Martingale optimality and cross hedging of insurance derivatives S. - PowerPoint PPT Presentation

Martingale optimality and cross hedging of insurance derivatives ∗ S. Ankirchner, Y. Hu, P . Imkeller, M. M¨ uller, A. Popier, G. Reis Universit´ e Rennes I, Humboldt-Universit¨ at zu Berlin, Universit´ e du Maine http://wws.mathematik.hu-berlin.de/ ∼ imkeller La Londe Les Maures, September 11, 2007 ∗ Supported by the DFG research center MATHEON at Berlin

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 1 1 Insurance derivatives Weather derivatives • underlyings: e.g. temperature, rainfall or snowfall indices • Aim: transfer exogenous risk caused by fluctuations in weather patterns to capital markets Catastrophe futures • underlyings: e.g. loss, windspeed index (hurricane bond) • Aim: transfer exogenous (insurer’s) risk of abnormal losses to other parts of economy • alternative to traditional catastrophe reinsurance

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 2 2 Hedging of portfolios of insurance derivatives Problem: underlying is not tradable, but correlated with tradable assets Examples: temperature ← → heating oil futures, electricity futures loss index ← → stock price of insurance companies Mutual hedging of weather derivatives rainfall in Spain ← → rainfall in Scandinavia Aims: determine utility indifference price determine explicit derivative hedge , i.e. optimal cross hedging strategy describe reduction of risk by cross hedging compare dynamic with static risk interpret pricing by marginal utility

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 3 3 The financial market model Index process, e.g. temperature dR t = ρ ( t, R t ) dW t + b ( t, R t ) dt, b : [0 , T ] × R m → R m , ρ : [0 , T ] × R m → R m × d deterministic functions, globally Lipschitz and of sublinear growth. R Markov process, R t,r s : start at t in r Insurance derivative F ( R T ) , F : R m → R bounded Correlated financial market, k risky assets with price process: dS i t = β i ( t, R t ) dW t + α i ( t, R t ) dt = β i ( t, R t )[ dW t + θ t dt ] , i = 1 , . . . , k, S i t α : [0 , T ] × R m → R k , β : [0 , T ] × R m → R k × d , θ = β ∗ [ ββ ∗ ] − 1 α . W d − dimensional Brownian motion, ββ ∗ uniformly elliptic

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 4 4 The optimal investment problem (N. El Karoui, R. Rouge ’00; J. Sekine ’02; J. Cvitanic, J. Karatzas ’92, Kramkov, Schachermayer ’99,...) investment strategy λ : value of portfolio fraction invested in risky assets wealth gain on [ t, s ] � s � s k � dS i u G λ,t λ i = = λ u β u [ dW u + θ u du ] , s u S i t u t i =1 utility function: U ( x ) = − e − ηx ( 0 < η risk aversion); maximal expected utility from terminal wealth without and with derivative: λ ∈A t,r EU ( v + G λ,t,r λ ∈A t,r EU ( v + G λ,t,r − F ( R t,r V 0 ( t, v, r ) = sup ) , V F ( t, v, r ) = sup T )) T T λ 0 resp. λ F optimal strategies ∆ = λ F − λ 0 derivative hedge

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 5 5 Optimization under non-convex constraints interpretation as maximization problem with constraints π ( t, r ) = λ ( t, r ) β ( t, r ) ∈ C ( t, r ) = { xβ ( t, r ) : x ∈ R k } here: C ( t, r ) convex Aim: construct solution using BSDE, even for non-convex constraints (N. El Karoui, R. Rouge ’00 for convex constraints) C ⊂ R k closed ˜ ˜ A set of strategies λ such that - λ ∈ ˜ P ⊗ l -a.s. ( l Lebesgue measure) C � τ dS s - { exp( − η 0 λ s S s ) : τ stopping time in [0 , T ] } uniformly integrable

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 6 5 Optimization under non-convex constraints F = F ( R T ) insurance derivative For simplicity t = 0 , G λ, 0 = G λ , V ( v ) = V F (0 , v, r ) , etc. First formulation: Find � T A E ( U ( G λ V ( v ) = sup λ ∈ ˜ T − F )) = sup λ ∈ ˜ A E ( U ( v + 0 λ s β s [ dW s + θ s ds ] − F )) . For simplicity: = π λβ, ˜ = C Cβ, ˜ A = A β. � t G π t = v + π s [ dW s + θ s ds ] , t ∈ [0 , T ] 0 Second formulation: Find � T E ( U ( G π V ( v ) = sup T − F )) = sup E ( − exp( − η ( x + π s [ dW s + θ s ds ] − F ))) . π ∈A π ∈A 0

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 7 6 A solution method based on BSDE Idea: Construct family of processes Q ( π ) such that Q ( π ) = constant , 0 Q ( π ) − exp( − η ( G π = T − F )) , form 1 T Q ( π ) supermartingale , π ∈ A , π ∗ ∈ A . Q ( π ∗ ) martingale, for (exactly) one Then E ( Q ( π ) E ( − exp( − η [ G π T − F ])) = T ) E ( Q π ≤ 0 ) = V ( v ) E ( Q ( π ∗ ) = ) 0 E ( − exp( − η [ G ( π ∗ ) = − F ])) . T Hence π ∗ optimal strategy.

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 8 6 A solution method based on BSDE Introduction of BSDE into problem Find generator f of BSDE � T � T Y t = F − Z s dW s − f ( s, Z s ) ds, Y T = F, t t such that with Q ( π ) = − exp( − η [ G π t − Y t ]) , t ∈ [0 , T ] , t we have Q ( π ) = − exp( − η ( v − Y 0 )) 0 = constant , (fulfilled) Q ( π ) form 2 − exp( − η ( G π = T − F )) (fulfilled) T Q ( π ) supermartingale , π ∈ A , π ∗ ∈ A . Q ( π ∗ ) martingale, for (exactly) one This gives solution of valuation problem.

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 9 7 Construction of generator of BSDE How to determine f : Suppose f generator of BSDE. Then Q ( π ) − exp( − η [ G π = t − Y t ]) t � t � t = − exp( − η [ v − Y 0 ]) · exp( − η [ ( π s − Z s ) dW s − [ f ( s, Z s ) − π s θ s ] ds ]) 0 0 � t � t ( π s − Z s ) dW s − η 2 ( π s − Z s ) 2 ds ) = exp( − η [ v − Y 0 ]) · exp( − η 2 0 0 � t [ ηf ( s, Z s ) − ηπ s θ s + η 2 2 ( π s − Z s ) 2 ] ds ) ·− exp( 0 M ( π ) · A ( π ) = , t t with M ( π ) nonnegative martingale. Q ( π ) satisfies (form 2) iff for q ( · , π, z ) = f ( · , z ) − πθ + η 2( π − z ) 2 , π ∈ A , z ∈ R , we have

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 10 7 Construction of generator of BSDE q ( · , π, z ) ≥ 0 , π ∈ A (supermartingale cond.) form 3 π ∗ ∈ A q ( · , π ∗ , z ) = 0 , for (exactly) one (martingale cond.) . Now f ( · , z ) − πθ + η 2( π − z ) 2 q ( · , π, z ) = f ( · , z )+ η 2( π − z ) 2 − ( π − z ) · θ + 1 − zθ − 1 2 ηθ 2 2 ηθ 2 = 2[ π − ( z + 1 − zθ − 1 f ( · , z )+ η ηθ )] 2 2 ηθ 2 . = Under non-convex constraint π ∈ C : [ π − ( z + 1 ηθ )] 2 ≥ d 2 ( C, z + 1 ηθ ) . with equality for at least one possible choice of p ∗ due to closedness of C . Hence (form 3) is solved by the choice

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 11 7 Construction of generator of BSDE − η 2 d 2 ( C, z + 1 η θ )+ z · θ + 1 2 η θ 2 f ( · , z ) = (supermartingale) form 4 d ( C, z + 1 η θ ) = d ( π ∗ , z + 1 π ∗ such that η θ ) (martingale) . Problem: Let Π C ( v ) = { π ∈ R d : d ( C, v ) = d ( π, v ) } . Find measurable selection π ∗ t from Π C t ( Z t + 1 η θ t ) . Solved by classical measurable selection method . p * Measurable selection t 1 π ( ) from z+ α θ t C t C t * p t 1 z+ α θ t 1 θ π ( ) z+ C α t t

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 12 8 Main result Thm 1 ( Y, Z ) unique solution of BSDE � T � T Y t = F − t Z s dW s − t f ( s, Z s ) ds, t ∈ [0 , T ] , with f ( t, Z t ) = − η 2 d 2 ( C t , Z t + 1 η θ t )+ Z t · θ t + 1 2 η θ 2 t . Then value function of utility optimization problem under constraint π ∈ A given by V ( v ) = − exp( − η [ v − Y 0 ]) . There exists an (non-unique) optimal trading strategy π ∗ ∈ A such that t ∈ Π C t ( Z t + 1 π ∗ ηθ t ) , t ∈ [0 , T ] . Proof: - existence, uniqueness for BSDE with quadratic non-linearity in z (M. Kobylanski ’00) measurable selection theorem for Π C t ( Z t + 1 - η θ t ) � � π ∗ - BMO properties of the martingales Z s dW s , s dW s for uniform integrability of exponentials (regularity of coefficients) •

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 13 9 Calculation of derivative hedge generalization to [ t, T ] instead of [0 , T ] , cond. on R t = r : ( Y t,r , Z t,r ) ,π t,r (without F ) resp. ( ˆ π t,r (with F ) instead of ( Y, Z ) ,π Y t,r , ˆ Z t,r ) , ˆ yields V 0 ( t, v, r ) = − exp( − η ( v − Y t,r V F ( t, v, r ) = − exp( − η ( v − ˆ Y t,r )) , )) , t t instead of V ( v ) = − exp( v − Y 0 ) . due to linearity of C ( t, r ) projections unique and linear, hence + 1 + 1 = Π C ( t,r ) [ ˆ π t,r = Π C ( t,r ) [ Z t,r ηθ ( s, R t,r π t,r Z t,r ηθ ( s, R t,r s )] , ˆ s )] , s s s s and so s ) = Π C ( t,r ) [ ˆ ∆ β ( s, R t,r Z t,r − Z t,r s ] . s

O PTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 14 10 Markov property and its consequences Markov property of R implies (Kobylanski ’00, El Karoui, Peng, Quenez ’97): Thm 2 There are measurable (deterministic) functions u and ˆ u such that ˆ Y t,r = u ( s, R t,r Y t,r u ( s, R t,r s ) , = ˆ s ) . s s There are measurable (deterministic) functions v and ˆ v such that Z t,r = ˆ ˆ Z t,r = vρ ( s, R t,r vρ ( s, R t,r s ) , s ) . s Corollary 1 p ( t, r ) := Y t,r Y t,r − � = u ( t, r ) − ˆ u ( t, r ) t t is the indifference price , i.e. V F ( t, v − p ( t, r ) , r ) = V 0 ( t, v, r ) . p depends only on R , not on S Aim: Explicit description of ∆