Set-Valued Risk Measures and Bellman s Principle

Transcription

1 Set-Valued Risk Measures and Bellman s Principle Zach Feinstein Electrical and Systems Engineering, Washington University in St. Louis Joint work with Birgit Rudloff (Vienna University of Economics and Business) University of Southern California March 7, 2016

2 Outline 1 Set-valued risk measures 2 Time consistency 3 Recursive algorithm 4 Computation 5 Examples Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 2 / 54

3 2. Set-valued risk measures Set-Valued Risk Measures: Primal Representation Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 3 / 54

4 2. Set-valued risk measures: Setup Probability space (Ω, (F t ) t [0,T ], P) d assets (may include different currencies) Portfolio vectors in physical units (numéraire free), i.e. number of units in d assets Claim: X L p := L p T (Rd ) payoff (in physical units) at time T Eligible portfolios M = R m {0} d m, linear subspace of R d of portfolios that can be used to compensate risk (e.g. Dollars & Euros) Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 4 / 54

5 2.1 Set-valued risk measures: Primal representation M t := L p t (M), M t,+ := M t L p t,+ P(Z; C) := {A Z A = A + C} G(Z; C) := {A Z A = cl co(a + C)} Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 5 / 54

6 2.1 Set-valued risk measures: Primal representation M t := L p t (M), M t,+ := M t L p t,+ P(Z; C) := {A Z A = A + C} G(Z; C) := {A Z A = cl co(a + C)} Conditional Set-Valued Risk Measure A set-valued function R t : L p P(M t ; M t,+ ) is a conditional risk measure if 1 Finite at zero: Rt (0) M t ; 2 Mt translative: R t (X + m) = R t (X) m for any m M t ; 3 L p + monotone: if X Y L p + then R t (X) R t (Y ). Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 5 / 54

7 2.1 Set-valued risk measures: Primal representation M t := L p t (M), M t,+ := M t L p t,+ P(Z; C) := {A Z A = A + C} G(Z; C) := {A Z A = cl co(a + C)} Conditional Set-Valued Risk Measure A set-valued function R t : L p P(M t ; M t,+ ) is a conditional risk measure if 1 Finite at zero: Rt (0) M t ; 2 Mt translative: R t (X + m) = R t (X) m for any m M t ; 3 L p + monotone: if X Y L p + then R t (X) R t (Y ). Normalized: for every X L p t : R t(x) = R t (X) + R t (0). Normalized version: R t (X) := R t (X). R t (0) = {u M t R t (0) + u R t (X)}. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 5 / 54

8 2.1 Set-valued risk measures: Primal representation (Conditionally) convex : for all X, Y L p, for all 0 λ 1 (λ L t (R) such that 0 λ 1) R t (λx + (1 λ)y ) λr t (X) + (1 λ)r t (Y ). (Conditionally) positive homogeneous: for all X L p, for all λ > 0 (λ L t (R ++ )) R t (λx) = λr t (X). (Conditionally) coherent: if it is (conditionally) convex and (conditionally) positive homogeneous. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 6 / 54

9 2.1 Set-valued risk measures: Primal representation K-compatible: for some set K L p if there exists a risk measure R such that R t (X) = k K R t (X k). Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 7 / 54

10 2.1 Set-valued risk measures: Primal representation K-compatible: for some set K L p if there exists a risk measure R such that R t (X) = k K R t (X k). Closed: if the graph of R t is closed in the product topology, i.e., graph R t := {(X, u) L p M t u R t (X)} is closed. (Conditionally) convex upper continuous [(c.)c.u.c.]: if for any closed (conditionally) convex set D G(M t ; M t, ) the inverse image is closed. R 1 t (D) := {X L p R t (X) D } Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 7 / 54

11 2.1 Set-valued risk measures: Primal representation Conditional Acceptance Set A set A t L p is an acceptance set at time t if Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 8 / 54

12 2.1 Set-valued risk measures: Primal representation Conditional Acceptance Set A set A t L p is an acceptance set at time t if 1 At M t M t ; 2 At + L p + A t Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 8 / 54

13 2.1 Set-valued risk measures: Primal representation Conditional Acceptance Set A set A t L p is an acceptance set at time t if 1 At M t M t ; 2 At + L p + A t Acceptance set: A t = {X L p 0 R t (X)} Risk measure: R t (X) = {u M t X + u A t } Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 8 / 54

14 2.1 Set-valued risk measures: Primal representation Acceptance set: A t = {X L p 0 R t (X)} Risk measure: R t (X) = {u M t X + u A t } Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 9 / 54

15 2.1 Set-valued risk measures: Primal representation Acceptance set: A t = {X L p 0 R t (X)} Risk measure: R t (X) = {u M t X + u A t } Properties Risk measure Acceptance set (Conditionally) convex (Conditionally) convex (Conditionally) coherent (Conditionally) convex cone Closed graph Closed B L p B-monotone A t + B = A t C M t R t (X) : L p P(M t ; C) A t + C A t R t (X) X L p L p = A t + M t R t (X) M t X L p L p = (L p \A t ) + M t Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 9 / 54

16 2. Set-valued risk measures Set-Valued Risk Measures: Dual Representation Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 10 / 54

17 2.2 Set-valued risk measures: Dual representation Dual variables: W t := where wt s (Q, w) = w E ( [ ] w 1 E dq1 dp F s { ( (Q, w) M d M t,+ + \M t Q = P Ft, w T t (Q, w) L q +} ; [ dq dp,..., w d E [ dqd dp ] F s = ]) T. F s Halfspace: G t (w) := { u L p t E [ w T u ] 0 } Conditional Halfspace: Γ t (w) := { u L p t wt u 0 a.s. } ) Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 11 / 54

18 2.2 Set-valued risk measures: Dual representation Dual variables: W t := where wt s (Q, w) = w E ( [ ] w 1 E dq1 dp F s { ( (Q, w) M d M t,+ + \M t Q = P Ft, w T t (Q, w) L q +} ; [ dq dp,..., w d E [ dqd dp ] F s = ]) T. F s Halfspace: G t (w) := { u L p t E [ w T u ] 0 } Conditional Halfspace: Γ t (w) := { u L p t wt u 0 a.s. } Set subtraction: A. B := {m M t B + m A} ) Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 11 / 54

19 2.2 Set-valued risk measures: Dual representation G t (w) := { u L p t E [ w T u ] 0 } Convex Risk Measures A function R t : L p G(M t ; M t,+ ) is a closed convex risk measure if and only if R t (X) = [( ) ] E Q [ X F t ] + G t (w) M t. β t (Q, w), (Q,w) W t where β t is the minimal penalty function given by βt min (Q, w) = ( ) E Q [ Y F t ] + G t (w) Y A t M t. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 12 / 54

20 2.2 Set-valued risk measures: Dual representation G t (w) := { u L p t E [ w T u ] 0 } Coherent Risk Measures A function R t : L p G(M t ; M t,+ ) is a closed coherent risk measure if and only if ( ) R t (X) = E Q [ X F t ] + G t (w) M t, (Q,w) W max t where W max t is the maximal set of dual variables given by W max t = { (Q, w) W t w T t (Q, w) A + t }. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 13 / 54

21 2.2 Set-valued risk measures: Dual representation Γ t (w) := { u L p t wt u 0 a.s. } Conditionally Convex and Coherent Risk Measures A function R t : L p G(M t ; M t,+ ) is a closed conditionally convex risk measure if and only if [( R t (X) = E Q [ X F t ] + Γ t (w) ) M t. α t (Q, w) ], (Q,w) W t where α t is the conditional penalty function given by α t (Q, w) = ( E Q [Y F t ] + Γ t (w) ) M t. Y A t R t is additionally conditionally coherent if and only if ( R t (X) = E Q [ X F t ] + Γ t (w) ) M t. (Q,w) W max t Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 14 / 54

22 3. Time consistency Time Consistency: Multiportfolio Time Consistency Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 15 / 54

23 3.1 Time consistency: Multiportfolio time consistency Multiportfolio Time Consistency A dynamic risk measure (R t ) T t=0 is multiportfolio time consistent if the relation R s (X) R s (Y ) R t (X) R t (Y ) Y Y Y Y for any times t < s, any X L p and any Y L p Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 16 / 54

24 3.1 Time consistency: Multiportfolio time consistency Multiportfolio Time Consistency A dynamic risk measure (R t ) T t=0 is multiportfolio time consistent if the relation R s (X) R s (Y ) R t (X) R t (Y ) Y Y for any times t < s, any X L p and any Y L p Y Y Multiportfolio time consistency implies time consistency defined by R s (X) R s (Y ) R t (X) R t (Y ) for any times t < s and X, Y L p. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 16 / 54

25 3.1 Time consistency: Multiportfolio time consistency Multiportfolio Time Consistency If (R t ) T t=0 is normalized (R t(x) = R t (X) + R t (0) for every X and t) then the following are equivalent: R s (X) Y Y R s(y ) R t (X) Y Y R t(y ); Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 17 / 54

26 3.1 Time consistency: Multiportfolio time consistency Multiportfolio Time Consistency If (R t ) T t=0 is normalized (R t(x) = R t (X) + R t (0) for every X and t) then the following are equivalent: R s (X) Y Y R s(y ) R t (X) Y Y R t(y ); R t (X) = Z R s(x) R t( Z) =: R t ( R s (X)); Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 17 / 54

27 3.1 Time consistency: Multiportfolio time consistency Multiportfolio Time Consistency If (R t ) T t=0 is normalized (R t(x) = R t (X) + R t (0) for every X and t) then the following are equivalent: R s (X) Y Y R s(y ) R t (X) Y Y R t(y ); R t (X) = Z R s(x) R t( Z) =: R t ( R s (X)); A t = A s + A t,s where A t,s := A t M s. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 17 / 54

28 3.1 Time consistency: Multiportfolio time consistency Multiportfolio Time Consistency If (R t ) T t=0 is normalized (R t(x) = R t (X) + R t (0) for every X and t) then the following are equivalent: R s (X) Y Y R s(y ) R t (X) Y Y R t(y ); R t (X) = Z R s(x) R t( Z) =: R t ( R s (X)); A t = A s + A t,s where A t,s := A t M s. If discrete time t, s {0, 1,..., T } then sufficient to have any of these conditions with s = t + 1. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 17 / 54

29 3.1 Time consistency: Multiportfolio time consistency Multiportfolio Time Consistency If (R t ) T t=0 is normalized, c.u.c., and convex then the following are equivalent: (R t ) T t=0 is multiportfolio time consistent; β t (Q, w) = cl ( β t,s (Q, w) + E Q [β s (Q, wt s (Q, w)) F t ] ) ; [ ] V (Q,w) t (X) E Q V (Q,ws t s (Q,w)) (X) F t for every X L p where V (Q,w) t (X) := cl [R t (X) + β t (Q, w)]. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 18 / 54

30 3.1 Time consistency: Multiportfolio time consistency Multiportfolio Time Consistency If (R t ) T t=0 is normalized, c.u.c., and convex then the following are equivalent: (R t ) T t=0 is multiportfolio time consistent; β t (Q, w) = cl ( β t,s (Q, w) + E Q [β s (Q, wt s (Q, w)) F t ] ) ; [ ] V (Q,w) t (X) E Q V (Q,ws t s (Q,w)) (X) F t for every X L p where V (Q,w) t (X) := cl [R t (X) + β t (Q, w)]. When (R t ) T t=0 is coherent then { G t (w) M t β t (Q, w) = if (Q, w) Wt max else. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 18 / 54

31 3.1 Time consistency: Multiportfolio time consistency Multiportfolio Time Consistency If (R t ) T t=0 is normalized, c.u.c., and conditionally convex with dual representation defined on Wt e := {(Q, w) W t Q P} then the following are equivalent: (R t ) T t=0 is multiportfolio time consistent; α t (Q, w) = cl ( α t,s (Q, w) + E Q [α s (Q, wt s (Q, w)) F t ] ) ; [ ] V (Q,w) t (X) cl E Q V (Q,ws t s (Q,w)) (X) F t for every X L p where V (Q,w) t (X) := cl [R t (X) + α t (Q, w)]. When (R t ) T t=0 is conditionally coherent then { Γ t (w) M t if (Q, w) Wt max α t (Q, w) = else. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 19 / 54

32 3.1 Time consistency: Multiportfolio time consistency Let Q, R M d then S = Q s R if [ ] / [ ] ds i dp := E dqi dp F s dr i dp E dri dp F s [ ] E dqi F s dp on else { [ ] } E dri dp F s > 0, Stability A set W t W t is stable at time t with respect to W t,s and W s if (Q, w) W t implies (Q, w s t (Q, w)) W s and (Q, w) W t,s and R M d such that (R, w s t (Q, w)) W s implies (Q s R, w) W t Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 20 / 54

33 3.1 Time consistency: Multiportfolio time consistency Multiportfolio Time Consistency If (R t ) T t=0 is normalized, c.u.c., and coherent then the following are equivalent then (R t ) T t=0 is multiportfolio time consistent if and only if any of the following equivalent properties holds for all times t < s Wt max Wt max Wt max is stable with respect to W max t,s = { (Q s R, w) (Q, w) W max t,s and W max = Wt,s max Ht s (Ws max ) where Ht s (W ) := {(Q, w) W t : (Q, wt s (Q, w)) W }. s ;, (R, w s t (Q, w)) W max s } ; Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 21 / 54

34 3. Time consistency Time Consistency: Composition of Risk Measures Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 22 / 54

35 3.2 Time consistency: Composition of risk measures Composition of One-Step Risk Measures ( ) T Let (R t ) T t=0 be a risk measure then Rt is the multiportfolio time t=0 consistent version if R T (X) := R T (X); Rt (X) := R t ( Z) Z R t+1 (X) Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 23 / 54

36 3.2 Time consistency: Composition of risk measures Composition of One-Step Risk Measures ( ) T Let (R t ) T t=0 be a risk measure then Rt is the multiportfolio time t=0 consistent version if R T (X) := R T (X); Rt (X) := R t ( Z) Also given by: Z R t+1 (X) Ã t := A t,t+1 + ( Ãt+1; β [ ]) t (Q, w) := cl βt,t+1(q, min w) + E Q βt+1 (Q, wt t+1 (Q, w)) F t ; ( α t (Q, w) := cl αt,t+1(q, min w) + E Q [ α t+1 (Q, wt t+1 (Q, w)) ] ) Ft ; W t := W max t,t+1 H t+1 t ( W t+1 ). Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 23 / 54

37 4. Recursive algorithm Recursive Algorithm Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 24 / 54

38 4. Recursive algorithm: Setting Discrete time t {0, 1,..., T } Finite probability space (Ω, (F t ) T t=0, P) Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 25 / 54

39 4. Recursive algorithm: Setting Discrete time t {0, 1,..., T } Finite probability space (Ω, (F t ) T t=0, P) Let Ω t be the set atoms in F t Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 25 / 54

40 4. Recursive algorithm: Setting Discrete time t {0, 1,..., T } Finite probability space (Ω, (F t ) T t=0, P) Let Ω t be the set atoms in F t The set of successor nodes for ω t Ω t is given by succ(ω t ) = {ω t+1 Ω t+1 : ω t+1 ω t } Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 25 / 54

41 4. Recursive algorithm: Setting Discrete time t {0, 1,..., T } Finite probability space (Ω, (F t ) T t=0, P) Let Ω t be the set atoms in F t The set of successor nodes for ω t Ω t is given by succ(ω t ) = {ω t+1 Ω t+1 : ω t+1 ω t } R t (X)[ω t ] = {u(ω t ) : u R t (X)} Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 25 / 54

42 4. Recursive algorithm: Setting R t (X)[ω t ] = {u(ω t ) : u R t (X)} R t (X)[ω t ] behaves like static risk measure Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 26 / 54

43 4. Recursive algorithm: Setting R t (X)[ω t ] = {u(ω t ) : u R t (X)} R t (X)[ω t ] behaves like static risk measure Pointwise Representation If R t has closed and conditionally convex images then u R t (X) if and only if u(ω t ) R t (X)[ω t ] for every ω t Ω t. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 26 / 54

44 4. Recursive algorithm: Setting R t (X)[ω t ] = {u(ω t ) : u R t (X)} R t (X)[ω t ] behaves like static risk measure Pointwise Representation If R t has closed and conditionally convex images then u R t (X) if and only if u(ω t ) R t (X)[ω t ] for every ω t Ω t. If R t is closed and conditionally convex, then R t has closed and conditionally convex images. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 26 / 54

45 4. Recursive algorithm: Setting R t (X)[ω t ] = {u(ω t ) : u R t (X)} R t (X)[ω t ] behaves like static risk measure Pointwise Representation If R t has closed and conditionally convex images then u R t (X) if and only if u(ω t ) R t (X)[ω t ] for every ω t Ω t. If R t is closed and conditionally convex, then R t has closed and conditionally convex images. R t is local if 1 D R t (X) = 1 D R t (1 D X) for every D F t Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 26 / 54

46 4. Recursive algorithm: Setting R t (X)[ω t ] = {u(ω t ) : u R t (X)} R t (X)[ω t ] behaves like static risk measure Pointwise Representation If R t has closed and conditionally convex images then u R t (X) if and only if u(ω t ) R t (X)[ω t ] for every ω t Ω t. If R t is closed and conditionally convex, then R t has closed and conditionally convex images. R t is local if 1 D R t (X) = 1 D R t (1 D X) for every D F t If R t is local then R t (X)[ω t ] = R t (1 ωt X)[ω t ] Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 26 / 54

47 4.1 Recursive algorithm: Bellman s Principle Recursive Algorithm: Bellman s Principle Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 27 / 54

48 4.1 Recursive algorithm: Bellman s Principle Want: The closed multi-portfolio time consistent version of R t Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 28 / 54

49 4.1 Recursive algorithm: Bellman s Principle Want: The closed multi-portfolio time consistent version of R t Pointwise Representation R T (X)[ω T ] = cl(r T (X)[ω T ]) R t (X)[ω t ] = cl {R t,t+1 ( Z)[ω t ] : ω t+1 succ(ω t ) : Z(ω t+1 ) R t+1 (X)[ω t+1 ] } Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 28 / 54

50 4.1 Recursive algorithm: Bellman s Principle Dynamic set optimization: R t (X)[ω t ] = inf Z Zt+1 [ω t] R t,t+1( Z)[ω t ] with Z t+1 [ω t ] = { Z L p t+1 : ω t+1 succ(ω t ) : Z(ω t+1 ) R t+1 (X)[ω t+1 ] } Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 29 / 54

51 4.1 Recursive algorithm: Bellman s Principle Dynamic set optimization: R t (X)[ω t ] = inf Z Zt+1 [ω t] R t,t+1( Z)[ω t ] with Z t+1 [ω t ] = { Z L p t+1 : ω t+1 succ(ω t ) : Z(ω t+1 ) R t+1 (X)[ω t+1 ] } Equivalent to vector optimization problem: R t (X)[ω t ] = inf Γ(Z, Y ) (Z,Y ) Z t+1 [ω t] for Γ(Z, Y ) = Y and Z t+1 [ω t ] = { (Z, Y ) Z t+1 [ω t ] M : Y R t,t+1 ( Z)[ω t ] } Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 29 / 54

52 4.1 Recursive algorithm: Bellman s Principle Interpretation as Bellman s Principle: Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 30 / 54

53 4.1 Recursive algorithm: Bellman s Principle Interpretation as Bellman s Principle: The at t truncated optimal solution (Z s ) T s=t obtained at time 0 from a given Z 0 R 0 (X) is still optimal at any later time point t {0,..., T }. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 30 / 54

54 4.1 Recursive algorithm: Bellman s Principle Interpretation as Bellman s Principle: The at t truncated optimal solution (Z s ) T s=t obtained at time 0 from a given Z 0 R 0 (X) is still optimal at any later time point t {0,..., T }. Meaning: For the risk compensating portfolio holding Z t R t (X), (Z s ) T s=t satisfies the conditions Z s R s (X) and Z s 1 R s 1 ( Z s ), s {t,..., T }. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 30 / 54

55 5. Computation Computation Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 31 / 54

56 5. Computation Approximation: Not always possible or feasible to find the risk measure exactly Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 32 / 54

57 5. Computation Approximation: Not always possible or feasible to find the risk measure exactly Approximation Rt δ is a δ-approximation of R t if for every X L p Rt δ (X) + δm1 R t (X) Rt δ (X) for a fixed m int(m + ) Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 32 / 54

58 5. Computation Approximation: Not always possible or feasible to find the risk measure exactly Approximation Rt δ is a δ-approximation of R t if for every X L p Rt δ (X) + δm1 R t (X) Rt δ (X) for a fixed m int(m + ) Question: How do errors grow with time steps? Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 32 / 54

59 5. Computation Propogation of Errors: Errors propogate linearly Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 33 / 54

60 5. Computation Propogation of Errors: Errors propogate linearly If R ɛ t+1 (X) is an ɛ-approximation of R t+1 (X) then composed backwards R ɛ t(x) is an ɛ-approximation of R t (X) Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 33 / 54

61 5. Computation Propogation of Errors: Errors propogate linearly If R ɛ t+1 (X) is an ɛ-approximation of R t+1 (X) then composed backwards R ɛ t(x) is an ɛ-approximation of R t (X) ɛ,γ R t R ɛ,γ t If (X) is a γ-approximation of ɛ-approximation R t(x), ɛ then (X) is an (ɛ + γ)-approximation of R t (X) Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 33 / 54

62 5. Computation Polyhedral risk measures: Linear vector optimization R t,t+1 ( Z)[ω t ] = {P t z + M t,+ : A t Z + B t z b t } dim(m)-dimensional problem with d succ(ω t ) + z -dimensional pre-image space Benson s algorithm can be applied directly Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 34 / 54

63 5. Computation Convex risk measures: Convex vector optimization R t,t+1 ( Z)[ω t ] = {P t (z) + M t,+ : g(z, z) 0} To calculate: a polyhedral δ-approximation of R t by recursively calculating backwards in time dim(m)-dimensional problem with d succ(ω t ) + z -dimensional pre-image space Convex extension of Benson s algorithm can be applied directly with a given approximation error desired Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 35 / 54

64 6. Examples Examples: Superhedging Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 36 / 54

65 6.1 Examples: Superhedging Convex transaction costs at time t: closed convex set R d + K t [ω] R d (solvency region), positions transferable into non-negative portfolios. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 37 / 54

66 6.1 Examples: Superhedging Convex transaction costs at time t: closed convex set R d + K t [ω] R d (solvency region), positions transferable into non-negative portfolios. Self-Financing Portfolio Process A self-financing portfolio process (V t ) T t=0 is a stochastic process of portfolio vectors (of physical units ) if starting with no assets you can trade for V t from the portfolio V t 1. t = 0,..., T : V t V t 1 K t with V 1 = 0 and K t is a solvency region. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 37 / 54

67 6.1 Examples: Superhedging Convex transaction costs at time t: closed convex set R d + K t [ω] R d (solvency region), positions transferable into non-negative portfolios. Self-Financing Portfolio Process A self-financing portfolio process (V t ) T t=0 is a stochastic process of portfolio vectors (of physical units ) if starting with no assets you can trade for V t from the portfolio V t 1. t = 0,..., T : V t V t 1 K t with V 1 = 0 and K t is a solvency region. Let K t,s := s r=t Lp r(k r ) denote the portfolios reachable from time t at time s. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 37 / 54

68 6.1 Examples: Superhedging SHP t (X) := {u L p t X + u K t,t }. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 38 / 54

69 6.1 Examples: Superhedging SHP t (X) := {u L p t X + u K t,t }. If the market model (K t ) T t=0 satisfies proper no-arbitrage argument (robust no scalable arbitrage) then the superhedging portfolios can be found via the dual representation with penalty function: α SHP t (Q, w) = T {u L p t s=t ess sup w T E Q [k F t ] w T u k L p s(k s) } Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 38 / 54

70 6.1 Examples: Superhedging SHP t (X) := {u L p t X + u K t,t }. If the market model (K t ) T t=0 satisfies proper no-arbitrage argument (robust no scalable arbitrage) then the superhedging portfolios can be found via the dual representation with penalty function: α SHP t (Q, w) = T {u L p t s=t ess sup w T E Q [k F t ] w T u k L p s(k s) } This is multiportfolio time consistent, but not necessarily self-recursive. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 38 / 54

71 6.1 Examples: Superhedging If solvency cones (K t is a.s. a cone) then the superhedging portfolios are conditionally coherent with dual variables defined by W {t,...,t } = {(Q, w) W t s {t,..., T } : w s t (Q, w) L q s(k + s ) } Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 39 / 54

72 6.1 Examples: Superhedging If solvency cones (K t is a.s. a cone) then the superhedging portfolios are conditionally coherent with dual variables defined by W {t,...,t } = {(Q, w) W t s {t,..., T } : w s t (Q, w) L q s(k + s ) } This is self-recursive. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 39 / 54

73 6. Examples Examples: Average Value-at-Risk Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 40 / 54

74 6.2 Examples: Average Value-at-Risk Level λ t L t bounded away from 0. Average Value-at-Risk at time t is closed and conditionally coherent risk measure defined by the dual variables: { Wt λ := (Q, w) W t 0 w dq } dp w/λt Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 41 / 54

75 6.2 Examples: Average Value-at-Risk Level λ t L t bounded away from 0. Average Value-at-Risk at time t is closed and conditionally coherent risk measure defined by the dual variables: { Wt λ := (Q, w) W t 0 w dq } dp w/λt Average Value-at-Risk is not time consistent. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 41 / 54

76 6.2 Examples: Average Value-at-Risk Consider M := R d and p = + Multiportfolio time consistent version has dual variables: W t λ := {(Q, w) W t s {t, t + 1,..., T 1} : wt s (Q, w)/λ s wt s+1 (Q, w) } = {(Q, w) W t s {t, t + 1,..., T 1} i : ( [ ] dqi P E dp F s+1 1 [ ] ) } dqi λ s E i dp F s or w i = 0 = 1 i Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 42 / 54

77 6. Examples Examples: Entropic Risk Measure Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 43 / 54

78 6.3 Examples: Entropic risk measure Utility based shortfall risk measures: R u t (X) := {m M t E [u(x + m) F t ] C t } for some vector utility function u and C t G(L p t ; Lp t,+ ) Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 44 / 54

79 6.3 Examples: Entropic risk measure Utility based shortfall risk measures: R u t (X) := {m M t E [u(x + m) F t ] C t } for some vector utility function u and C t G(L p t ; Lp t,+ ) Restrictive entropic risk measure: M = R d, u i (x) = 1 exp( λ ix) λ i and C t = L p t,+ Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 44 / 54

80 6.3 Examples: Entropic risk measure Utility based shortfall risk measures: R u t (X) := {m M t E [u(x + m) F t ] C t } for some vector utility function u and C t G(L p t ; Lp t,+ ) Restrictive entropic risk measure: M = R d, u i (x) = 1 exp( λ ix) λ i and C t = L p t,+ Closed convex risk measure with penalty function αt ent (Q, w) = H t (Q P)/λ + Γ t (w) (Q, w) = H t (Q P)/λ + G t (w) β ent t where Ĥt(Q P) = E Q [ log( dq dp ) Ft ] c.u.c., normalized, and multiportfolio time consistent. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 44 / 54

81 6. Examples: 2 assets, proportional transaction costs Examples: 2 assets, proportional transaction costs Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 45 / 54

82 6. Examples: 2 assets, proportional transaction costs 1 Superhedging Average Value at Risk 0.5 Risky asset Risk free asset Figure: The superhedging risk measure and composed average value-at-risk of an at-the-money European put option Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 46 / 54

83 6. Examples: 3 assets, proportional transaction costs Examples: 3 assets, proportional transaction costs Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 47 / 54

84 6. Examples: 3 assets, proportional transaction costs Figure: The composed relaxed worst case risk measure of an outperformance option Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 48 / 54

85 6. Examples: 3 assets, proportional transaction costs Risky asset Risky asset 1 Figure: Contour plot of the efficient frontier of the at differing levels of captial in the risk-less asset Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 49 / 54

86 6. Examples: 2 assets, convex transaction costs Examples: 2 assets, convex transaction costs Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 50 / 54

87 4. Examples: 2 assets, convex transaction costs Superhedging Entropic: C = [[1;0],[0;1]] Entropic: C = [[1; 0.90],[ 0.90;1]] Risky asset Risk free asset Figure: The superhedging risk measure and composed entropic risk measures of a binary option Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 51 / 54

88 6. Examples: 2 assets, convex transaction costs Superhedging Entropic: C = [[1;0],[0;1]] Entropic: C = [[1; 0.90],[ 0.90;1]] 100 Risky asset Risk free asset Figure: The superhedging risk measure and composed entropic risk measures of a binary option; zoomed-in Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 52 / 54

89 6. Examples: 2 assets, convex transaction costs Superhedging Entropic: C = [[1;0],[0;1]] Entropic: C = [[1; 0.90],[ 0.90;1]] Risky asset Risk free asset Figure: The superhedging risk measure and composed entropic risk measures of a binary option; near 0 Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 53 / 54

90 Thank you Zachary Feinstein and Birgit Rudloff. Time consistency of dynamic risk measures in markets with transaction costs. Quantitative Finance, 13(9): , Zachary Feinstein and Birgit Rudloff. Multi-portfolio time consistency for set-valued convex and coherent risk measures. Finance and Stochastics, 19(1):67 107, Zachary Feinstein and Birgit Rudloff. A recursive algorithm for multivariate risk measures and a set-valued Bellman s principle. Preprint, Zachary Feinstein and Birgit Rudloff. A supermartingale relation for multivariate risk measures Preprint. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 54 / 54