Set-Valued Risk Measures and Bellman s Principle

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

6. Examples: 2 assets, proportional transaction costs 1 Superhedging Average Value at Risk 0.5 Risky asset 0 0.5 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 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

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

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

6. Examples: 3 assets, proportional transaction costs 20 15 10 10 30 10 Risky asset 2 5 0 5 20 30 0 0 10 20 10 10 20 20 10 0 0 10 40 15 30 20 10 50 20 20 15 10 5 0 5 10 15 20 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

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

4. Examples: 2 assets, convex transaction costs 10000 9000 8000 7000 6000 Superhedging Entropic: C = [[1;0],[0;1]] Entropic: C = [[1; 0.90],[ 0.90;1]] Risky asset 5000 4000 3000 2000 1000 0 1000 1000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 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

6. Examples: 2 assets, convex transaction costs 200 150 Superhedging Entropic: C = [[1;0],[0;1]] Entropic: C = [[1; 0.90],[ 0.90;1]] 100 Risky asset 50 0 50 100 150 150 100 50 0 50 100 150 200 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

6. Examples: 2 assets, convex transaction costs 5 4 3 Superhedging Entropic: C = [[1;0],[0;1]] Entropic: C = [[1; 0.90],[ 0.90;1]] Risky asset 2 1 0 1 2 3 3 2 1 0 1 2 3 4 5 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

Thank you Zachary Feinstein and Birgit Rudloff. Time consistency of dynamic risk measures in markets with transaction costs. Quantitative Finance, 13(9):1473 1489, 2013. Zachary Feinstein and Birgit Rudloff. Multi-portfolio time consistency for set-valued convex and coherent risk measures. Finance and Stochastics, 19(1):67 107, 2015. Zachary Feinstein and Birgit Rudloff. A recursive algorithm for multivariate risk measures and a set-valued Bellman s principle. Preprint, 2015. Zachary Feinstein and Birgit Rudloff. A supermartingale relation for multivariate risk measures. 2015. Preprint. Z. Feinstein Set-Valued Risk Measures and Bellman s Principle 54 / 54