Pathwise Finance: Arbitrage and Pricing-Hedging Duality

Pathwise Finance: Arbitrage and Pricing-Hedging Duality Marco Frittelli Milano University Based on joint works with Matteo Burzoni, Z. Hou, Marco Maggis and J. Obloj CFMAR 10th Anniversary Conference, UCSB Santa Barbara, May 20, 2017 Marco Frittelli Milano University () Pathwise Finance SB 2017 1 / 40

Outline Superhedging Duality Classical results Quasi-sure analysis Pathwise setting Example of duality gap Our approach in the Pathwise setting in discrete time Superhedging duality Tuning arbitrage conditions: Arbitrage de la Classe S Tuning from model-free to model specification: selection of universe of scenarios Arbitrage theory Marco Frittelli Milano University () Pathwise Finance SB 2017 2 / 40

Three approaches 1 We are completely sure about the reference probability measure P: The classical set up. 2 Knightian uncertainty: We 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 of priors. This leads to the theory of Quasi-sure Stochastic Analysis; or in the Decision Theory literature: MinMax Expected Utility (Gilboa Schmeidler (89)); Variational Preferences (Maccheroni, Marinacci, Rustichini (06)). 3 We face complete uncertainty about any probabilistic model and therefore we describe our model independently by any probability: Pathwise approach. Marco Frittelli Milano University () Pathwise Finance SB 2017 3 / 40

Literature On classical Superhedging (a probability P is fixed): El Karoui and Quenez (95); Karatzas (97);... Robust hedging: Breeden Litzenberger (78), Rubinstein (94), Dupire (94), Derman Kani (98), Hobson (98), Brown Hobson Rogers (01), Carr Madan (04), Davis Hobson (07), Hobson (09), Hobson (11), Cox Obloj (11), Riedel (11), Hou Obloj (15),... Optimal martingale transport: Beiglböck, Cox, Dolinsky, Galichon, Henry-Labordère, Hobson, Hou, Huesmann, Juillet, Nutz, Obloj, Penker, Rogers, Soner, Spoida, Tan, Touzi... Superhedging with respect to a non dominated class of probability measures: Beissner (12), Bouchard Nutz (15), Biagini S. Bouchard Kardaras Nutz (14), Bayraktar (15), (16). Superhedging via model-free Arbitrage: Riedel (15), Acciaio Beiglböck Penker Schachermayer (16), Burzoni (16), Cheridito Kupper Tangpi (16) Marco Frittelli Milano University () Pathwise Finance SB 2017 4 / 40

The Pricing-Hedging Duality The classical case: Let (S t ) t=0,...t be an F-adapted d-dimensional stochastic process on a probability space (Ω, F, P). Let: (H S) t := t H u (S u S u 1 ) = u=1 d j=1 t u=1 H j u(s j u S j u 1 ). Theorem Let g be an F T -measurable contingent claim. sup E Q [g] = inf {x R H H s.t. x + (H S) T g P a.s.} Q M e (P) where M e (P) is the class of equivalent martingale measures and H is the class of predictable processes respect to F. Marco Frittelli Milano University () Pathwise Finance SB 2017 5 / 40

The Pricing-Hedging Duality The classical case: Let (S t ) t=0,...t be an F-adapted d-dimensional stochastic process on a probability space (Ω, F, P). Let: (H S) t := t H u (S u S u 1 ) = u=1 d j=1 t u=1 H j u(s j u S j u 1 ). Theorem Let g be an F T -measurable contingent claim. sup E Q [g] = inf {x R H H s.t. x + (H S) T g P a.s.} Q M e (P) where M e (P) is the class of equivalent martingale measures and H is the class of predictable processes respect to F. What about if we do not know P? Marco Frittelli Milano University () Pathwise Finance SB 2017 5 / 40

The Pricing-Hedging Duality The classical case: Let (S t ) t=0,...t be an F-adapted d-dimensional stochastic process on a probability space (Ω, F, P). Let: (H S) t := t H u (S u S u 1 ) = u=1 d j=1 t u=1 H j u(s j u S j u 1 ). Theorem Let g be an F T -measurable contingent claim. sup E Q [g] = inf {x R H H s.t. x + (H S) T g P a.s.} Q M e (P) where M e (P) is the class of equivalent martingale measures and H is the class of predictable processes respect to F. What about if we do not know P? Model (Knightian) Uncertainty Marco Frittelli Milano University () Pathwise Finance SB 2017 5 / 40

Dealing with Uncertainty Some possible choices Replace P with a class of probabilities P Quasi-sure analysis Remove P and provide a pathwise description Pathwise approach Theorem Let g be an F T -measurable contingent claim. sup E Q [g] = inf {x R H H s.t. x + (H S) T g P a.s.} Q M e (P) where M e (P) is the class of equivalent martingale measures and H is the class of predictable processes respect to F. Marco Frittelli Milano University () Pathwise Finance SB 2017 6 / 40

Dealing with Uncertainty Some possible choices Replace P with a class of probabilities P Quasi-sure analysis Remove P and provide a pathwise description Pathwise approach Theorem Let g be an F T -measurable contingent claim. sup E Q [g] = inf {x R H H s.t. x + (H S) T g Q M e (P) P q.s.} where M e (P) is the class of martingale measures Q for which there exists P P satisfying Q P. where, by definition, an inequality holds P-q.s. if it holds P-a.s. P P. Marco Frittelli Milano University () Pathwise Finance SB 2017 6 / 40

Dealing with Uncertainty Some possible choices Replace P with a class of probabilities P Quasi-sure analysis Remove P and provide a pathwise description Pathwise approach Theorem Let g be an F T -measurable contingent claim. sup E Q [g] = inf {x R H H s.t. x + (H S) T g Q M e (P) P q.s.} where M e (P) is the class of martingale measures Q for which there exists P P satisfying Q P. where, by definition, an inequality holds P-q.s. if it holds P-a.s. P P. Does this Theorem hold? Marco Frittelli Milano University () Pathwise Finance SB 2017 6 / 40

Dealing with Uncertainty Some possible choices Replace P with a class of probabilities P Quasi-sure analysis Remove P and provide a pathwise description Pathwise approach Theorem Let g be an F T -measurable contingent claim. sup E Q [g] = inf {x R H H s.t. x + (H S) T g Q M e (P) P q.s.} where M e (P) is the class of martingale measures Q for which there exists P P satisfying Q P. where, by definition, an inequality holds P-q.s. if it holds P-a.s. P P. Does this Theorem hold? Yes, see Bouchard, Nutz (15) under some assumptions: Ω is a product space, some technical conditions on the class P. Marco Frittelli Milano University () Pathwise Finance SB 2017 6 / 40

Dealing with Uncertainty Some possible choices Replace P with a class of probabilities P Quasi-sure analysis Remove P and provide a pathwise description Pathwise approach Theorem Let g be an F T -measurable contingent claim. sup E Q [g] = inf {x R H H s.t. x + (H S) T g ω Ω} Q M where M is the class of all probability measures Q for which the price process is an F-martingale. Does this Theorem hold? Marco Frittelli Milano University () Pathwise Finance SB 2017 6 / 40

Dealing with Uncertainty Some possible choices Replace P with a class of probabilities P Quasi-sure analysis Remove P and provide a pathwise description Pathwise approach Theorem Let g be an F T -measurable contingent claim. sup E Q [g] = inf {x R H H s.t. x + (H S) T g ω Ω} Q M where M is the class of all probability measures Q for which the price process is an F-martingale. Does this Theorem hold? No in general, We are about to show a counterexample. Positive answer under market assumptions and restrictions on the contingent claim, see Acciaio et al. (16) and Cheridito Kupper Tangpi (16). Marco Frittelli Milano University () Pathwise Finance SB 2017 6 / 40

An example In the next 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 sup Q M E Q [g], unless some additional 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 one martingale measure Q M). Marco Frittelli Milano University () Pathwise Finance SB 2017 7 / 40

An example 1/4 Consider a one period market with a non-risky asset S 0 = [1, 1] and three risky assets defined on (R +, B(R + )) as follows: Canonical process S 1 (ω) := [1, ω] Marco Frittelli Milano University () Pathwise Finance SB 2017 8 / 40

An example 1/4 Consider a one period market with a non-risky asset S 0 = [1, 1] and three risky assets defined on (R +, B(R + )) as follows: Canonical process S 1 (ω) := [1, ω] Butterfly option S 2 (ω) := [0, φ(ω)] with φ(x) := (x K 0 ) + 2(x (K 0 + 1)) + + (x (K 0 + 2)) + Marco Frittelli Milano University () Pathwise Finance SB 2017 8 / 40

An example 1/4 Consider a one period market with a non-risky asset S 0 = [1, 1] and three risky assets defined on (R +, B(R + )) as follows: Canonical process S 1 (ω) := [1, ω] Butterfly option S 2 (ω) := [0, φ(ω)] Power option S 3 (ω) := [1, Φ(ω)] with φ(x) := (x K 0 ) + 2(x (K 0 + 1)) + + (x (K 0 + 2)) + and Φ(x) := (x 2 K 1 ) + Marco Frittelli Milano University () Pathwise Finance SB 2017 8 / 40

An example 2/4 Remark: S 2 is an arbitrage respect to any probability measure such that P(K 0, K 0 + 2) > 0 any martingale measure satisfies Q(K 0, K 0 + 2) = 0. Marco Frittelli Milano University () Pathwise Finance SB 2017 9 / 40

An example 2/4 Remark: S 2 is an arbitrage respect to any probability measure such that P(K 0, K 0 + 2) > 0 any martingale measure satisfies Q(K 0, K 0 + 2) = 0. Suppose we want to calculate the superhedging price of the digital option: g := 1 (K0,K 0 +2)(S 1 T (ω)) = 1 (K0,K 0 +2)(ω) We have two possible mechanisms via martingale measures E Q [g] = 0 sup Q M E Q [g] = 0 Marco Frittelli Milano University () Pathwise Finance SB 2017 9 / 40

An example 2/4 Remark: S 2 is an arbitrage respect to any probability measure such that P(K 0, K 0 + 2) > 0 any martingale measure satisfies Q(K 0, K 0 + 2) = 0. Suppose we want to calculate the superhedging price of the digital option: g := 1 (K0,K 0 +2)(ST 1 (ω)) = 1 (K0,K 0 +2)(ω) We have two possible mechanisms via martingale measures E Q [g] = 0 sup Q M E Q [g] = 0 via the cheapest superhedging strategy (x, H) R H such that x + (H S) T (ω) g(ω) ω Ω Marco Frittelli Milano University () Pathwise Finance SB 2017 9 / 40

An example 2/4 Remark: S 2 is an arbitrage respect to any probability measure such that P(K 0, K 0 + 2) > 0 any martingale measure satisfies Q(K 0, K 0 + 2) = 0. Suppose we want to calculate the superhedging price of the digital option: g := 1 (K0,K 0 +2)(ST 1 (ω)) = 1 (K0,K 0 +2)(ω) We have two possible mechanisms via martingale measures E Q [g] = 0 sup Q M E Q [g] = 0 via the cheapest superhedging strategy (x, H) R H such that x + (H S) T (ω) g(ω) ω Ω we show that such infimum is > 0 and hence a duality gap exists. Marco Frittelli Milano University () Pathwise Finance SB 2017 9 / 40

An example 3/4 The following Figures show how S 2 and S 3 are useless for superhedging g Figure: αs 2 does not dominate g on (K 0, K 0 + ε) for any α with ε = ε(α) Marco Frittelli Milano University () Pathwise Finance SB 2017 10 / 40

An example 3/4 The following Figures show how S 2 and S 3 are useless for superhedging g Figure: αs 2 does not dominate g on (K 0, K 0 + ε) for any α with ε = ε(α) Figure: S 3 has no positive wealth on (K 0, K 0 + 2). Marco Frittelli Milano University () Pathwise Finance SB 2017 10 / 40

An example 4/4 We can conclude that it is possible to superhedge g by strategies that involve only the non-risky asset S 0 and S 1. An easy calculation shows then that the superhedging price of g is worth { min 1, S 0 1 } > 0 = sup E Q [g] K 0 Q M Remark In this example all the assumptions of Acciaio et al. (16) are satisfied except for the upper-semi continuity of g. If we consider the usc digital option g := 1 [K0,K 0 +2] then the duality gap disappear. Marco Frittelli Milano University () Pathwise Finance SB 2017 11 / 40

Relaxation The supremum on the left-hand side is taken over the whole set of martingale measure: we cannot increase its value So in order to find a duality Theorem in the general case: We need to reduce the set of trajectories for superhedging. We show that, without any additional assumptions on the market or on the contingent claim g, there is No duality gap. The proof of this result is not based on functional analytic duality theorems, but rely on the pathwise analysis of No Arbitrage. Marco Frittelli Milano University () Pathwise Finance SB 2017 12 / 40

The setup - without statically traded options Let (Ω, F := B(Ω), F := (F t ) t I ) a filtered space with a d-dimensional process S := (S t ) t I such that: I = {0,..., T } i.e. discrete time Marco Frittelli Milano University () Pathwise Finance SB 2017 13 / 40

The setup - without statically traded options Let (Ω, F := B(Ω), F := (F t ) t I ) a filtered space with a d-dimensional process S := (S t ) t I such that: I = {0,..., T } i.e. discrete time Ω a Polish space We are not making any particular assumption on S t : Ω R d so that it may represent generic financial securities. F is the universal completion of F S i.e. F t = P F S t N P t where Nt P is the collection of Ft S -measurable P-null sets and the intersection is with respect to all probability measures on (Ω, B(Ω)) Marco Frittelli Milano University () Pathwise Finance SB 2017 13 / 40

Towards a correct duality Let us now introduce the set Ω := {ω Ω Q({ω}) > 0 for some Q M} where M := {Q P S is a (Q, F)-martingale} We might call Ω the support of the class of martingale measure: any event in Ω has positive probability for at least one martingale measure. Proposition Ω is an analytic set and hence belongs to F T This will be important also for the extensions that we will see by the end of the talk, where the theory is developed replacing Ω with any (analytic) subset of feasible paths: Ω X Marco Frittelli Milano University () Pathwise Finance SB 2017 14 / 40

Towards a correct duality Question: is the set Ω non-empty? Marco Frittelli Milano University () Pathwise Finance SB 2017 15 / 40

Towards a correct duality Question: is the set Ω non-empty? Theorem (Burzoni, F., Maggis (16)) M = Ω = No Strong Arbitrage where a strong arbitrage is a strategy H H such that (H S) T (ω) > 0 ω Ω. Caution: here one needs a filtration enlargement. Something more can be said assuming stronger notions of No Arbitrage. A one point arbitrage consists in a strategy H H such that (H S) T (ω) 0 ω Ω and ω Ω s.t. (H S) T ( ω) > 0. Proposition Ω = Ω iff No one point arbitrage holds. Marco Frittelli Milano University () Pathwise Finance SB 2017 15 / 40

The superhedging duality Theorem The set Ω is crucial for obtaining the duality since it represents the set of efficient trajectories where the hedging strategy should be considered. Theorem Let g be an F-measurable contingent claim. sup E Q [g] = inf {x R H H s.t. x + (H S) T g ω Ω } Q M where M is the class of probability measures Q for which the price process is an F-martingale. When finite, the inf is attained. 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. No additional assumptions on the market or on the claim g. Marco Frittelli Milano University () Pathwise Finance SB 2017 16 / 40

The superhedging duality Theorem We obtain a similar result also admitting dynamic trading in S and static trading a finite number of options: Φ = {φ i } k i=1. Here we set: H := (α, H) R k H) Theorem (Superhedging with options) Let g be an F-measurable contingent claim. Let Φ = {φ i } k i=1. } sup E Q [g] = inf {x R H s.t. x + ( H S) T + α Φ g ω Ω Φ Q M Φ where M Φ is the class of probability Q with finite support for which the price process is an F-martingale and E Q [φ] = 0 for all φ Φ and Ω Φ := {ω Ω Q({ω}) > 0 for some Q M Φ } This formulation of the theorem is in a joint paper with Burzoni, Hou, Maggis, Obloj (16) and is not a trivial consequence of the case without options. Marco Frittelli Milano University () Pathwise Finance SB 2017 17 / 40

Key result for the proof One of the main technical results of the paper is the proof that the set Ω is an analytic set, and so our findings show that the natural setup for studying this problem is (Ω, S, F,H), with F = {F t } t the Universal filtration F t = P F S t N P t H := {F-predictable processes}. as the universal filtration contains the analytic sets. Marco Frittelli Milano University () Pathwise Finance SB 2017 18 / 40

Results based on BFM2016 The results presented so far can be found in: Burzoni M., F., Maggis M., Model-free Superhedging Duality, 2016, Ann. Appl. Prob. 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. 1/2016, 1-50 which we now sketch. Marco Frittelli Milano University () Pathwise Finance SB 2017 19 / 40

On Arbitrage The market does not admit arbitrage opportunities if it is not possible to create a portfolio of financial securities in such a way that the initial investment is zero (or even negative) and the final outcome is always non negative and in some cases strictly positive. Seminal papers by: B. de Finetti, Fondamenti logici del pensiero probabilistico, Bollettino Unione Matematica Italiana, 1930. B. de Finetti, Sul significato soggettivo della probabilita, Fundamenta Mathematicae, 1931 B. de Finetti, Theory of Probability, 1970: Coherence and previsions S. Ross 1976, G. Huberman: 1976 Arbitrage Pricing Theory Harrison and Pliska 1979: FTAP in finite state space Marco Frittelli Milano University () Pathwise Finance SB 2017 20 / 40

The elementary version of the FTAP Theorem Let Ω = (ω 1,..., ω n ) be a finite state space, S 0 = (s 1,..., s d ) be the time zero price of d financial assets, S(ω) = (S 1 (ω),..., S d (ω)), be the final random value of the d assets. H R d s.t. H S 0 0 and H S(ω) 0 ω, with strict > for some ω Q s.t. E Q [S j ] = s j and Q(ω i ) > 0 i = 1,..., n; j = 1,..., d, where Q is a probability measure on Ω. In the finite setting, there is no need to specify a reference probability. Marco Frittelli Milano University () Pathwise Finance SB 2017 21 / 40

Extension of the FTAP to general state space Such linear pricing rules Q consistent with the observed prices (s 1,..., s d ) turn out to be martingale probability measures with full support, i.e. they assign positive measure to any state of the world. The formulation of the previous theorem can not hold for uncountable infinite dimensional Ω, as Q(ω) > 0 ω Ω is impossible. By introducing a reference probability measure P with full support and defining an arbitrage as: H S 0 0 and P(H S(ω) 0) = 1, P(H S(ω) > 0) > 0 the previous theorem could be restated as: No Arbitrage Q P s.t. E Q [S j ] = s j j = 1,..., d. Indeed, this formulation is a non trivial extension of the FTAP to the case of a general infinite dimensional Ω. Dalang Morton Willinger 1990 - W. Schachermayer 1994 - C. Rogers 1994 - Kabanov Kramkov 1995... Marco Frittelli Milano University () Pathwise Finance SB 2017 22 / 40

Arbitrage de la Classe S Let: V + H := {ω Ω (H S) T (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 (H S) T (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((H S) T ) > 0) > 0 When a subset P of probability measures is given, one replace the P-a.s. conditions with P-q.s conditions, similar to BN (15). Marco Frittelli Milano University () Pathwise Finance SB 2017 23 / 40

Examples of Arbitrage de la Classe S H is a 1p (one point) Arbitrage when S = {C F C = } Marco Frittelli Milano University () Pathwise Finance SB 2017 24 / 40

Examples of Arbitrage de la Classe S 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 Milano University () Pathwise Finance SB 2017 24 / 40

Examples of Arbitrage de la Classe S 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 of probabilities. H is a P-a.s. Arbitrage when S = {C F P(C ) > 0} for fixed probability P. H is a Strong Arbitrage when when S = {Ω} Obviously, No 1p A No A de la Classe S No Strong Arbitrage and A de la Classe S are not necessarily related to a probabilistic model. Marco Frittelli Milano University () Pathwise Finance SB 2017 24 / 40

Tuning between arbitrage conditions The class S has the role of tuning between two possible extreme choices for the arbitrage conditions: 1p Arbitrage Strong Arbitrage S = {C F C = } S S = {Ω} Compare with P-q.s. analysis: P Arb. 1p Arb. Strong Arb. P = {P} P P = all prob. S S = {Ω} S = {C F C = } In the theory of arbitrage for (non-dominated) sets of priors, versions of the FTAP are provided by Bouchard and Nutz (15) and Bayraktar (15), (16). We provide versions of the FTAP in the pathwise setting for arbitrage de la Classe S (for any class S) Marco Frittelli Milano University () Pathwise Finance SB 2017 25 / 40

On FTAP and Arbitrage de la Classe S Let N the family of polar sets of M. Theorem ( FTAP) No Arb. de la Classe S M = and N does not contain sets of S Marco Frittelli Milano University () Pathwise Finance SB 2017 26 / 40

On FTAP and Arbitrage de la Classe S Let N the family of polar sets of M. Theorem ( FTAP) No Arb. de la Classe S M = and N does not contain sets of S A particular example is: Corollary (FTAP for Open Arbitrage) No Open Arbitrage there are no open M-polar sets M + = where M + is the class of martingale measure with full support. Marco Frittelli Milano University () Pathwise Finance SB 2017 26 / 40

On FTAP and Arbitrage de la Classe S Let N the family of polar sets of M. Theorem ( FTAP) No Arb. de la Classe S M = and N does not contain sets of S A particular example is: Corollary (FTAP for Open Arbitrage) No Open Arbitrage there are no open M-polar sets M + = where M + is the class of martingale measure with full support. The above hold only under an enlargement of the filtration! Marco Frittelli Milano University () Pathwise Finance SB 2017 26 / 40

A more general perspective We observed the need to superhedge on a (analytic) subset of Ω The class S has the role of tuning between arbitrage conditions Our next aim is to interpolate between Model Specific and Model Free settings to obtain FTAP (and Pricing-Hedging duality) with or without statically traded options. Scenario based analysis: agent s beliefs are represented by selecting a set of admissible scenarios which we denote by Ω, a subset of all possible paths. Remark (Selection of Paths) This selection may be formulated in terms of behaviour of some market observable quantities and may reflect both the information the agent is endowed with, as well as the modelling assumptions. Marco Frittelli Milano University () Pathwise Finance SB 2017 27 / 40

Model free vs Model Specific The first part of this paper concentrates on laying the foundations for a rational theory of option pricing. It is an attempt to derive theorems about the properties of option prices based on assumptions sufficiently weak to gain universal support. To the extent it is successful the resulting theorems become necessary conditions to be satisfied by any rational option pricing theory. As one might expect, assumptions weak enough to be accepted by all are not sufficient to determine uniquely a rational theory of warrant pricing. To do so, more structure must be added to the problem through additional assumptions at the expense of loosing some agreement. R.C. Merton, Theory of rational option pricing, 1973. Marco Frittelli Milano University () Pathwise Finance SB 2017 28 / 40

Model formulations Modelling financial Probabilistic model markets specification P Models with universal support Marco Frittelli Milano University () Pathwise Finance SB 2017 29 / 40

Model formulations Modelling financial Probabilistic model Financial crises markets specification P Model-risk Models with universal support Marco Frittelli Milano University () Pathwise Finance SB 2017 29 / 40

Model formulations Modelling financial Probabilistic model Financial crises markets specification P Model-risk Models with Probabilistic model universal support with multiple priors P Marco Frittelli Milano University () Pathwise Finance SB 2017 29 / 40

On Arbitrage More structure must be added to the problem through additional assumptions at the expense of loosing some agreement. The paradox of Arbitrage: Universal principle: all agents agree we can not price rationally Ambiguous implementation: agents may disagree on the way they should create arbitrage Marco Frittelli Milano University () Pathwise Finance SB 2017 30 / 40

A curious example Davis, Hobson 07 [..] 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 C 1, C 2 on the same underlying with same initial price p 1 = p 2 but strikes K 2 > K 1. { 0 if ST K Strategy A: C 1 C 2 V T (H) = 1 > 0 if S T > K 1 what if P(S T > K 1 ) = 0? Marco Frittelli Milano University () Pathwise Finance SB 2017 31 / 40

A curious example Davis, Hobson 07 [..] 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 C 1, C 2 on the same underlying with same initial price p 1 = p 2 but strikes K 2 > K 1. { 0 if ST K Strategy A: C 1 C 2 V T (H) = 1 > 0 if S T > K 1 what if P(S T > K 1 ) = 0? Strategy B: C 1 V T (H) = p 1 > 0 if S T K 1 which happens with probability one. Marco Frittelli Milano University () Pathwise Finance SB 2017 31 / 40

THE FUNDAMENTAL THEOREM OF ASSET PRICING Burzoni, Hou, F, Maggis, Obloj, Pointwise Arbitrage Pricing Theory in Discrete Time, (2016) Arxiv Marco Frittelli Milano University () Pathwise Finance SB 2017 32 / 40

The Setup: semistatic trading with options Discrete time I := {0,..., T }; (X, d) Polish space; Ω X, Ω F A, where F A is the sigma-algebra generated by the analytic sets. numeraire asset S 0 1; S 1,..., S d assets available for dynamic trading; Φ := {φ 1,..., φ k } a finite set of F A -measurable statically traded options; A Φ (F) is the set of admissible strategies (α, H) such that α R k and H H so that the value is given by: α Φ + (H S) T. M Ω,Φ (F) := { Q P S is an F-martingale under Q, Q(Ω) = 1 and E Q [φ] = 0 φ Φ } ; is the set of calibrated martingale measures. Marco Frittelli Milano University () Pathwise Finance SB 2017 33 / 40

The Setup: semistatic trading with options Discrete time I := {0,..., T }; (X, d) Polish space; Ω X, Ω F A, where F A is the sigma-algebra generated by the analytic sets. numeraire asset S 0 1; S 1,..., S d assets available for dynamic trading; Φ := {φ 1,..., φ k } a finite set of F A -measurable statically traded options; A Φ (F) is the set of admissible strategies (α, H) such that α R k and H H so that the value is given by: α Φ + (H S) T. M f Ω,Φ (F) := { Q P S is an F-martingale under Q, Q(Ω) = 1 and E Q [φ] = 0 φ Φ } ; is the set of calibrated finite support martingale measures. Marco Frittelli Milano University () Pathwise Finance SB 2017 33 / 40

Arbitrage notions on the subset Ω We now introduce the notions of arbitrage opportunities on Ω. Definition Fix a filtration F, Ω F A and a set of statically traded options Φ. A One-Point Arbitrage on Ω (1p-Arbitrage) is a strategy (α, H) A Φ (F) such that α Φ + (H S) T 0 on Ω with a strict inequality for some ω Ω. A Strong Arbitrage on Ω is a strategy (α, H) A Φ (F) such that α Φ + (H S) T > 0 on Ω. Marco Frittelli Milano University () Pathwise Finance SB 2017 34 / 40

The set of efficient paths Let F M be the completion of F S with respect to the set of calibrated martingale measures with finite support M f Ω,Φ : F M t = P M f Ω,Φ F S t N P t Ω Φ F A is the set of scenarios (efficient paths) charged by martingale measures in M f Ω,Φ Ω Φ := { ω Ω Q M f Ω,Φ such that Q(ω) > 0 } = supp(q) Q M f Ω,Φ Our FTAP: Universal principle which characterizes the existence of efficient paths. We look for a unique object to aggregate ambiguous implementations Marco Frittelli Milano University () Pathwise Finance SB 2017 35 / 40

Pathwise FTAP on Ω X with options Theorem (Pathwise FTAP on Ω X ) Fix Ω F A and Φ a finite set of F A -measurable statically traded options. Then, there exists a filtration F which aggregates arbitrage views in that: No Strong Arbitrage in A Φ ( F) on Ω M Ω,Φ (F S ) = Ω Φ = and F S F F M. Further, Ω Φ F A and there exists an Arbitrage Aggregator (α, H ) A Φ ( F) such that α Φ + (H S) T 0 on Ω and Ω Φ = {ω Ω α Φ(ω) + (H S) T (ω) = 0}. Marco Frittelli Milano University () Pathwise Finance SB 2017 36 / 40

Pathwise FTAP on Ω X with options Some remark about the FTAP with statically traded options We provide a new simplified proof ( w.r.to the case with no options of Burzoni, Frittelli, M. 16), which relies on a conditional construction of the Arbitrage Aggregator via Castaing representation. The case with a finite number of options is interconnected with the proof of Pricing-Hedging duality with options. By fixing P and by selecting a particular set Ω P X we obtain the Dalang-Morton-Willinger FTAP We also recover the Acciao et al. (2016) result on the pricing-hedging duality when a superlinear option is available for trading. Marco Frittelli Milano University () Pathwise Finance SB 2017 37 / 40

Classical model specification - DMW setting Fix a probability measure P and Φ = 0 (no options). The filtration F can be chosen to be the P-completion of F S ; The set Ω is constructed as follows. For 1 t T, we denote χ t 1 the (random) support of the conditional distribution of S t. Consider now the set T U := {ω X S t (ω) χ t 1 (ω)} t=1 which satisfies U B X and P(U) = 1. Define now { Ω P U if P(U ) > 0 := U \ U if P(U ) = 0, which satisfies Ω P F A and P(Ω P ) = 1. Marco Frittelli Milano University () Pathwise Finance SB 2017 38 / 40

Recovering the DMW Theorem Consider a probability P P and let M P := {Q M Q P}. Proposition There exists a set of scenarios Ω P F A and a filtration F such that F S F F M and No Strong Arbitrage in A( F) on Ω P M P =. Further, ( No P arbitrage P (Ω P ) ) = 1 M P =, where M P := {Q M Q P}. where No P-Arbitrage means the absence of the classical arbitrage opportunity with respect to (P, F S ). Marco Frittelli Milano University () Pathwise Finance SB 2017 39 / 40

Thank you for the attention Marco Frittelli Milano University () Pathwise Finance SB 2017 40 / 40