Additional information and pricing-hedging duality in robust framework

Transcription

1 Additional information and pricing-hedging duality in robust framework Anna Aksamit based on joint work with Zhaoxu Hou and Jan Ob lój London - Paris Bachelier Workshop on Mathematical Finance Paris, September 29-30, 2016

2 Pricing and hedging problems How to determine the prices of exotic options? How to hedge the positions in exotic options by using underlying assets and vanilla options? Model-specific approach: the price process of the underlying assets (S t ) t T are modelled by some parametric family of stochastic processes. Model-independent approach: many possible models, weaker economic assumptions Quasi-sure approach Pathwise approach 2 / 25

3 Robust approach an active field of research Explicit bounds LB PO T UB and robust super-/sub- hedges Arbitrage considerations and robust FTAP Pricing-hedging duality Pathspace restrictions A Acciaio, Bayraktar, Beiglböck, Biagini, Bouchard, Brown, Burzoni, Cheridito, Cox, Davis, Denis, Dolinsky, Dupire, Frittelli, Galichon, Gassiat, Guo, Henry-Labordère, Hobson, Hou, Huesmann, Källblad, Kardaras, Klimmek, Kupper, Maggis, Martini, Mykland, Nadtochiy, Neuberger, Neufeld, Nutz, Ob lój, Penker, Perkowski, Possamaï, Prömel, Raval, Riedel, Rogers, Schachermayer, Soner, Spoida, Tan, Tangpi, Temme, Touzi, Wang... 3 / 25

4 Additional information F regular agent/ common knowledge/ public information G F informed agent with additional information represented by the entire filtration Hedging ξ: how much the additional information is worth? Super-hedging price: Price F (ξ) Price G (ξ) inf{x : (x, γ)-super-hedges ξ} vs market model price: sup E P (ξ) P M No (duality) gap between these two values 4 / 25

5 Set-up Yan Dolinsky and Mete Soner Martingale Optimal Transport and Robust Hedging in Continuous Time, Probability Theory and Related Fields (2014), Zhaoxu Hou and Jan Ob lój On robust pricing-hedging duality in continuous time. 5 / 25

6 Traded assets and information Stock price S is the canonical process on := {ω C([0, T ], R d +) : ω(0) = 1} F is the filtration generated by S, i.e., F t := σ(s s : s t) X 0, X 1,..., X n statically traded options which have prices P(X i ) at time 0, X 0 = 1 and P(X 0 ) = 1 G is the enlarged filtration G t := F t H t, where H := (H t ) t T is another filtration G is called the initial enlargement of F with random variable Z if H t = σ(z) 6 / 25

7 Equivalence relation and atoms (, F T ) measurable space and G sub-σ-field of F T, ω, ω ω and ω are G-equivalent, ω G ω, if 1 G (ω) = 1 G ( ω) holds G G Note that ω Ft ω ω u = ω u for each u t, and ω σ(z) ω Z(ω) = Z( ω) [ω] G denotes the equivalence class, or atom, in where ω belongs to: [ω] G = {A : A G, ω A} (, G) is countably generated if there exists a sequence (B n ) n N G such that σ((b n ) n N ) = G. In this case each atom is G-measurable 7 / 25

8 Trading strategies integral of g : [0, T ] R d of finite variation w.r.t. ω : t 0 t g(u)dω(u) := g(t)ω(t) g(0)ω(0) ω(u)dg(u) 0 γ : D([0, T ], R d ) is G-adapted if γ t is G t -measurable, i.e. if ω Gt ω implies γ(ω) t = γ( ω) t and it is G-admissible strategy if moreover it has finite variation and t 0 γ(ω) u ds u (ω) M(ω) ω, t for some M L 0 (, G 0 ) A G-admissible semi-static strategy is a pair (X, γ) where X = A 0 + n i=1 A ix i for some G 0 -measurable random variables A i and G-admissible strategy γ. Initial cost of such a strategy is P(X ) = A 0 + m i=1 A ip(x i ). The set of all G-admissible semi-static strategies is denoted by A(G). 8 / 25

9 The super-hedging price G-super-hedging price of ξ on A F T : V G A (ξ)(ω) := inf{p(x )(ω) : X ( ω) + T 0 (X, γ) A(G) such that γ( ω) u ds u ( ω) ξ( ω) for all ω A} Proposition The G-super-hedging price on is constant on each [ω] and given by V G (ξ)(ω) = V G [ω] (ξ) where [ω] denotes the G 0 -equivalence class containing ω. It holds that V G (ξ) V F (ξ) 9 / 25

10 The market model price The set of G-calibrated martingale measures concentrated on A F T : M G A := {P : S is a (P, G)-martingale, P(A) = 1 and E P (X i G 0 ) = P(X i ) for all i {1,..., n} P-a.s.} G-market price of ξ on A F T : PA G (ξ)(ω) := sup E P (ξ G 0 )(ω) P M G A Proposition Assume each element of G is countably generated. Let P M G X,P,. Then, there exists a set P G 0 with P( P ) = 1 and a version {P ω } of the regular conditional probabilities of P with respect to G 0 such that for each ω P, P ω M G X,P,[ω] G0. 10 / 25

11 The market model price Let A F T. The G-market price of ξ on A is defined by P G A (ξ)(ω) := sup P M G X,P,A Ē Pω (ξ), ω, where Ē Pω (ξ) = E Pω (ξ) for ω P and Ē Pω (ξ) = for ω \ P. Proposition The G-market price on is constant on each [ω] and given by P G (ξ)(ω) = P G [ω] (ξ) where [ω] is G 0 -equivalent class containing ω. It holds that P G (ξ) PF (ξ) 11 / 25

12 Easy inequality in the pricing-hedging duality Lemma The G-super-hedging price V G (ξ) and the G-market model price PG (ξ) of ξ on satisfy V G (ξ)(ω) PG (ξ)(ω) ω. Proof: G-super-replicating portfolio (X, γ) A M (G) on [ω] G0 measure P M[ω] G G0. and {P v } regular conditional probabilities of P with respect to G 0 Since P(M const) = 1, T ) E P (ξ) E P (X + γ u ds u E P (X ) / 25

13 Duality 13 / 25

14 Atoms and path restriction Let G = F σ(z) Additional information arrives entirely at time 0 Atoms of G 0 are simply atoms of σ(z) Each atom can be seen as path restriction since on each atom the filtration G and F coincide, i.e., for each ω G G t F F t s.t. [ω] G0 G = [ω] G0 F 14 / 25

15 Duality for G = F σ(z) Theorem Let G = F σ(z) and assume that for each value c of Z we have P F {Z=c} (ξ) = V F {Z=c} (ξ) for any bounded uniformly continuous ξ. Suppose that assumptions of Theorem Hou Ob lój are satisfied. Then, duality in G holds, i.e., V G (ξ)(ω) = PG (ξ)(ω) for any bounded uniformly continuous ξ. Proof: One can show that: P F {Z=c} (ξ) = PG {Z=c} (ξ) V G {Z=c} (ξ) = V F {Z=c} (ξ) 15 / 25

16 [HO] Beliefs: approximate pricing-hedging duality Approximation of A: A ε = {ω : inf v A ω v ε}. Ṽ F A (ξ) := inf{p(x ) : (X, γ) A(F), ε > 0 s.t. X + T 0 γ u ds u ξ on A ε } P A (ξ) := lim ε 0 sup P M F,ε A E P (ξ) where M F,ε A := {P : S is a (P, F)-mart., P(Aε ) = 1 ε and E P (X i ) P(X i ) ε i} Theorem (Hou & Ob lój) Assume that all payoffs X i are bounded and uniformly continuous and that for all ε > 0 there exists P M F,ε A. Then for any bounded uniformly continuous ξ : R This theorem implies that Ṽ A (ξ) = P A (ξ). P A (ξ) V A (ξ) ṼA(ξ) = P A (ξ) 16 / 25

17 Duality for G = F σ(z) Theorem Let G = F σ(z) and assume that for each value c of Z we have P F {Z=c} (ξ) = P F {Z=c} (ξ) for any bounded uniformly continuous ξ. Suppose that assumptions of Theorem Hou Ob lój are satisfied. Then, duality in G holds, i.e., V G (ξ)(ω) = PG (ξ)(ω) for any bounded uniformly continuous ξ. Example: Assume no options and d = 1. No duality gap in G holds: Z = sup t [0,T ] ln S t Z = 1 {a<st<b t [0,T ]} where a < 1 < b 17 / 25

18 Dynamic programming principle 18 / 25

19 Dynamic approach Additional information σ(z) is disclosed at time T 1 (0, T ): G t = F t for t [0, T 1 ) and G t = F t σ(z) for t [T 1, T ] Assume Z satisfies { ( ) ω [T1 Z,T ] ω Z(ω) = T1 if ω T1 > 0 1 if ω T1 = 0 for r.v. Z on [T1,T ]. This encodes the idea that the additional information concerns only the evolution of prices after time T 1 irrespectively of the prices before time T 1. Theorem Duality in G holds, i.e., V G(ξ) = PG (ξ) holds for any bounded uniformly continuous ξ. 19 / 25

20 Dynamic approach Firstly solve the problem for each atom of G T1 by the same arguments as for F σ(z) on [T 1, T ] separately Secondly aggregate atoms of G T1 Apply dynamic principle into atoms of F T1 Thus the problem is now reduced to [0, T 1 ] and trading w.r.t F V G,[0,T ] (ξ) = V F,[0,T 1] (V G,[T 1,T ] (ξ)) = V F,[0,T 1] (P G,[T 1,T ] = P F,[0,T 1] (P G,[T 1,T ] (ξ)) = P G,[0,T ] (ξ). (ξ)) 20 / 25

21 Dynamic programming principle Proposition Let ξ be uniformly continuous. (i) Define T V G,[T 1,T ] (ξ)(ω) := inf{x : γ A(G) s.t. x + γ u ds u ξ on [ω] FT1 }. T 1 Then, V G,[T 1,T ] (ξ) is u.c. and ( V G,[0,T ] (ξ) = V F,[0,T 1] V G,[T 1,T ] ) (ξ) (ii) Define P G,[T 1,T ] (ξ)(ω) := sup G,[T P M 1,T ] E P (ξ). Then, P G,[T 1,T ] (ξ) is [ω] FT1 u.c. and ( ) P G,[0,T ] (ξ) = P F,[0,T 1] P G,[T 1,T ] (ξ) 21 / 25

22 Conclusions Formulation of the duality problem for a general filtration with possibly non trivial initial σ-field Translating original problem to the path restriction language from Hou & Ob lój in case of an initial enlargement Disclosure of an additional information after initial time and dynamic programming principle 22 / 25

23 Thank You! 23 / 25

24 Dynamic programming principle The path modification mapping α v,ṽ by v [0,T1 ] v T 1 ṽ T1 ω [T1,T ] α(ω) := ṽ [0,T1 ] ṽt 1 v T1 ω [T1,T ] ω ω Bṽ ω B v ω / B v Bṽ If the strategy γ super-replicates on B v, the strategy v T 1 ṽ T1 γ α + δ 1 4 ṽ T1 1 [T1, τ) super-replicates on Bṽ and ξ(ṽ) ξ(v) + e ξ( v ṽ ). If P M G,[T 1,T ] B then P = P α M G,[T 1,T ] v B and v E P (ξ) E P (ξ) e ξ( v ṽ ) 24 / 25

25 Informational metrics Quantification of the value of the information Initial enlargement distances between σ-fileds d(g, H): sup G G inf H H sup 0 ξ 1 sup P(G H) sup P M H H sup P M E P sup Q M [ω] G E Q (ξ) inf G G sup P(G H) P M sup Q M [ω] H E Q (ξ) 25 / 25