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(·) = (X1(·), · · · , Xn(·))′ 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(·) := X1(·) + · · · + Xn(·) is the total capitalization, and Z1(·) := X1(·) X(·) , · · · , Zn(·) := Xn(·) X(·) , the corresponding relative market weights.

The vector Z(·) = (Z1(·), · · · , Zn(·))′ of these weights is a semi- martingale with values in the interior ∆o of the simplex ∆ :=

• (z1, · · · , zn)′ = z ∈ [0, 1]n :

n

• i=1

zi = 1

• ;

Γ := ∆ \ ∆o will be the boundary of ∆ .

• 2. PORTFOLIO

π(·) = (π1(·), · · · , πn(·))′ is an F−predictable process with n

i=1 π1(·) ≡ 1 . Collection Π .

Here πi(t) stands for the proportion of wealth V v,π(t) that gets invested at time t > 0 in the ith asset, for each i = 1, · · · , n .

Dynamics of wealth corresponding to portfolio π(·) is d V v,π(t) V v,π(t) =

n

• i=1

πi(t) dXi(t) Xi(t) , V v,π(0) = v . Portfolios with values in ∆ , that is with π1(·) ≥ 0 , · · · , πn(·) ≥ 0 , will be called “long-only”. The most conspicuous long-only port- folio is the Market Portfolio Z(·) = (Z1(·), · · · , Zn(·))′ 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

P

• V 1,π(T) ≥ V 1,ρ(T)
• = 1

and

P

• V 1,π(T) > V 1,ρ(T)
• > 0 .
• When in fact

P

• V 1,π(T) > V 1,ρ(T)
• = 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(·) X(·) = V qX(0),π(·) V X(0),Z(·) , 0 < q < ∞ ; 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 d Y q,π(t) Y q,π(t) =

n

• i=1

πi(t) dZi(t) Zi(t) , Y q,π(0) = q . (1)

It is convenient to describe π(·) in terms of its scaled relative weights ψ(·) = (ψ1(·), · · · , ψn(·))′ , with ψi(·) := πi(·) / Zi(·) . Then the relative performance dynamics (1) become d Y q,π(t) Y q,π(t) =

n

• i=1

ψi(t) dZi(t) = ψ′(t) dZ(t) ,

• Y q,π(0) = q .

Since n

i=1 Zi(t) ≡ 1 , the vector ψ(t) of scaled portfolio weights

need be specified only modulo a scalar factor; then we can re- cover from ψ1(t), · · · , ψn(t) the ordinary portfolio weights πi(t) = Zi(t)

• ψi(t) + 1 −

n

• j=1

Zj(t)ψj(t)

• ,

i = 1, · · · , n . (2)

• 4. RELATIVE ARBITRAGE FUNCTION

The smallest amount of relative initial wealth U(T, z) := inf

• q > 0 : ∃

π(·) ∈ Π s.t.

P

• Y q,

π(T) ≥ 1

• = 1
• 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 πq(·) ∈ Π such that Y q,πq(T) ≥ 1 ; equivalently, V 1,πq(T) V 1,Z(T) ≥ 1 q > 1 , holds P − a.s. 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

Hybrid Markovian-Itˆ

• process model for the ∆o−valued relative

market weight Z(·) =

• Z1(·), · · · , Zn(·)

′ process, of the form

dZ(t) = s

• Z(t)

dW(t) + ϑ(t) dt

• ,

Z(0) = z ∈ ∆o. (3) Here W(·) is an n−dimensional P−Brownian motion; ϑ(·) is

F−progressively measurable and

T

• ϑ(t)
• 2dt < ∞

holds P − a.s. for every T ∈ (0, ∞) ; whereas s(·) = (siν(·))1≤i,ν≤n a volatility matrix-valued function with siν : ∆ → R continuous and

n

• i=1

siν(·) ≡ 0 ν = 1, · · · , n .

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

• f every sub-simplex d ⊂ Γ in k dimensions, k = 1, · · · , n − 1 .

The main “actor” here is the volatility structure s(·) = (siν(·))1≤i,ν≤n . The relative drift (“relative market price of risk”) process ϑ(·) plays a “supporting” rˆ

• le; 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

n

• i=1

zi

aii(z)

z 2

i

• ≥ h ,

∀

z ∈ ∆o .

(5) The weighted relative variance of log-returns in (5) is a measure

• f 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, πi(t) = Zi(t)(c − log Zi(t))

n

j=1 Zj(t)(c − log Zj(t)) ,

i = 1, · · · , n .

• 6. A CONCRETE EXAMPLE

A concrete example where (5) is satisfied concerns the model d log Xi(t) =

• κ/Zi(t)
• dt +
• 1/
• Zi(t)
• dWi(t)

for the log-capitalizations log Xi(t) , i = 1, · · · , n with some con- stant κ ≥ 0 , or equivalently dZi(t) = 2κ

• 1 − n Zi(t)
• dt +
• Zi(t) dWi(t) − Zi(t)

n

• k=1
• Zk(t) dWk(t)

for the market weight process Z(·) . (Stabilization by Volatility.) The variances turn out to be of the Wright-Fisher type aii(z) = zi(1 − zi) ; 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)

i

(·) = πi(·)/Zi(·) and ψ(ν)

i

(·) = νi(·)/Zi(·) . Itˆ

• ’s rule

d

• Y r,π(t)

Y r,ν(t)

• =
• Y r,π(t)

Y r,ν(t) ψ(π)(t)−ψ(ν)(t)

′

dZ(t)−a

• Z(t)
• ψ(ν)(t)dt
• .

(6) The finite-variation part of this expression vanishes, if ν(·) has scaled relative weights ψ(ν)

1

(·), · · · , ψ(ν)

n

(·) that satisfy

• s(Z(·))

′ ψ(ν)(·) = ϑ(·) .

(7)

With ν(·) ≡ νP(·) selected this way, the ratio Y r,π(·)/Y r,νP(·) is, for any portfolio π(·) ∈ Π , a positive local martingale, thus also a supermartingale. We express this by saying that the portfolio νP(·) 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.

• With ν(·) ≡ νP(·) selected as in (2) via (7), i.e.,

νi(t) Zi(t) = ψ(ν)

i

(t) + 1 −

n

• j=1

Zj(t)ψ(ν)

j

(t) with (s(Z(·)))′ψ(ν)(·) = ϑ(·), the expression (6) gives d log

• Y r,π(t)

Y r,νP(t)

• =
• ψt − ψ(ν)

t

′ s(Zt) dW(t)

− 1 2

• ψt − ψ(ν)

t

′ a(Zt)

• ψt − ψ(ν)

t

• dt
• .

We deduce the relative log-optimality property of νP(·) : for every portfolio π(·) ∈ Π and (T, r) ∈ (0, ∞)2 , we have

E P

• log Y r,π(T)

r

• ≤ E P
• log Y r,νP(T)

r

• = 1

2 E P

T

0 ϑ(t)2 dt .

We also have that the so-called deflator process, the performance

• f the market relative to the the num´

eraire portfolio νP(·) , i.e., 1 Λ(·) = exp

• −

·

0 ϑ′(t) dW(t) − 1

2

·

0 ||ϑ(t)||2 dt

• ≡

r Y r,νP(·) is a strictly positive local martingale and supermartingale.

• We need not assume a priori that this local martingale is a

true martingale.

• But we are assuming it is strictly positive: for every T ∈ (0, ∞) ,

T

• ϑ(t)
• 2dt < ∞

holds P − a.s. Thanks to this assumption: NoUPBR.

• 8. U(·, ·) AS SMALLEST SUPERSOLUTION OF PDE

Under regularity conditions on the covariance structure a(·) of (4), and on the relative drift ϑ(·) , the arbitrage function U(· , ·) is of class C1,2 on (0, ∞) × ∆o and satisfies there the equation DτU(τ, z) = 1 2

n

• i=1

n

• j=1

aij(z) D2

ijU(τ, z) ,

• r DτU =

1 2 Tr

• a D2U
• ; and that U(· , ·) is also the smallest

nonnegative supersolution of this equation, subject to U(0, z) ≡ 1 , ∀

z ∈ ∆o.

• 9. U(·, ·) AND THE F ¨

OLLMER “EXIT MEASURE” Under “canonical” conditions on the filtered space (Ω, F), F = {F(t)}0≤t<∞ , there exists a probability measure

Q , the so-called

F¨

• llmer exit measure, such that: The process
• W(t) := W(t) +

t

0 ϑ(s) ds ,

0 ≤ t < ∞ , is a

• Q−Brownian motion; the exponential

Λ(t) := exp

t

0 ϑ′(s) d

W(s) − 1 2

t

0 ||ϑ(s)||2 ds

• ,

t ≥ 0 , the reciprocal of our deflator process, is a

• Q−martingale, indeed

P(A) =

• A Λ(t) d

Q ,

A ∈ F(t) ;

whereas the relative market weight process Z(·) is a

Q−martingale

and Markov process, with values in ∆ and “purely diffusive” dy- namics Z(t) = s

• Z(t)
• d

W(t) , Z(0) = z ∈ ∆o. (8) Thus, viewed as a portfolio, Z(·) has the num´ eraire property under

• Q : Z(·) ≡ ν

Q(·) .

• If we consider the first time

T := inf

• t ≥ 0 : Z(t) ∈ Γ
• that Z(·) reaches the boundary of the simplex ∆ , we can rep-

resent the arbitrage function as U(T, z) =

Qz

T > T

• ,

(T, z) ∈ (0, ∞) × ∆o .

The arbitrage function emerges thus as the probability, under the F¨

• llmer measure, that Z(·) has not reached the boundary

Γ of the simplex by time t = T .

• Please think of the passage from the original measure P to

the F¨

• llmer measure
• Q , as a Girsanov-like change of probability

that “removes the drift” in (3), when the deflator process 1 Λ(·) = exp

• −

·

0 ϑ′(t) dW(t) − 1

2

·

0 ||ϑ(t)||2 dt

• ≡

r Y r,νP(·) is a strict local martingale under P, i.e., when U(T, z) < 1.

The process Λ(·) can reach the origin with positive

Q−probability,

so this is in general not an equivalent change of measure. Nonetheless, the market weight process Z(·) is a

• Q−martingale,

so we can think of the F¨

• llmer measure
• Q as a “martingale

measure” for the model under consideration.

• It can be shown that

T = inf{t ≥ 0 : Λ(t) = 0} holds

• Q−a.s., and that the arbitrage function admits also the

representation U(T, z) = E Pz 1 Λ(T)

• .

• 10. CONDITIONING, CLASS P

Let us consider the class P of probability measures P that satisfy

P(Z(t) ∈ ∆o, ∀ 0 ≤ t ≤ T) = 1 . An element of this space is P⋆(A) := Q ( A | T > T ) ,

A ∈ F(T) , (9) the conditioning of the F¨

• llmer measure
• Q on the event that

Z(·) has not reached the boundary Γ of the simplex by time T. One computes,

Q−a.s.:

d P⋆ d

Q

• F(t)

= U(T − t, Z(t)) U(T, z)

1{T >t} =

• Y (t)
• Y (0)

=: ΛP⋆(t) .

Here with q = U(T, z) we have set

• Y (t) := U(T − t, Z(t)) 1{T >t} ≡ Y q,

π(·) ,

0 ≤ t ≤ T and π(·) is the functionally-generated portfolio

• πi(t)

Zi(t) = Di log U

• T − t, Z(t)
• + 1 −

n

• j=1

Zj(t) Dj log U

• T − t, Z(t)
• .

(10) This portfolio has the num´ eraire property under the measure P⋆

• f (9), to wit,
• π(·) ≡ νP⋆(·) ; and if U(T, z) < 1 , it implements

the optimal arbitrage under the original probability measure.

• 11. A STOCHASTIC GAME

The pair (P⋆ , π(·)) of (9), (10) is a saddle point in P × Π for the zero-sum stochastic game with value log(1/U(T, z)) = EP⋆

• log Y q,

π(T)

q

• =

min

P ∈ P

max

π(·) ∈ Π EP

• log Y q,π(T)

q

• =

max

π(·) ∈ Π min

P ∈ P EP

• log Y q,π(T)

q

• ;

in particular, for every (P , π(·)) ∈ P×Π and q ∈ (0, ∞) , we have

EP

• log Y q,

π(T)

q

• ≥ EP⋆
• log Y q,

π(T)

q

• ≥ EP⋆
• log Y q,π(T)

q

• .

• 12. A RECIPE

To recapitulate: we can characterize the portfolio

• π(·) of (10)

that implements optimal arbitrage over [0, T] as follows:

• Given the market weight covariance structure, start with a

probability measure

Q that generates driftless, Markovian market

weights as in dZ(t) = s

• Z(t)
• d

W(t) , Z(0) = z ∈ ∆o ;

• then construct the measure P⋆ by conditioning

Q on {T > T}

as in

P⋆(A) := Q ( A | T > T ) ,

A ∈ F(T);

• finally, find a portfolio
• π(·) that maximizes expected log-

returns under P⋆ .

• 13. MINIMAL ENERGY AND ENTROPY

With HT( P |

Q ) := EP

• log
• d P/d

Q

• F(T)
• = 1

2 EP

T

• ϑ P(t)
• 2 dt

we have the “minimum entropy and energy” properties log

• 1/U(T, z)
• = HT( P⋆ |

Q ) = min

P ∈ P HT( P |

Q )

= 1 2 E P⋆

T

• ϑ P⋆(t)
• 2 dt = min

P ∈ P

1 2 E P

T

• ϑ P(t)
• 2 dt .

We call P⋆ “minimal energy” measure in P : it corresponds to the relative risk process ϑ(·) that keeps the markets weight strictly positive throughout [0,T] with the minimal effort.

• 14. BIBLIOGRAPHY

DELBAEN, F. & SCHACHERMAYER, W. (1995) Arbitrage pos- sibilities in Bessel processes and their relations to local martin-

• gales. Probability Theory & Related Fields 102, 357-366.

DELBAEN, F. & SCHACHERMAYER, W. (1995) The no-arbitrage property under a change of num´

• eraire. Stochastics & Stochas-

tics Reports 53, 213-226. DELBAEN, F. & SCHACHERMAYER, W. (2006) The Mathe- matics of Arbitrage. Springer Verlag, New York.

FERNHOLZ, D. & KARATZAS, I. (2010) Optimal arbitrage. Annals of Applied Probability, to appear. FERNHOLZ, D. & KARATZAS, I. (2010) Probabilistic aspects

• f arbitrage. Volume in Honor of Professor Eckhard Platen, to

be published by Springer. F ¨ OLLMER, H. (1972) The exit measure of a supermartingale.

• Zeit. Wahrscheinlichkeitstheorie verw. Gebiete

21, 154-166. F ¨ OLLMER, H. (1973) On the representation of semimartingales. Annals of Probability 1, 580-589.

FERNHOLZ, E.R. (2002) Stochastic Portfolio Theory. Springer Verlag, New York. FERNHOLZ, E.R. & KARATZAS, I. (2009) Stochastic Portfo- lio Theory: A Survey. Handbook of Numerical Analysis, special volume on Mathematical Modeling and Numerical Methods in Fi- nance (A. Bensoussan and Q. Zhang, eds.), pp. 89-168. Elsevier Publishers BV, Amsterdam. PAL, S. & PROTTER, Ph. (2008) Strict local martingales, bubbles, and no early exercise. Preprint, Cornell University. RUF, J. (2010) Hedging under arbitrage. Mathematical Finance, under revision.

THANK YOU, WALTER, FOR ALL YOU HAVE DONE FOR OUR FIELD.