ ∆ o . d Z ( t ) = s Z ( t ) dW ( t ) + ϑ ( t ) dt , (3) Here W ( · ) is an n − dimensional P − Brownian motion; ϑ ( · ) is F − progressively measurable and � T � � 2 dt < ∞ � � � ϑ ( t ) holds P − a . s . � 0 for every T ∈ (0 , ∞ ) ; whereas s( · ) = (s iν ( · )) 1 ≤ i,ν ≤ n a volatility matrix-valued function with s iν : ∆ → R continuous and n � s iν ( · ) ≡ 0 ν = 1 , · · · , n . i =1
We shall assume that the corresponding covariance matrix a( z ) := s( z ) s ′ ( z ) , z ∈ ∆ (4) has rank n − 1 , ∀ z ∈ ∆ o , as well as rank k − 1 in the interior d o of every sub-simplex d ⊂ Γ in k dimensions, k = 1 , · · · , n − 1 . The main “actor” here is the volatility structure s( · ) = (s iν ( · )) 1 ≤ i,ν ≤ n . The relative drift (“relative market price of risk”) process ϑ ( · ) plays a “supporting” rˆ ole; more on this distinction below....
¶ With such a model, a sufficient condition for U ( T, z ) < 1 is that there exist a real constant h > 0 for which � a ii ( z ) � n � z ∈ ∆ o . z i ≥ h , ∀ (5) z 2 i =1 i The weighted relative variance of log-returns in (5) is a measure of the market’s “intrinsic volatility”; condition (5) posits a pos- itive lower bound on this quantity as sufficient for U ( T, z ) < 1 . Under this condition, very simple long-only portfolios lead to ar- bitrage relative to the market. For instance, under the condition (5) and for c > 0 sufficiently large, Z i ( t )( c − log Z i ( t )) π i ( t ) = i = 1 , · · · , n . j =1 Z j ( t )( c − log Z j ( t )) , � n
6. A CONCRETE EXAMPLE A concrete example where (5) is satisfied concerns the model � � � � � d log X i ( t ) = κ/Z i ( t ) dt + 1 / Z i ( t ) dW i ( t ) for the log-capitalizations log X i ( t ) , i = 1 , · · · , n with some con- stant κ ≥ 0 , or equivalently � n � � � � dZ i ( t ) = 2 κ 1 − n Z i ( t ) dt + Z i ( t ) dW i ( t ) − Z i ( t ) Z k ( t ) dW k ( t ) k =1 for the market weight process Z ( · ) . (Stabilization by Volatility.) The variances turn out to be of the Wright-Fisher type a ii ( z ) = z i (1 − z i ) ; so (5) holds as equality for h = n − 1 ≥ 1.
7. NUM´ ERAIRE AND LOG-OPTIMALITY PROPERTIES Two portfolios π ( · ) , ν ( · ) with corresponding scaled relative weights ψ ( π ) ψ ( ν ) ( · ) = π i ( · ) /Z i ( · ) and ( · ) = ν i ( · ) /Z i ( · ) . i i Itˆ o’s rule � � � � � Y r,π ( t ) Y r,π ( t ) � ′ � � � � ψ ( π ) ( t ) − ψ ( ν ) ( t ) ψ ( ν ) ( t ) dt d = d Z ( t ) − a Z ( t ) . Y r,ν ( t ) Y r,ν ( t ) (6) The finite-variation part of this expression vanishes, if ν ( · ) has scaled relative weights ψ ( ν ) ( · ) , · · · , ψ ( ν ) ( · ) that satisfy n 1 � � ′ ψ ( ν ) ( · ) = ϑ ( · ) . s( Z ( · )) (7)
With ν ( · ) ≡ ν P ( · ) selected this way, the ratio Y r,π ( · ) /Y r,ν P ( · ) is, for any portfolio π ( · ) ∈ Π , a positive local martingale, thus also a supermartingale. We express this by saying that the portfolio ν P ( · ) has the “num´ e- raire property”. No arbitrage relative to such a portfolio is possible over ANY finite time-horizon. And if ϑ ( · ) ≡ 0 , then the market portfolio Z ( · ) itself has the num´ eraire property. ¶ Indeed, “You cannot beat the market” portfolio, if it has the num´ eraire property.
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