A Robust Predictable L -Martingale Representation Property for - PowerPoint PPT Presentation

A Robust Predictable L ∞ -Martingale Representation Property for Marked Point Processes and Super-Additive Insurance Markets Johannes Leitner TU Vienna December 2, 2008

Content 1. The Economic Model 2. Marked Point Process / Random Measures 3. Super-Additive Markets, No-Arbitrage 4. Irreversible / Dynamic Insurance Markets 5. Completeness / Hedging, Replication 1

The Economic Model • New information arrives at stopping times, • Super-additive (insurance) market, • No-arbitrage, completeness, replication of contingent claims. 2

Marked Point Process • (Ω , F , P ) probability space. ( E, B ) separable metric space, E := E ∪ { ∆ } , ∆ �∈ E . E.g. E = R d \ { 0 } . ¯ • Sequence R + × ¯ E -valued random variables ( T n , X n ) n ≥ 1 with 1. Points: T 0 := 0 < T n < T n +1 on { T n < ∞} , n ≥ 1, 2. Marks: { X n = ∆ } = { T n = ∞} , n ≥ 1. • Interpretation: price, interest rate, etc., jumps by X n at T n . 3

Random Measure Corresponding random measure: � µ ( ω, dt, dx ) = ǫ ( T n ( ω ) ,X n ( ω )) ( dt, dx ) 1 { T n ( ω ) < ∞} , ω ∈ Ω , n ≥ 1 ǫ ( t,x ) probability measure concentrated in ( t, x ) ∈ R + × ¯ E . µ is optional w.r.t. � � F t := F 0 ∨ σ X n 1 [ T n , ∞ ) ( t ) , n ≥ 1 , t ≥ 0. 4

Stochastic Integrals w.r.t. Random Measures H predictable (optional) process, define pathwise (if integral exists): � H ∗ µ t ( ω ) := [0 ,t ] × E H ( ω, s, x ) µ ( ω ; ds, dx ) , ω ∈ Ω . Marked point process: On [0 , sup n ≥ 1 T n ) � H ∗ µ = H ( T n , X n ) 1 [ T n , ∞ ) . n ≥ 1 5

Compensator of a Random Measures Let µ be optional, σ -finite. There exists a unique predictable random measure ˆ µ such that H ∗ µ − H ∗ ˆ µ is a local martingale for all predictable H such that H ∗ µ is of locally integrable total variation. µ is the predictable compensator of µ . ˆ ∃ kernel K from (Ω × R + , P ) into ( E, B ) and non-decreasing predictable process A such that: µ ( ω ; dt, dx ) = K ( ω, t, dx ) dA t ( ω ) . ˆ 6

n ≥ 1 1 [ T n , ∞ ) , Compensator ˆ Counting process N := � N = A Assume throughout • ˆ N is continuous. • ˆ N ∞ is uniformly bounded. 7

Insurance Market K ( t, dx ) describes the law of X n given F t − and T n = t . H ( ω, t, x ) = x 1 x> 0 , λ ≥ 0 security loading : � � � � · ˆ H ∗ µ − H ∗ (1 + λ ) · ˆ = H ∗ µ − E K [ H · ](1 + λ ) µ N is a risk process . 8

Spaces of predictable integrands H : Ω × R + × E → R measurable w.r.t. P ⊗ B (predictable) � � Insurance claim equals H ( ω, t, x ) if T n ( ω ) , X n ( ω ) = ( t, x ) G ∞ := { H predictable } ∩ L ∞ , and for p < ∞ � E | H ( · ; x ) | p K ( · ; dx ) ∈ L ∞ } . G p := { H predictable | 9

Change of measure For q conjugate to p set G ++ := { Y ∈ G q | Y > 0 } . q and M Y := ( Y − 1) ∗ ( µ − ˆ µ ) define Q Y by For Y ∈ G ++ 1 dQ Y = E ( M Y ) ∞ > 0 . dP µ Q Y of µ is given as Y · ˆ W.r.t. Q Y the compensator ˆ µ . 10

Integrability Condition ( H , Y ) := ( G p , G ++ ) satisfies the following condition q (INT) For all ( H, Y ) ∈ H × Y , we have Q Y ∼ P and µ Q Y ) = H ∗ µ − ( HY ) ∗ ˆ H ∗ ( µ − ˆ µ is a uniformly integrable Q Y -martingale. 11

Super-Additive Markets Linear space V 0 ⊆ L 1 ( F 0 ) of initial capitals, H linear space (of actions in a market) Functional W : V 0 × H → L 0 , ( v, H ) �→ W H such that W 0 v = v and v W H + ˆ v + W ˆ H ≥ W H H v . ˆ v +ˆ v 12

EMM Condition Z � = ∅ , Z > 0 for all Z ∈ Z . (EMM) For all ( v, H ) ∈ V 0 × H there exists a Z ∈ Z such that ∈ L 1 , (i) ZW H v (ii) E [ ZW H v | F 0 ] = v . 13

Abstract No-Arbitrage v, ˜ Let ( v, H ) , (˜ H ) ∈ V 0 × H . Under Condition (EMM) we have Proposition. ≤ W ˜ = W ˜ W H H v and W H H and v ≥ ˜ v imply v = ˜ v . v ˜ v ˜ v I.e. a no-domination property holds: ≤ W ˜ < W ˜ W H H and P ( W H H v ) > 0 imply P ( v < ˜ v ) > 0 . v ˜ v ˜ v In particular the no-arbitrage property holds: ≥ v = W 0 W H v implies W H = v. v v 14

Worst Case Scenario Condition Let H × Y ⊆ G p × G ++ , (INT). q (WCS) For all H ∈ H there exists a Y H ∈ Y such that ( HY H ) ∗ ˆ µ ∞ ≤ ( HY ) ∗ ˆ µ ∞ for all Y ∈ Y . 15

The Irreversible Insurance Market Assume ( H , Y ) to satisfy property (WCS). For ( v, H ) ∈ L p ( F 0 ) × H , define v + ( H ∗ µ ) ∞ − ( HY − H ) ∗ ˆ W H := µ ∞ v �� � E HY − H dK · ˆ = v + ( H ∗ µ ) ∞ − N ∞ . 16

Let ( H , Y ) satisfy Conditions (INT) and (WCS). Set Z Y := { dQ Y dP | Y ∈ Y} : Z Y satisfies property (EMM) . I.e. the insurance Proposition. market described by { W H v | v ∈ L p ( F 0 ) , H ∈ H} satisfies the no- domination condition. 17

Irreversible Contracts Y − H does in general not equal Y − H 1 [0 ,t ] on [0 , t ]. Change of contract not possible. Pricing in general not compatible with starting and stopping. 18

Decomposability We say that Y is P - decomposable if Y � = ∅ and Definition. for all A ∈ P and Y, ˜ Y ∈ Y , 1 A Y + 1 A c ˜ Y ∈ Y holds. Y H ∈ Y| � H, Y H � K = essinf Y ∈Y � H, Y � K > −∞ � � Set Y H := for all H ∈ H . If H ⊆ L p and Y is P -decomposable and weakly Proposition. compact in L q , then it satisfies Condition (WCS) and Y H � = ∅ for all H ∈ H . Furthermore, under Condition (INT) no-arbitrage holds. 19

Dynamic Insurance Markets Assume Y to be P -decomposable weakly compact in L q . For ( v, H ) ∈ L p ( F 0 ) × L p choose Y − H ∈ Y − H and define the semimartingale value process v + H ∗ µ − ( HY − H ) ∗ ˆ V v,H := µ �� � E HY − H dK · ˆ = v + H ∗ µ − N. 20

Starting and Stopping Assume Y to be P -decomposable, weakly com- Proposition. pact in L q . Then for all ( v, H ) ∈ L p ( F 0 ) × L p and all stopping times τ 0 ≤ τ 1 , equals V 0 ,H 1 ( τ 0 ,τ 1] on [ τ 0 , τ 1 ] . V v,H − V v,H τ 0 21

Robustness and Uniqueness Assume ( H , Y ) to satisfy (INT), and Y to be P -decomposable, weakly compact in L q . For all ( v, H ) ∈ L p ( F 0 ) ×H , V v,H is a local Q Y -super- Theorem. martingale for all Y ∈ Y and there exists a Y ∈ Y such that V v,H is a uniformly integrable Q Y -martingale. No-arbitrage holds and v, ˜ = V ˜ uniqueness: V v,H H H ) ∈ L p ( F 0 ) × H implies v, ˜ for ( v, H ) , (˜ ∞ ∞ V v,H = V ˜ v, ˜ H . 22

Robust Compensator Define a time-additive/spatially super-additive random measure Y · ˆ µ by µ ) := essinf Y ∈Y � H, Y � K · ˆ H ∗ ( Y · ˆ H ∈ H . N, Y · ˆ µ can be interpreted as a robust compensator for µ w.r.t. the probability measures in the closed convex hull of { Q Y | Y ∈ Y} . V v,H = v + H ∗ µ − H ∗ ( Y · ˆ µ ) is a local Q Y - super-martingale for all Y ∈ Y , resp. a uniformly integrable Q Y -martingale for all Y ∈ Y H . 23

Example Assume insurance contracts, described by H i ∈ H , 1 ≤ i ≤ N , to be given. Consider a market where trading in V i := V 0 ,H i is possible under a short-sale restriction : We assume for all W i ∈ L ∞ i =1 W i · V i is an at- + , that V := � N i =1 V 0 ,W i H i ≤ V 0 ,H for = � N tainable value process. Since V H := � N i =1 W i H i , the resulting market is still arbitrage free, an investor never loses and possibly gains, buying the insurance H instead of trading in the single contracts V i . 24

Main Result to be P -decomposable and Z Y (or Q Y := Assume Y ⊆ G ++ 1 { Q Y | Y ∈ Y} ) to be weakly compact in L 1 ( Ω ), and ( F t ) t ∈ R + to equal the internal filtration generated by µ and F 0 : µ has the robust predictable martingale representa- Theorem. tion property for L ∞ ( F ∞ ) with respect to the closed convex hull of Q Y . 25

Hedging in Dynamic Insurance Markets. We say that Y is P - additive if Y is P -decomposable Definition. and if the predictable process Λ := � 1 , Y � K does not depend on Y ∈ Y . 1 , Q Y -compensator of N : P -additive Y ⊆ G + N Q Y = Λ · ˆ ˆ Y ∈ Y . N, The law of N , resp. ( T i ) i ≥ 1 , under Q Y does not depend on Y ∈ Y . 26

Coherent Risk Measures Y P -additive, weakly compact in G ++ , H ∈ G p : q ρ · ( H ) := − essinf Y ∈Y � H, Y � K Λ − 1 ∈ L 1 (ˆ Ω ) , ( v, H ) ∈ L p ( F 0 ) × G p : V v,H = v + H ∗ µ − ρ · ( − H ) · (Λ · ˆ N ) . 27

Random Set Theory: ρ t : L p ( dK t ) → R : H · ( x ) := H ( · ; x ) , x ∈ E , ˜ ρ · ( H · ) = ρ · ( H ) . ˜ E.g. law invariant risk measure : � 1 0 F ← ρ · ( H · ) := − ˜ ( u ) g · (1 − u ) du, · 28

Representation: Finite Jump Case Ω ) × G ++ H × Y ⊆ L ∞ (˜ . v ∈ L ∞ ( F 0 ) and H ∈ L ∞ (˜ Ω ) 1 Consider the following SDE: V = v + ( H − V − ) ∗ µ + essinf Y ∈Y � V − − H, Y � K · ˆ (1) N. with terminal condition V ∞ = Z ∈ L ∞ . Translation invariance of ρ implies Y − H = Y V − − H ! 29

Linear Inhomogeneous ODE On [ T i , T i +1 ], wlog i = 0, ( T, X ) := ( T 1 , X 1 ): v + ( H − V − ) ∗ µ − � H, Y − H � K · ˆ N + V − · (Λ · ˆ = N ) V � � · (Λ · ˆ v + ( H − V − ) ∗ µ − ρ · ( − H ) − V − = N ) . V T = V T − + ∆ V T = H ( T, X ) on { T < ∞} . We can try to choose v such that V ∞ = Z on { T = ∞} too ! 30