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

OPTIMAL 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

OPTIMAL 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

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 3

3 The financial market model

Index process, e.g. temperature dRt = ρ(t, Rt)dWt + b(t, Rt)dt, b : [0, T] × Rm → Rm, ρ : [0, T] × Rm → Rm×d deterministic functions, globally Lipschitz and of sublinear growth. R Markov process, Rt,r

s : start at t in r

Insurance derivative F(RT), F : Rm → R bounded Correlated financial market, k risky assets with price process: dSi

t

Si

t

= βi(t, Rt)dWt + αi(t, Rt)dt = βi(t, Rt)[dWt + θtdt], i = 1, . . . , k, α : [0, T] × Rm → Rk, β : [0, T] × Rm → Rk×d, θ = β∗[ββ∗]−1α. W d−dimensional Brownian motion, ββ∗ uniformly elliptic

OPTIMAL 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] Gλ,t

s

=

k

• i=1

s

t

λi

u

dSi

u

Si

u

= s

t

λuβu[dWu + θudu], utility function: U(x) = −e−ηx (0 < η risk aversion); maximal expected utility from terminal wealth without and with derivative: V 0(t, v, r) = sup

λ∈At,r EU(v + Gλ,t,r T

), V F(t, v, r) = sup

λ∈At,r EU(v + Gλ,t,r T

− F(Rt,r

T ))

λ0 resp. λF optimal strategies ∆ = λF − λ0 derivative hedge

OPTIMAL 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 ∈ Rk} 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 ⊂ Rk closed ˜ A set of strategies λ such that

• λ ∈ ˜

C P ⊗ l-a.s. (l Lebesgue measure)

• {exp(−η

τ

0 λs dSs Ss ) : τ

stopping time in [0, T]} uniformly integrable

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 6

5 Optimization under non-convex constraints

F = F(RT) insurance derivative For simplicity t = 0, Gλ,0 = Gλ, V (v) = V F(0, v, r), etc. First formulation: Find V (v) = supλ∈ ˜

A E(U(Gλ T − F)) = supλ∈ ˜ A E(U(v +

T

0 λsβs[dWs + θsds] − F)).

For simplicity: π = λβ, C = ˜ Cβ, A = ˜ Aβ. Gπ

t = v +

t πs[dWs + θsds], t ∈ [0, T] Second formulation: Find V (v) = sup

π∈A

E(U(Gπ

T − F)) = sup π∈A

E(− exp(−η(x + T πs[dWs + θsds] − F))).

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 7

6 A solution method based on BSDE

Idea: Construct family of processes Q(π) such that form 1 Q(π) = constant, Q(π)

T

= − exp(−η(Gπ

T − F)),

Q(π) supermartingale, π ∈ A, Q(π∗) martingale, for (exactly) one π∗ ∈ A. Then E(− exp(−η[Gπ

T − F]))

= E(Q(π)

T )

≤ E(Qπ

0)

= V (v) = E(Q(π∗) ) = E(− exp(−η[G(π∗)

T

− F])). Hence π∗ optimal strategy.

OPTIMAL 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 Yt = F − T

t

ZsdWs − T

t

f(s, Zs)ds, YT = F, such that with Q(π)

t

= − exp(−η[Gπ

t − Yt]),

t ∈ [0, T], we have form 2 Q(π) = − exp(−η(v − Y0)) = constant, (fulfilled) Q(π)

T

= − exp(−η(Gπ

T − F))

(fulfilled) Q(π) supermartingale, π ∈ A, Q(π∗) martingale, for (exactly) one π∗ ∈ A. This gives solution of valuation problem.

OPTIMAL 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)

t

= − exp(−η[Gπ

t − Yt])

= − exp(−η[v − Y0]) · exp(−η[ t (πs − Zs)dWs − t [f(s, Zs) − πsθs]ds]) = exp(−η[v − Y0]) · exp(−η t (πs − Zs)dWs − η2 2 t (πs − Zs)2ds) ·− exp( t [ηf(s, Zs) − ηπsθs + η2 2 (πs − Zs)2]ds) = M (π)

t

· A(π)

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

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 10

7 Construction of generator of BSDE

form 3 q(·, π, z) ≥ 0, π ∈ A (supermartingale cond.) q(·, π∗, z) = 0, for (exactly) one π∗ ∈ A (martingale cond.). Now q(·, π, z) = f(·, z)−πθ + η 2(π − z)2 = f(·, z)+η 2(π − z)2 − (π − z) · θ + 1 2ηθ2 −zθ− 1 2ηθ2 = f(·, z)+η 2[π − (z + 1 ηθ)]2 −zθ − 1 2ηθ2. Under non-convex constraint π ∈ C: [π − (z + 1 ηθ)]2 ≥ d2(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

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 11

7 Construction of generator of BSDE

form 4 f(·, z) = −η

2d2(C, z + 1 ηθ)+z · θ + 1 2ηθ2

(supermartingale) π∗ such that d(C, z + 1

ηθ) = d(π∗, z + 1 ηθ)

(martingale). Problem: Let ΠC(v) = {π ∈ Rd : d(C, v) = d(π, v)}. Find measurable selection π∗

t from

ΠCt(Zt + 1

ηθt). Solved by classical measurable selection method.

1 α θt z+ p

t *

p

t *

t

C

1 α θt z+ π (

)

Measurable selection from

t

C π (

)

t

C

1 θ z+

t

α

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 12

8 Main result

Thm 1 (Y, Z) unique solution of BSDE Yt = F − T

t ZsdWs −

T

t f(s, Zs)ds,

t ∈ [0, T], with f(t, Zt) = −η

2d2(Ct, Zt + 1 ηθt)+Zt · θt + 1 2ηθ2 t.

Then value function of utility optimization problem under constraint π ∈ A given by V (v) = − exp(−η[v − Y0]). There exists an (non-unique) optimal trading strategy π∗ ∈ A such that π∗

t ∈ ΠCt(Zt + 1

ηθt), t ∈ [0, T]. Proof:

• existence, uniqueness for BSDE with quadratic non-linearity in z

(M. Kobylanski ’00)

• measurable selection theorem for ΠCt(Zt + 1

ηθt)

• BMO properties of the martingales
• ZsdWs,
• π∗

sdWs

for uniform integrability of exponentials (regularity of coefficients)•

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 13

9 Calculation of derivative hedge

generalization to [t, T] instead of [0, T], cond. on Rt = r: (Y t,r, Zt,r),πt,r (without F) resp. ( ˆ Y t,r, ˆ Zt,r),ˆ πt,r (with F) instead of (Y, Z),π yields V 0(t, v, r) = − exp(−η(v − Y t,r

t

)), V F(t, v, r) = − exp(−η(v − ˆ Y t,r

t

)), instead of V (v) = − exp(v − Y0). due to linearity of C(t, r) projections unique and linear, hence πt,r

s

= ΠC(t,r)[Zt,r

s

+ 1 ηθ(s, Rt,r

s )],

ˆ πt,r

s

= ΠC(t,r)[ ˆ Zt,r

s

+ 1 ηθ(s, Rt,r

s )],

and so ∆β(s, Rt,r

s ) = ΠC(t,r)[ ˆ

Zt,r

s

− Zt,r

s ].

OPTIMAL 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 Y t,r

s

= u(s, Rt,r

s ),

ˆ Y t,r

s

= ˆ u(s, Rt,r

s ).

There are measurable (deterministic) functions v and ˆ v such that Zt,r

s

= vρ(s, Rt,r

s ),

ˆ Zt,r = ˆ vρ(s, Rt,r

s ).

Corollary 1 p(t, r) := Y t,r

t

− Y t,r

t

= u(t, r) − ˆ u(t, r) 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 ∆

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 15

11 Differentiability

Thm 3 (Parameter Differentiability) smoothness conditions on F, f There exists a version of ( Y t,r

s

, Zt,r

s ) such that a.s.

Y t,r

s

is continuous in s and cont. differentiable in r (classical sense)

Zt,r is differentiable in a weak sense (norm topology)

• (∇r

Y t,r, ∇r Zt,r) solves the BSDE ∇r Y r

t

= ∇rF(Rt,r

s )∇rRt,r s

− T

t ∇r

Zt,r

s dWs

+ T

t

• ∇rf(s, Rt,r

s ,

Zt,r

s )∇rRt,r s

+∇zf(s, Rt,r

s ,

Zt,r

s )∇r

Zt,r

s

• ds.

Proof uses norm inequalities, and inverse H¨

• lder inequalities, based on BMO

properties of the stochastic integral processes of ˆ Zt,r Thm 4 (Malliavin Differentiability) Dϑ Y t,r

s

= ∇r u(s, Rt,r

s )DϑRt,r s

and

• Zt,r

s

= Ds Y t,r

s

= ∇r u(s, Rt,r

s )ρ(s, Rt,r s )

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 16

12 Explicit description of derivative hedge

Properties of the BSDEs = ⇒ Thm 5 The indifference price p(t, r) = Y t,r

t

− Y t,r

t

is differentiable in r. Thm 6 The derivative hedge ∆ at time t depends only on Rt, and ∆(t, r)β(t, r) = ΠC(t,r)[ Zt,r

t

− Zt,r

t ]

= ΠC(t,r)[∇r( Y t,r

t

− Y t,r

t

)ρ(t, r)] = −ΠC(t,r)[∇rp(t, r)ρ(t, r)]. Remarks:

• complete case: ∆ = ’delta hedge’
• where is the risk aversion η?

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 17

13 Example: Heating degree days

• common underlying of weather derivatives
• Ti = average of the maximum and the minimum temperature on day i at a

specific location

• HDDi = max (0, 18 − Ti)

Cumulative heating degree days cHDDt =

30

• i=1

HDDt−i Derivatives:

• Option:

(cHDD − K)+

• Swap:

b(cHDD − K)

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 18

13 Example: Heating degree days

cHDD:

• statistical analysis shows: cHDDs are log-normally distributed

(M. Davis ’01)

• cHDD can be modeled as a geometric Brownian motion

dXt = µXtdt + νXtdWt (moving average) Other indices: cooling degree days CDDi = min (0, 18 − Ti)

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 19

13 Example: Heating degree days

• R = cHDDs (geometric Brownian Motion)
• d = 2
• 1-dim market + index: k = m = 1
• index volatility: ρ =

α

• price volatility: β =

β1 β2

• with α, β1, β2 ∈ R \ {0}

Then ∆(t, r) = −α∂p(t, r) ∂r β1 β2

1 + β2 2

.

OPTIMAL CROSS HEDGING OF INSURANCE DERIVATIVES USING BSDE 20

14 Example: Heating degree days; diversification pressure

derivative hedge: ∆(t, r) = −α∂p(t, r) ∂r β1 β2

1 + β2 2

. Call option: F(RT) = (RT − K)+ = ⇒ ∂p(t,r)

∂r

> 0 Comparison of the optimal strategies:

• β1α < 0 (negative correlation)

= ⇒ F(XT) diversifies portfolio = ⇒ ∆ > 0 = ⇒

• π > π
• β1α > 0 (positive correlation)

= ⇒ F(XT) amplifies portfolio = ⇒ ∆ < 0 = ⇒

• π < π