EXPLOSIONS AND ARBITRAGE IOANNIS KARATZAS. Joint work with Daniel FERNHOLZ and Johannes RUF. Talk at the University of Michigan

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1 EXPLOSIONS AND ARBITRAGE IOANNIS KARATZAS Department of Mathematics, Columbia University, New York INTECH Investment Management LLC, Princeton Joint work with Daniel FERNHOLZ and Johannes RUF Talk at the University of Michigan Ann Arbor, October 2014 For Information Purposes Only

2 PART ONE: A CLASSICAL SETTING DISTRIBUTION OF THE TIME-TO-EXPLOSION FOR LINEAR DIFFUSIONS

3 I.1: STOCHASTIC DIFFERENTIAL EQUATION dx(t) = s(x(t)) [ dw (t) + b(x(t))dt ], X(0) = ξ I The state-space is an open subinterval I = (l, r) R of the real line. Here W ( ) is standard Brownian motion, and b : I R, s : I R \ {0} are measurable functions. Standing Assumption: The function 1/s 2 ( ) and the local mean/over/variance (or signal-to-noise ratio ) function f( ) := b( ) s( ) are locally integrable over I. = b( ) s( ) s 2 ( )

4 . Under these conditions, there exists a weak solution of the above SDE, defined up until the so-called explosion time S := lim n S n, S n = inf{t 0 : X(t) / (l n, r n )} for l n l, r n r. This solution is unique in distribution. (ENGELBERT & SCHMIDT 1984, 1991.) We know that P(S = ) = 1 holds under the familiar linear growth conditions of the Itô theory, when I = R.

5 More generally, fixing a reference point c I and introducing the Feller function x y ( y ) dz v(x) := exp 2 f(u)du c c z s 2 (z) dy, x I, we have: P(S = ) = 1 if and only if v(l+) = v(r ) =. This is the classical FELLER test for explosions. QUESTION (posed to us by Marc YOR):. If this condition fails and P(S < ) > 0, what can we say about the distribution function P(S T ), 0 < T < of the explosion time?

6 I.2: A GENERALIZED-GIRSANOV / McKEAN IDENTITY Let us consider the diffusion in natural scale dx o (t) = s(x o (t)) dw o (t), X(0) = ξ I with explosion time S o ; clearly, Q(S o = ) = 1 if I = R. Here W o ( ) is Brownian motion under another probability measure Q. Suppose that the mean/variance function f( ) is locally squareintegrable on I, and define the exponential Q local martingale L( ; X o ) := exp = exp { { 0 b(xo (t)) dw o (t) f(xo (t)) dx o (t) b2 (X o (t)) dt 0 b2 (X o (t)) dt } } on [0, S o ).

7 Then for T (0, ) and bounded, B T measurable f T : Ω R, E P[ f T (X) 1 {S>T } ] = E Q [ L(T ; X o ) f T (X o ) 1 {S o >T } ]. A couple of early lessons from this identity:. The exponential process L( ; X o ) is a true Q martingale, if both X( ) and X o ( ) are non-explosive, that is, if P(S = ) = Q(S o = ) = 1.. Suppose only X( ) is non-explosive: P(S = ) = 1. Then holds, and E P[ f T (X) ] = E Q[ L(T ; X o ) f T (X o ) 1 {S o >T } E P ( 1 L(T ; X) ) = Q ( S o > T ), T (0, ). ]

8 Please note that, always under the assumption P(S = ) = 1, the exponential process 1 L( ; X) = exp = exp { 0 f(x(t)) dx(t) { b(x(t)) dw (t) b2 (X(t)) dt } 0 b2 (X(t)) dt } is a strictly positive P local martingale (and supermartingale), with Q ( S o > T ) ( ) = E P 1 ; L(T ; X) it is a true P martingale, if and only if Q(S o = ) = 1.

9 . When f( ) is actually continuous and continuously differentiable on I, the above expression gives [ P ξ (S > T ) = E Q exp where ( X o (T ) ξ f(z) dz T 0 V (Xo (t))dt V (x) := 1 2 s2 (x) ( f 2 (x) + f (x) ). ) 1 {S o >T } ]. And in a totally symmetrical fashion: Q ξ (S o > T ) = E P [ exp ( X(T ) ξ f(z) dz + T 0 V (X(t))dt ) 1 {S>T } ].

10 I.3: RESULTS: We have the following, general results. PROPOSITION 1: Positivity, Full Support. The function [0, ) I (T, ξ) U(T, ξ) := P ξ (S > T ) (0, 1] is (strictly positive and) continuous; as well as strictly decreasing in T ( ), when P ξ (S < ) > 0. (***) Last result (strict decrease) needs the local square-integrability of 1/s 2 ( ) on I (with the possible exception of finitely many points). This assumption guarantees that the diffusion can reach far away points fast, with positive probability.

11 PROPOSITION 2: The continuous function U(, ) of [0, ) I (T, ξ) U(T, ξ) := P ξ (S > T ) (0, 1] is dominated by every nonnegative, classical (super)solution of the Cauchy problem U τ (τ, x) = s2 (x) 2 2 U U (τ, x) + b(x)s(x) (τ, x), x2 x τ > 0, x I U(0, x) = 1, x I.. Please note that this characterization is completely impervious to the boundary behavior of the diffusion X( ) at the endpoints of its state-space I = (l, r).

12 PROPOSITION 3: Minimality. Suppose that both functions s( ), b( ) are locally Hölder-continuous on I.. Then U(, ) solves this Cauchy problem, and is its smallest nonnegative classical (super)solution.. And if U(, ) 1 (i.e., if our SDE is non-explosive), then the above Cauchy problem has a unique bounded classical solution, namely, U(, ) 1. RECENT IMPORTANT WORK: Vast generalizations of these results, in the viscosity and generalized solution framework, when the functions s( ), b( ) are simply continuous, have been carried out and in several dimensions by Yinghui WANG (2014).

13 PROPOSITION 4: A Generalized FELLER Test. The following conditions are equivalent: (i) The diffusion X( ) has no explosions, i.e., P(S = ) = 1 ; (ii) v(l+) = v(r ) = hold for the Feller test function; (iii) The truncated exponential Q-supermartingale ( L ( ; X o ) = exp 0 b(xo (t))dw o (t) 1 2 is a true Q-martingale. 0 b2 (X o (t))dt ) 1 {S o > }. If the functions s( ) and b( ) are locally Hölder-continuous on I, then the conditions (i) (iii) are equivalent to: (iv) The smallest nonnegative classical solution of the above Cauchy problem is U(, ) 1; (iv) The unique bounded classical solution of the Cauchy problem is U(, ) 1.

14 I.4: AN EXAMPLE: Bessel Process in dimension δ (1, 2). dx(t) = δ 1 2X(t) dt + dw (t), X(0) = ξ I = (0, ). The solution of this equation does not explode to infinity, but reaches the origin in finite time: P(S < ) = 1. We have f(x) = 1/2 ν x, V (x) = ν2 1/4 2 x 2 for ν = 1 (δ/2). With X o (t) = ξ + W (t), S o = inf{t 0 : X o (t) = 0}, the representation [ P ξ (S > T ) = E Q exp ( X o (T ) ξ f(z) dz T 0 V (Xo (t))dt ) 1 {S o >T } ]

15 gives P(S > T ) = E Q ( X o (T ) ξ ( X o (T ) ξ ) 2ν ) ν+1/2 exp ( 1/4 ν 2 2 T 0 ) dt (X o (t)) 2 1 {S o >T } = E Qν ( X o (T ) ξ ) 2ν. Here Q ν is the probability measure under which the process X o ( ) = ξ + W ( ) is Bessel in dimension 2ν + 2 = 4 δ > 2,

16 and with the modified Bessel function of the second type this gives P(S > T ) = I ν (u) := 1 T ξν exp n N 0 ( ξ 2 2T (u/2)ν+2n n! Γ(n + ν + 1) ) 0 x1 ν exp ( x 2 2T ) ( ) ξx I ν dx, T. Algebraic manipulation leads now to a simple proof of ( ) ( ) ( ) U(T, ξ) = P ξ S > T = P G < ξ2 ξ 2 = H, 2T 2T a result of Ronald GETOOR (1979), where H(u) := 1 Γ(ν) u 0 t ν 1 exp( t) dt.

17 The resulting function U(T, ξ) = P ξ ( S > T ) = 1 Γ(ν) ξ 2 /(2T ) 0 t ν 1 exp( t) dt is the smallest nonnegative classical solution of the Cauchy problem U T (T, ξ) = U δ 1 (T, ξ) + ξ2 2 ξ U (T, ξ), (T, ξ) (0, ) I, ξ U(0+, ξ) = 1, ξ I.

18 Many more such (one-dimensional) examples are possible; a small parlor game.. A somewhat amusing corollary of one of these examples goes as follows (extends and simplifies a result of DELBAEN and SHIRAKAWA (2002)): PROPOSITION 5. Suppose that the function 1/s( ) is locally square-integrable on I = (0, ), and that the diffusion X o ( ) in natural scale dx o (t) = s(x o (t)) dw o (t), X(0) = ξ I becomes absorbed at the origin the first time it gets there. Then X o ( ) is a nonnegative local martingale and supermartingale; it is a true martingale, IF AND ONLY IF 0 z dz s 2 (z) =.

19 PART TWO: A MORE ELABORATE SETTING OPTIMAL ARBITRAGE RELATIVE TO THE MARKET PORTFOLIO

20 II.1: PRELIMINARIES Filtered probability space (Ω, F, P), F = {F(t)} 0 t<. Vector X( ) = (X 1 ( ),, X n ( )) of strictly positive semimartingales; these represent the capitalizations of assets in an equity market, say n = 7, 000. Then X( ) := X 1 ( ) + + X n ( ) is the total capitalization, and Z 1 ( ) := X 1( ) X( ),, Z n( ) := X n( ) X( ), the corresponding relative market weights.

21 The vector Z( ) = (Z 1 ( ),, Z n ( )) of these weights is a semimartingale with values in the interior o of the simplex := {(z 1,, z n ) [ 0, 1 ] n n } : z i = 1 ; Γ := \ o will be the boundary of. We shall denote (z 1,, z n ) =: z. i=1 II.2: PORTFOLIO π( ) = (π 1 ( ),, π n ( )) is a bounded, F predictable process with n i=1 π i ( ) 1. We denote the resulting collection by Π. Here π i (t) stands for the proportion of wealth V π (t) that gets invested at time t > 0 in the i th asset, for each i = 1,, n.

22 Dynamics of wealth corresponding to portfolio π( ) is multiplicative in the initial wealth, and given by d V π (t) V π (t) = n i=1 π i (t) dx i(t) X i (t), V π (0) = 1$.. Portfolios with values in, that is, with will be called long-only. π 1 ( ) 0,, π n ( ) 0,. The most conspicuous long-only portfolio 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 Z ( ) = X( )/X(0).

23 II.3: ARBITRAGE Given a horizon T (0, ) and two portfolios π( ) and ρ( ), we say that π( ) is arbitrage relative to ρ( ) over [0, T ], if P ( V π (T ) V ρ (T ) ) = 1 and P ( V π (T ) > V ρ (T ) ) > 0. When in fact P ( V π (T ) > V ρ (T ) ) = 1, we call such relative arbitrage strong.

24 As we shall be interested in performance with respect to the market, we consider for any given portfolio π( ) Π its relative performance Y π ( ) := V π ( ) V Z ( ), with d Y π (t) Y π (t) = n Equivalently, write d Y π (t) Y π (t) = n i=1 i=1 π i (t) dz i(t) Z i (t) = n with the portfolio proportions expressed as π i (t) = Z i (t) Ψ i (t) + 1 n j=1 i=1 π i (t) dz i(t) Z i (t). Ψ i (t) dz i (t), Ψ j (t) Z j (t), i = 1,, n. The process Ψ( ) = (Ψ 1 ( ),, Ψ n ( )) in this scheme of things generates the portfolio process π( ) = (π 1 ( ),, π n ( )).

25 II.4: RELATIVE ARBITRAGE FUNCTION The smallest amount of relative initial wealth 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.: U(T, z) := inf { q (0, 1] : π( ) Π s.t. P ( q Y π (T ) 1 ) = 1 }.. 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 characterizations of this function.

26 The strict inequality U(T, z) > 0 is a consequence of conditions to be imposed below. These amount to NUPBR (No Unbounded Profits with Bounded Risk): Absence of Egregious arbitrages. When U(T, z) = 1, it is not possible to outperform ( beat ) the market over [0, T ]. When U(T, z) < 1, there exists for every q [ U(T, z), 1) a portfolio π q ( ) Π such that q Y πq (T ) 1, i.e., V πq (T ) V Z (T ) 1 q > 1, holds P a.s. 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 : i.e., some specification of dynamics.

27 II.5: MARKET WEIGHT MODEL Hybrid Markovian-Itô process dymanics for the o valued relative market weights Z( ) = ( Z 1 ( ),, Z n ( ) ), of the form dz(t) = s ( Z(t) ) ( dw (t) + ϑ(t) dt ), Z(0) = z o. Here W ( ) is an n dimensional P Brownian motion; the relative drift process ϑ( ) is F progressively measurable and satisfies T 0 ϑ(t) 2 dt <, P a.s. for every T (0, ) ;

28 whereas s( ) = (s iν ( )) 1 i,ν n s iν : R continuous, is a matrix-valued function with n i=1 s iν ( ) 0, ν = 1,, n.. We shall assume that the corresponding covariance matrix a(z) := s(z) s (z), z 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.

29 II.6: PREVIEW Within such a framework, we shall characterize the arbitrage function (0, ) o (T, z) U(T, z) (0, 1] in various equivalent ways, and provide necessary and sufficient conditions for U(T, z) < 1. We shall also use this function U(, ) to describe the portfolio that implements the optimal (i.e., best achievable) arbitrage. The quantity U(T, z) is a number in the interval (0,1]. So it is the probability of some event. Which event? Under what probability measure? We shall try to answer these questions.

30 II.7: NUMÉRAIRE PORTFOLIO, LOG-OPTIMALITY Recall the relative portfolio dynamics in the form d Y π (t) Y π (t) = n i=1 π i (t) dz i(t) Z i (t) = n i=1 Ψ (π) i (t) dz i (t) where we are expressing the portfolio proportions as π i (t) = Z i (t) Ψ (π) i (t) + 1 n j=1 Ψ (π) j (t) Z j (t), i = 1,, n. The market portfolio π( ) Z( ) is generated by Ψ (π) ( ) 0.

31 Now, for any two portfolios π( ), ν( ) with corresponding scaled relative weights Ψ (π) i ( ) and Ψ (ν) i ( ) as above, simple calculus gives d ( Y π (t) Y ν (t) ) = ( Y π ) (t) (Ψ (π) Y ν (t) Ψ (ν) (t) ) [ dz(t) a ( Z(t) ) ] Ψ (ν) (t) dt. (t) The finite-variation part of this expression vanishes, if the portfolio ν( ) has scaled relative weights that satisfy the perfect balance condition ( s(z( )) ) Ψ (ν) ( ) = ϑ( ).

32 With ν( ) ν P ( ) selected this way, namely ( s(z( )) ) Ψ (ν) ( ) = ϑ( ) :. For any given portfolio π( ) Π, the ratio Y π ( )/Y νp ( ) is a positive local martingale thus also a supermartingale. We say that this portfolio ν P ( ) has the numéraire property, and that 1/Y νp ( ) is a deflator in this market. No arbitrage relative to a portfolio with the numéraire property is possible, over ANY finite time-horizon.

33 . And if ϑ( ) 0, i.e., dz(t) = s ( Z(t) ) dw (t), then the market portfolio Z( ) itself has the numéraire property. Because then we can take Ψ (ν) ( ) 0, thus ν( ) Z( ). Indeed: You cannot beat the market portfolio, when it has the numéraire property. But this property is (very) special.

34 Log-Optimality of the numéraire portfolio ν P ( ) : For every portfolio π( ) Π and time-horizon T (0, ), we have [ ] E P log Y π (T ) E P [ ] log Y νp (T ) = 12 E P T ϑ(t) 2 dt. 0. The deflator process 1 Y νp ( ) exp { 0 ϑ (t) dw (t) ϑ(t) 2 dt } =: 1 L( ), to wit, the performance of the market relative to the numéraire portfolio ν P ( ), is a strictly positive P local martingale and a supermartingale.

35 . We need not assume and are not assuming a priori, that this local martingale is a true martingale. But we are assuming that it is strictly positive. This is guaranteed by the assumption that, for every T (0, ), T 0 ϑ(t) 2 dt < holds P a.s. Thanks to this assumption there is in this model, as we shall see, No Unbounded Profit with Bounded Risk. No Arbitrage of the First Kind, No Egregious Arbitrage, No Scalable Arbitrage.

36 II.8: U(, ) AND THE FÖLLMER 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 FÖLLMER exit measure, under which is Brownian motion. W o ( ) := W ( ) + 0 ϑ(t) dt. And the performance of the numéraire portfolio ν P ( ) relative to the market, i.e., the reciprocal of our deflator process Y νp ( ) L( ) = exp { is a Q martingale; indeed, P(A) = A 0 ϑ (t) dw o (t) ϑ(t) 2 dt L(T ) dq, A F(T ) ; T (0, ). },

37 . Whereas the relative-market-weight process Z( ) is a Q martingale and Markov process, with values in and purely diffusive Q dynamics (zero drift) dz(t) = s ( Z(t) ) dw o (t), Z(0) = z o. Thus, the market portfolio Z( ) has the numéraire property under the exit measure Q : Z( ) ν Q ( ). If we consider the first time ( explosion, or rather implosion) S := inf { t 0 : Z(t) Γ } that Z( ) reaches the boundary Γ of the unit simplex, the arbitrage function is represented in the already familiar form [ ] U(T, z) = E P 1 ( ) z = Q z S > T, (T, z) (0, ) o. L(T )

38 The relative arbitrage function U(T, z) emerges as the probability under the FÖLLMER measure, that Z( ) has not reached the boundary Γ of the simplex by time t = T, when started at initial configuration z. Tail-distribution of explosion time. Please think of the passage from the original measure P to the FÖLLMER measure Q, as a Girsanov-like change of probability that removes the drift in the dynamics dz(t) = s ( Z(t) ) ( dw (t) + ϑ(t) dt ), when all we can say about the exponential ( deflator ) process 1 L( ) = exp { 0 ϑ (t) dw (t) ϑ(t) 2 dt } 1 Y νp ( ) is that it is a local martingale under P (strict, when U(T, z) < 1).

39 The process L( ) can reach the origin with positive Q probability, so this is in general not an equivalent change of measure: We have P Q, but not necessarily Q P.. Nonetheless, the process Z( ) of market weights is a Q martingale with values in the unit simplex. (Thus, we can think of the FÖLLMER measure Q as an Ersatz martingale measure for the model under consideration.) It can be shown that S = inf{t 0 : L(t) = 0} = inf holds Q a.s. { t 0 : } 1 Y νp ( ) =

40 II.9: U(, ) AS SMALLEST SUPERSOLUTION Under regularity conditions on the covariance structure a( ) and on the relative drift ϑ( ), the arbitrage function U(, ) is of class C 1,2 on (0, ) o and satisfies there the equation D τ U(τ, z) = 1 2 n n i=1 j=1 a ij (z) Dij 2 U(τ, z), or D τ U = 1 2 Tr( a D 2 U ) ; and U(, ) is also the smallest nonnegative supersolution of this equation, subject to U(0+, ) 1.

41 Please note that this equation D τ U = 1 2 Tr( a D 2 U ) involves only the covariance structure of the assets.. The only rôle the relative drift ϑ( ) plays in this context, is to keep the market weight process Z( ) in the interior of the unit simplex. (Once again, this characterization is completely impervious to boundary conditions on the faces of the simplex.). With Knightian uncertainty about the covariance a( ) and the relative drift ϑ( ), this equation becomes fully nonlinear (of HJB-PUCCI type), as in the work of Terry LYONS (1995).. Great generalizations of these results, in the context of viscosity solutions of the fully nonlinear PDE s, appear in very recent work by our own Yinghui WANG (2014).

42 II.10: CONDITIONING, CLASS P Let us consider the collection P of probability measures P Q with P(Z(t) o, 0 t T ) = 1. (Our original measure P belongs to this collection.) We single out an element of P via P (A) := Q ( A S > T ), A F(T ). (1) This is the conditioning of the FÖLLMER measure Q on the set {Z( ) has not reached the boundary of the simplex by time T }. Elementary computations give, Q a.s.: d P d Q = F(t) U(T t, Z(t)) U(T, z) 1 {S>t} =: Ŷ (t) Ŷ (0), 0 t T

43 d P d Q = F(t) U(T t, Z(t)) U(T, z) 1 {S>t} =: Ŷ (t) Ŷ (0), 0 t T with the Q martingale Ŷ (t) := U ( T t, Z(t) ) 1 {S>t} q Y π (t) for q = U(T, z), and with the functionally-generated portfolio π i (t) Z i (t) = D i log U ( T t, Z(t) ) + 1 n j=1 Z j (t) D j log U ( T t, Z(t) ). This portfolio has the numéraire property under the conditioning P of the FÖLLMER measure: π( ) ν P ( ). (2)

44 . Whenever U(T, z) < 1, this portfolio implements the best achievable arbitrage under the original probability measure P ; that is, V π (T ) V Z (T ) = 1 U(T, z) > 1 holds P a.s. II.11: A RECIPE We can characterize the portfolio π( ) of (2) that implements the optimal arbitrage over a given time-horizon [0, T ] as follows, given the market weight covariance structure under the original probability P (and nothing else...):

45 Find a probability measure Q under which the market weights are martingales, as in dz(t) = s ( Z(t) ) d W o (t), Z(0) = z o. Then construct the measure P by conditioning Q on the event {S > T } as in P (A) := Q ( A S > T ), A F(T ). Finally, construct the portfolio π( ) that maximizes expected log-returns (equivalently, has the numéraire property) under P. This portfolio is generated by the vector process of log-derivatives D i log U(T, Z( )), i = 1,, n, i.e., is given by the recipe π i (t) Z i (t) = D i log U ( T t, Z(t) ) + 1 n j=1 Z j (t) D j log U ( T t, Z(t) ).

46 II.12: MINIMAL ENERGY AND ENTROPY With H T ( P Q ) := E P [ ( (d ) )] log P/d Q = 12 T F(T ) EP 0 we have the minimum entropy and energy properties log ( 1/U(T, z) ) = H T ( P Q ) = min P P H T ( P Q ) ϑ P (t) 2 dt = 1 2 E P T 0 ϑ P (t) 2 dt = min P P 1 T 2 E P 0 ϑ P (t) 2 dt. We call P minimal energy measure in P : it corresponds to the relative risk process ϑ( ) that keeps the market weights strictly positive throughout [0,T ] by expending minimal energy.

47 This minimal entropy function H(τ, z) := log ( 1 / U(T, z) ) = H T ( P Q ) solves the HJB equation for this problem D τ H(τ, z) = 1 2 Tr( a(z) D 2 H(τ, z) ) [ (DH(τ, ) s(z) 1 + min z) θ + θ R n 2 θ 2 ], which is of course a semilinear equation D τ H(τ, z) = 1 2 Tr( a(z) D 2 H(τ, z) ) 1 2 ( DH(τ, z) ) s(z) ( DH(τ, z) ).

48 II.13: A STOCHASTIC GAME The pair (P, π( )) of (1), (2) is a saddle point in P Π for the zero-sum stochastic game with value log ( 1/U(T, z) ) = E P [ log Y π (T ) ] = = min P P [ max π( ) Π EP log Y π (T ) ] [ = max min π( ) Π P P EP log Y π (T ) ] ; and for every (P, π( )) P Π we have the saddle E P [ log Y π (T ) ] E P [ log Y π (T ) ] = = log ( 1/U(T, z) ) [ ] E P log Y π (T ).

49 II.14: A SUFFICIENT CONDITION It can be shown that a sufficient condition for U(T, z) < 1 is that there exist a real constant h > 0 for which n i=1 z i ( aii (z) z 2 i ) h, z o. (3) The weighted relative variance of log-returns in (3) is a measure of the market s intrinsic (or average relative ) variance; condition (3) posits a positive lower bound on this quantity as sufficient for U(T, z) < 1.. Under the condition (3), very simple long-only portfolios can be designed, that lead to arbitrage over sufficiently long horizons.

50 For instance, given any real number T > (2 log n)/h, there is c > 0 sufficiently large, so that the portfolio π i (t) = Z i (t)(c log Z i (t)) nj=1 Z j (t)(c log Z j (t)), i = 1,, n is strong arbitrage relative to the market portfolio Z( ) over the time-horizon [0, T ].. OPEN QUESTION: Is arbitrage relative to the market possible under condition (3) over arbitrary time-horizons? (A few additional examples exist, under different structural conditions, and with the equally-weighted portfolio playing a very important rôle. Would be nice to have more of them....)

51 II.15: A CONCRETE TOY-EXAMPLE A concrete example where the condition n i=1 a ii (z) z i h, z o of (3) is satisfied concerns the Volatility-Stabilized Model d log X i (t) = ( κ/z i (t) ) dt + ( 1 / Z i (t) ) dw i (t), i = 1,, n with some constant κ 0, or equivalently for the market weights dz i (t) = 2κ ( 1 n Z i (t) ) dt + = 2κ ( 1 n Z i (t) ) dt + Z i (t) dw i (t) Z i (t) n k=1 Z i (t) 1 Z i (t) dw i # (t). Z k (t) dw k (t)

52 The variances in this last diffusion equation dz i (t) = 2κ ( 1 n Z i (t) ) dt + Z i (t) 1 Z i (t) dw i # (t) (in which the W i # ( ), i = 1,, n are correlated BM s) are of WRIGHT-FISHER type a ii (z) = z i (1 z i ) ; so the condition n i=1 a ii (z) z i h, z o of (3) holds as equality, in fact with h = n 1 1.

53 . Here, and indeed in any setting of the form d log X i (t) = β i (t) dt + ( 1 / Z i (t) ) dw i (t), i = 1,, n, the market CAN be outperformed over arbitrary time horizons (A. BANNER & D. FERNHOLZ (2008), R. PICKOVÁ (2014)). In this case, one can compute the relative arbitrage function [ ] [ ] U(T, z) = E P z 1 z n E P e (n 1)(γ T +W (T )), Z 1 (T ) Z n (T ) because S. PAL (2011) has computed the joint distribution of the weights Z 1 (T ),, Z n (T ) fairly explicitly (Dirichlet). Here γ = 1 ( (1 ) ) + 2κ n 1. 2

54 Under the FÖLLMER measure Q, each weight Z i ( ) is a Wright-Fisher diffusion in natural scale, and reaches an endpoint of (0,1) in finite expected time S i = inf{t 0 : Z i (t) = 0} : dz i (t) = 2κ ( 1 n Z i (t) ) dt + = Z i (t) Z i (t) 1 Z i (t) dwi o (t). 1 Z i (t) dw # i (t) For us, of course, the time of interest is S = min S i. 1 i n (Eventually all but one of the Z i ( ) s vanish, and one of them emerges as the survivor think of a catalytic reaction with n compounds, or of a gladiatorial fight in the Colosseum.)

55 SOURCES FOR THIS TALK KARATZAS, I. & RUF, J. (2013) Distribution of the time to explosion for one-dimensional diffusions. Preprint, available at Probability Theory and Related Fields, under revision. FERNHOLZ, D. & KARATZAS, I. (2010) On optimal arbitrage. Annals of Applied Probability 20, FERNHOLZ, D. & KARATZAS, I. (2010) Probabilistic aspects of arbitrage. In Contemporary Quantitative Finance: Essays in Honor of Eckhard Platen (C. Chiarella & A. Novikov, Eds.), FERNHOLZ, D. & KARATZAS, I. (2011) Optimal arbitrage under model uncertainty. Annals of Applied Probability 21,

56 FURTHER BIBLIOGRAPHY FERNHOLZ, E.R. (2002) Stochastic Portfolio Theory. Springer Verlag, New York. FÖLLMER, H. (1972) The exit measure of a supermartingale. Zeit. Wahrscheinlichkeitstheorie verw. Gebiete 21, FÖLLMER, H. (1973) On the representation of semimartingales. Annals of Probability 1, PERKOWSKI, N. & RUF, J. (2014) Supermartingales as Radon- Nikodým densities, and related measure extensions. Annals of Probability, to appear.

57 BAYRAKTAR, E. & XING, HAO (2010) On the uniqueness of classical solutions to Cauchy problems. Proc. Am. Math. Soc. 128, KARATZAS, I. & KARDARAS, C. (2007) The numéraire portfolio and arbitrage in semimartingale markets. Finance & Stochastics 11, KARDARAS, C. & ROBERTSON, S. (2012) Robust maximization of asymptotic growth. Annals of Applied Probability 22, PAL, S. (2011) Analysis of market weights under volatility stabilized models. Annals of Applied Probability 21,

58 PICKOVÁ, R. (2014) Generalized volatility stabilized processes. Annals of Finance 10, RUF, J. (2013) Hedging under arbitrage. Mathematical Finance 32, WANG, Y. (2014) Viscosity characterization of the explosion time distribution for diffusions. Preprint WANG, Y. (2014) Viscosity characterization of the optimal arbitrage function under model uncertainty. In Preparation. THANK YOU FOR YOUR ATTENTION