Recent results in game theoretic mathematical finance Nicolas - PowerPoint PPT Presentation

Recent results in game theoretic mathematical finance Nicolas Perkowski Humboldt–Universit¨ at zu Berlin May 31st, 2017 Thera Stochastics In Honor of Ioannis Karatzas’s 65th Birthday Based on joint work with R. � Lochowski (Warsaw), D. Pr¨ omel (Zurich) Nicolas Perkowski Game-theoretic math finance 1 / 31

Motivation Game theoretic approach formulates probability / math finance without measure theory. Kolmogorov’s approach powerful but sometimes not well justified (frequentist vs subjective probability). Martingales usually introduced as “fair games”: ◮ not obvious from definition; ◮ which parts of martingale theory come from “fair game” description, which from measure theoretic modelling? Model free math finance also eliminates reference probability ⇒ connections to game-theoretic approach. Nicolas Perkowski Game-theoretic math finance 2 / 31

Scope of Vovk’s approach Vovk’s Vovk ’08 approach convenient book-keeping for model free math finance. qualitative properties of “typical price paths”: variation regularity Vovk ’11 , quadratic variation Vovk ’12, Vovk ’15, omel ’16 , local times P.-Pr¨ omel ’15 , rough paths P.-Pr¨ omel ’16 . � Lochowski-P.-Pr¨ measure free stochastic calculus: omel ’16 , P.-Pr¨ omel ’16, � Lochowski ’15, Vovk ’16, � Lochowski-P.-Pr¨ quantitative results: pathwise Dambis Dubins-Schwarz theorem Vovk ’12 ; model free pricing-hedging duality Beiglb¨ ock-Cox-Huesmann-P.-Pr¨ omel ’15, omel-Tangpi ’17 . Bartl-Kupper-Pr¨ Nicolas Perkowski Game-theoretic math finance 3 / 31

Outline Definition and basic properties 1 Overview of some nice results 2 Measure free stochastic calculus 3 Pathwise stochastic calculus 4 Nicolas Perkowski Game-theoretic math finance 4 / 31

Vovk’s approach Ω := C ([0 , ∞ ) , R ) (or C ([0 , T ] , R ), D + ([0 , T ] , R d ), . . . ); S t ( ω ) = ω ( t ); F t = σ ( S s : s ≤ t ); simple strategy H : ◮ stopping times 0 = τ 0 < τ 1 < . . . ◮ F τ n -measurable F n : Ω → R . Well-defined integral: ∞ � ( H · S ) t ( ω ) = F n ( ω )[ S τ n +1 ∧ t ( ω ) − S τ n ∧ t ( ω )] n =0 H is λ -admissible ( ∈ H λ ) if ( H · S ) t ( ω ) ≥ − λ ∀ ω, t . Definition (Vovk ’09 / P-Pr¨ omel ’15) Outer measure P of A ⊆ Ω is � � λ : ∃ ( H n ) n ⊆ H λ s.t. lim inf n →∞ ( λ +( H n · S ) ∞ ( ω )) ≥ 1 A ( ω ) ∀ ω P ( A ) := inf . Game-theoretic martingales are the capital processes λ + ( H · S ), H ∈ H λ . Nicolas Perkowski Game-theoretic math finance 5 / 31

Link with measure-theoretic martingales Lemma (Vovk ’12) sup P ( A ) ≤ P ( A ) , A ∈ F ∞ . P MM For λ > P ( A ) we find ( H n ) ⊆ H λ with n →∞ ( λ + ( H n · S ) ∞ ( ω )) . 1 A ( ω ) ≤ lim inf Throw martingale measure P at both sides: � n →∞ ( λ + ( H n · S ) ∞ ) � P ( A ) ≤ E P lim inf n →∞ E P [( λ + ( H n · S ) ∞ )] ≤ λ. ≤ lim inf Nicolas Perkowski Game-theoretic math finance 6 / 31

Link with (NA1) By scaling: P ( A ) = 0 iff ∃ ( H n ) ⊂ H 1 with n →∞ (1 + ( H n · S ) ∞ ) ≥ ∞ · 1 A . lim inf Recall: P satisfies (NA1) (= (NUPBR)) if { 1 + ( H · S ) ∞ : H ∈ H 1 } bounded in P -probability. sup P (NA1) P ( A ) �≤ P ( A ), but: Lemma (P-Pr¨ omel ’15) Let A ∈ F ∞ . If P ( A ) = 0 , then P ( A ) = 0 for all P with (NA1). (NA1) is minimal assumption any market model should fulfill. (Ankirchner ’05, Karatzas-Kardaras ’07, Ruf ’13, Fontana-Runggaldier ’13, Imkeller-P. ’15...) Nicolas Perkowski Game-theoretic math finance 7 / 31

Outline Definition and basic properties 1 Overview of some nice results 2 Measure free stochastic calculus 3 Pathwise stochastic calculus 4 Nicolas Perkowski Game-theoretic math finance 8 / 31

Typical price paths Property (P) holds for typical price paths if it is violated on a null set. Observations due to Vovk: Typical price paths have no points of increase. Typical price paths have finite p -variation for p > 2. Typical price paths have a quadratic variation [ S ]. Observations due to P.-Pr¨ omel: Typical price paths are rough paths in the sense of Lyons. Typical price paths have nice local times. Nicolas Perkowski Game-theoretic math finance 9 / 31

Typical price paths have quadratic variation ∞ [ S ] n � k ∧ t ) 2 t := ( S τ n k +1 ∧ t − S τ n k =0 ∞ = S 2 t − S 2 � 0 − 2 S τ n k ∧ t ( S τ n k +1 ∧ t − S τ n k ∧ t ) k =0 0 − 2( S n · S ) t = S 2 t − S 2 Deterministic τ n k : no chance for convergence. Set τ n 0 = 0, τ n k +1 = inf { t ≥ τ n k | ≥ 2 − n } ; k : | S t − S τ n t = 2(( S n − S n +1 ) · S ) t . [ S ] n +1 − [ S ] n t Bounds on ( S n − S n +1 ) and S τ n k +1 − S τ n k + a priori control on # { τ n k : k } + pathwise Hoeffding inequality: convergence of [ S ] n ( ω ) for typical price paths ω Vovk ’12 (continuous paths or bounded jumps). Nicolas Perkowski Game-theoretic math finance 10 / 31

Pathwise Dambis Dubins-Schwarz Theorem Ω = C ([0 , ∞ ) , R ), define time-change operator t : Ω → Ω: [ t ( ω )] t = t , t ∈ [0 , ∞ ) . Theorem (Vovk ’12) W Wiener measure, F ≥ 0 measurable, c ∈ R : � E [( F ◦ t ) 1 { S 0 = c , [ S ] ∞ = ∞} ] = F ( c + ω ) W ( d ω ) , Ω where � n →∞ ( λ +( H n · S ) ∞ ( ω )) ≥ F ( ω ) ∀ ω � λ : ∃ ( H n ) n ⊆ H λ s.t. lim inf E ( F ) := inf . Nicolas Perkowski Game-theoretic math finance 11 / 31

Outline Definition and basic properties 1 Overview of some nice results 2 Measure free stochastic calculus 3 Pathwise stochastic calculus 4 Nicolas Perkowski Game-theoretic math finance 12 / 31

Model free concentration of measure Ω = C ([0 , T ] , R d ). Want “stochastic integral”. For step functions F ok. Extension? Lemma (� Lochowski-P.-Pr¨ omel ’16) F adapted step function, then � T √ � � ≤ 2 e − a 2 / 2 . F ⊗ 2 P � F · S � ∞ ≥ a b , d [ S ] t ≤ b t 0 Pathwise Hoeffding: a 1 , . . . , a n ∈ R with | a n | ≤ c , then ∀ λ there exist b ℓ = b ℓ ( a 1 , . . . , a ℓ − 1 , c , λ ) with ℓ ℓ a k − λ 2 � 2 ℓ c 2 � � � 1 + b k a k ≥ exp λ ∀ ℓ. k =1 k =1 Now discretize S and apply Hoeffding. Nicolas Perkowski Game-theoretic math finance 13 / 31

Topologies on path space � � T � 0 ( F t − G t ) ⊗ 2 d [ S ] t ∧ 1 d QV ( F , G ) := E : complete metric space of integrands. d ∞ ( X , Y ) := E ( � X − Y � ∞ ∧ 1): complete metric space of (possible) integrals. F �→ F · S continuous on (step functions, d QV ), extends to closure. No idea how closure looks like. Need to localize: �� � T ∞ � � � 2 − n E ( F t − G t ) ⊗ 2 d [ S ] t ∧ 1 d QV , loc ( F , G ) := . 1 [ S ] T ≤ n 0 n =1 Now closure contains c` agl` ad paths, open problem if also bounded predictable processes. Convergence of integrals for typical price paths, Itˆ o’s formula, integral is independent of approximating sequence, . . . Nicolas Perkowski Game-theoretic math finance 14 / 31

What about jumps? Strategy H is λ -admissible if ∞ � ( H · S ) t ( ω ) = F n ( ω )[ S τ n +1 ∧ t ( ω ) − S τ n ∧ t ( ω )] ≥ − λ ∀ t , ω. n =0 Ω = D ([0 , T ] , R d ): no admissible H ! Ω paths with bounded jumps: Vovk ’12 . Canonical: D + ([0 , T ] , R d ) (positive c` adl` ag paths). means no short-selling; want [ S ], but all constructions of [ S ] use short-selling. Way out: relax problem to allow “little bit of short-selling”. Take relaxation away ⇒ [ S ] ex for typical positive c` adl` ag price paths � Lochowski-P.-Pr¨ omel ’16. Nicolas Perkowski Game-theoretic math finance 15 / 31

Integration with jumps Ω = D S 0 , + ([0 , T ] , R d ). Again canonical definition of F · S for step functions F . Extension? Pathwise Hoeffding no longer works: F τ k ( S τ k +1 − S τ k ) unbounded. Instead: pathwise B-D-G inequality of Beiglb¨ ock-Siorpaes ’15 a 1 , . . . , a n ∈ R , then there exist b ℓ = b ℓ ( a 1 , . . . , a ℓ − 1 ) with � ℓ m ℓ � � � � � � � a 2 b k a k ≥ max a k � − 6 ∀ ℓ. � � � k � m ≤ ℓ k =1 k =1 k =1 From here: √ � T ≤ (1 + | S 0 | )6 b + 2 c � � F ⊗ 2 P � F · S � ∞ ≥ a , d [ S ] t ≤ b , � F � ∞ ≤ c . t a 0 Extension to c` agl` ad F as before. Nicolas Perkowski Game-theoretic math finance 16 / 31

Outline Definition and basic properties 1 Overview of some nice results 2 Measure free stochastic calculus 3 Pathwise stochastic calculus 4 Nicolas Perkowski Game-theoretic math finance 17 / 31

Pathwise stochastic calculus Measure free calculus excludes “nontypical price paths” at every step ⇒ not pathwise. ollmer ’81 : pathwise Itˆ o calculus. F¨ Lyons ’98 and Gubinelli ’04 : generalization to rough paths. Can we implement / extend this here? Nicolas Perkowski Game-theoretic math finance 18 / 31

Pathwise Itˆ o formula (no probability) Consider f ∈ C 2 ( R , R ), partition π . Taylor expansion: � f ( S ( t )) − f ( S (0)) = f ( S ( t j +1 )) − f ( S ( t j )) t j ∈ π f ′ ( S ( t j ))( S ( t j +1 ) − S ( t j )) + 1 � � f ′′ ( S ( t j ))( S ( t j +1 ) − S ( t j )) 2 = 2 t j ∈ π t j ∈ π � ϕ ( | S ( t j +1 ) − S ( t j ) | )( S ( t j +1 ) − S ( t j )) 2 . + t j ∈ π Nicolas Perkowski Game-theoretic math finance 19 / 31