MESURES DE RISQUE DYNAMIQUES DYNAMIC RISK MEASURES

Transcription

1 from BMO martingales MESURES DE RISQUE DYNAMIQUES DYNAMIC RISK MEASURES CNRS - CMAP Ecole Polytechnique March 1, / 45

2 OUTLINE from BMO martingales 1 INTRODUCTION 2 DYNAMIC RISK MEASURES Time Consistency Regularity of the paths Classical examples 3 DYNAMIC RISK MEASURES FROM BMO MARTINGALES from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales 4 BID-ASK DYNAMIC PRICING PROCEDURE Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options 5 CONCLUSION 2/ 45

3 OUTLINE from BMO martingales 1 INTRODUCTION 2 Time Consistency Regularity of the paths Classical examples 3 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales 4 Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options 5 3/ 45

4 INTRODUCTION from BMO martingales MONETARY RISK MEASURES: Coherent risk measures: Artzner, Delbaen, Eber, Heath (1999) Convex monetary risk measures: Föllmer and Schied (2002)and Frittelli and Rosazza Gianin (2002) CONDITIONAL RISK MEASURES on a probability space : Detlefsen Scandolo (2005) in a context of uncertainty : Bion-Nadal (preprint 2004) DYNAMIC RISK MEASURES Coherent : Delbaen (2003) and Artzner, Delbaen, Eber, Heat, Ku (2004), and Riedel (2004) Convex dynamic risk measures considered in many recent papers, among them: Frittelli and Rosaza Gianin (2004), Klöppel, Schweizer (2005), Cheredito, Delbaen, Kupper (2006), Bion-Nadal (2006), Föllmer and Penner (2006) g expectations or Backward Stochastic Differential Equations : Peng (2004), Rosazza Gianin (2004) and Barrieu El Karoui (2005) 4/ 45

5 from BMO martingales DYNAMIC RISK MEASURES MAIN RESULTS FOR DYNAMIC RISK MEASURES Characterization of Time Consistency by a cocycle condition on the minimal penalty Regularity of paths for time consistent dynamic risk measures Families of dynamic risk measures constructed from right continuous BMO martingales generalizing B.S. D. E. and allowing for jumps. 5/ 45

6 from BMO martingales APPLICATION TO DYNAMIC PRICING Develop an axiomatic approach to associate to any financial product a dynamic ask price process and a dynamic bid price process taking into account liquidity risk and transaction costs. The ask price process is convex. MAIN RESULT: A time consistent pricing procedure has No Free Lunch if and only if there is an equivalent probability measure R such that for any X the martingale E R (X F t ) is between the bid price process and the ask price process associated with X. Furthermore the ask price process is then a R supermartingale with cadlag modification This generalizes the result of Jouiny and Kallal which was obtained for sublinear ask prices. 6/ 45

7 OUTLINE from BMO martingales Time Consistency Regularity of the paths Classical examples 1 2 DYNAMIC RISK MEASURES Time Consistency Regularity of the paths Classical examples 3 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales 4 Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options 5 7/ 45

8 from BMO martingales DYNAMIC RISK MEASURES Time Consistency Regularity of the paths Classical examples FRAMEWORK Consider a filtered probability space (Ω, F, (F t ) t IR +, P). Assume that the filtration (F t ) is right continuous and that F 0 is the σ-algebra generated by the P null sets of F so that L (Ω, F 0, P) = IR. We work on stopping times. For a stopping time τ, recall that F τ = {A F t IR + A {τ t} F t }. 8/ 45

9 from BMO martingales DYNAMIC RISK MEASURE Time Consistency Regularity of the paths Classical examples DEFINITION A Measure on (Ω, F, (F t ) t IR +, P) is a family of maps (ρ σ,τ ) 0 σ τ, (σ τ are stopping times) defined on L (F τ ) with values into L (F σ ) satisfying 1 monotonicity: if X Y then ρ σ,τ (X) ρ σ,τ (Y) 2 translation invariance: 3 convexity: Z L (F σ ), X L (F τ ) ρ σ,τ (X + Z) = ρ σ,τ (X) Z (X, Y) (L (F τ )) 2 λ [0, 1] ρ σ,τ (λx + (1 λ)y) λρ σ,τ (X) + (1 λ)ρ σ,τ (Y) 9/ 45

10 CONTINUITY from BMO martingales Time Consistency Regularity of the paths Classical examples DEFINITION A Measure (ρ σ,τ ) is continuous from above (resp below) if for any decreasing (resp increasing) sequence X n of elements of L (Ω, F τ, P) converging to X, for any σ τ, the increasing (resp decreasing) sequence ρ σ,τ (X n ) converges to ρ σ,τ (X) Continuity from above is equivalent to the existence of a dual representation in terms of probability measures. Continuity from below implies continuity from above. DEFINITION A Measure (ρ σ,τ ) is normalized if σ τ, ρ σ,τ (0) = 0 10/ 45

11 from BMO martingales DUAL REPRESENTATION Time Consistency Regularity of the paths Classical examples PROPOSITION Any Measure continuous from above has a dual representation : Let σ τ X L (F τ ) ρ σ,τ (X) = ess sup Q M 1 σ,τ (P)[(E Q ( X F σ ) α m σ,τ (Q)] where α m σ,τ (Q) = Q ess sup X L (Ω,F τ,p)[e Q ( X F σ ) ρ σ,τ (X)] M 1 σ,τ (P) = {Q on (Ω, F τ ), Q P, Q Fσ = P and E P (α m σ,τ (Q)) < } 11/ 45

12 from BMO martingales Time Consistency Regularity of the paths Classical examples DUAL REPRESENTATION IN CASE OF CONTINUITY FROM BELOW PROPOSITION Any Measure continuous from below has the following dual representation X L (F τ ) ρ σ,τ (X) = ess max Q Mσ,τ (P)[(E Q ( X F σ ) α m σ,τ (Q)] where M σ,τ (P) = {Q P, Q Fσ = P and α m σ,τ (Q) L (Ω, F σ, P} 12/ 45

13 from BMO martingales TIME CONSISTENCY Time Consistency Regularity of the paths Classical examples TIME CONSISTENCY DEFINITION A Measure (ρ σ,τ ) 0 σ τ is time consistent if 0 ν σ τ X L (F τ ) ρ ν,σ ( ρ σ,τ (X)) = ρ ν,τ (X). The notion of time consistency first appeared in the work of Peng. For coherent dynamic risk measures: Delbaen (2003) In a discrete time setting: Cheridito, Delbaen, Kupper: characterization by a condition on the acceptance sets and characterization by a concatenation condition. Denote for Q P A ν,τ (Q) = {Y L (Ω, F τ, P) ρ ν,τ (Y) 0 Q a.s.} 13/ 45

14 from BMO martingales COCYCLE CONDITION Time Consistency Regularity of the paths Classical examples THEOREM Let (ρ σ,τ ) be a Measure continuous from above. The three following properties are equivalent. Let ν σ τ I) The dynamic risk measure is time consistent i.e. ρ ν,τ (X) = ρ ν,σ ( ρ σ,τ (X)) X L (Ω, F τ ) II) For any probability measure Q absolutely continuous with respect to P, A ν,τ (Q) = A ν,σ (Q) + A σ,τ (P) III) For any probability measure Q absolutely continuous with respect to P, the minimal penalty function satisfies the cocycle condition α m ν,τ (Q) = α m ν,σ(q) + E Q (α m σ,τ (Q) F ν ) Q a.s. 14/ 45

15 from BMO martingales CADLAG MODIFICATION Time Consistency Regularity of the paths Classical examples Denote M 0 0,τ = {Q P α m 0,τ (Q) = 0} THEOREM Let (ρ σ,τ ) σ τ be a normalized time consistent dynamic risk measure continuous from above. Assume that M 0 0,τ. For any X L (F τ ), for any R M 0 0,τ, there is a cadlag R-supermartingale process Y such that for any finite stopping time σ τ, ρ σ,τ (X) = Y σ R a.s. The proof follows that of Delbaen for coherent dynamic risk measures and is based on theorems of Dellacherie Meyer. LEMMA Same hypothesis. Then the process (ρ σ,τ (X)) σ is a R-supermartingale: ν σ τ ρ ν,τ (X) E R (ρ σ,τ (X) F ν ) R a.s. 15/ 45

16 from BMO martingales CADLAG MODIFICATION Time Consistency Regularity of the paths Classical examples LEMMA Same hypothesis. Let σ τ, σ finite. Consider a decreasing sequence of finite stopping times σ n τ converging to σ. Then E R (ρ σn,τ (X)) converges to E R (ρ σ,τ (X)), and ρ σn,τ (X) tends to ρ σ,τ (X) in L 1 (Ω, F, R). DEFINITION A Normalized Measure (ρ στ ) is called non degenerate if A F, ρ 0, (λ1 A ) = 0 λ IR + implies P(A) = 0. COROLLARY Let (ρ σ,τ ) be time consistent, non degenerate and continuous from below. Then M 0 0,. Any R M0 0, is equivalent to P and X L (F ) there is a cadlag R-supermartingale process Y, such that for all finite σ, ρ σ, (X) = Y σ L (Ω, F σ, P) 16/ 45

17 from BMO martingales EXAMPLE:ENTROPIC RISK Time Consistency Regularity of the paths Classical examples ENTROPIC DYNAMIC RISK MEASURE A s,t = {Y L (F t ) E(e [ αy] F s ) 1} ρ s,t (X) = essinf{y L (F s ) X + Y A s,t } = essinf{y L (F s ) / E(e [ α(x+y)] F s ) 1} = 1 α ln E(e αx F s ) αs,t(q) m = 1 α (E P(ln( dq dp )dq dp F s) The entropic dynamic risk measure is time-consistent. Also: Barrieu el Karoui, Cheredito Kupper, Detlefsen Scandolo. Cheredito Kupper: Expected utilities lead to time-consistent dynamic risk measures iff the utility fonction is linear or exponential. 17/ 45

18 from BMO martingales EXAMPLE:BSDE Time Consistency Regularity of the paths Classical examples BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS F t is the augmented filtration of a d dimensional Brownian motion. Assume that g(t, z) is convex (in z) and satisfies the condition of quadratic growth. The associated BSDE, dy t = g(t, Z t )dt Z t db t Y T = X has a solution which gives rise to a time-consistent dynamic risk measure ρ s,t ( X) = Y s Peng. Also Barrieu El Karoui, Klöppel Schweizer, Roosazza Gianin... The paths of dynamic risk measures associated with B.S.D.E. are continuous. 18/ 45

19 OUTLINE from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales 1 2 Time Consistency Regularity of the paths Classical examples 3 DYNAMIC RISK MEASURES FROM BMO MARTINGALES from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales 4 Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options 5 19/ 45

20 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales STABLE SET OF PROBABILITY MEASURES Conditions on Q and α so that ρ σ,τ (X) = esssup Q Q {E Q ( X F σ )) α σ,τ (Q)} defines a time consistent Measure. DEFINITION A set Q of probability measures all equivalent to P is stable if for every stopping times, ν σ τ, for every Q Q, for every R Q, there is S Q such that i.e. X L (F τ ), E S (X F ν ) = E R (E Q (X F σ ) F ν ) P.a.s. where ( ds dp ) t means E( ds dp F t) dp ) τ ( ds dp ) τ = ( dq ( dq dp ) σ ( dr dp ) σ 20/ 45

21 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales CONDITIONS ON THE PENALTY FUNCTION DEFINITION A penalty function α σ,τ defined on Q with values in L is local if for every stopping times σ τ, for every A F σ -measurable, if E Q1 (X1 A F σ ) = E Q2 (X1 A F σ ) P.a.s. X L (Ω, F τ, P), then 1 A α σ,τ (Q 1 ) = 1 A α σ,τ (Q 2 ) P.a.s. satisfies the cocycle condition if for every Q in Q, for every stopping times ν σ τ, α ν,τ (Q) = α ν,σ (Q) + E Q (α σ,τ (Q) F ν ) 21/ 45

22 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales SUFFICIENT CONDITION FOR TIME CONSISTENCY The following theorem allows for the construction of many examples of time consistent : THEOREM Let Q be a stable set of probabilities all equivalent to P. Assume that α is local, satisfies the cocycle condition and esssup Q Q ( α σ,τ (Q)) L (F σ ). Then the Measure (ρ σ,τ ) 0 σ τ defined by is time-consistent. ρ σ,τ (X) = esssup Q Q {E Q ( X F σ )) α σ,τ (Q)} In the case of penalty identically equal to 0 on Q this result was first proved by Delbaen. 22/ 45

23 from BMO martingales STOCHASTIC EXPONENTIAL from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales (M) t martingale, M 0 = 0. Stochastic exponential of M: unique solution of E(M) t = 1 + t 0 E(M) s dm s E(M) t = exp(m t 1 2 ([M, M]c ) t )Π s t (1 + M s )e Ms MARTINGALES WITH BOUNDED QUADRATIC VARIATION Q 1 = {Q M ; dq M dp = E(M) M continuous P martingale ; [M, M] L (Ω, F, P)} is a stable set of probability measures all equivalent to P. 23/ 45

24 from BMO martingales BMO MARTINGALES from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales BMO MARTINGALES(Doleans-Dade Meyer) a right continuous square integrable martingale M is BMO if there is a constant c such that for any stopping time S, E([M, M] [M, M] S F S ) c 2 The smallest c is M BMO CASE OF CONTINUOUS BMO MARTINGALES Q 2 = {Q M ; dq M dp = E(M) M continuous martingale ; M BMO < } is a stable set of probability measures all equivalent to P. 24/ 45

25 STABLE SETS from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales RIGHT CONTINUOUS BMO MARTINGALES For right continuous BMO martingales: restrictions on the BMO norms. LEMMA Let M 1,..., M j be strongly orthogonal square integrable right continuous martingales in (Ω, F, (F t ) 0 t, P). Let Φ be a non negative predictable process such that Φ.M i is a BMO martingale of BMO norm m i. Let M = { H i.m i, H i predictable H i Φ} 1 i j Any M M is a BMO martingale with BMO norm less than ( 1 i j (mi ) 2 ) 1 2 = m. If m < 1 8, Q(M) = {(Q M) M M dqm dp stable set of equivalent probability measures. No restriction on m if each M i is continuous. = E(M)} is a 25/ 45

26 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales TIME-CONSISTENT DYNAMIC RISK MEASURES FROM BMO RIGHT CONTINUOUS MARTINGALES THEOREM Let M be a set of BMO martingales; Assume that Q(M) = {Q M dqm dp = E(M), M M} is a stable set of probability measures all equivalent to P. Assume { M BMO M M} is bounded (by 1 16 in right continuous case) Let H a bounded predictible process. α σ,τ (Q M ) = E QM ((H.M) τ (H.M) σ F σ ) ρ σ,τ (X) = esssup QM Q(M) (E Q M ( X F σ ) α σ,τ (Q M )) defines a time consistent dynamic risk measure. 26/ 45

27 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales DYNAMIC RISK MEASURES FROM BMO MARTINGALES THEOREM Let Q(M) = {Q M dqm dp = E(M), M M} be a stable set of probability measures such that any M M is BMO. τ α σ,τ (Q M ) = E QM ( σ b u d[m, M] u ) F σ ) If b bounded and { M BMO } bounded (by 1 16 in right continuous case) or if b is non negative and 0 M ρ σ,τ (X) = esssup QM Q(M) (E Q M ( X F σ ) α σ,τ (Q M )) defines a time-consistent dynamic risk measure. 27/ 45

28 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales DYNAMIC RISK MEASURES FROM BMO MARTINGALES GENERALIZE BSDE PROPOSITION Let (M i ), (M0 i = 0) be strongly orthogonal continuous martingales. Let M = { 1 i j H i.m i H i.m l BMO (i, l)}. Let b i (s, x 1,..., x k ) Borel functions of quadratic growth. For M = 1 i j H i.m i, define α s,t (Q M ) = E QM ( Assume that t 1 i j s b i (u, (H 1 ) u, (H 2 ) u,..., (H j ) u )d[m i, M i ] u F s ) ρ s,t (X) = essmax M M ((E QM ( X F s ) α s,t (Q M )) Then ρ s,t is a time-consistent dynamic risk measure. 28/ 45

29 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales DYNAMIC RISK MEASURES FROM BMO MARTINGALES GENERALIZE BSDE N. El Karoui and P. Barrieu have computed the dual representation associated withe BSDE when g is strictly convex of quadratic growth. (M i ) 1 i j are independent Brownian motions. The filtration is the augmented filtration of the (M i ) 1 i j. Thus M as in proposition is M = { 1 i k H i.m i sup S E( H i (u) 2 du F S ) < }. The penalty function is t α s,t (Q M ) = E QM ( where G is the Fenchel transform of g. s S G(u, (H 1 ) u, (H 2 ) u,..., (H k ) u )du F s ) 29/ 45

30 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales RISK MEASURES FROM RIGHT CONTINUOUS BMO MARTINGALES PROPOSITION Let M = { 1 i j H i.m i H i Φ} m = ( Φ.M i 2 BMO ) 1 2 < Define α s,t (Q M ) = E QM ( t 1 i j s b i (u, (H 1 ) u, (H 2 ) u,..., (H j ) u )d[m i, M i ] u F s ) ρ s,t (X) = esssup M M ((E QM ( X F s ) α s,t (Q M )) If any b i is non negative and b i (s, 0, 0,, 0) = 0, or If any b i is of quadratic growth ρ s,t is a time-consistent dynamic risk measure. 30/ 45

31 OUTLINE from BMO martingales Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options 1 2 Time Consistency Regularity of the paths Classical examples 3 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales 4 BID-ASK DYNAMIC PRICING PROCEDURE Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options 5 31/ 45

32 from BMO martingales ECONOMIC MODEL Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options In financial markets, for any traded security you observe a book of orders. For a financial product X: bid and ask prices associated with nx (n IN). For n large the ask price of nx is greater than n times the ask price of X. DYNAMIC PRICING Financial position at time τ: X L (F τ ) Goal: Assign to X a dynamic ask price process (Π σ,τ (X)) σ τ and a dynamic bid price process ( Π σ,τ ( X)) σ τ (selling X is the same as buying X). Taking into account the impact of diversification, and liquidity risk, this leads to the convexity of the ask price. Develop an axiomatic approach taking inspiration from that of Jouiny Kallal for sublinear pricing. 32/ 45

33 from BMO martingales TIME CONSISTENT PRICING PROCEDURE Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options DEFINITION A Time Consistent Pricing Procedure, TCPP, (Π σ,τ ) 0 σ τ is a family of maps (Π σ,τ ) : L (F τ ) L (F σ ) continuous from below, satisfying 1 monotonicity: if X Y then Π σ,τ (X) Π σ,τ (Y) 2 translation invariance: Z L (F σ ), X L (F τ ) 3 convexity: (X, Y) (L (F τ )) 2 Π σ,τ (X + Z) = Π σ,τ (X) + Z Π σ,τ (λx + (1 λ)y) λπ σ,τ (X) + (1 λ)π σ,τ (Y) 4 normalization: Π σ,τ (0) = 0 5 time consistency: X L (F τ ), Π ν,σ (Π σ,τ (X)) = Π ν,τ (X) 33/ 45

34 from BMO martingales TIME CONSISTENCY Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options REMARK 1 ρ σ,τ (X) = Π σ,τ ( X) is a normalized time consistent Measure continuous from above. 2 For any stopping time τ for any X L (Ω, F τ ) Π σ,τ ( X) Π σ,τ (X) 34/ 45

35 from BMO martingales Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options NO FREE LUNCH PRICING PROCEDURE DEFINITION Set financial products atteignable at zero cost via self financing simple strategies: K 0 = {X = X 0 + (Z i Y i ), (X 0, Z i, Y i ) L (F ) 1 i n Π 0, (X 0 ) 0; Π τi, (Z i ) Π τi, ( Y i ) 1 i n} where 0 τ 1... τ n < are stopping times. DEFINITION The TCPP has No Free Lunch if K L + (Ω, F, P) = {0} where K is the weak* closure of K = {λx, (λ, X) IR + K 0 } 35/ 45

36 from BMO martingales FIRST FUNDAMENTAL THEOREM Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options THEOREM Let (Π σ,τ ) σ τ be a TCPP. The following conditions are equivalent: i) The Dynamic Pricing Procedure has No Free Lunch. ii) There is a probability measure R equivalent to P with zero penalty α m 0, (R) = 0 iii) There is a probability measure R equivalent to P such that X L (Ω, F τ, P) σ τ Π σ,τ ( X) E R (X F σ ) Π σ,τ (X) (1) 36/ 45

37 from BMO martingales CADLAG PRICING PROCESSES Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options COROLLARY Let (Π σ,τ ) σ τ be a No Free Lunch TCPP. For any X L (Ω, F, P), for any probability measure R equivalent to P with zero penalty, there is a cadlag modification of (Π σ,τ (X)) σ τ which is a R-supermartingale process (resp. a cadlag modification of (Π σ,τ ( X)) σ τ which is a R-submartingale process), and for any stopping time σ τ. Π σ,τ ( X) E R (X F σ ) Π σ,τ (X) (2) 37/ 45

38 from BMO martingales STRONG ADMISSIBILITY Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options DEFINITION A TCPP (Π σ,τ ) 0 σ τ is strongly admissible with respect to the reference family ((S k ) 0 k d, (Y l ) 1 l p ) and the observed bid (resp ask) prices (C l bid, C l ask) 1 l p if it extends the dynamics of the process (S k ) 0 k d if S k τ L (F τ ) then Π σ,τ (ns k τ ) = ns k σ it is compatible with the observed bid and ask prices for the (Y l ) 1 l p 1 l p C l bid Π 0,τl ( Y l ) Π 0,τl (Y l ) C l ask 38/ 45

39 from BMO martingales Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options CHARACTERIZATION OF STRONG ADMISSIBILITY PROPOSITION A TCPP is strongly admissible with respect to the reference family ((S k ) 0 k d, (Y l ) 1 l p ) and the observed bid (resp ask) prices (C l bid, C l ask) 1 l p if and only if Any probability measure R P such that α0, m (R) < is a local martingale measure with respect to any process S k. For any probability measure R P, for any stopping time τ, α m 0,τ (R) sup(0, sup {l τ l τ} (C l bid E R (Y l ), E R (Y l ) C l ask) (3) Let Q 0 be an equivalent local martingale measure for S k such that l C l bid E Q0 (Y l ) C l ask 39/ 45

40 from BMO martingales Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options ADMISSIBLE PRICING FROM BMO MARTINGALES THEOREM Let M = {M = 1 i j H i.m i, H i Φ}, set of BMO martingales either continuous, or right continuous and of BMO norm less than m < 1 16 Q(M) = {Q M dqm dq 0 = E(M), M M} Let b s be a bounded non negative predictable process. Let τ α σ,τ (Q M ) = E QM ( σ b u d[m, M] u ) F σ ) Π σ,τ (X) = ess inf QM Q(M)(E QM (X F σ ) + α σ,τ (Q M )) defines a No Free Lunch time TCPP. If any M i is strongly orthogonal to any martingale S k, there is B IR + such that the procedure is strongly admissible with respect to (S k, Y l ) if b u B. 40/ 45

41 OUTLINE from BMO martingales 1 2 Time Consistency Regularity of the paths Classical examples 3 from BMO martingales from a stable set of probability measures Stable Sets of Probability Measures from BMO Martingales 4 Markets with transaction costs and liquidity risk No Free Lunch Pricing Procedure Pricing Procedure compatible with observed bid ask prices for options 5 CONCLUSION 41/ 45

42 from BMO martingales CONCLUSION Time consistency is a key property for. It is characterized by the cocycle condition of the minimal penalty function. For normalized dynamic risk measures, the process associated with any financial instrument is a supermartingale with respect to any equivalent measure with zero penalty. It has a cadlag modification. Using the theory of right continuous BMO martingales, we construct time-consistent dynamic risk measures, generalizing those coming from B.S.D.E.. Starting with right continuous BMO martingales with jumps this leads naturally to dynamic risk measures with jumps. 42/ 45

43 from BMO martingales CONCLUSION A TCPP (Π σ,τ ) σ τ associates to any financial instrument a ask (resp bid) price process Π σ,τ (X) (resp Π σ,τ ( X)). First Frundamental Theorem: A TCPP has No Free Lunch If and only if there is a probability measure R equivalent to P such that for any financial instrument X, the martingale process E R (X F σ ) is between the bid price process Π σ,τ ( X) and the ask price process Π σ,τ (X). Using BMO martingales we are able to construct No Free Lunch time-consistent pricing procedures extending the dynamics of reference assets and compatible with the observed bid and ask prices for reference options. 43/ 45

44 from BMO martingales REFERENCES Bion-Nadal J. ( March 2006), Dynamic risk measuring: discrete time in a context of uncertainty and continuous time on a probability space, preprint CMAP 596. Bion-Nadal J. (July 2006), Time Consistent Processes. Cadlag modification, Preprint arxiv: math PR/ v1 Bion-Nadal J. ( February 2007),Bid-Ask Dynamic Pricing in Financial Markets with Transaction Costs and Liquidity Risk bionnada 44/ 45

45 from BMO martingales REFERENCES Delbaen F.: The structure of m-stable sets and in particular of the set of risk neutral measures, Seminaire de Probabilités XXXIX, Lecture Notes in Mathematics 1874, (2006) Doléans-Dade C. and Meyer P.A.: Une caractérisation de BMO, Séminaire de probabilités XI, Université de Strasbourg, Lecture notes in mathematics 581, (1977) Föllmer H. and Schied A.: Stochastic Finance, An in Discrete Time. De Gruyter Studies in Mathematics 27: (2002). Jouini E. and Kallal H. (1995), Martingales and Arbitrage in Secrities Markets with Transaction Costs. Journal of Economic Theory 66, Kazamaki N. (1994): Continuous Exponential Martingales and BMO. Lecture notes in mathematics / 45