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Arthur CHARPENTIER - Pricing catastrophe options in incomplete market.

Pricing catastrophe optionsin incomplete market

Arthur Charpentier

[email protected]

Actuarial and Financial Mathematics Conference

Interplay between finance and insurance, February 2008

based on some joint work with R. Elie, E. Quemat & J. Ternat.

1


Agenda

A short introduction

• From insurance valuation to financial pricing• Financial pricing in complete markets

Pricing formula in incomplete markete

• The model : insurance losses and financial risky asset• Classical techniques with Levy processes

Indifference utility technique

• The framework• HJM : primal and dual problems• HJM : the dimension problem

Numerical issues

2


Agenda







Numerical issues

3


A general introduction : from insurance to finance

Finn and Lane (1995) : “there are no right price of insurance, there is simply thetransacted market price which is high enough to bring forth sellers and lowenough to induce buyers”.

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securitization

traditional indemnity indust parametric ilreinsurance securitization loss securitization derivatives

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4


A general introduction

Indemnity versus payoffIndemnity of an insurance claim, with deductible d : X = (S − d)+ooPayoff of a call option, with strike K : X = (ST −K)+oo

Pure premium versus pricePure premium of insurance product : π = EP [(S − d)+]ooPrice of a call option, in a complete market : V0 = EQ [(ST −K)+|F0]oo

Alternative to the pure premiumExpected Utility approach, i.e. solving U(ω − π) = EP(U(ω −X))ooYaari’s dual approach, i.e. solving ω − π = Eg◦P(ω −X)oo

Essher’s transform, i.e. π = EQ(X) =EP(X · eαX)

EP(eαX)oo

5


Financial pricing (complete market)

Price of a European call with strike K and maturity T is V0 = EQ [(ST −K)+|F0]where Q stands for the risk neutral probability measure equivalent to P.

• the market is not complete, and catastrophe (or mortality risk) cannot bereplicated by financial assets,

• the guarantees are not actively traded, and thus, it is difficult to assumeno-arbitrage,

• the hedging portfolio should be continuously rebalanced, and there should belarge transaction costs,

• if the portfolio is not continuously rebalanced, we introduce an hedging error,• underlying risks are not driven by a geometric Brownian motion process.

=⇒ Market is incomplete and catastrophe might be only partially hedged.

6


Impact of WTC 9/11 on stock prices (Munich Re and SCOR)

2001 2002

3035

4045

5055

60

250

300

350

Mun

ich

Re

stoc

k pr

ice

Fig. 1 – Catastrophe event and stock prices (Munich Re and SCOR).

7


Agenda







Numerical issues

8


The model : the insurance loss process

Under P, assume that (Nt)t>0 is an homogeneous Poisson process, withparameter λ.

Let (Mt)t>0 be the compensated Poisson process of (Nt)t>0, i.e. Mt = Nt − λt.

The ith catastrophe has a loss modeled has a positive random variableFTi-measurable denoted Xi. Variables (Xi)i>0 are supposed to be integrable,independent and identically distributed.

Define Lt =∑Nti=1Xi as the loss process, corresponding to the total amount of

catastrophes occurred up to time t.

9


The model : the financial asset process

Financial market consists in a free risk asset, and a risky asset, with price (St)t>0.

The value of the risk free asset is assumed to be constant (hence it is chosen as anumeraire).

The price of the risky asset is driven by the following diffusion process,

dSt = St−

µdt︸︷︷︸trend

+ σdWt︸︷︷︸volatiliy

+ ξdMt︸︷︷︸jump

with S0 = 1

where (Wt)t>0 is a Brownian motion under P, independent of the catastropheoccurrence process (Nt)t>0. Note that ξ is here constant.

Note that the stochastic differential equation has the following explicit solution

St = exp[(µ− σ2

2− λξ

)t+ σWt

](1 + ξ)Nt .

10


0 2 4 6 8 10

02

46

810

Underlying and loss processes

Time

Fig. 2 – The loss index and the price of the underlying financial asset.

11


Financial pricing (incomplete market)

Let (Xt)t≥0 be a Levy process.

The price of a risky asset (St)t≥0 is St = S0 exp (Xt).

Xt+h −Xt has characteristic function φh with

log φ(u) = iγu− 12σ2u2 +

∫ +∞

−∞

(eiux − 1− iux1{|x|<1}

)ν(dx),

where γ ∈ R, σ2 ≥ 0 and ν is the so-called Levy measure on R/{0}.

The Levy process (Xt)t≥0 is characterized by triplet (γ, σ2, ν).

12


Get a martingale measure : the Esscher transform

A classical premium in insurance is obtained using Esscher transform,

π = EQ(X) =EP(X · eαX)

EP(eαX)

Given a Levy process (Xt)t≥0 under P with characteristic function φ or triplet(γ, σ2, ν), then under Esscher transform probability measure Qα, (Xt)t≥0 is still aLevy process with characteristic function φα such that

log φα(u) = log φ(u− iα)− log φ(−iα),

and triplet (γα, σ2α, να) for X1, where σ2

α = σ2, and

γα = γ + σ2α+∫ +1

−1

(eαx − 1)ν(dx) and να(dx) = eαxν(dx),

see e.g. Schoutens (2003). Thus, V0 = EQα [(ST −K)+|F0]

13


Agenda







Numerical issues

14


Indifference utility pricing

As in Davis (1997) or Schweizer (1997), assume that an investor has a utilityfunction U , and initial endowment ω.

The investor is trading both the risky asset and the risk free asset, forming adynamic portfolio δ = (δt)t>0 whose value at time t isΠt = Π0 +

∫ t0δudSu. = Π0 + δ · S)t where (δ · S) denotes the stochastic integral of

δ with respect to S.

A strategy δ is admissible if there exists M > 0 such thatP(∀t ∈ [0, T ], (δ · S)t > −M

)= 1, and further if EP

[∫ T0δ2t S

2t−dt

]< +∞.

∀x ∈ R∗+, UL(x) = log(x) : logarithmic utility

∀x ∈ R∗+, UP (x) =xp

pwhere p ∈]−∞, 0[∪]0, 1[ : power utility

∀x ∈ R, UE(x) = − exp(− x

x0

): exponential utility.

15



If X is a random payoff, the classical Expected Utility based premium is obtainby solving

u(ω,X) = U(ω − π) = EP(U(ω −X)).

Consider an investor selling an option with payoff X at time T• either he keeps the option, uδ?(ω+π, 0o) = supδ∈AEP

[U(ω + (δ · S)T−X)

],

• either he sells the option,o uδ?(ω + π,X) = supδ∈A EP

[U(ω + (δ · S)T −X)

].

The price obtained by indifference utility is the minimum price such that the twoquantities are equal, i.e.

π(ω,X) = inf {π ∈ R such that uδ?(ω + π,X)− uδ?(ω, 0) > 0} .

This price is the minimal amount such that it becomes interesting for the sellerto sell the option : under this threshold, the seller has a higher utility keeping theoption, and not selling it.

16



Set Vδ,ω,t = ω + (δ · S)t.

A classical idea to obtain a fair price is to use some marginal rate of substitutionargument, i.e. π is a fair price if diverting of his funds into it at time 0 will haveno effect on the investor’s achievable utility.

Hence, the idea is to find π, solution of

∂

∂εmax

{EP

[U (Vδ?,ω−ε,T + εX/π)

]}∣∣∣∣ε=0

= 0,

under some differentiability conditions of the function, where δ? is an optimalstrategy.

Then (see Davis (1997)), the price of a contingent claim X is

π =EP(U ′(Vδ?,ω,T )X)

U ′(ω),

where again, the expression of the optimal strategy δ? is necessary.

17


The model : the dimension issue

Set Y t = (t,Πt, St, Lt), taking values in S = [0, T ]× R× R2+.

The control δ takes values in U = R. Assume that random variables (Xi)’s have adensity (with respect to Lebesgue’s measure) denoted ν, and denote m itsexpected value. The diffusion process for Y t is

dY t = b (Y t, δt) dt︸︷︷︸trend

+ Σ (Y t, δt) dWt︸︷︷︸volatility

+∫

Rγ (Y t− , δt− , x)M(dt, dx)︸︷︷︸

jumps

18


The model : the dimension issue

where functions b, Σ and γ are defined as follows

b : R4 × U → R4 Σ : R4 × U → R4t

π

s

l

, δ

7→

1µδs

µs

λm

t

π

s

l

, δ

7→

0σδs

σs

0

γ : R4 × U × R → R4

t

π

s

l

, δ , x

7→

0ξδs

ξs

x

,

and where M(dt, dx) = N(dt, dx)− ν(dx)λdt, and N(dt, dx) denotes the pointmeasure associated to the compound Poisson process (Lt) (see Øksendal andSulem (2005)).

19


Solving the optimization problem

The Hamilton-Jacobi-Bellman equation related to this optimal control problemof a European option with payoff X = φ(ST ) is then{

supδ∈AA(δ)u(y) = 0u(T, π, s, l) = U(π − φ(l))

(1)

where

A(δ)ϕ(y) =∂ϕ

∂t(y) + s(µ− ξλ)

(δ(y)

∂ϕ

∂π(y) +

∂ϕ

∂s(y))

+12s2σ2

(δ(y)2

∂2ϕ

∂π2(y) + 2δ(y)

∂2ϕ

∂π∂s(y) +

∂2ϕ

∂s2(y))

+ λ

∫R

[ϕ(t, π + sξδ(y), s(1 + ξ), l + x)− ϕ(t, π, s, l)

]ν(dx).

20


Solving the optimization problem

Define C0 as the convex cone of random variables dominated by a stochasticintegral, i.e.

C0 = {X | X 6 (δ · S)T for some admissible portfolio δ}

and set C = C0 ∩ L∞ the subset of bounded random variables. Note thatfunctional u introduced earlier can be written

u(·) solution of u(x, φ) = supX∈C0

EP [U (x+X − φ(LT ))] . (2)

21


Solving the optimization problem : the dual version

Consider the pricing of an option with payoff X = φ(ST ).

Define the conjugate of utility function U , V : R+ → R defined asV (y) = supx∈DU [U(x)− xy].

Denote by (L∞)? the dual of bounded random variables, and define

D ={Q ∈ (L∞)? | ‖Q‖ = 1 and (∀X ∈ C)(〈Q,X〉 6 0)

}.

so that the dual of Equation (2) is then

v(·) solution of v(y, φ) = infQ∈D

{EP

[V

(ydQr

dP

)− yQφ(LT )

]}. (3)

∀x ∈ R∗+, UL(x) = log(x)

∀x ∈ R∗+, UP (x) =xp

p

∀x ∈ R, UE(x) = − exp(− x

x0

) i.e.

∀y ∈ R?+, VL(y) = − log(y)− 1

∀y ∈ R?+, VP (y) = −yq

q, q =

p

p− 1∀y ∈ R?+, VE(y) = y(log(y)− 1).

22


Merton’s problem, without jumps

Assume that dSt = St−(µdt+ σdWt) (i.e. ξ = 0) then

for logarithm utility function

uL(t, π) = UL

[π exp

(α2

2σ2(T − t)

)].

for power utility function

uP (t, π) = UP

[π exp

(α2

2(1− p)σ2(T − t)

)].

for exponential utility function

uE(t, π) = UE

[π +

α2x0

2σ2(T − t)

].

23


Merton’s problem, with jumps

Assume that dSt = St−(µdt+ σdWt + ξdMt) (i.e. ξ 6= 0) then

for logarithm utility function uL(t, π) = UL(πe(T−t)C

)where C satisfies{

C = (α− ξλ)D − 12σ

2D2 + λ log(1 + ξD)0 = (α− ξλ− σ2D)(1 + ξD) + λξ

for power utility function uP (t, π) = UP(πe(T−t)C

)where C satisfies{

C = (α− ξλ)D + 12σ

2(p− 1)D2 + λp [(1 + ξD)p − 1]

0 = (α− ξλ) + σ2(p− 1)D + λξ(1 + ξD)p−1

for exponential utility function uE(t, π) = UE (π + (T − t)C) where C satisfies{C = αx0

ξ + (α− σ2

ξ − ξλ)D − 12x0

σ2D2

0 = ξλ− α+ σ2

x0D − ξλ exp

[− ξDx0

]

24


Assume that the investor has an exponential utility, U(x) = − exp(−x/x0),

Theorem 1. Let φ denote a C2 bounded function. If utility is exponential, thevalue function associated to the primal problem,

u(t, π, s, l) = maxδ∈A

EP

[U(

ΠT − φ(LT ))| Ft

]does not depend on s and can be expressed as u(t, π, l) = U

(π −C(t, l)

), where C

is a function independent of π satisfying0 = ξλ− µ+

σ2sδ?

x0− ξλ exp

[− ξsδ? + C(t, l)

x0

]EP

(e

1x0C(t,l+X)

)∂C

∂t(t, l) = µx0

ξ + (µ− σ2

ξ − ξλ)sδ? − 12x0

σ2(sδ?)2

C(T, l) = φ(l)

where δ? denotes the optimal control.

Demonstration. Theorem 19 in Quema et al. (2007).

25


Theorem 2. Further, given K > 0 and a distribution for X such that

EP

[exp

(4x0KX

)]<∞

if we consider the set A′ of admissible controls δ satisfying inequality

EP

(∫ T0δ2t S

2t−dt

)2

<∞, the previous results holds for φ(x) = Kx and

C(t, l) = Kl − (T − t)C, where C is a constant solution0 = ξλ− µ+

σ2

x0sδ? − ξλ exp

[−ξsδ

?

x0

]EP

(e

1x0KX)

C =µx0

ξ+ (µ− σ2

ξ− ξλ)sδ? − 1

2x0σ2(sδ?)2.

Demonstration. Theorem 19 in Quema et al. (2007).

26


Agenda







Numerical issues

27


Practical and numerical issues

The main difficulty of Theorem 1 is to derive C(t, l) and δ?(t, l) characterized byintegro-differential system

0 = ξλ− µ+σ2sδ?

x0− ξλ exp

[− ξsδ? + C(t, l)

x0

] ∫R+

e1x0C(t,l+x)f(x)dx

∂C(t, l)∂t

= µx0ξ + (µ− σ2

ξ − ξλ)sδ? − 12x0

σ2(sδ?)2

C(T, l) = φ(l)

If payoff φ has a threshold (i.e. there exists B ≥ 0 such that φ is constant oninterval [B,+∞)) ; it is possible to use a finite difference scheme. Hence, given twodiscretization parameters Nt and Ml, Cni ' C(tn, li) and Dn

i ' sδ?(tn, li) where{tn = n∆t where Nt∆t = T and n ∈ [0, Nt] ∩ Nli = i∆l where Nl∆l = B and i ∈ [0, Nl] ∩ N

28


Practical and numerical issues, with threshold payoff

Calculation of the integral can be done simply (and efficiently) using thetrapezoide method. Note that can restrict integration on the interval [0, B], since

I(tn, li) = Ini =∫

R+

e1x0C(tn,li+x)f(x)dx

=∫ B−li

0

e1x0C(tn,li+x)f(x)dx+

∫ ∞B−li

e1x0C(tn,li+x)f(x)dx

'Nl−i−1∑k=0

12

{exp

[lk+ix0

]f(lk+i) + exp

[lk+i+1

x0

]f(lk+i+1)

}+ exp

[CnNlx0

]F(lNl−i

)

29


Practical and numerical issues with threshold payoff

For the control, the goal is to calculate Dni ' sδ?(tn, li), where sδ?(tn, li) is the

solution of

0 = ξλ− µ+σ2sδ?(tn, li)

x0− ξλ exp

[− ξsδ?(rn, li) + C(tn, li)

x0

]I(tn, li).

Thus, Dni is the solution (obtained using Newton’s method) of G(x) = 0 where

G(x) = ξλ− µ+σ2

x0x− ξλ exp

[− ξx+ Cni

x0

]Ini

30


Practical and numerical issues, with affine payoff

In the case of an affine payoff, we have seen already that the system isdegenerated, and that its solution is simply C(t, l) = K +Kl − (T − t)C, whereC is a constant solution of C =

µx0

ξ+ (µ− σ2

ξ− ξλ)sδ? − 1

2x0σ2(sδ?)2

0 = ξλ− µ+ σ2sδ?

x0− ξλ exp

[− 1

x0ξsδ?

]EP

(e

1x0KX).

If the Laplace transform of X is unknown, it is possible to approximate it usingMonte Carlo techniques. And the second equation can be solved using Newton’smethod, as in the previous section : we can derive sδ? and then get immediatelyC.

31


2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

0 2 4 6 8 10 12 14 16 18

Loi Exponentielle(1)

Loi Pareto(1,2)

Fig. 3 – Price as a function of the risk aversion coefficient x0 with T = 1, µ = 0,σ = 0.12, λ = 4, ξ = 0.05 and B = 4

32


0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10 12 14 16 18

Loi Exponentielle(1)

Loi Pareto(1,2)

Fig. 4 – Nominal amount at time t = 0 of the optimal edging, as a function of x0

with T = 1, µ = 0, σ = 0.12, λ = 4, ξ = 0.05 and B = 4

33


Properties of optimal strategies

Two nice results have been derived in Quema et al. (2007),

Lemma 3. C(t, ·) is increasing if and only if φ is increasing.

Lemma 4. If φ is increasing and µ > 0, then the optimal amount of risky assetto be hold when hedging is bounded from below by a striclty positive constant.

34


Davis, M.H.A (1997). Option Pricing in Incomplete Markets, in Mathematics ofDerivative Securities, ed. by M. A. H.. Dempster, and S. R. Pliska, 227-254.

Finn, J. and Lane, M. (1995). The perfume of the premium... or pricinginsurance derivatives. in Securitization of Insurance Risk : The 1995 BowlesSymposium, 27-35.

Øksendal, B. and Sulem, A. (2005). Applied Stochastic Control of JumpDiffusions. Springer Verlag.

Quema, E., Ternat, J., Charpentier, A. and Elie, R. (2007). Indifferenceprices of catastrophe options. submitted.

Schoutens, W. (2003). Levy Processes in Finance, pricing financial derivatives.Wiley Interscience.

Schweizer, M. (1997). From actuarial to financial valuation principles.Proceedings of the 7th AFIR Colloquium and the 28th ASTIN Colloquium,261-282.

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