Model-free Arbitrage and Superhedging in Discrete Time Marco - - PowerPoint PPT Presentation
Model-free Arbitrage and Superhedging in Discrete Time Marco Frittelli Universit` a di Milano Joint with Matteo Burzoni and Marco Maggis Advanced Methods in Mathematical Finance Angers Conference, Sept 1, 2015 Marco Frittelli Universit` a di
Marco Frittelli Universit` a di Milano Joint with Matteo Burzoni and Marco Maggis Advanced Methods in Mathematical Finance Angers Conference, Sept 1, 2015
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 1 / 32
We have two extreme cases. 1 We are completely sure about the reference probability measure P. 2 We face complete uncertainty about any probabilistic model and therefore we describe our model independently by any probability: Model-free approach
Hobson 1998, Brown Hobson Rogers 2001, Davis Hobson 2007, Cox Obloj 2011, Riedel 2011, Acciaio, Beiglb¨
2013.
Between cases 1. and 2., there is the possibility to accept that the model could be described in a probabilistic setting, but we cannot assume the knowledge of a specific reference probability measure but at most of a set
3 Quasi-sure Stochastic Analysis: Peng, Touzi, Zhang, Dolinski, Soner, Kardaras, Bouchard, Nutz, Biagini S., Denis, Martini, Bion-Nadal, Cohen...
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 2 / 32
Theorem
Let Ω be Polish, t ∈ {0, 1, ..., T < ∞} , g : Ω → R be FT-measurable: inf {x ∈ R | ∃H ∈ H such that x + (H · S)T ≥ g M-q.s.} = inf {x ∈ R | ∃H ∈ H such that x + (H · S)T(ω) ≥ g(ω) ∀ω ∈ Ω∗} = sup
Q∈Mf
EQ[g] = sup
Q∈M
EQ[g], (H · S)t := ∑t
u=1 Hu(Su − Su−1) = ∑d j=1 ∑t u=1 Hj u(Sj u − Sj u−1);
Ft :=
P∈P(F S t ∨ N P t ), with: N P t := {N ⊆ A ∈ F S t | P(A) = 0};
H := {all F-predictable proc.} , F := {Ft}t Universal Filtration; P := {all probabilities on (Ω, B(Ω)}; M : =
Mf : = {Q ∈ M | Q has finite support} ; Ω∗ := {ω ∈ Ω | ∃Q ∈ M s.t. Q(ω) > 0} .
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 3 / 32
No reference to any a priori assigned probability measure and the notions of M, H and Ω∗ only depend on the measurable space (Ω, F) and the price process S. In general, the class M is not dominated.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 4 / 32
No reference to any a priori assigned probability measure and the notions of M, H and Ω∗ only depend on the measurable space (Ω, F) and the price process S. In general, the class M is not dominated. In an example, we show that the initial cost of the cheapest portfolio that dominates a contingent claim g on every possible path inf {x ∈ R | ∃H ∈ H such that x + (H · S)T(ω) ≥ g(ω) ∀ω ∈ Ω} can be strictly greater than supQ∈M EQ[g], unless some artificial assumptions are imposed on g or on the market. To avoid such restrictions it is crucial to select the correct set of paths (i.e. the set Ω∗ of those ω ∈ Ω which are weighted by at least
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 4 / 32
The family of M-polar sets is: N := {N ⊆ A ∈ F | Q(A) = 0 ∀Q ∈ M} Recall that a property is said to hold quasi surely (q.s.) if it holds
We show the existence of the maximal M-polar set N∗, namely a set N∗ ∈ N containing any other set N ∈ N . Moreover Ω∗ = (N∗)C. The inequality x + (H · S)T ≥ g M-q.s. is therefore equivalent to the inequality x + (H · S)T(ω) ≥ g(ω) ∀ω ∈ Ω∗, which justifies the first equality in the Theorem.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 5 / 32
The set Ω∗ can be equivalently determined via the set Mf of martingale measures with finite support: Ω∗ := {ω ∈ Ω | ∃Q ∈ Mf s.t. Q(ω) > 0} , a property that turns out to be crucial in several proofs. One of the main technical results of the paper is the proof that the set Ω∗ is an analytic set (it can be written as the nucleous of a Souslin scheme), and so our findings show that the natural setup for studying this problem is (Ω, S, F,H), with
F = {Ft}t the Universal filtration Ft :=
P∈P(FS t ∨ N P t ),
H := {F-predictable processes}.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 6 / 32
The set Ω∗ can be equivalently determined via the set Mf of martingale measures with finite support: Ω∗ := {ω ∈ Ω | ∃Q ∈ Mf s.t. Q(ω) > 0} , a property that turns out to be crucial in several proofs. One of the main technical results of the paper is the proof that the set Ω∗ is an analytic set (it can be written as the nucleous of a Souslin scheme), and so our findings show that the natural setup for studying this problem is (Ω, S, F,H), with
F = {Ft}t the Universal filtration Ft :=
P∈P(FS t ∨ N P t ),
H := {F-predictable processes}.
Financial interpretation: Ω∗ is the set of points where is not possible to build 1p Arbitrage opportunities.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 6 / 32
No need to assume M = ∅ (we shall discuss this when dealing with No Arbitrage)
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 7 / 32
No need to assume M = ∅ (we shall discuss this when dealing with No Arbitrage) No restriction on S, so that it may describe stocks and/or options. However, in the above Theorem the class H of admissible trading strategies requires dynamic trading in all assets. In the theorem below we easily extend this setup to the case of semi-static trading on a finite number of options.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 7 / 32
Theorem
Let g : Ω → R be the FT measurable claim and φj : Ω → R, j = 1, ..., k, be FT measurable random variables representing the payoff of k given
πΦ(g) = sup
Q∈MΦ
EQ[g]. where ΩΦ : = {ω ∈ Ω | ∃Q ∈ MΦ s.t. Q(ω) > 0} ⊆ Ω∗ MΦ : = {Q ∈ Mf | EQ(φj) = 0 ∀j = 1, ..., k} ⊆ Mf πΦ(g) := inf
x + (H · S)T(ω) + hΦ(ω) ≥ g(ω) ∀ω ∈ ΩΦ
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 8 / 32
On classical Superhedging (a probability P is fixed): El Karoui and Quenez (95); Karatzas (97); .... Model free set up and robust hedging: Hobson (98), Brown Hobson Rogers (01), Davis Hobson (07), Hobson (09), Hobson (11), Cox Obloj (11), Riedel (11), ... Optimal mass transport: Beiglb¨
Henry-Labord` ere, Hobson, Hou, Nutz, Obloj, Penker, Rogers, Soner, Spoida, Tan, Touzi... Superhedging with respect to a non dominated class of probability measures P′ ⊆ P: Bouchard Nutz (13), Biagini S. Bouchard Kardaras Nutz (14) Superhedging via model-free Arbitrage: Acciaio Beiglb¨
Schachermayer (13).
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 9 / 32
Superhedging Duality Theorem w.r.to a family P′ ⊆ P. If g : Ω → R is upper semianalytic (Borel measurable) then inf
= sup
Q∈M(P′)
EQ[g]. where M(P′) := {Q ∈ M | ∃P ∈ P′ s.t. Q ≪ P}. The theorem is obtained under two technical hypothesis: Ω = ΩT
1 , where Ω1 is Polish and Ωt 1 is the t-fold product space;
The set of priors P′ have the form P := P0 ⊗ . . . ⊗ PT where every Pt is a measurable selector of a certain random class P′
t ⊆ P(Ω1).
P′
t(ω) is the set of possible models, given state ω at time t and the
graph(P′
t) must be an analytic subset of Ωt 1 × P(Ω1).
In our setting we do not impose restrictions on the state space Ω and even in the case of Ω = ΩT
1 , the class of martingale probability measures M is
endogenously determined by the market and we do not require that it satisfies any additional restrictions.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 10 / 32
Same discrete time market as ours, but S is a one dimensional canonical process on the path space Ω = [0, ∞)T.
Theorem
Assume No Model Independent Arbitrage. Let φj = f j(ST), with f j : R+ → R, j = 0, 1, ... be the payoff of options traded at zero price and let f 0 be convex and super linear. If g = f (ST) with f : R+ → R upper semicountinuous then:
= sup
Q∈MΦ
EQ[g] MΦ := {Q ∈ M | EQ(φj) = 0 ∀j}
x + (H · S)T(ω) + hΦ(ω) ≥ g(ω) ∀ω ∈ Ω
Example where this duality doesn’t hold if f is not usc, a property with no financial meaning.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 11 / 32
To prove our theorem we need the following aggregation results:
Proposition
Let g : Ω → R be FT measurable and define π(g)
πQ(g)
Q- a.s. } . Then π(g) = sup
Q∈Mf
πQ(g) In particular, if π(g) < +∞ the inf is a min.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 12 / 32
Corollary (Super replication)
If for every Q ∈ Mf there exists HQ ∈ H such that x + (HQ · S)T ≥ g Q-a.s, then there exists H ∈ H such that x + (H · S)T(ω) ≥ g(ω) for every ω ∈ Ω∗.
Corollary (Replication)
If for every Q ∈ Mf there exists HQ ∈ H, xQ ∈ R such that xQ + (HQ · S)T = g Q − a.s. then there exists H ∈ H, x ∈ R such that x + (H · S)T(ω) = g(ω) for every ω ∈ Ω∗,
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 13 / 32
The results presented so far can be found in: Burzoni M., Frittelli M., Maggis M., Model-free Superhedging Duality, 2015. and are based on the theory about Model-free arbitrage in discrete time developed in: Burzoni M., Frittelli M., Maggis M., Universal Arbitrage Aggregator in Discrete Time Markets under Uncertainty, Fin. Stoch., forthcoming. which we now illustrate.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 14 / 32
In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class of significant sets S, which we call Arbitrage de la classe S.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 15 / 32
In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class of significant sets S, which we call Arbitrage de la classe S. The choice of S reflects into the intrinsic properties of the class of polar sets of martingale measures. In particular for
S = {Ω}, absence of Model Independent Arbitrage ⇐ ⇒ M = ∅ S = {open sets}, absence of “Open Arbitrage” is equivalent to the existence of full support martingale measures.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 15 / 32
In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class of significant sets S, which we call Arbitrage de la classe S. The choice of S reflects into the intrinsic properties of the class of polar sets of martingale measures. In particular for
S = {Ω}, absence of Model Independent Arbitrage ⇐ ⇒ M = ∅ S = {open sets}, absence of “Open Arbitrage” is equivalent to the existence of full support martingale measures.
We provide the dual representation of “Open Arbitrage” in terms of weakly open sets of probability measures, which highlights the robust nature of such concept.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 15 / 32
In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class of significant sets S, which we call Arbitrage de la classe S. The choice of S reflects into the intrinsic properties of the class of polar sets of martingale measures. In particular for
S = {Ω}, absence of Model Independent Arbitrage ⇐ ⇒ M = ∅ S = {open sets}, absence of “Open Arbitrage” is equivalent to the existence of full support martingale measures.
We provide the dual representation of “Open Arbitrage” in terms of weakly open sets of probability measures, which highlights the robust nature of such concept. These results are obtained by adopting a technical filtration enlargement and by constructing a universal arbitrage aggregator of all arbitrage opportunities. We introduce the notion of market feasibility and characterize it.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 15 / 32
We consider the market model as initially described: (Ω, I, S, F, H) and we assume that Ω is a Polish space, F = B(Ω) is the Borel sigma algebra, I = {0, ..., T} , S is a d-dimensional stochastic process, F = FS, the trading strategies H ∈ H are Rd-valued F-predictable stoc. proc.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 16 / 32
We consider the market model as initially described: (Ω, I, S, F, H) and we assume that Ω is a Polish space, F = B(Ω) is the Borel sigma algebra, I = {0, ..., T} , S is a d-dimensional stochastic process, F = FS, the trading strategies H ∈ H are Rd-valued F-predictable stoc. proc. The value process is given by: V0(H) = 0, Vt(H) = (H · S)t =
t
u=1
Hu(Su − Su−1) =
d
j=1 t
u=1
Hj
u(Sj u − Sj u−1).
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 16 / 32
We consider the market model as initially described: (Ω, I, S, F, H) and we assume that Ω is a Polish space, F = B(Ω) is the Borel sigma algebra, I = {0, ..., T} , S is a d-dimensional stochastic process, F = FS, the trading strategies H ∈ H are Rd-valued F-predictable stoc. proc. The value process is given by: V0(H) = 0, Vt(H) = (H · S)t =
t
u=1
Hu(Su − Su−1) =
d
j=1 t
u=1
Hj
u(Sj u − Sj u−1).
Set: P+ : = {Q ∈ P | Q has full support} M : = {Q ∈ P | S is a martingale under Q} M+ : = M ∩ P+, M+ is the set of martingale probability measures with full support. No reference probability measure is required.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 16 / 32
Let: V+
H := {ω ∈ Ω | VT(H)(ω) > 0} .
Definition
Let S be a class of measurable subsets of Ω such that ∅ / ∈ S. A trading strategy H ∈ H is an Arbitrage de la classe S if V0(H) = 0 and VT(H)(ω) ≥ 0 ∀ω ∈ Ω and V+
H contains a set de la classe S.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 17 / 32
Let: V+
H := {ω ∈ Ω | VT(H)(ω) > 0} .
Definition
Let S be a class of measurable subsets of Ω such that ∅ / ∈ S. A trading strategy H ∈ H is an Arbitrage de la classe S if V0(H) = 0 and VT(H)(ω) ≥ 0 ∀ω ∈ Ω and V+
H contains a set de la classe S.
The class S has the role to translate mathematically the meaning of a “true gain”: there is a “true gain” if the set V+
H contains a set
considered significant. When a “reference Probability” P is given, then a true gain is: P(VT(H) > 0) > 0 When a subset P′ of probability measures is given, one replace the P-a.s. conditions with P′-q.s conditions, as in [BN13].
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 17 / 32
H is a 1p (one point) Arbitrage when S = {C ∈ F | C = ∅}
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 18 / 32
H is a 1p (one point) Arbitrage when S = {C ∈ F | C = ∅} H is an Open Arbitrage if S = {C ∈ F | C open non-empty}
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 18 / 32
H is a 1p (one point) Arbitrage when S = {C ∈ F | C = ∅} H is an Open Arbitrage if S = {C ∈ F | C open non-empty} H is a P′-q.s. Arbitrage when S = {C ∈ F | P(C) > 0 for some P ∈ P′}, for a fixed family P′ ⊆ P. H is a P-a.s. Arbitrage when S = {C ∈ F | P(C) > 0} for fixed P ∈ P.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 18 / 32
H is a 1p (one point) Arbitrage when S = {C ∈ F | C = ∅} H is an Open Arbitrage if S = {C ∈ F | C open non-empty} H is a P′-q.s. Arbitrage when S = {C ∈ F | P(C) > 0 for some P ∈ P′}, for a fixed family P′ ⊆ P. H is a P-a.s. Arbitrage when S = {C ∈ F | P(C) > 0} for fixed P ∈ P. H is a model independent Arbitrage when when S = {Ω} Obviously, No 1p A ⇒ No A de la Classe S ⇒ No Model Ind. A and A de la Classe S are not necessarily related to a probabilistic model.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 18 / 32
In order to show in which context the FTAP holds true we need to build up a filtration enlargement F := ( Ft)t∈I which follows directly from the market structure and It preserves the sets of martingale measures M( F) ⇄ M(F)
The restriction of any Q ∈ M( F) to FT belongs to M(F). Any Q ∈ M(F) can be uniquely extended to an element of M( F).
It ensure the existence of an F-predictable multifunction Ht which is an aggregator for all P-a.s. arbitrage opportunities. We provide many examples where there is no equivalence between “No Arbitrage” and existence of martingale measures with reasonable properties, without the enlargement of the filtration.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 19 / 32
Davis Hobson 1997 “[..] a weak arbitrage opportunity is a situation where we know there must be an arbitrage but we cannot tell, without further information, what strategy will realize it.”
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 20 / 32
Davis Hobson 1997 “[..] a weak arbitrage opportunity is a situation where we know there must be an arbitrage but we cannot tell, without further information, what strategy will realize it.”
Example
Consider two call options C1, C2 on the same underlying asset with same initial price p1 = p2 but strikes K2 > K1.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 20 / 32
Davis Hobson 1997 “[..] a weak arbitrage opportunity is a situation where we know there must be an arbitrage but we cannot tell, without further information, what strategy will realize it.”
Example
Consider two call options C1, C2 on the same underlying asset with same initial price p1 = p2 but strikes K2 > K1. Strategy A: C1 − C2 VT(H) = if ST ≤ K1 > 0 if ST > K1
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 20 / 32
Davis Hobson 1997 “[..] a weak arbitrage opportunity is a situation where we know there must be an arbitrage but we cannot tell, without further information, what strategy will realize it.”
Example
Consider two call options C1, C2 on the same underlying asset with same initial price p1 = p2 but strikes K2 > K1. Strategy A: C1 − C2 VT(H) = if ST ≤ K1 > 0 if ST > K1 what if P(ST > K1) = 0?
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 20 / 32
Davis Hobson 1997 “[..] a weak arbitrage opportunity is a situation where we know there must be an arbitrage but we cannot tell, without further information, what strategy will realize it.”
Example
Consider two call options C1, C2 on the same underlying asset with same initial price p1 = p2 but strikes K2 > K1. Strategy A: C1 − C2 VT(H) = if ST ≤ K1 > 0 if ST > K1 what if P(ST > K1) = 0? Strategy B: −C1 VT(H) = p1 > 0 if ST ≤ K1 which happens with probability one.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 20 / 32
Theorem (Model Free FTAP)
Let (Ω, FT, ( Ft)t∈I) be the enlarged filtered space. Let N the family of polar sets of M. Then No Arbitrage de la classe S in H ⇐ ⇒ N does not contain sets of S
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 21 / 32
Theorem (Model Free FTAP)
Let (Ω, FT, ( Ft)t∈I) be the enlarged filtered space. Let N the family of polar sets of M. Then No Arbitrage de la classe S in H ⇐ ⇒ N does not contain sets of S In particular for S = {open non-empty sets} we obtain:
Corollary (FTAP for Open Arbitrage)
Let (Ω, FT, ( Ft)t∈I) as before. No Open Arbitrage in H ⇐ ⇒ there are no open M-polar sets ⇐ ⇒ M+ = ∅ where M+ is the class of martingale measure with full support.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 21 / 32
Riedel 2011, pointed out the relevance of the concept of full support martingale measures in the model-free setting: He proved, in one-period model and under the assumption that the price process is continuous with respect to the state variable ω, that: No 1p arbitrage ⇐ ⇒ M+ = ∅ In such setting, from the continuity of S one also deduces that No open arbitrage in H ⇐ ⇒ No 1p arbitrage ⇐ ⇒ M+ = ∅ However, both equivalences are no longer true in the multi-period setting (even with S continuous in ω). To recover the equivalence between No open arbitrage and M+ = ∅
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 22 / 32
For S = {Ω} we obtain:
Corollary (Model Independent FTAP)
Let (Ω, FT, ( Ft)t∈I) as before: No Model Independent Arbitrage in H ⇐ ⇒ M = ∅
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 23 / 32
For any trajectory z ∈ Mat(d, T + 1), study the level set of the price process S i.e. Σz
t−1 := {ω ∈ Ω | S0:t−1(ω) = z0:t−1} ∈ F S t−1
If 0 / ∈ ri(co(conv(∆St(Σz
t−1))) ∪ {0}) we may decompose the level set as
Σz
t−1 = B∗ ∪ β i=1 Bi.
The difference between the sets Bi and B∗: Restricted to the time interval [t − 1, t], a probability measure whose mass is concentrated on B∗ admits an equivalent martingale measure For those probabilities that assign positive mass to at least one Bi an arbitrage opportunity can be constructed.
Hi · ∆St(ω) > 0 ∀ω ∈ Bi Hi · ∆St(ω) ≥ 0 ∀ω ∈ ∪β
j=iBj
Remark: β is finite and β ≤ d
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 24 / 32
1
Ω∗ =
T
B∗
t,z
and M = ∅ ⇐ ⇒ Ω∗ = ∅ ⇐ ⇒ M ∩ Pf = ∅,
2 Fix t ∈ I and Q ∈ M. The set
Bt :=
βt,z
Bi
t,z
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 25 / 32
Let (Ω, F) = (R+, B(R+)). Consider the market S0 = [1, 1, 1] and St = [S1
t , S2 t , S3 t ] with
S1
1(ω) =
ω ∈ R+ \ Q 1 ω ∈ Q ∩ (1/2, +∞) 2 ω ∈ Q ∩ (0, 1/2) S2
1(ω) =
1 ω ∈ R+ \ Q ω2 ω ∈ Q ∩ (1/2, +∞) ω2 ω ∈ Q ∩ (0, 1/2) S3
1(ω) =
1 + ω2 ω ∈ R+ \ Q 1 ω ∈ Q ∩ (1/2, +∞) 1 ω ∈ Q ∩ (0, 1/2)
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 26 / 32
Set Yi := Si
1 − Si
In this example: B1 = R+ \ Q B2 = Q ∩ (0, 1/2) B∗ = Q ∩ (1/2, +∞)
1 −1 Y1 Y3 Y2
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 27 / 32
Let B∗, Bi, i = 1, . . . , β the disintegration of the level set Σz
t−1.
Ht(ω) :=
ω) ≥ 0 for any ω ∈ ∪β
j=i(ω)Bj
Theorem
If P is not absolutely continuous w.r.to some Q ∈ M then there exists an FP-predictable trading strategy HP which is a P-Classical Arbitrage and HP(ω) ∈ H(ω) P-a.s. where F P
t denote the P-completion of Ft and FP := {F P t }t∈I.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 28 / 32
Proposition
H is an Open Arbitrage iff there exists a non empty σ(P, Cb)-open set U ⊆ P such that VT(H) ≥ 0 P-a.s. ∀ P ∈ U and P(VT(H) > 0) > 0 ∀ P ∈ U. (1)
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 29 / 32
Proposition
H is an Open Arbitrage iff there exists a non empty σ(P, Cb)-open set U ⊆ P such that VT(H) ≥ 0 P-a.s. ∀ P ∈ U and P(VT(H) > 0) > 0 ∀ P ∈ U. (1) No a priori fixed class of probability measures If (H, U) satisfy (1) and we disregard any finite subset of probabilities then H still satisfy (1). If (H, U) satisfy (1), U will contain a full support probability P under which H is a P-Arbitrage. If the σ(P, Cb) topology is replaced by the one induced by · we
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 29 / 32
How large is the set of probability measures such that (Ω, F, F, P) satisfy No Classical Arbitrage (w.r.to P) ?
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 30 / 32
How large is the set of probability measures such that (Ω, F, F, P) satisfy No Classical Arbitrage (w.r.to P) ? Let P0 be the set of models for which there is no classical arbitrage (NA(P)) for the price process S: P0 = {P ∈ P | NA(P) holds} = {P ∈ P | Me(P) = ∅}
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 30 / 32
How large is the set of probability measures such that (Ω, F, F, P) satisfy No Classical Arbitrage (w.r.to P) ? Let P0 be the set of models for which there is no classical arbitrage (NA(P)) for the price process S: P0 = {P ∈ P | NA(P) holds} = {P ∈ P | Me(P) = ∅} We introduce the following:
Definition
The market is feasible if P0
σ(P,Cb) = P.
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 30 / 32
We can relate the notion of feasibility and No arbitrage as follows:
Theorem
Let (Ω, FT, ( Ft)t∈I) be the enlarged filtered space. TFAE:
1 M+ = ∅; 2 No Open Arbitrage holds w.r.to the strategies in
H.
3 The market is feasible: P0
σ(P,Cb) = P
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 31 / 32
Marco Frittelli Universit` a di Milano () Model Free Finance Angers 2015 32 / 32