Slides brussels

Arthur CHARPENTIER - Pricing catastrophe options in incomplete market.

Pricing catastrophe options
in incomplete market
Arthur Charpentier

arthur.charpentier@univ-rennes1.fr

Actuarial and Financial Mathematics Conference
Interplay between ﬁnance and insurance, February 2008

´
based on some joint work with R. Elie, E. Qu´mat & J. Ternat.
e

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 L´vy processes
e
Indifference utility technique
• The framework
• HJM : primal and dual problems
• HJM : the dimension problem
Numerical issues

2


Agenda
e
• The framework
Numerical issues

3


A general introduction : from insurance to ﬁnance
Finn and Lane (1995) : “there are no right price of insurance, there is simply the
transacted market price which is high enough to bring forth sellers and low
enough to induce buyers”.
traditional indemni industry parametr il
reinsurance securitiza loss securitiza derivatives
securitization

traditional indemnity indust parametric il
reinsurance securitization loss securitization derivatives
securitiza

4


A general introduction
Indemnity versus payoﬀ
Indemnity of an insurance claim, with deductible d : X = (S − d)+ oo
Payoﬀ of a call option, with strike K : X = (ST − K)+ oo

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

Alternative to the pure premium
Expected Utility approach, i.e. solving U (ω − π) = EP (U (ω − X))oo
Yaari’s dual approach, i.e. solving ω − π = Eg◦P (ω − X)oo
EP (X · eαX )
Essher’s transform, i.e. π = EQ (X) = αX )
oo
EP (e

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 be
replicated by ﬁnancial assets,
• the guarantees are not actively traded, and thus, it is diﬃcult to assume
no-arbitrage,
• the hedging portfolio should be continuously rebalanced, and there should be
large 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)
60
55

350
Munich Re stock price
50
45

300
40
35

250
30

2001 2002

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

7


Agenda
e
• The framework
Numerical issues

8


The model : the insurance loss process
Under P, assume that (Nt )t 0 is an homogeneous Poisson process, with
parameter 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 variable
FTi -measurable denoted Xi . Variables (Xi )i 0 are supposed to be integrable,
independent and identically distributed.
t N
Deﬁne Lt = i=1 Xi 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 a
numeraire).
The price of the risky asset is driven by the following diffusion process,
 

dSt = St−  µdt + σdWt + ξdMt  with S0 = 1
trend volatiliy jump

where (Wt )t 0 is a Brownian motion under P, independent of the catastrophe
occurrence process (Nt )t 0 . Note that ξ is here constant.
Note that the stochastic differential equation has the following explicit solution
σ2
St = exp µ− − λξ t + σWt (1 + ξ)Nt .
2

10


Underlying and loss processes
10
8
6
4
2
0

0 2 4 6 8 10

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 L´vy process.
e
The price of a risky asset (St )t≥0 is St = S0 exp (Xt ).
Xt+h − Xt has characteristic function φh with
+∞
1
log φ(u) = iγu − σ 2 u2 + eiux − 1 − iux1{|x|<1} ν(dx),
2 −∞

where γ ∈ R, σ 2 ≥ 0 and ν is the so-called L´vy measure on R/{0}.
e
The L´vy process (Xt )t≥0 is characterized by triplet (γ, σ 2 , ν).
e

12


Get a martingale measure : the Esscher transform
A classical premium in insurance is obtained using Esscher transform,
EP (X · eαX )
π = EQ (X) =
EP (eαX )

Given a L´vy process (Xt )t≥0 under P with characteristic function φ or triplet
e
(γ, σ 2 , ν), then under Esscher transform probability measure Qα , (Xt )t≥0 is still a
L´vy process with characteristic function φα such that
e

log φα (u) = log φ(u − iα) − log φ(−iα),
2 2
and triplet (γα , σα , να ) for X1 , where σα = σ 2 , and
+1
γα = γ + σ 2 α + (eαx − 1)ν(dx) and να (dx) = eαx ν(dx),
−1

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

13


Agenda
e
• The framework
Numerical issues

14


Indiﬀerence utility pricing
As in Davis (1997) or Schweizer (1997), assume that an investor has a utility
function U , and initial endowment ω.
The investor is trading both the risky asset and the risk free asset, forming a
dynamic portfolio δ = (δt )t 0 whose value at time t is
t
Πt = Π0 + 0 δu dSu . = 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 that
T 2 2
P ∀t ∈ [0, T ], (δ · S)t −M = 1, and further if EP 0
δt St− dt < +∞.


 ∀x ∈ R∗ , UL (x) = log(x) : logarithmic utility
 +
xp


∗
∀x ∈ R+ , UP (x) = where p ∈] − ∞, 0[∪]0, 1[ : power utility

p
 ∀x ∈ R, UE (x) = − exp − x : exponential utility.




x0

15


If X is a random payoff, the classical Expected Utility based premium is obtain
by 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∈A EP 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 two
quantities 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 seller
to sell the option : under this threshold, the seller has a higher utility keeping the
option, 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 substitution
argument, i.e. π is a fair price if diverting of his funds into it at time 0 will have
no effect on the investor’s achievable utility.
Hence, the idea is to find π, solution of
∂
max EP U (Vδ ,ω−ε,T + εX/π) = 0,
∂ε ε=0

under some differentiability conditions of the function, where δ is an optimal
strategy.
Then (see Davis (1997)), the price of a contingent claim X is
EP (U (Vδ ,ω,T )X)
π= ,
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 a
density (with respect to Lebesgue’s measure) denoted ν, and denote m its
expected value. The diﬀusion process for Y t is

dY t = b (Y t , δt ) dt + Σ (Y t , δt ) dWt + γ (Y t− , δt− , x) M (dt, dx)
R
trend volatility jumps

18


The model : the dimension issue
where functions b, Σ and γ are deﬁned as follows
b: R4 × U → R4 Σ: R4 × U → R4
       
t 1 t 0
 π   µδs   π   σδs 
, δ → , δ →
       
     
 s   µs   s   σs 
l λm l 0

γ: R4 × U × R → R4
   
t 0
 π   ξδs  ,
, δ , x → 
   
 
 s   ξs 
l x
and where M (dt, dx) = N (dt, dx) − ν(dx)λdt, and N (dt, dx) denotes the point
measure associated to the compound Poisson process (Lt ) (see Øksendal and
Sulem (2005)).

19


Solving the optimization problem
The Hamilton-Jacobi-Bellman equation related to this optimal control problem
of a European option with payoﬀ X = φ(ST ) is then

supδ∈A A(δ) u(y) = 0
(1)
u(T, π, s, l) = U (π − φ(l))

where
(δ) ∂ϕ ∂ϕ ∂ϕ
A ϕ(y) = (y) + s(µ − ξλ) δ(y) (y) + (y)
∂t ∂π ∂s
2
1 2 2 2∂ ϕ ∂2ϕ ∂2ϕ
+ s σ δ(y) (y) + 2δ(y) (y) + 2 (y)
2 ∂π 2 ∂π∂s ∂s

+ λ ϕ(t, π + sξδ(y), s(1 + ξ), l + x) − ϕ(t, π, s, l) ν(dx).
R

20


Solving the optimization problem
Deﬁne C0 as the convex cone of random variables dominated by a stochastic
integral, i.e.

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

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

u(·) solution of u(x, φ) = sup EP [U (x + X − φ(LT ))] . (2)
X∈C0

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 as
V (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 0) .

so that the dual of Equation (2) is then
dQr
v(·) solution of v(y, φ) = inf EP V y − yQφ(LT ) . (3)
Q∈D dP


 ∀x ∈ R∗ , UL (x) = log(x) 
 +  ∀y ∈ R+ , VL (y) = − log(y) − 1
xp

yq
 
∗ p
∀x ∈ R+ , UP (x) =
 
p i.e. ∀y ∈ R+ , VP (y) = − , q =
q p−1
 ∀x ∈ R, UE (x) = − exp − x

 

∀y ∈ R+ , VE (y) = y(log(y) − 1).
 

x0
22


Merton’s problem, without jumps
Assume that dSt = St− (µdt + σdWt ) (i.e. ξ = 0) then
for logarithm utility function
α2
uL (t, π) = UL π exp 2
(T − t) .
2σ

for power utility function
α2
uP (t, π) = UP π exp 2
(T − t) .
2(1 − p)σ

for exponential utility function
α 2 x0
uE (t, π) = UE π+ 2
(T − t) .
2σ

23


Merton’s problem, with jumps
Assume that dSt = St− (µdt + σdWt + ξdMt ) (i.e. ξ = 0) then
for logarithm utility function uL (t, π) = UL πe(T −t)C where C satisfies
1
C = (α − ξλ)D − 2 σ 2 D2 + λ log(1 + ξD)
0 = (α − ξλ − σ 2 D)(1 + ξD) + λξ

for power utility function uP (t, π) = UP πe(T −t)C where C satisfies
1
C = (α − ξλ)D + 2 σ 2 (p − 1)D2 + λ [(1 + ξD)p − 1]
p
0 = (α − ξλ) + σ 2 (p − 1)D + λξ(1 + ξD)p−1

for exponential utility function uE (t, π) = UE (π + (T − t)C) where C satisfies
αx0 σ2 1 2 2
C= ξ + (α − ξ − ξλ)D − 2x0 σ D
σ2
0 = ξλ − α + x0 D − ξλ exp − ξD
x0

24


Assume that the investor has an exponential utility, U (x) = − exp(−x/x0 ),
Theorem 1. Let φ denote a C 2 bounded function. If utility is exponential, the
value function associated to the primal problem,

u(t, π, s, l) = max EP U ΠT − φ(LT ) | Ft
δ∈A

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 satisfying
σ 2 sδ ξsδ + C(t, l)

1
x0 C(t,l+X)

 0 = ξλ − µ + − ξλ exp − EP e

 x0 x0
∂C µx0 σ2 1
 ∂t
 (t, l) = ξ + (µ − ξ − ξλ)sδ − 2x0 σ 2 (sδ )2


C(T, l) = φ(l)

where δ denotes the optimal control.

D´monstration. Theorem 19 in Qu´ma et al. (2007).
e e

25


Theorem 2. Further, given K > 0 and a distribution for X such that
4
EP exp KX <∞
x0
if we consider the set A of admissible controls δ satisfying inequality
T 2
2 2
EP 0
δt St− dt < ∞, the previous results holds for φ(x) = Kx and
C(t, l) = Kl − (T − t)C, where C is a constant solution

 0 = ξλ − µ + σ2 ξsδ 1
x0 KX
sδ − ξλ exp − EP e


x0 x0
µx0 σ2 1 2
 C = + (µ − − ξλ)sδ − σ (sδ )2 .


ξ ξ 2x0

D´monstration. Theorem 19 in Qu´ma et al. (2007).
e e

26


Agenda
e
• The framework
Numerical issues

27


Practical and numerical issues
The main difficulty of Theorem 1 is to derive C(t, l) and δ (t, l) characterized by
integro-differential system
σ 2 sδ

ξsδ + C(t, l) 1
x0 C(t,l+x)
0 = ξλ − µ + − ξλ exp − e f (x)dx


x0 x0


 R+
∂C(t, l) µx0 σ2 1
 = ξ + (µ − ξ − ξλ)sδ − 2x0 σ 2 (sδ )2


 ∂t
C(T, l) = φ(l)


If payoff φ has a threshold (i.e. there exists B ≥ 0 such that φ is constant on
interval [B, +∞)) ; it is possible to use a finite difference scheme. Hence, given two
n n
discretization parameters Nt and Ml , Ci C(tn , li ) and Di sδ (tn , li ) where

tn = n∆t where Nt ∆t = T and n ∈ [0, Nt ] ∩ N
li = 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 the
trapezoide method. Note that can restrict integration on the interval [0, B], since
1
n C(tn ,li +x)
I(tn , li ) = Ii = e x0
f (x)dx
R+
B−li 1
∞ 1
C(tn ,li +x) C(tn ,li +x)
= e x0
f (x)dx + e x0
f (x)dx
0 B−li
Nl −i−1
1 lk+i lk+i+1
exp f (lk+i ) + exp f (lk+i+1 )
2 x0 x0
k=0
n
CNl
+ exp ¯
F lNl −i
x0

29


Practical and numerical issues with threshold payoﬀ
n
For the control, the goal is to calculate Di sδ (tn , li ), where sδ (tn , li ) is the
solution of
σ 2 sδ (tn , li ) ξsδ (rn , li ) + C(tn , li )
0 = ξλ − µ + − ξλ exp − I(tn , li ).
x0 x0
n
Thus, Di is the solution (obtained using Newton’s method) of G(x) = 0 where
σ2 n
ξx + Ci n
G(x) = ξλ − µ + x − ξλ exp − Ii
x0 x0

30


Practical and numerical issues, with affine payoff
In the case of an affine payoff, we have seen already that the system is
degenerated, and that its solution is simply C(t, l) = K + Kl − (T − t)C, where
C is a constant solution of

µx0 σ2 1 2
 C = + (µ − − ξλ)sδ − σ (sδ )2

ξ ξ 2x0
2 1
 0 = ξλ − µ + σ sδ − ξλ exp − 1 ξsδ EP e x0 KX .

x0 x0

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

31


4

Loi Exponentielle(1)

3.8

Loi Pareto(1,2)

3.6

3.4

3.2

3

2.8

2.6

2.4
0 2 4 6 8 10 12 14 16 18

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


3.5

3

2.5

2

1.5

1

Loi Exponentielle(1)
0.5

Loi Pareto(1,2)

0
0 2 4 6 8 10 12 14 16 18

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 Qu´ma et al. (2007),
e
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 asset
to 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 of
Derivative 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 pricing
insurance derivatives. in Securitization of Insurance Risk : The 1995 Bowles
Symposium, 27-35.

Øksendal, B. and Sulem, A. (2005). Applied Stochastic Control of Jump
Diffusions. Springer Verlag.
´
Quema, E., Ternat, J., Charpentier, A. and Elie, R. (2007). Indifference
´
prices of catastrophe options. submitted.

Schoutens, W. (2003). L´vy Processes in Finance, pricing financial derivatives.
e
Wiley Interscience.

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

35

Slides brussels

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (6)

Similar to Slides brussels

Similar to Slides brussels (20)

More from Arthur Charpentier

More from Arthur Charpentier (20)

Slides brussels