PROBABILISTIC ASPECTS OF ARBITRAGE IOANNIS KARATZAS INTECH - PowerPoint PPT Presentation

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.