Optimal investment strategies for an index-linked insurance payment process with stochastic intensity
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1 for an index-linked insurance payment process with stochastic intensity Warsaw School of Economics Division of Probabilistic Methods
2 Probability space ( Ω, F, P ) Filtration F = (F(t)) 0 t T satisfies the usual conditions of: completeness, right continuity. Complete natural filtration generated by processes
3 Financial market ds 0 (t) S 0 (t) = rdt, S 0(0) = 1, ds(t) = µdt + dl(t), S(0) = 1, S(t ) where L := (L(t), 0 t T) denotes a zero-mean Lévy process, F-adapted with càdlàg sample paths, which satisfies the Lévy-Itô decomposition L(t) = σw(t) + z ( M(ds dz) ν(dz)ds ). (0,t]
4 Financial market W := (W(t), 0 t T) denotes a Brownian motion, M denotes a Poisson random measure, defined on Ω B((0, T]) B( {0}) M(dt dz) = s>0 1 (s, L(s)) (dt, dz)1 ( L(s) =0) (s), M denotes a compensated martingale measure M(dt dz) = M(dt dz) ν(dz)dt, ν is a Lévy measure, z <1 z2 ν(dz) <, ν({0}) = 0, W and M independent.
5 Lévy processes Brownian motion compound Poisson process subordinators Variance Gamma process Normal Inverse Gaussian hiperbolic distribution tempered stable process
6 Financial market r, µ, σ are non-negative constants, µ > r, ν is defined on ( 1, z), where µ r σ 2 + z 1 z2 ν(dz) z < 1, z z, σ > 0 or β (0, 2)liminf ǫ 0 ǫ β ǫ ǫ log(1 + z) 2 ν(dz) > 0.
7 Point process N denotes a random measure, defined on Ω B((0, T]) B( {0}), N(dt dy) = 1 (Tn,ζ n)(dt, dy)1 {ζn 0}, n=1 associated with a sequence (T n, ζ n ) n 1, where 1. (T n ) n 1 is an increasing sequence of F-stopping times with T n P a.s., 2. (ζ n ) n 1 is a sequence of random variables taking values in, 3. ζ n is F Tn -measurable, N and M independent.
8 Point process Ñ denotes a compensated martingale measure Ñ(dt dy) = N(dt dy) ξ(t, λ(t))k(dy)dt, 1. k is a probability measure on B() with k({0}) = 0, 2. ξ : Ω [0,T] + is a predictable mapping, 3. yk(dy) < and E[ T 0 ξ(s,λ(s))ds] <.
9 Default process dλ(t) = a(t, λ(t))dt + b(t, λ(t))db(t), λ(0) = λ B := (B(t), 0 t T) denotes an F-adapted Brownian motion, a : [0, T] (0, ), b : [0, T] (0, ) (0, ) are continuous functions, locally Lipschitz continuous in λ, uniformly in t, P( s [t,t] λ(s) (0, ) λ(t) = λ) = 1 for all starting points (t, λ) [0, T] (0, ), sup t [0,T] E[ λ(t) 2+ǫ ] < for some ǫ > 0, E [ T 0 b(s, λ(s)) 2+ǫ ds ] < for some ǫ > 0, B and L independent.
10 Stochastic mortality models Milevsky, Promislow (2001) λ(t) = Ae Bt+Y (t), dy (t) = ay(t)dt + bdb(t) Dahl (2004) dλ(t) = (a(t) b(t)λ(t))dt + c(t) λ(t)db(t) Schrager (2004) n λ(t) = a(t) + Y i (t)b i (t) i=1 Y i (t) are n positive affine stochastic processes Luciano, Vigna (2005) dλ(t) = aλ(t)dt + c λ(t)db(t)
11 Default probabilities p(t, s, λ) = E [ e s t λ(u)du λ(t) = λ ] p(t, s, λ) = E [ e s t λ(u)du λ(s) λ(t) = λ ] m(t, s, λ) = E [ λ(s) λ(t) = λ ] p λ (t, s, λ) is uniformly bounded p(t, s, λ) K(1 + λ) m(t, s, λ) K(1 + λ) p λ (t, s, λ) K(1 + λ)
12 Index-linked payment process P(t) = t 0 H(s)ds + G(s, y)n(ds dy) + F1 t=t (0,t] H : Ω [0, T] + is a predictable mapping such that E [ T 0 H(t) 2 dt ] <, G : Ω [0, T] + is a predictable mapping such that E [ T 0 G(t, y) 2 ξ(t, λ(t))k(dy)dt ] <, F : Ω + is a F T -measurable mapping such that E [ F 2] <.
13 Examples of payment processes in insurance Life insurance t P(t) = (n N(s ))h(s, S(s ))ds + g(s, S(s ))dn(s) 0 (0,t] +(n N(T))f (T, S(T))1 t=t Non-life insurance P(t) = (0,t] g(s, S(s ), y)n(ds dy) functions h, f are globally Lipschitz continuous in (t, s), g(t, s, z) g( t, s, z) K(z)( t t + s s ) and g(t, s, z) K(z)(1 + s) with K(z)ν(dz) <.
14 Insurer s wealth process dx π (t) = π(t) ds(t) S(t ) + ( X π (t ) π(t) ) ds 0 (t) S 0 (t) dp(t) = π(t) ( µdt + σdw(t) + z M(dt dz) ) + ( X π (t ) π(t) ) rdt H(t)dt G(t, z)n(dt dz) Definition A strategy (π(t), 0 < t T) is admissible, π A, if it satisfies the following conditions: 1. π : (0, T] Ω is a predictable mapping, 2. E [ T 0 π(t) 2 ] dt <, 3. there exists a unique solution X π on [0, T].
15 Optimization problem T inf E[ ( θ X π (t) (t) ) 2 ( dt + X π (T) (T) ) 2 ] π A 0 (t) = P(t) + (t) = P(t) + E P [ e r(s t) dp(t) ] Ft (t,t] P P is an equivalent martingale measure induced by the market
16 Areas of applications life annuities minimum guaranteed benefits products unit-linked products structured products etc.
17 Process spaces Definition Let S 2 denote the space of F-adapted -valued on [0, T] processes for which E [ sup U(t) 2 dt ] <. t [0,T] Let L 2 denote the space of F-predictable -valued on [0, T] processes for which E [ T U(t) 2 dt ] <. 0 Let L 2 M denote the space of F-predictable -valued on [0, T] processes for which E [ T U(t,z) 2 ν(dz)dt ] <. 0 Let L 2 N denote the space of F-predictable -valued on [0, T] processes for which E [ T U(t,z) 2 ξ(t, λ(t))k(dz)dt ] <. 0
18 Backward stochastic differential equations (1) d p(t) β(t)dw (t) γ(t)db(t) (1) κ(t,z) M(dt dz) η(t,z)ñ(dt dz) +2 p(t)rdt + θdt ( p(t)(µ r) + β(t)σ 2 + ) 2 κ(t,z)zν(dz) p(t)(σ 2 + z2 ν(dz)) + κ(t,z)z2 ν(dz) dt = 0 p(t) = 1
19 Backward stochastic differential equation (2) dp(t) β(t)dw (t) γ(t)db(t) (2) κ(t,z) M(dt dz) η(t,z)ñ(dt dz) +p(t)rdt 2θ (t)dt 2 η(t, z)g(t, z)ξ(t, λ(t))k(dz)dt ( ) 2 p(t) H(t) + G(t, z)ξ(t, λ(t))k(dz) dt ( p(t)(µ r) + β(t)σ 2 + ) κ(t,z)zν(dz) p(t)(σ 2 + z2 ν(dz)) + κ(t,z)z2 ν(dz) ( ) p(t)(µ r) + β(t)σ 2 + κ(t, z)zν(dz) dt = 0 p(t) = 2 (T)
20 Solution of the optimization problem Theorem Assume that the backward stochastic differential equations (1), (2) have solutions (p, β, γ, κ, η), ( p, β, γ, κ, η) S 2 L 2 L 2 L 2 M L2 N. Then, the strategy ˆπ(t) = p(t)(µ r) + β(t)σ 2 + κ(t, z)zν(dz) p(t)(σ 2 + z2 ν(dz)) + κ(t, z)z2 ν(dz) X ˆπ (t ) p(t)(µ r) + β(t)σ2 + κ(t, z)zν(dz) 2 p(t)(σ 2 + z2 ν(dz)) + 2 κ(t, z)z2 ν(dz) is optimal in the class of admissible strategies.
21 Backward stochastic differential equation (1) d p(t) dt β = γ = κ = η = 0 + (δ + r) p(t) + θ = 0 p(t) = 1 δ = r (µ r) 2 σ 2 + z2 ν(dz)
22 Optimal investment strategy µ r ˆπ(t) = 2 σ 2 + z2 ν(dz)) X ˆπ (t ) p(t)(µ r) + β(t)σ2 + κ(t,z)zν(dz) p(t)(σ 2 + z2 ν(dz))
23 Backward stochastic differential equation (2) dp(t) β(t)dw (t) γ(t)db(t) (3) κ(t,z) M(dt dz) η(t,z)ñ(dt dz) +p(t)δdt 2θ (t)dt ( ) 2 p(t) H(t) + G(t, z)ξ(t, λ(t))k(dz) dt µ r ( ) σ 2 + β(t)σ 2 + κ(t, z)zν(dz) dt = 0 z2 ν(dz) p(t) = 2 (T)
24 Change of measure Z(t) = E ( (µ r)σ σ 2 + (µ r)z dw (t) z2 ν(dz) σ 2 + z2 ν(dz) M(dt dz) ) dq F T = Z(T) dp Q is an equivalent martingale measure, W (t) + (µ r)σ σ 2 + t is Q-Brownian motion, z2 ν(dz) (µ r)z M(dt dz) = M(dt dz) (1 σ 2 + )ν(dz)dt is z2 ν(dz) Q-compensated Poisson random measure.
25 Backward stochastic differential equation (2) Theorem There exists a unique solution (p, β, γ, κ, η) S 2 L 2 L 2 L 2 F L2 M of the backward stochastic differential equation (3) and p is given by p(t) = 2E Q[ (T)e T (δ r)(t t) + e (δ r)(s t)( θ (s) t + p(s) ( H(s) + G(s, z)ξ(s, λ(s))k(dz) )) ds F t ].
26 Martingale representation Theorem The solution (β, γ, κ, η) L 2 L 2 L 2 M L2 N representation of satisfies the martingale 2E Q[ (T) T + e δ(t s)( θ (s) 0 + p(s) ( H(s) + G(s, z)ξ(s, λ(s))k(dz) )) ds ] Ft t = M(0) + + t 0 0 e δ(t s) β(s)dw Q (s) + t e δ(t s) κ(s, z) M Q (ds dz) + 0 e δ(t s) γ(s)db(s) t 0 e δ(t s) η(s, y)ñ(ds dy).
27 Special case The pricing measure coincides with the minimal martingale measure P = Q emark The general case can still be handled.
28 Martingale representation ˆπ(t) = µ r 2 σ 2 + z2 ν(dz)) X ˆπ (t ) p(t)(µ r) + β(t)σ2 + κ(t, z)zν(dz) p(t)(σ 2 + z2 ν(dz)) Let φ h (t, T, s), φ g (t, T, s), φ f (t, T, s) denote prices at time t (under Q) of contingent claims expiring at time T with pay-out functions h, g, f under the current value of S.
29 Life insurance payment process { T β(t) = 2(n N(t ))S(t )σ (e δ(s t) p(s) + θ eδ(s t) e r(s t) ) t δ + r ( ) p(t,s, λ(t))φ h s(t,s,s(t )) + p(t,s, λ(t))φ g s (t, s,s(t )) ds + (e δ(t t) + θ eδ(t t) e δ + r r(t t) ) } p(t,t, λ(t))φ f s(t,t, S(t )) { T κ(t,z) = 2(n N(t )) (e δ(s t) p(s) + θ eδ(s t) e r(s t) ) t δ + r (p(t,s, λ(t)) ( φ h (t, s,s(t ) + z) φ h (t,s, S(t )) ) + p(t, s, λ(t)) ( φ g (t,s, S(t ) + z) φ g (t,s,s(t )) )) ds + (e δ(t t) + θ eδ(t t) r(t t) e ) p(t,t, λ(t)) δ + r ( } φ f (t,t, S(t ) + z) φ f (t,t, S(t ))
30 Non-life insurance payment process T β(t) = 2S(t )σ (e δ(s t) p(s) + θ eδ(s t) e r(s t) ) t δ + r m(t, s, λ(t)) φ g s (t, s, S(t ), y)f (dy)ds T κ(t, z) = 2 t (e δ(s t) p(s) + θ eδ(s t) e r(s t) δ + r ) m(t, s, λ(t)) ( φ g (t, s, S(t ) + z, y) φ g (t, s, S(t ), y) ) f (dy)ds
31 Thank you very much Lukasz Delong Institute of Econometrics, Division of Probabilistic Methods Warsaw School of Economics lukasz.delong@sgh.waw.pl homepage : akson.sgh.waw.pl/ delong/
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