On Transaction Costs in Insurance some work in progress Stefan - PowerPoint PPT Presentation

On Transaction Costs in Insurance some work in progress Stefan Thonhauser Johann Radon Institute for Computational and Applied Mathematics (RICAM) Special Semester on Stochastics with Emphasis on Finance Concluding Workshop

Outline Introduction Risk Model Dividends Optimization Problem General Characterization of a Solution Iterated Optimal Stopping Quasi-Variational Inequalities Examples A Good Guess Ideas - General Case 2 / 18

Classical Risk Model Let (Ω , F , P ) be a probability space carrying all stochastic quantities The classical risk reserve process R = ( R t ) t ≥ 0 is described by N t � R t = x + ct − Y k k =1 Components: ◮ x ≥ 0 . . . deterministic initial capital ◮ c > 0 . . . deterministic premium rate ◮ N = ( N t ) t ≥ 0 . . . claim counting process, homogeneous Poisson process with intensity λ > 0 ◮ { Y k } k ∈ N . . . claim amounts, sequence of iid distributed positive random variables with continuous distribution function F Y We assume that N and Y 1 are independent 3 / 18

Dividends - General Strategies Extension allows dividend payments from portfolio (De Finetti 1957) Definition A process L = ( L t ) t ≥ 0 is called admissible dividend strategy if ◮ predictable, c` agl` ad ( L t − = L t ), non-decreasing ◮ no payments after ruin, L t = L τ L for t > τ L ◮ paying dividends can not lead to ruin, L t + − L t ≤ R L t L t denotes the cumulated dividends up to time t The controlled reserve process R L = ( R L t ) t ≥ 0 given by � τ L e − δ t dL t ) , R L t = R t − L t , V L ( x ) = E x ( 0 where τ L = inf { t > 0 | R L t < 0 } denotes the time of ruin of R L 4 / 18

Transaction Costs - Impulse Control Introduce fixed and proportional transaction costs: Dividend payment z is charged by kz − K with k ∈ (0 , 1) , K > 0 Considering optimization problems it’s enough to consider impulse controls S = { ( τ i , Z i ) } i ∈ N ◮ intervention times τ i and associated actions Z i ◮ 0 ≤ τ i < tau i +1 a.s. for all i ∈ N ◮ τ i stopping time w.r.t. F t = σ { R S s − | s ≤ t } for t ≥ 0 ◮ Z i is F τ i measurable ◮ P (lim i →∞ τ i ≤ T ) = 0 for all T ≥ 0 ◮ a := K k < Z i ≤ R τ i S denotes set of admissible impulse controls S n admissible controls with at most n interventions 5 / 18

Optimization Problem The controlled process R S = ( R S t ) t ≥ 0 for S ∈ S , is defined by N t ∞ R S � � t = x + ct − Y n − I { τ i < t } Z i . n =1 i =1 τ S is its time of ruin Value of strategy S � ∞ � � e − δτ i u ( Z i ) I { τ i <τ S } V S ( x ) = E x , i =1 γ ( kz − K ) γ and γ ∈ (0 , 1] with u ( z ) = 1 Value function of maximization problem: V ( x ) = sup V S ( x ) S ∈S 6 / 18

Properties of V ◮ V is absolutely continuous, increasing and linearly bounded ◮ Let f be continuous, increasing an linearly bounded Value of optimal intervention: Mf ( x ) = sup { u ( y ) + f ( x − y ) } , y ∈ ( a , x ] Mf is (like f ) continuous, increasing and linearly bounded (in x ) For the first characterization we need: Associated optimal stopping operator � e − δθ Mf ( R θ ) � M f ( x ) = sup E x . θ ∈T M f ( x ) has the same properties like f 7 / 18

Iterated optimal Stopping Define { v n } n ∈ N 0 by ◮ v 0 ( x ) = 0 (no intervention) ◮ v n by v n ( x ) = M v n − 1 ( x ) (at most n interventions) ◮ We get v n ( x ) = sup V S ( x ) S ∈S n Indeed V is the limit of iterated optimal stopping Theorem (1st Characterization) Let { v n } n ∈ N be defined as above. Then the following holds n →∞ v n ( x ) = V ( x ) and V ( x ) = M V ( x ) . lim V is the smallest fixed point of M . 8 / 18

Motivation of Quasi-Variational Inequalities At some point x ≥ 0 we can choose between ◮ intervention, it’s optimal if MV ( x ) − V ( x ) = 0 ◮ no intervention in open intervall around x �� x � L V ( x ) = cV ′ ( x ) + λ V ( x − y ) dF Y ( y ) − V ( x ) − δ V ( x ) = 0 0 Therefore we expect V to fulfill QVI: max {L V , MV − V } = 0 . Proposition V is differentiable on (0 , a ) and fulfills the QVI a.e. . 9 / 18

Abstract Strategy - QVI Control Let g be an increasing, absolutely continuous and linearly bounded solution to the QVI Define the (admissible) impulse control S g ∈ S , the QVI control, by � � t > 0 | Mg ( R S g t ) = g ( R S g τ g 1 = inf t ) , � � Z g u ( z ) + g ( R S g 1 − z ) | z ∈ ( a , R S g 1 = argmax 1 ] τ g τ g and for n ≥ 2 � n − 1 | Mg ( R S g t ) = g ( R S g � t > τ g τ g n = inf t ) , � � u ( z ) + g ( R S g n − z ) | z ∈ ( a , R S g Z g n = argmax n ] . τ g τ g { x ≥ 0 | g ( x ) = Mg ( x ) } . . . intervention region { x ≥ 0 | g ( x ) > Mg ( x ) } . . . continuation region 10 / 18

Some Sort of Verification We have Theorem (2nd Characterization) The strategy S V is admissible and optimal,i.e. V ≥ V S for all S ∈ S . Further we have that V is the smallest increasing, absolutely continuous and linearly bounded solution to the QVI. From the proof we get: Every solution to QVI corresponds one to one to a fixed point of M How to calculate V ? ◮ explicit solutions to L g = 0 simplify things ◮ what to do if there is no explicit solution to QVI there is no initial value ◮ general structure of optimal strategies ? 11 / 18

Easy Strategies R Fix some b 1 , b 2 with x ◮ 0 ≤ b 1 < b 2 − a b2 ◮ below level b 2 > a the process behaves uncontrolled ◮ hitting b 2 pay immediately b1 b 2 − b 1 ◮ above b 2 take individual t τ argmax such that process jumps Figure: Path - easy strategy below b 2 Advantage: below b 2 explicit solutions in special cases 12 / 18

Value of such a strategy b = { b 1 , b 2 } given by f ( x ) u ( b 2 − b 1 ) � x ≤ b 2 , 1 − f ( b 1 ) V b ( x ) = sup { x − b 2 ≤ z ≤ x & z > a } u ( z ) + f ( x − z ) u ( b 2 − b 1 ) x > b 2 1 − f ( b 1 ) where f is unique solution to L f = 0 and f ( b 2 ) = 1 e − δθ ( b 2 ) | θ ( b 2 ) < τ � � Because f ( x ) = E x V b ( b 2 ) = u ( b 2 − b 1 ) V b ( x ) = f ( x ) V b ( b 2 ) , 1 − f ( b 1 ) First idea maximize u ( b 2 − b 1 ) 1 − f ( b 1 ) over b 1 , b 2 Justification: ◮ if b 1 , b 2 > 0 like smooth fit at b 2 , ( f ′ ( b 2 ) = f ′ ( b 1 )) ◮ optimality results for diffusion and L´ evy models (Ronnie’s preprint) 13 / 18

Example with exponential claims c = 2 . 5 , α = 2 , λ = 1 , δ = 0 . 03 , k = 0 . 99 , K = 0 . 1 , γ = 0 . 7 130 120 0.30 0.25 110 0.20 100 0.15 0.10 90 0.05 20 40 60 80 100 2 4 6 8 10 Figure: Value function Figure: Difference of sup and simple b ∗ = (4 . 61 , 4 . 95) is optimal by numerical verification , V b ∗ fulfills QVI It is worth to continue with the argmax 14 / 18

Optimal Strategies in the classical model The HJB equation is max {L V , 1 − V ′ } = 0 General optimal strategies are of band type ◮ A = { x ∈ [0 , ∞ ) | IDE( x ) = 0 & V ′ ( x − ) = 1 } stay at this level, pay out continuously ◮ B = { x ∈ (0 , ∞ ) | V ′ ( x ) = 1 & IDE( x ) < 0 } pay out lump sums ◮ C = ( A ∪ B ) c do nothing The sets partition R + , the areas can alternate C − A − B − C − A − B (Schmdili 2008, Albrecher and T. 2008) 15 / 18

Examplary Sample Path and Value Function R(t) V � x � x 20 B x1=12.96 15 C 10 a=0.96 5 B x0=0 x 0.96 5 10 12.96 15 t τ Figure: Value function for i = 0 . 02 Figure: Path of R L ∗ and i = 0 16 / 18

Conjectures - future Program Assume a similar band structure The following algorithm could work ◮ starting in [0 , a ) calculate V b ∗ ◮ find [ c 1 , b 3 ) where again it is optimal to do nothing, L V = 0 ◮ find [ b 3 , c 2 ) where it is optimal to pay out, MV = V ◮ and so on... ◮ by construction it should be a QVI control ◮ but a detailed characterization of interval boundaries is needed, even after an succesfull construction you have to verify numerically Alternative: use discretization and iterated optimal stopping approach 17 / 18

Thanks for your Attention 18 / 18