Random G -Expectations

Transcription

1 Random G -Expectations Marcel Nutz ETH Zurich New advances in Backward SDEs for nancial engineering applications Tamerza, Tunisia, Marcel Nutz (ETH) Random G-Expectations 1 / 17

2 Outline 1 Random G -Expectations 2 Axiomatic Framework and Superhedging (joint work with Mete Soner) Marcel Nutz (ETH) Random G-Expectations 2 / 17

3 Outline 1 Random G -Expectations 2 Axiomatic Framework and Superhedging (joint work with Mete Soner) Marcel Nutz (ETH) Random G-Expectations 2 / 17

4 Peng's G -Expectation Intuition: Volatility is uncertain, but prescribed to lie in a xed interval D = [a, b]. E G (X) is the worst-case expectation of a random variable X over all these scenarios for the volatility. Ω: canonical space of continuous paths on [0, T ]. B: canonical process. P G = martingale laws under which d B t /dt D, then E G 0 (X ) = sup P P G E P [X ], X L 0 (F T ) regular enough. E G 0 is a sublinear functional. Marcel Nutz (ETH) Random G-Expectations 3 / 17

5 Peng's G -Expectation Intuition: Volatility is uncertain, but prescribed to lie in a xed interval D = [a, b]. E G (X) is the worst-case expectation of a random variable X over all these scenarios for the volatility. Ω: canonical space of continuous paths on [0, T ]. B: canonical process. P G = martingale laws under which d B t /dt D, then E G 0 (X ) = sup P P G E P [X ], X L 0 (F T ) regular enough. E G 0 is a sublinear functional. Marcel Nutz (ETH) Random G-Expectations 3 / 17

6 Conditional G -Expectation Extension to a conditional G -expectation E G t for t > 0. Nontrivial as measures in P G are singular. Peng's approach: for X = f (B T ) with f Lipschitz, dene E G t (X ) = u(t, B t ) where u t G(u xx ) = 0, u(t, x) = f ; G(x) := 1 2 sup xy. Extend to X = f (B t1,..., B tn ) and pass to completion. Time-consistency property: E G s E G t = E G s for s t. y D Marcel Nutz (ETH) Random G-Expectations 4 / 17

7 Short (and Incomplete) History Uncertain volatility model: Avellaneda, Levy, Parás (95), T. Lyons (95). BSDEs and g-expectations: Pardoux, Peng (90),... Capacity-based analysis of volatility uncertainty: Denis, Martini (06). G -expectation and related calculus were introduced by Peng (07). Dual description via P G is due to Denis, M. Hu, Peng (10). Constraint on law of B T Galichon, Henry-Labordère, Touzi (1?). Relations to 2BSDEs Cheridito, Soner, Touzi, Victoir (07), Soner, Touzi, J. Zhang (10). Bion-Nadal, Kervarec (10). Marcel Nutz (ETH) Random G-Expectations 5 / 17

8 Random G -Expectation Allow for updates of the volatility bounds. Take into account historical volatility: path-dependence. Replace D = [a, b] by a stochastic interval D t (ω) = [a t (ω), b t (ω)]. (In general: a progressive set-valued process.) This corresponds to a random function G (possibly innite). At t = 0 dene E 0 (X ) = sup P P E P [X ] for corresponding set P. Marcel Nutz (ETH) Random G-Expectations 6 / 17

9 Random G -Expectation Allow for updates of the volatility bounds. Take into account historical volatility: path-dependence. Replace D = [a, b] by a stochastic interval D t (ω) = [a t (ω), b t (ω)]. (In general: a progressive set-valued process.) This corresponds to a random function G (possibly innite). At t = 0 dene E 0 (X ) = sup P P E P [X ] for corresponding set P. Marcel Nutz (ETH) Random G-Expectations 6 / 17

10 Our Approach At t > 0: want E t (X ) = sup P P E P [X F t ]. We shall construct E t (X ) such that [ E t (X ) = ess sup P E P X F P t P(t,P) where P(t, P) := {P P : P = P on F t }. Non-Markov problem: PDE approach not suitable. ] P-a.s. for all P P, Pathwise denition: condition D, P, X on ω up to t: E t (X )(ω) := sup E P [X t,ω ], ω Ω. P P(t,ω) Benets: Control arguments, D need not be bounded. Time consistency corresponds to dynamic programming principle. Approach follows Soner, Touzi, Zhang (10). Here P(t, ω) is path-dependent. Regularity of ω P(t, ω) is needed. Marcel Nutz (ETH) Random G-Expectations 7 / 17

11 Our Approach At t > 0: want E t (X ) = sup P P E P [X F t ]. We shall construct E t (X ) such that [ E t (X ) = ess sup P E P X F P t P(t,P) where P(t, P) := {P P : P = P on F t }. Non-Markov problem: PDE approach not suitable. ] P-a.s. for all P P, Pathwise denition: condition D, P, X on ω up to t: E t (X )(ω) := sup E P [X t,ω ], ω Ω. P P(t,ω) Benets: Control arguments, D need not be bounded. Time consistency corresponds to dynamic programming principle. Approach follows Soner, Touzi, Zhang (10). Here P(t, ω) is path-dependent. Regularity of ω P(t, ω) is needed. Marcel Nutz (ETH) Random G-Expectations 7 / 17

12 Our Approach At t > 0: want E t (X ) = sup P P E P [X F t ]. We shall construct E t (X ) such that [ E t (X ) = ess sup P E P X F P t P(t,P) where P(t, P) := {P P : P = P on F t }. Non-Markov problem: PDE approach not suitable. ] P-a.s. for all P P, Pathwise denition: condition D, P, X on ω up to t: E t (X )(ω) := sup E P [X t,ω ], ω Ω. P P(t,ω) Benets: Control arguments, D need not be bounded. Time consistency corresponds to dynamic programming principle. Approach follows Soner, Touzi, Zhang (10). Here P(t, ω) is path-dependent. Regularity of ω P(t, ω) is needed. Marcel Nutz (ETH) Random G-Expectations 7 / 17

13 Strong Formulation of Volatility Uncertainty P 0 Wiener measure, F raw ltration of B. P S = { P α := P 0 ( α 1/2 db) 1, α > 0, T 0 α dt < }. Dene B and â = d B /dt simultaneously under all P P S. (E.g. by Föllmer's (81) pathwise calculus.) P := { P P S : â Int δ D ds P-a.e. for some δ > 0 }, where Int δ D := [a + δ, b δ] for δ > 0. Marcel Nutz (ETH) Random G-Expectations 8 / 17

14 Strong Formulation of Volatility Uncertainty P 0 Wiener measure, F raw ltration of B. P S = { P α := P 0 ( α 1/2 db) 1, α > 0, T 0 α dt < }. Dene B and â = d B /dt simultaneously under all P P S. (E.g. by Föllmer's (81) pathwise calculus.) P := { P P S : â Int δ D ds P-a.e. for some δ > 0 }, where Int δ D := [a + δ, b δ] for δ > 0. Marcel Nutz (ETH) Random G-Expectations 8 / 17

15 Conditioning and Regularity To condition X to ω up to t, set X t,ω ( ) := X (ω t ), where t is the concatenation at t. X t,ω is an r.v. on the space Ω t of paths starting at time t. On Ω t we have B t, P t, 0 ât, P t S,... as for t = 0. P(t, ω) := { P P t S : â t Int δ D t,ω ds P-a.e. on [t, T ] Ω t, δ > 0 }. Dene E t (X ) as the value function E t (X )(ω) := sup E P [X t,ω ], ω Ω. P P(t,ω) Regularity: X UC b (Ω) and D uniformly continuous: for all δ > 0 and (t, ω) [0, T ] Ω there exists ε = ε(t, ω, δ) > 0 s.t. ω ω t ε Int δ D t,ω s ( ω) Int ε D t,ω s ( ω) (s, ω) [t, T ] Ω t. Marcel Nutz (ETH) Random G-Expectations 9 / 17

16 Conditioning and Regularity To condition X to ω up to t, set X t,ω ( ) := X (ω t ), where t is the concatenation at t. X t,ω is an r.v. on the space Ω t of paths starting at time t. On Ω t we have B t, P t, 0 ât, P t S,... as for t = 0. P(t, ω) := { P P t S : â t Int δ D t,ω ds P-a.e. on [t, T ] Ω t, δ > 0 }. Dene E t (X ) as the value function E t (X )(ω) := sup E P [X t,ω ], ω Ω. P P(t,ω) Regularity: X UC b (Ω) and D uniformly continuous: for all δ > 0 and (t, ω) [0, T ] Ω there exists ε = ε(t, ω, δ) > 0 s.t. ω ω t ε Int δ D t,ω s ( ω) Int ε D t,ω s ( ω) (s, ω) [t, T ] Ω t. Marcel Nutz (ETH) Random G-Expectations 9 / 17

17 Consequences of Uniform Continuity ω E t (X )(ω) is F t -measurable and LSC for X UC b(ω). Theorem (DPP, time consistency) Let X UC b (Ω) and 0 s t T. Then E s (X )(ω) = sup E P[ E t (X ) s,ω] for all ω Ω, P P(s,ω) [ E s (X ) = ess sup P E P Et (X ) F ] P s P-a.s. for all P P, P(s,P) where P(s, P) := {P P : P = P on F s }. On the proof: Main problem due to stochastic D: admissibility of pastings. Regularity of E t (X ) turns out not to be a problem. Marcel Nutz (ETH) Random G-Expectations 10 / 17

18 Consequences of Uniform Continuity ω E t (X )(ω) is F t -measurable and LSC for X UC b(ω). Theorem (DPP, time consistency) Let X UC b (Ω) and 0 s t T. Then E s (X )(ω) = sup E P[ E t (X ) s,ω] for all ω Ω, P P(s,ω) [ E s (X ) = ess sup P E P Et (X ) F ] P s P-a.s. for all P P, P(s,P) where P(s, P) := {P P : P = P on F s }. On the proof: Main problem due to stochastic D: admissibility of pastings. Regularity of E t (X ) turns out not to be a problem. Marcel Nutz (ETH) Random G-Expectations 10 / 17

19 Extension to Completion of UC b (Ω) L 1 P = space of r.v. X such that X L 1 := sup P P X L P 1 (P) <. L 1 P = closure of UC b L 1 P (can be described explicitly). DPP implies that E t is 1-Lipschitz wrt. L 1, hence extends to P E t : L 1 P L 1 P(F t ). Theorem. For X L 1 P the DPP holds: [ E s (X ) = ess sup P E P Et (X ) F ] P s P(s,P) P-a.s. for all P P. In particular, E s (X ) is characterized by [ E s (X ) = ess sup P E P X F ] P s P(s,P) P-a.s. for all P P. Marcel Nutz (ETH) Random G-Expectations 11 / 17

20 Extension to Completion of UC b (Ω) L 1 P = space of r.v. X such that X L 1 := sup P P X L P 1 (P) <. L 1 P = closure of UC b L 1 P (can be described explicitly). DPP implies that E t is 1-Lipschitz wrt. L 1, hence extends to P E t : L 1 P L 1 P(F t ). Theorem. For X L 1 P the DPP holds: [ E s (X ) = ess sup P E P Et (X ) F ] P s P(s,P) P-a.s. for all P P. In particular, E s (X ) is characterized by [ E s (X ) = ess sup P E P X F ] P s P(s,P) P-a.s. for all P P. Marcel Nutz (ETH) Random G-Expectations 11 / 17

21 Outline 1 Random G -Expectations 2 Axiomatic Framework and Superhedging (joint work with Mete Soner) Marcel Nutz (ETH) Random G-Expectations 11 / 17

22 Axiomatic Setup For the random G -expectations, we had: a set P P S with a time consistency is a property, [ ] an aggregated r.v. for ess sup P E P X F P s, P P, P(s,P) for X in a subspace L 1 P L1 P. Axiomatic approach: start with some set P P S. P is assumed to be stable under F -pasting ( time consistency): for all P P and P 1, P 2 P(F t, P) and Λ F t, P( ) := E P[ P 1 ( F t )1 Λ + P 2 ( F t )1 Λ c ] P. aggregated r.v. E [ ] s (X ) = ess sup P E P X F P P(F,P) s P-a.s., P P s for all X in some subspace H L 1 P. Marcel Nutz (ETH) Random G-Expectations 12 / 17

23 Axiomatic Setup For the random G -expectations, we had: a set P P S with a time consistency is a property, [ ] an aggregated r.v. for ess sup P E P X F P s, P P, P(s,P) for X in a subspace L 1 P L1 P. Axiomatic approach: start with some set P P S. P is assumed to be stable under F -pasting ( time consistency): for all P P and P 1, P 2 P(F t, P) and Λ F t, P( ) := E P[ P 1 ( F t )1 Λ + P 2 ( F t )1 Λ c ] P. aggregated r.v. E [ ] s (X ) = ess sup P E P X F P P(F,P) s P-a.s., P P s for all X in some subspace H L 1 P. Marcel Nutz (ETH) Random G-Expectations 12 / 17

24 Getting Path Regularity ˆF = { ˆF t } 0 t T, where ˆF t := F t+ N P and N P = P-polar sets. Take right limits of {E t (X ), t [0, T ]}: Theorem For X H, there exists a unique càdlàg ˆF-adapted process Y, Y t = E t+(x ) P-q.s. for all t. Y is the minimal ( ˆF, P)-supermartingale with Y T = X. Y t = ess sup P E P [X ˆF t ] P-a.s. for all P P. P P( ˆF t,p) Y is a P-modication of {E t (X ), t [0, T ]} in regular cases but there are counterexamples. The process E(X ) := Y is called the (càdlàg) E-martingale associated with X H. Marcel Nutz (ETH) Random G-Expectations 13 / 17

25 Stopping Times and Optional Sampling Typically, the construction of E is not compatible with stopping times (e.g. G -expectation). But we can easily dene E at a stopping time. Theorem Let 0 σ τ T be ˆF-stopping times and X H. Then E σ (X ) = E σ (X ) = ess sup E P [X ˆF σ ] P-a.s. for all P P; P P( ˆF σ,p) ess sup E P [E τ (X ) ˆF σ ] P-a.s. for all P P. P P( ˆF σ,p) Marcel Nutz (ETH) Random G-Expectations 14 / 17

26 Decomposition of E-Martingales Theorem Let X H. There exist such that an ˆF-progressive process Z X a family (K P ) P P of F P -pred. increasing processes, E P [ K P ] <, T E t (X ) = E 0 (X ) + (P) t 0 Z X s db s K P t, P-a.s. for all P P. Z X does not depend on P, but the integral may do. Cf. optional decomposition: El Karoui, Quenez (95), Kramkov (96). Construction as in the theory of 2BSDEs: Soner, Touzi, Zhang (10) Here we only need Doob-Meyer decomposition + martingale represent. + pathwise integration (Bichteler 81). More precise results for G -expectation: Peng (07), Xu, B. Zhang (09), Soner, Touzi, Zhang (10), Song (10), Y. Hu, Peng (10). Marcel Nutz (ETH) Random G-Expectations 15 / 17

27 Decomposition of E-Martingales Theorem Let X H. There exist such that an ˆF-progressive process Z X a family (K P ) P P of F P -pred. increasing processes, E P [ K P ] <, T E t (X ) = E 0 (X ) + (P) t 0 Z X s db s K P t, P-a.s. for all P P. Z X does not depend on P, but the integral may do. Cf. optional decomposition: El Karoui, Quenez (95), Kramkov (96). Construction as in the theory of 2BSDEs: Soner, Touzi, Zhang (10) Here we only need Doob-Meyer decomposition + martingale represent. + pathwise integration (Bichteler 81). More precise results for G -expectation: Peng (07), Xu, B. Zhang (09), Soner, Touzi, Zhang (10), Song (10), Y. Hu, Peng (10). Marcel Nutz (ETH) Random G-Expectations 15 / 17

28 Superhedging Interpretation for decomposition X = E T (X ) = E 0 (X ) + (P) T 0 Z X s db s K P T : E 0 (X ) = ˆF 0 -superhedging price, Z X K P T = superhedging strategy, = overshoot for the scenario P Minimality of the overshoot: [ ess inf P E P K P ] K P P T t ˆF t = 0 P( ˆF t,p) P-a.s. for all P P. replicable claims correspond to K P 0 for all P P. Marcel Nutz (ETH) Random G-Expectations 16 / 17

29 2BSDE for E(X ) (Y, Z) is a solution of the 2BSDE if there exists a family (K P ) P P of F P -adapted increasing processes satisfying E P [ K P ] < such that T Y t = X (P) T t Z s db s + K P T K P t, 0 t T, P-a.s. for all P P and such that [ ess inf P E P K P ] K P P T t ˆF t = 0 P( ˆF t,p) P-a.s. for all P P. Theorem (X H) (E(X ), Z X ) is the minimal solution of the 2BSDE. If (Y, Z) is a solution of the 2BSDE such that Y is of class (D,P), then (Y, Z) = (E(X ), Z X ). In particular, if X H p for some p (1, ), then (E(X ), Z X ) is the unique solution of the 2BSDE in the class (D,P). Marcel Nutz (ETH) Random G-Expectations 17 / 17

30 Pasting and Time Consistency P is maximally chosen for H if P contains all P P S such that E P [X ] sup P P E P [X ] for all X H. P is time-consistent on H if [ ] ess sup P E P ess sup P E P [X F P P(F s,p) P P(F t ] F s = ess sup P E P [X F t,p ) P P(F s ] s,p) P-a.s. for all P P, 0 s t T and X H. Theorem stability under pasting time consistency. If P is maximally chosen: time consistency stability under pasting Similar results by Delbaen (06) for classical risk measures. Marcel Nutz (ETH) Random G-Expectations 18 / 17

31 Time Consistency of Mappings Consider a family (E t ) 0 t T of mappings E t : H L 1 P (F t ). H t := H L 1 P (F t ). Denition (E t ) 0 t T is called time-consistent if E s (X ) ( ) E s (φ) for all φ H t such that E t (X ) ( ) φ and (H t -) positively homogeneous if E t (X φ) = E t (X )φ for all bounded nonnegative φ H t for all 0 s t T and X H. Marcel Nutz (ETH) Random G-Expectations 19 / 17

32 More on L 1 P By arguments of Denis, Hu, Peng (10): { L 1 P = X L 1 P X is P-quasi uniformly continuous, lim n X 1 { X n} L 1 P = 0 } If D is uniformly bounded, we retrieve the space of Denis, Hu, Peng: L 1 P is also the closure of Cb L 1 P, `quasi uniformly continuous' = `quasi continuous'. If D is uniformly bounded, E t maps L 1 P into L1 P (F t ). Hence time consistency can be expressed as E s E t = E s. Marcel Nutz (ETH) Random G-Expectations 20 / 17