Pricing catastrophe options in incomplete market
Arthur Charpentier††

CREM-Université Rennes 1, 7 Place Hoche, F-35000 Rennes, France
Email: arthur.charpentier@univ-rennes1.fr

Abstract
In complete markets, pricing financial products is easy (at least from a theoretical point ofview). In incomplete markets (e.g. when the underlying process has jumps with random size,
such has an insurance loss process), the price is no longer unique. So on the one hand, it
becomes difficult to provide a tractable price of insurance-linked derivatives. On the other
hand, when facing catastrophic losses, using the pure premium as a price might not be relevant(e.g. for solvency issues). Both financial market and (re)insurance industry have proposed
techniques to price identical hedging products that can be related (e.g. Esscher transform andmore generally distorted risk measures in insurance, Gerber-Shiu transform in finance). In
this paper, we focus on indifference utility techniques, assuming that stock prices have jumps,
related to major catastrophic losses, and thus, partial hedging should then be possible.

1. Introduction and motivations
1.1. Notations and definition
The buyer of an insurance contract buys the right to get reimbursed - by the insurance company all the losses which occurred during a given period of time, (for which the loss amount exceeded a
deductible, if any). The buyer of a call options buys the right to buy the underlying stock from theseller to capture its increased value above the strike price.
Both (financial) options and insurance policies have the objective to transfer a risk from onepart to another, against a specific payment (called premium in insurance). But classical techniques
in insurance (based on the use of the pure premium, EP ((X − d)+ )) and finance (based on theassumption of complete market and no-arbitrage, so that the price of a call option is EQ ((X−K)+ ))are no longer relevant.
On the one hand, most of the techniques designed to price insurance contracts have been developed for standard risks, not to hedge against catastrophes. Pricing reinsurance, where events arerare and with high severity, is more challenging, and the use is the pure premium might not be relevant, for solvency issues. On the other hand, the closed-form model for pricing financial options
1

A. Charpentier2

obtained in the beginning of the 70’s, assumed that a volatility of the underlying stock was available, known and constant, and that the underlying price was continuous. Those two assumption
assumptions were related to the idea of complete markets.
The challenge in insurance-linked derivatives is find a price for those financial products, and to
relate them to classical insurance covers, since question asked by any risk manager is “which risk
transfer technique is the cheapest one ?”. But, as mentioned in Finn and Lane (1995), one has tokeep in mind, “there are no right price of insurance, there is simply the transacted market pricewhich is high enough to bring forth sellers and low enough to induce buyers”. From a terminology
point of view, Holtan (2007) suggested to distinguish the price of an option or the premium of an
insurance contract, and the so-called value of those products. The difference depending mostly onmarket conditions.

1.2. Trading insurance risks
Insurance risks are traded as long as there are insurance contracts, buyers and sellers, but they aretraded within the (re)insurance market only. To compare with the financial market, derivatives aretraded on structured market, as well as the underlying stock, which will make replication possible(and therefore hedging and pricing derivatives). In the case of insurance risks, we can image
that some standard contracts could be - somehow - concluded with financial companies, but theunderlying risk (cumulated insurance claims for indemnity covers, or weather related index) is nottraded on financial market: in that case, there is few chance that insurance risks could be replicated,
and therefore classical techniques to price are no longer valid.
Assuming that financial markets integrate information about catastrophes (are more generally
any insurance related information), it might be possible to hedge insurance risks on financial markets. But most of the assumptions underlying the Black & Scholes assumptions are usually not
fulfilled with insurance-linked derivatives• the market is not complete, and catastrophe (or mortality risk) cannot be replicated,• 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• equities prices are not driven by a geometric Brownian motion process
The goal of this paper is to focus on catastrophe options and to find a price for those financial
products.

1.3. Outline of the paper
In Section 2, classical results on financial pricing will be recalled, focusing on assumptions underlying the fundamental theorem of asset pricing. Then, in Section 3, classical insurance pricing

Pricing catastrophe options in incomplete market

3

methods will be presented (based either on expected utility principles or using distorted risk measures). In Section 4, classical financial method to avoid drawbacks of uncompleteness will bepresented (and related to insurance pricing), to find a possible martingale measure.
And finally, in Section 5, we will study a model based on indifference utility pricing. The underlying idea is that financial markets can be affected by shocks related to major insurance losses.
As mentioned in Shimpi (1995) with a qualitative point of view, “from an insurance industry perspective, the closer the index is to the loss experience, the better the ability to hedge the loss
exposure of insurers”. Even, if a stock price is not perfectly correlated with insurance losses, atleast its discontinuous part can be. The goal here will be to see if those jumps in financial prices
can be used to hedge again catastrophes.

2. Pricing financial products in complete markets

Harrison and Pliska (1981) said that a market is complete if there is only one equivalent martingale
measure to the underlying stock price. Insurance markets would be complete if the would bea unique price for each risk, and if each contract could perfectly be hedged in the market. As
mentioned in Embrechts and Meister (1997) market uncompleteness can be explained by jumps
in the underlying stochastic process, with random size, by stochastic volatility, or by the existenceof transaction costs (or more generally any friction). Hence, in complete markets, all relevantmarket information is supposed to be known and integrated in the price: no investor will expect
a higher return than the risk free rate of return. The technique is to tune the historical probability
P into an equivalent probability measure Q, so that the price process of the underlying financial
asset becomes a martingale under probability Q, i.e. EQ (St+h |Ft ) = St . Hence, it becomes
impossible to use history of the stock to earn money: all possible relevant information is alreadyincluded in the its spot price. This link between no-arbitrage assumption and martingale processes
is the fundamental theorem of asset pricing (see Delbaen and Schachermayer (1994)): the priceof a contingent claim X (e.g. the payoff of the European call with strike K and maturity T is
X = (ST − K)+ ) is π(X) = EQ (e−rT X), assuming constant risk free rate r, where Q stands forthe risk neutral probability measure equivalent to P.
The Black & Scholes model assumes that the price of a risky asset (St )t≥0 satisfies St =S0 exp (Xt ) where (Xt )t≥0 is a Brownian process, i.e. St = S0 exp (µ + σXt0 ) where (Xt0 )t≥0
is a standard Brownian motion. Having a geometric Brownian motion reflecting uncertainty on
financial markets (for stock prices (St )) yield simple and nice pricing formulas. The most difficult
practical issue is that the only unknown valuation parameter is the stock volatility σ, making option
dealers simply “volatility dealers”: the value of the a financial option depends on the volatility of
the underlying financial stock, and not its expected return (which has to be equal to the risk freerate of return), leading to a “risk neutral” pricing.

A. Charpentier

4
3. Pricing insurance products

The basic principle of insurance is the law of large numbers: if the premium asked is EP (X), thenthe insurance company makes a null profit, on average. Feller (1945) called EP (X) the fair price(of a game, in his terminology). In the terms of d’Alembert, the pure premium is the “inner productof probabilities and losses”. Thus, EP (X) is called pure premium, but using it as the price of arisk, the company is very likely to lose money (since the balance is only on average. Therefore,
traditional premium calculation principles areπ(X) = EP (X): equivalence principle (pure premium) π(X) = E (X) + λE (X): expected value principle
P
P
π(X)
=
E
(X)
+
λV

P

parP (X): variance principle
π(X) = EP (X) + λ V arP (X): standard-deviation principle

Remark 3.1 For the standard-deviation principle, if X has a Gaussian distribution, then π(X) is
simply a quantile of X.

3.1. Pricing using expected utility principles
The fact that the pure premium might not be appropriate has been mention starting from SaintPetersburg’s paradox. One of the answer was to introduce a moral utility of X. A utility function
U is an increasing twice differentiable function on R, strictly increasing (U ′ (·) > 0, i.e. “more isbetter”) and concave (U ′′ (·) < 0, i.e. “marginal utility decreases”). Concavity is related to risk
aversion; since we assume that the agent is willing to transfer a risk, it is relevant to assume that U
is concave.
Example 3.1 Three types of expected utility are frequently used in the context of expected utility,⋆
logarithmic utility ∀x ∈ R+ , UL (x) = log(x):xp⋆∀x ∈ R+ , UP (x) = p where p ∈] − ∞, 0[∪]0, 1[: power utility∀x ∈ R, UE (x) = − exp(− xx0 ): exponential utility.

Functions UP and UL belong to the set of functions have constant relative risk aversion (CRRA).
Functions UE belong to the set of functions have constant absolute risk aversion (CARA).

Given utility function U , the premium π that an agent is willing to pay to transfer loss X is asolution of the following equation
U (ω − π) = EP (U (ω − X))(1)where ω denotes initial wealth of the insured. Using Jensen’s inequality (since U is assumed to beconcave), note that π ≥ EP (X).
Example 3.2 Assuming exponential utility, i.e. U (x) = −e−x/x0 (with constant risk aversion
1/x0 ), then π = x0 log EP (eX/x0 ) (also called entropy measure).

Pricing catastrophe options in incomplete market

5

Borch (1962) observed that the price of reinsurance contracts obtained with HARA utility
functions (more general than CARA and CRRA) is quite similar to the financial formulas: “Thisindicates that the theory of insurance premiums and the theory of asset prices are special cases ofa more general theory”. This emphazises the idea that it should be possible to relate insurance andfinance valuation techniques.

3.2. Pricing using distorted risk measures
Using the duality principle (see Yaari (1987)), instead of distorting losses using a utility function,an alternative is to use a distortion of probabilities (leading to the dual approach, since the expectedvalue can be seen as an inner product, as mentioned already by d’Alembert). Hence, the agent solve
the dual version of Equation (1), i.e. (with an abuse of notation to highlight duality, see Remark
3.2)Zω − π = Eg◦P (ω − X) =or equivalently, π =Z

xg ◦ P(dx) =Z(ω − x)g ◦ P(dx),(2)

g(P(X > x))dx in the case X is a positive randomvariable, where g is a distortion measure, i.e. an increasing function on [0, 1], with g(0) = 0 andg(1) = 1.
Remark 3.2 Note that this probability distortion does not necessarily define a probability measure, but only a capacity: if Q = g ◦ P, Q(∅) = 0 (since g(0) = 0), Q(Ω) = 1 (since g(1) = 1),
and Q(A) ≤ Q(B) if A ⊂ B (since g is an increasing function). Hence, in Equation (2) Eg◦P is
not an expected value, but a Choquet integral with respect to (nonadditive) measure g ◦ P.
Example 3.3 If g(x) = 1(x > α), then π = F −1 (1 − α), α ∈ (0, 1) and F (x) = P(X ≤ x).
As a particular case of distorted probabilities, an important principle is the use of the Esschertransform,
EP (X · eαX )π = EQ (X) =,EP (eαX )for some α > 0. More generally, Delbaen and Haezendonck (1989) considered the followingchange of measure, so that cumulative distribution function of the Radon-Nikodym derivative
dP/dQ isZ x1
exp(β(y))dF (y), x ≥ 0,
G(x) =EP (eβ(X) ) 0
where F is the distribution function of X under P, and β(·) : [0, ∞) → (−∞, ∞) satisfies
EP (eβ(X) ) < ∞ and EP (Xeβ(X) ) < ∞.
Example 3.4 If β(x) = log (1 + b (x − EP (X)))), then π = EQ (X) = EP (X)+bV arP (X), which
EP (X · eαX).
is the variance principle. If β(x) = αx−log EP (eαX ), for some α > 0, then π = EQ (X) =EP (eαX )A. Charpentier

6
4. Pricing financial products in incomplete markets
4.1. A natural framework based on Lévy processes

As mentioned in Section 2, market uncompleteness arises when the underlying stochastic process
has jumps with random size. Hence, in a general framework, assume that the price of a risky
asset (St )t≥0 satisfies St = S0 exp (Xt ) where (Xt )t≥0 is a Lévy process. Recall that (Xt )t≥0
has independent, infinitely divisible and stationary increments, thus Xt+h − Xt has characteristic
function φh . The cumulant characteristic function satisfies the Lévy-Khintchine formula, i.e.
1ψ(u) = log φ(u) = iγu − σ 2 u2 +
2

Z +∞
−∞


eiux − 1 − iux1{|x|<1} ν(dx),where γ ∈ R, σ 2 ≥ 0 and ν is the so-called Lévy measure on R/{0}. Hence, the Lévy process
is characterized either by φ (the characteristic function of X1 ), or by the triplet (γ, σ 2 , ν) in theLévy-Khintchine formula.

Remark 4.1 Again, except the case when (Xt )t≥0 is a (pure) Poisson process or a Brownian motion, any Lévy model is an incomplete model.

Market completeness is related to the existence of a unique martingale measure, also called thepredictable representation property of a martingale: a martingale (Mt )t≥0 satisfies this property
if and only if for any square-integrable random
R T variable Z ∈ FT , there exists a Ft -predictableprocess (at )t∈[0,T ] such that Z = E(Z) + 0 as dMs . Actually, (at )t∈[0,T ] is related to the selfbalancing strategy. Nualart and Schoutens (2000) proved that under some weak assumptions, aLévy process (Xt )t≥0 can also have a predictable representation property of the form
Z = E(Z) +

∞ Z T
X
i=1

0

(i)
(i)
a(i)
s d(Hs − E(Hs )),

P
(i)
(i)
where the (at )t∈[0,T ] ’s are Ft -predictable processes, and Ht = 0<s≤t [Xs − Xs− ]i , where times
s are times where the Lévy process jumps. As mentioned in Schoutens (2003), the predicable(i)
integrands (at )t∈[0,T ] ’s appearing in this representation can be interpreted in terms of minimal
variance strategies. Hence, those processes correspond to the risk that cannot be hedged away.
(1)
The term (at )t≥0 leads the strategy that realizes the closest hedge to the claim.
A first idea, related to the classical pricing process in complete market is to find an equivalent martingale measure, and to use it to derive a price. Hence, in Section 4.2 we will mention
two ideas widely used to obtain one equivalent martingale measure Q: one based on Gerber and
Shiu (1994) (i.e. Esscher transform from insurance pricing) and the other one based on somemean-correcting martingale measure. The main problem in incomplete market is that there is no
replication portfolio. But it is still possible to super-replicate.

Pricing catastrophe options in incomplete market

7

4.2. Finding one risk neutral measure
4.2.1. U SING THE E SSCHER TRANSFORM
Following Gerber and Shiu (1994) we can - by using the Esscher transform - find in some cases
at least one equivalent martingale measure Q. More generally, Bühlmann et al. (1998) discussed
the Esscher transform for specific classes of semi-martingales, with applications in finance andinsurance.
Given a Lévy process (Xt )t≥0 under P with characteristic function φ or triplet (γ, σ 2 , ν), thenunder Esscher transform probability measure Qα (as defined in Section 3.2), (Xt )t≥0 is still a Lévy
process with characteristic function φα such thatlog φα (u) = log φ(u − iα) − log φ(−iα),and triplet (γα , σα2 , να ) for X1 , where σα2 = σ 2 , and2γα = γ + σ α +Z +1(eαx − 1)ν(dx) and να (dx) = eαx ν(dx),−1see e.g. Schoutens (2003).
Example 4.1 A particular case is given when (Xt )t≥0 is a Brownian motion under P, then if
α = (r − µ)/σ 2 , (Xt )t≥0 is still a Brownian motion under Qα .
Proposition 4.2 If the price of a risky asset (St )t≥0 satisfies St = S0 exp (Xt ), where (Xt )t≥0 is aLévy process, such that random variable X1 is non-degenerate and possesses a moment generating
function M (t) = EP (etX ) on some interval (a, b), and if there exists u ∈ (a, b − 1) such that
M (1 + u) = M (u), then (e−rt St )t≥0 is a Qu -martingale.
Proof. Shiryaev (1999)
In order to have unicity, additional assumptions are necessary (see Kallsen and Shiryaev (2002)).

4.2.2. A MEAN - CORRECTING MARTINGALE MEASURE
Another way to obtain an equivalent martingale measure is inspired from the Black & Scholesmodel, and is related to some mean-correcting martingale measure. The underlying idea is to notethat given a Lévy process (Xt )t≥0 under P with characteristic function φ and triplet (γ, σ 2 , ν), thenthe shifted process (Yt )t≥0 = (Xt − mt)t≥0 is also a Lévy process with characteristic function
2
φm (u) = eium φ(u) and triplet (γm , σm
, νm ) = (γ + m, σ 2 , ν) for X1 , see e.g. Schoutens (2003).
In the Black and Scholes model, we just switch from mean µ − σ 2 /2 to r − σ 2 /2. In the Lévy
model, the idea is to use the same kind of transform, mnew = mold + r − log φ(−i) (in the Blackand Scholes model, log φ(−i) = α). The choice of mnew will be such that the discounted price is amartingale.

A. Charpentier

8
5. IThe indifference utility approach

As point out in Swiss Re (1999), about the pricing of financial stop loss contracts “the risk-neutral
valuation technique traditionally used for the pricing of financial derivatives cannot be applied
directly”. Nevertheless, practitioners need a price for insurance-linked derivatives.
PNt
Let (St )t≥0 denote the accumulated insurance claim process, St =i=1 Xi . The classicalstop-loss contract (ST − K)+ . The payoff of a call option is also (ST − K)+ . Hence, those twocovers are identical for an insurance company, willing to transfer risk claims exceeding priority K.
The idea of the pricing model here is to assume that the price of the financial asset has jumps
related to the occurrence of catastrophes. This assumption can be validated by stylized facts, e.g.
stock price of reinsurance companies and WTC 9/11 in 2001 (see Figure 5, with Munich Re and
SCOR - European markets since Wall street has been closed after the catastrophe), oil price andKatrina in August 2005... etc.

350

50

300

45
40

250

35
30

Munich Re stock price55

60

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

2001

2002

Figure 1: Catastrophe event and stock prices.
The following model and results are based on Quéma et al. (2007).

5.1. Description of the model
The occurrence process is a (Ft )-adapted process denoted (Nt )t>0 . Under P, assume that (Nt )t>0
is an homogeneous Poisson process, with parameter λ, i.e. with stationary and independent increments. Further, recall that EP (NT ) = λT and V ar(NT ) = λT .
At time t that number of catastrophe that had already occurred is Nt . Define the sequence ofPricing catastrophe options in incomplete market

9stopping times (Tn )n>0 corresponding the dates of occurrence of catastrophes, i.e.
T0 = 0 and Tn+1 = inf {t | t > Tn , Nt 6= NTn } .
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
P t(Xi )i>0 are supposed to be integrable, independent and identically distributed. Define Lt = N
i=1 Xi as the loss process, corresponding to the total amount of catastrophes occurred
up to time t.
Assume that financial market satisfies the no-arbitrage assumption, and consists in a free risk
asset, and a risky asset, with price (St )t>0 . Without loss of generality, the value of the risk freeasset is assumed to be constant (hence it is chosen as a numeraire). The price of the risky asset isdriven by the following diffusion process,dSt = St− (µdt + σdWt + ξdMt ) with S0 = 1
where (Wt )t>0 is a Brownian motion under P, independent of the catastrophe occurrence process(Nt )t>0 . Parameters µ and σ 2 are respectively the trend and the volatility of the risky asset, pertime unit. Parameter ξ corresponds to the relative variation of the asset value when it jumps.
Note that the stochastic differential equation has the following explicit solution



σ2
St = exp µ −− λξ t + σWt (1 + ξ)Nt .
2
5.2. Indifference utility
As in Davis (1997) or Schweizer (1997), assume that an investor has a utility function U , andinitial endowment ω. The investor is trading both the risky asset andR t the risk free asset, forming a
dynamic portfolio δ = (δt )t>0 whose value at time t is 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 P ∀t ∈ [0, T ], (δ · S)t > −M = 1,
hR
i
T 2 2
and further if EP 0 δt St− dt < +∞.
If X is a random payoff, the classical Expected Utility based premium is obtain by solvingu(ω, X) = U (ω − π) = EP (U (ω − X)).
Consider an investor selling an option with payoff X at time T ,hi• either he keeps the option, uδ⋆ (ω, 0) = supδ∈A EP U (ω + (δ · S)T ) ,hi• either he sells the option, 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 areequal, i.e.
π(ω, X) = inf {π ∈ R such that uδ⋆ (ω + π, X) − uδ⋆ (ω, 0) > 0} .

10

A. CharpentierThis 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.
Based on optimal control results, Quéma et al. (2007) derived some analytical expression, thatcan be related to Merton (1976), in the case of exponential utility.

5.3. Following Merton’s work
Assume that the asset has no jump, i.e. dSt = St− (µdt + σdWt ) (i.e. ξ = 0), and that wewishh to price a derivativewith payoff φ(ST ), then in the case of exponential utility uE (t, π) =i
µ2 x 0UE π + 2σ2 (T − t) .
In the case where the asset has jump, i.e. dSt = St− (µdt + σdWt + ξdMt ) (i.e. ξ 6= 0), andthat we wish to price a derivative with payoff φ(ST ), then uE (t, π) = UE (π + (T − t)C) whereC(t) satisfies(
2
C(t) = αxξ 0 + (α − σξ − ξλ)D − 2x10 σ 2 D2hi,2
0 = ξλ − α + σx0 D − ξλ exp − ξDx0with also a border condition, C(T ) = φ(ST ), and where D is related to the optimal strategy, andis obtained also from the previous system.
Here, we wish to price a derivative with payoff φ(LT ), when the underlying asset has jumps.
Then, assuming that the investor has an exponential utility, U (x) = − exp(−x/x0 ),Theorem 5.1 Let φ denote a C 2 bounded function. If utility is exponential, the value function
associated to the primal problem,h iu(t, π, s, l) = max EP U ΠT − φ(LT ) | Ftδ∈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h ξsδ ⋆ + C(t, l) i  1
σ 2 sδ ⋆
C(t,l+X)

x0

E
e
−
ξλ
exp
−
0
=
ξλ
−
µ
+
P


x0
x0
2
µx0
σ
1 2 ⋆ 2
∂C(t, l) =+ (µ −− ξλ)sδ ⋆ −σ (sδ )



∂t
ξ
ξ
2x0

 C(T, l) = φ(l)where δ ⋆ denotes the optimal control.

Proof. Theorem 19 in Quéma et al. (2007).

5.4. Numerical issues and properties of optimal portfolios
From theorem 5.1 we have to find (C, δ), i.e. 2 functions solutions of an integro-differential equation. Hopefully, using a simple discretization on a finite grid, it is possible to obtain a (stable)
numerical approximation of C, and therefore of function C, and thus of the price of the derivative.
Note further that ywo nice results have been derived in Quéma et al. (2007),

Pricing catastrophe options in incomplete market

11

Lemma 5.2 C(t, ·) is increasing if and only if φ is increasing.
Lemma 5.3 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 strictly positive constant.
For a numerical example, assume that the trend is null (µ = 0), i.e. amounts hold are uniquely
explained by the hedging strategy. Prices are decreasing in x0 , and therefore, increasing withrisk aversion (the higher x0 , the lower risk aversion). When x0 → 0, risk aversion is infinite,
and thus, whatever appends, the agent wants to hedge against any losses: the price tends to thesuper-replication price i.e. kφk∞ , since if he holds underlying, he might loose money.
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

Figure 2: 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

Acknowledgments: the author thanks Émilie Quéma, Jérôme Ternat and Romuald Élie fordiscussions and comments.

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