evaluatinghybridproducts: theinterplaybetween ...references insuredloss(usd billions) year event...

References

Evaluating hybrid products: the interplay betweenfinancial and insurance markets

Francesca Biagini

Mathematisches InstitutLudwig–Maximilians–Universität München

March 31, 2011

Francesca Biagini Spring School Stochastic Models in Finance and Insurance 1/74

References

Introduction• A current issue in the theory and practice of insurance and reinsurancemarkets is to find alternative ways of securitizing risks.

• Insurance companies have tried to take advantage of the vastpotential of capital markets by introducing exchange-tradedinsurance-linked instruments such as mortality derivatives andcatastrophe insurance derivatives.

• At the same time, insurance products such as unit-linked life insurancecontracts, where the insurance benefits depend on the price of somespecific traded stocks, offer a combination of traditional life insuranceand financial investment.

• Furthermore, new kinds of insurance instruments, which insure againstrisks connected to macro-economic factors such as unemployment, arerecently offered on the market.


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• Insurance markets and financial markets may no longer be viewed assome disjoint objects:

I Insurance companies have the possibility to invest in financial marketsand therefore hedge against their risks with financial instruments,

I they can sell parts of their insurance risk by putting insurance linkedproducts on the financial markets.

• Hence, we can consider insurance and financial markets as onearbitrage-free market.


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• Here we will provide an introduction to how sophisticatedmathematical methods for pricing and hedging financial claims can becan be applied

I to the valuation of the hybrid productsI to premium determination, risk mitigation and claim reserve

management.

• Main issues: pricing (benchmark approach) and hedging (local riskminimization).


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Pricing of insurance claims

• The actuarial way of pricing usually considers the classical premiumcalculation principles that consist of net premium and safety loading.

• If C describes a random claim, which the insurance company has topay (eventually) in the future, then a premium P(C ) to be charged forthe claim is defined by

P(C ) = E[C ]︸︷︷︸net premium

+ A(C )︸︷︷︸safety loading

(1)

• Note that the net premium is the expected value of C with respect tothe real-world (or objective) probability measure.


References

• Possible safety loadings:I A(C ) = 0 (net premium principle),I A(C ) = a · E[C ] (expected value principle, where a ≥ 0),I A(C ) = a · Var(C ) (variance principle, where a > 0),I A(C ) = a ·

√Var(C ) (standard deviation principle, where a > 0), see

e.g. [17].

• The existence of a safety loading is justified by ruin arguments and therisk-averseness of the insurance company.


References

• Since riskless profits shall be excluded also competitive and liquidreinsurance market, in which insurance companies can “trade” theirrisks among each other, no-arbitrage theory can be also used tocalculate insurance premiums.

• There have been several attempts to connect actuarial premiumcalculation principles with the financial no-arbitrage theory, see [8] and[19].

• Both papers actually show that under some assumptions there existequivalent martingale measures, which explain premiums of the form(1), so that these principles provide arbitrage-free prices, too.


References

• Standard no-arbitrage theory requires to choose a numeraire and anequivalent (local) martingale measure.

• In general, insurance claims are not replicable by other financialinstruments, which implies that the hybrid market of financial andinsurance products is incomplete.

• Consequently there exist infinitely many equivalent martingalemeasures and one has to choose one of them by using a suitablecriterion.

• To avoid this problem, we choose the so called benchmark approachfor our pricing issue.


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The Benchmark approach

• The benchmark approach provides a pricing rule under the real-worldprobability measure P, even if no equivalent (local) martingalemeasure exists.

• For actuarial applications, other advantages in using this pricingmethod is

I to keep a close connection to the classical premium calculationprinciples, which also use the real-world probability measure,

I to benefit from the statistical advantages of working directly under thereal-world probability measure.


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The setting

• Let (Ω,F,P) be a complete probability space endowed with a filtrationF := (Ft)t≥0 satisfying the usual hypotheses.

• Market: d + 1 nonnegative, adapted tradable (primary) securityaccount processes, given by the càdlàg semimartingalesS j = (S j

t )0≤t<∞, j ∈ 0, 1, ..., d, d ≥ 1. Here S0t is the strictly

positive savings account at time t, t ≥ 0.• We interpret the securities S j to describe both stocks and insuranceclaims.


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• Let L(S) denote the space of Rd+1-valued, predictable strategies

δ = (δt)0≤t<∞ = (δ0t , δ1t , ..., δ

dt )tr

0≤t<∞ ,

for which the corresponding gain from trading in the assets, i.e.t∫0δs · dSs , exists for all t ∈ [0,∞).

• δjt represents the units of asset j held at time t by a market participant.• Portfolio value Sδt at time t ≥ 0:

Sδt = δt · St =d∑

j=0

δjtSjt .


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• A strategy δ ∈ L(S) is called self-financing if changes in the portfoliovalue are only due to changes in the assets and not due to in- oroutflow of money, i.e. if

Sδt = Sδ0 +

t∫0

δs · dSs , t ≥ 0 ,

or equivalently

dSδt = δt · dSt .

• Denote by V+x (Vx) the set of all strictly positive (nonnegative), finite

and self financing portfolios Sδ with initial capital Sδ0 = x .


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Numeraire Portfolio

• DefinitionA portfolio Sδ∗ ∈ V+

1 is called P-numeraire portfolio if every nonnegativeportfolio Sδ ∈ Vx , discounted (or benchmarked) with Sδ∗ , forms a(F,P)-supermartingale for every x ≥ 0. In particular, we have

E[SδσSδ∗σ

∣∣∣Fτ] ≤ SδτSδ∗τ

a.s. (2)

for all stopping times 0 ≤ τ ≤ σ <∞.• We choose the P-numéraire portfolio as benchmark.• Benchmarked value of a portfolio Sδ:

Sδt :=SδtSδ∗t

, t ≥ 0.


References

• If a P-numeraire portfolio exists, it is unique as can be easily seen withthe help of the supermartingale property and Jensen’s inequality, see[1].

• In most market models and applications, the P-numeraire portfolio isequal to the “growth optimal portfolio” (in short: GOP), which isdefined as the portfolio with the maximal growth-rate in the marketand which satisfies several other optimality criteria, see [1], [13] or[14].

AssumptionThe P-numeraire portfolio Sδ∗ ∈ V+

1 exists in our market.


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No-arbitrage

DefinitionA benchmarked nonnegative self-financing portfolio Sδ is a strong arbitrageif Sδ0 = 0, and at a later time t > 0 and P(Sδt > 0) > 0.

• With the existence of the P-numeraire portfolio and the correspondingsupermartingale property (2), arbitrage opportunities, as defined in theabove Definition, are excluded, see [13].

• There could still exist some weaker forms of arbitrage, which wouldrequire to allow for negative portfolios of total wealth, however.

• Principle of limited liability: negative portfolios of total wealth areexcluded in a realistic market model.


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Fair price

• Let us now consider a time horizon T ∈ (0,∞) and two portfoliosSδ ∈ Vx and Sδ

′ ∈ Vy with SδT = Sδ′

T P a.s.

• Let the benchmarked portfolio process Sδt , t ∈ [0,T ], be a martingaleand the benchmarked portfolio process Sδ

′t , t ∈ [0,T ], be a

supermartingale. Then

Sδt = E[SδT∣∣∣Ft

]= E

[Sδ′

T

∣∣∣Ft

]≤ Sδ

′t , ∀t ∈ [0,T ] , (3)

and in particular

x = Sδ0 ≤ Sδ′

0 = y .


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• Hence, a rational (risk-averse) investor would always invest in abenchmarked martingale portfolio (if it exists) and we can give thefollowing definition of “fair” wealth processes, see [13].

• DefinitionA portfolio process Sδ = (Sδt )t≥0 is called fair if its benchmarked valueprocess

Sδt =SδtSδ∗t

, t ≥ 0 ,

forms a (F,P)-martingale.


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Benchmarked contingent claims

DefinitionGiven some maturity T ∈ (0,∞), a T-contingent claim C is a

FT -measurable random variable with E[|C |Sδ∗

T

]<∞. We denote by

C :=CSδ∗T

(4)

the benchmarked payoff of the T -contingent claim C .


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Real-world pricing formula

DefinitionFor a T -contingent claim C the fair price Pt(C ) of C at time t ∈ [0,T ] isgiven by the real-world pricing formula

Pt(C ) := Sδ∗t E

[CSδ∗T

∣∣∣Ft

]= Sδ∗t E

[C∣∣Ft

]. (5)

• Hence the corresponding benchmarked fair price process(Pt)t∈[0,T ] =

(PtSδ∗

t

)t∈[0,T ]

forms a (F,P)-martingale.


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• The real world pricing formula provides the least expensive way toreplicate a benchmarked payoff HT that admits a replicatingself-financing portfolio δH with Sδ

H= HT .

• The benchmark approach allows other self-financing hedge portfoliosto exist for HT . However, these nonnegative portfolios are notP-martingales and, as supermartingales, therefore more expensive thanthe P-martingale

SδHt = E[HT

∣∣∣Ft

].


References

• Hybrid markets are incomplete. Therefore, not every T -contingentclaim, introduced to the market, can be replicated by a self-financingportfolio.

• For non-negative, replicable T -contingent claims, the real-worldpricing formula defines the claim’s minimal price for every t ≤ T .

• For any non-replicable claim, the real-world pricing method isconsistent with utility indifference pricing in a very general setting, see[14].


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Actuarial pricing formula

• If a T -contingent claim C and the value Sδ∗T at time T of theP-numeraire portfolio are independent, we get

Pt(C ) = Sδ∗t E[ CSδ∗T|Ft

]= Sδ∗t E

[ 1Sδ∗T|Ft

]E [C |Ft ]

= P(t,T )E [C |Ft ] , (6)

where P(t,T ) is the fair price at time t ≤ T of a zero coupon bondwith nominal value one, paid at time T .


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Application I: unemployment insurance products

• The product’s basic idea is that the insurance company compensatesto some extend the financial deficiencies, to which an unemployedinsured person is exposed.

• Example: Payment Protection Insurance (in short: PPI) productsagainst unemployment, which are linked to some payment obligationof an obligor to its creditor. The claim amount is hereby defined bythe (a priori known) instalments Ci , i = 1, · · · ,N, which are paid atpredefined payment dates T i , i = 1, · · · ,N.


References

• We focus on calculating single premiums.• This is again motivated by PPI unemployment products, which areoften sold as an add-on directly by the creditor.

• The insurance company then receives a single rate from the creditor,who in turn allocates this rate to the instalments.


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• Exclusion clauses of the contract:- The waiting period [0,W ] starts with the beginning of the contract. Ifan insured person becomes unemployed at any time of this period, he isnot entitled to receive any claim payments during the wholeunemployment time.

- The deferment period of length D starts with the first day ofunemployment. An insured person is not entitled to receive claimpayments until the end of this period.

- The requalification period of length R starts with the end of anyunemployment period that occurred during the contract’s duration. Ifan insured person becomes (again) unemployed at any time of therequalification period, he is not entitled to receive any claim paymentduring the whole time of unemployment.


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• Let Ci be the random insurance claim at the payment date Ti ,i = 1, · · · , n.

• Price Pt(Ci ) of Ci at time t ∈ [0,Ti ]:

Pt(Ci ) = Sδ∗t E[Ci

∣∣∣Gt

],

where Ci is the benchmarked value of Ci .• Overall insurance premium Pt at time t ∈ [0,T ]:

Pt =n∑

i=1

Pt(Ci ) =n∑

i=1

Sδ∗t E[Ci

∣∣∣Gt

].


References

• Random jump times, when the stochastic process X “jumps” from onestate to the other:

• τ0 := 0• τn := inft > τn-1 : Xt 6= Xt- n ≥ 1 , (7)

where Xt- := limst Xs .• Random insurance claim Ci at the payment date Ti :

Ci (ω) := ci IW<τ1≤Ti−D,τ2>Ti∪

∞⋃j=2τ2j-1−τ2j-2>R,W<τ2j-1≤Ti−D,τ2j>Ti

(ω).


References

• Denote

Axj := τj ≤ x,Bx

j := x < τj,Dxj := τj − τj-1 > x.

• Random insurance claim Ci :

Ci (ω) = ci(IBW

1 ∩ATi−D1 ∩BTi

2(ω) +

∞∑j=2

IDR

2j-1∩BW2j-1∩ATi−D

2j-1 ∩ATi2j

(ω)).


References

• Price Pt(Ci ) of Ci at time t ∈ [0,Ti ]

Pt(Ci ) = Sδ∗t ci(E[IBW

1 ∩ATi−D1 ∩BTi

2

∣∣∣Ft

]+∞∑j=2

E[IDR

2j-1∩BW2j-1∩ATi−D

2j-1 ∩BTi2j

∣∣∣Ft

])• The (overall) insurance premium Pt at time t ∈ [Tk-1,Tk), k ≥ 1,1

Pt =N∑

i=k

Sδ∗t ci

(E[IBW

1 ATi−D1 BTi

2

∣∣∣Ft

]+

∞∑j=2

E[IDR

2j-1BW2j-1A

Ti−D2j-1 BTi

2j

∣∣∣Ft

]).

1We set T0 := 0 here.Francesca Biagini Spring School Stochastic Models in Finance and Insurance 29/74

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Assumption

• Every (random) insurance claim Ci of the unemployment insurancecontract, paid at time Ti , is independent of the respective value Sδ∗Ti

ofthe P-numeraire portfolio at time Ti .

• There is the possibility of putting money on a bank account withconstant interest rate r > 0, i.e.

P(t,T ) = e−r(T−t) , (8)

• the employment-unemployment process X follows atime-homogeneous, strong Markov chain with respect to P and

Ft = FXt = σ(Xu, u ≤ t) .


References

• Under these hypotheses the sojourn times τj − τj−1, j ≥ 1 givenX0 = i0, i0 ∈ 0, 1, are conditionally independent and exponentiallydistributed, with parameters given by the intensity or generator matrix

Λ =

(λ0 −λ0−λ1 λ1

)(9)

of X . In particular, we have

P (τ1 − τ0 > t1, ..., τn − τn−1 > tn|X0 = i0) = e−λi0 t1 · ... · e−λin−1+1tn ,(10)

where i0, i1, ..., in−1 ∈ 0, 1 with ik = 1− ik−1, t1, ..., tn ∈ [0,∞), andλi0 , ..., λin−1 are defined by (9).


References

PropositionUnder the above Assumption, if t ∈ [Tk−1,W ), X0 = 0 and the first jumpto unemployment τ1 > t, than the insurance premium Pt is given by:

Pt =N∑

i=k

cie−r(Ti−t)( λ0

λ0 − λ1e−λ1(Ti−t)

(e−(λ0−λ1)(W−t) − e−(λ0−λ1)(Ti−D−t)

)+ λ2

0λ1e−λ1(Ti−t)

Ti−D−t∫(W−t)∧R

y∫R

y−x∫0

e−(λ0−λ1)(x+u)B0(2√λ0λ1u(y − x − u))dudxdy

),

(11)

where B0(x) is the modified first kind Bessel function of order α = 0 given by

Bα(x) =∞∑

m=0

1m!Γ(m + α + 1)

(x2

)2m+α

. (12)


References

Application II: Catastrophe Insurance Derivatives

• Over the past decades the rise in insured losses has exploded fromUSD 2.5 billions per year to an average value of the aggregatedinsurance losses of USD 30.4 billions per year (in prices of 2006).

• The following Table gives a summary of the five most expensivenatural catastrophes for the last 30 years.


References

Insured Loss (USDBillions)

Year Event Country

66.3 2005 HurricaneKatrina

U.S., Gulf of Mexico, Ba-hamas, North Atlantic

23.0 1992 HurricaneAndrew

U.S., Bahamas

21.4 2001 Terrorist at-tack

U.S.

19.0 1994 Northridgeearthquake

U.S.

13.7 2004 HurricaneIvan

U.S., Caribbean

Table: Top 5 Insured Catastrophe Losses (Source: Swiss Re, Sigma Nr. 2/2007).


References

• Introduction of exchange-traded catastrophe insurance derivatives:I Chicago Board of Trade (1992 - 1999)I Bermuda Commodities Exchange (1997, suspended)I New York Mercantile Exchange (NYMEX), since 2007.I Eurex, since 2010.

The Eurex catastrophe futures is settled against the Re-Ex loss index,created from the data supplied by an internationally recognized marketauthority on property losses in the USA (Property Claim Service,PCS).


References

• The derivatives are written on a loss index that evolves over twoperiods:

I a loss period [0,T1];I a development period [T1,T2], T1 < T2 <∞.

• The loss index provides thus at any t ∈ [0,T2] an estimation of theaccumulation of the final time-T2-amounts of catastrophe losses thathave occurred during the loss period.


References

• Let Nt , t ∈ [0,T1] denote the number of catastrophes up to time t,and Ui , i = 1, ...,Nt , the corresponding final amounts of the losses attime T2 (which are unknown at time 0 ≤ t < T2). Then the value Ltof the loss index can be expressed as

Lt =

Nt∧T1∑i=1

E [Ui |Ft ] , t ∈ [0,T2] , (13)

where the filtration Ft , t ∈ [0,T2] represents the informationavailable.


References

• We follow the approach of [5]: Loss index

Lt =

Nt∧T1∑j=1

YjAjt−τj , t ∈ [0,T2] , (14)

where(H1) Ns , s ∈ [0,T2], is a Poisson process with intensity λ > 0 and jump

times τj (number of catastrophes occurring during the loss period).(H2) Yj , j = 1, 2, . . . , are positive i.i.d. random variables with distribution

function FY , (first loss estimation at the time the j-th catastropheoccurs).


References

(H3) Ajs , s ∈ [0,T2], j = 1, 2, . . . , are positive i.i.d. martingales such that

Aj0 = 1, ∀j = 1, 2, . . . .

They represent the instantaneous unbiased restimation factors.(H4) Aj ,Yj , j = 1, 2, . . . , and N are independent.

• Re-estimation begins immediately after the occurrence of the j-thcatastrophe with initial loss estimate Yj , individually for eachcatastrophe.


References

• We here suppose that market participants observe the evolution of theindividual catastrophe losses.

• The market information filtration (Ft)0≤t≤T2 to be the rightcontinuous completion of the filtration generated by the catastropheoccurrences N, the corresponding initial loss estimates Yj , and thecorresponding re-estimation factors Aj .

• We consider a financial market endowed with a risk-free asset withdeterministic interest rate rt , and the possibility of trading catastropheinsurance derivatives, written on a loss index.


References

• Consider a derivative written on the loss index with maturity T2 andunderlying

h(LT2) > 0

for a measurable function h : R 7→ R+.• Set r ≡ 0 and assume that the numeraire portfolio and the loss indexare independent.

• Then the price process is given by

πt = E [h(LT2)|Gt ] , t ∈ [0,T2]. (15)


References

Futures pricing in the benchmark approach

• Let F := F (t,T ), t ∈ [0,T ], be the futures price of a futures contracton Y = h(LT ). The benchmarked cash-flow of the contract is due tothe streams of payments associated to the futures price variation

Σ(σ)t =

n∑i=1

1Sδ∗ti

(F (ti ,T )− F (ti−1,T ))

=n∑

i=1

1Sδ∗ti−1

(F (ti ,T )− F (ti−1,T ))

+n∑

i=1

(1

Sδ∗ti− 1

Sδ∗ti−1

)(F (ti ,T )− F (ti−1,T )) .


References

• Suppose Sδ∗càdlàg. Then if F is given by a semimartingale, then for

|σ| = supi |ti − ti−1| → 0

Σ(σ)t → Σ =

∫ T

t

1Sδ∗s−

dF (s,T ) +

∫ T

td[

1Sδ∗

,F]

s, (16)

in ucp.• Since the present value of the contract is 0, we have E [Σ|Ft ] = 0 and

F (t,T )

Sδ∗t= E

[F (T ,T )

Sδ∗T−∫ T

tF (s−,T )d

(1

Sδ∗s

)∣∣∣∣∣Ft

]


References

• If 1Sδ∗ is a martingale and F sufficiently integrable, then

F (t,T )

Sδ∗t= E

[F (T ,T )

Sδ∗T

∣∣∣∣Ft

]. (17)

• This happens iff r = 0, i.e. P(t,T ) = 1 for all t ∈, since by thereal-world pricing formula for the bond price, we have

P(t,T )

Sδ∗t= E

[1Sδ∗T

∣∣∣∣Ft

]. (18)

• Furthermore, if Y and the numeraire portfolio are independent,formula (17) reduces to

F (t,T ) = P(t,T )E [F (T ,T )|Ft ] = P(t,T )E [Y |Ft ] = E [Y |Ft ] .(19)


References

(C1) The payoff function h(·) is continuous.

(C2) h(·)− k ∈ L2(R) =g : R→ C measurable

∣∣∣ ∫∞−∞ |g(x)|2dx <∞

for some k ∈ R.(C3) h(·) ∈ L1(R), where

h(u) =12π

∫ ∞−∞

e−iuz(h(z)− k)dz , ∀u ∈ R,

is the Fourier transform of h(·)− k .

• Since (C2) and (C3) are in force, the following inversion formula holds:

h(x)− k =

∫ ∞−∞

e iux h(u)du . (20)


References

• By (20) and (C3) we obtain

πt = E[h(LT2)|Ft ] = E[h(LT2)− k |Ft ] + k

= E[∫ ∞−∞

e iuLT2 h(u)du|Ft

]+ k (21)

=

∫ ∞−∞

E[e iuLT2 |Ft

]︸︷︷︸

:=ct(u)

h(u)du + k , (22)

where in the last equation we could apply Fubini’s theorem, because(C3) holds.


References

• Essential task is to compute the conditional characteristic function ofLT2

ct(u) = E[e iuLT2 |Ft

]= E

expiu

NT1∑j=1

YjAjT2−τj

∣∣∣∣∣∣Ft

, u ∈ R,

(23)

for t ∈ [0,T2].• Define the conditional characteristic function of the re-estimationmartingale Aj as

ψAj

t (s, u) := E[e iuAj

s∣∣∣FAj

t

], 0 ≤ t ≤ s ≤ T2 , (24)

where FAjt := σ(Aj

s , 0 ≤ s ≤ t) is the filtration generated by the j-thre-estimation factor.


References

TheoremThe conditional characteristic function (23) of the loss index LT2 is given1. for t < T1 by

ct(u) = exp−λ(T1 − t)

(1− E

[ψA

0 (T2 − U, uY )])

·Nt∏j=1

ψAj

t−sj (T2 − sj , uyj)∣∣sj =τj , yj =Yj

, u ∈ R;

2. for t ∈ [T1,T2] by

ct(u) =

NT1∏j=1

ψAj

t−sj (T2 − sj , uyj)∣∣sj =τj , yj =Yj

, u ∈ R .

Here U is a uniformly distributed random variable on [t,T1], and Y is arandom variable with distribution function FY and independent of U.


References

Fourier transform for a call spread

• Consider the payoff function at maturity given by

h(x) =

0, if 0 ≤ x ≤ K1;x − K1, if K1 < x ≤ K2;K2 − K1, if x > K2.

• The integrability condition h(·)− k ∈ L2(R+) is satisfied fork := K2 − K1. In particular, h(·)− k ∈ L1(R+).

• To satisfy (C1) and (C3) we continuously extend h from R+ to R by

h(x) :=

h(−x), if x < 0;h(x), if x ≥ 0.


References

• Let

ˆh(u) :=12π

∫ ∞−∞

e−iuz(h(z)− k)dz , ∀u ∈ R,

be the Fourier transform of h − k .• Then

ˆh(u) =12π

[∫ −K1

−K2

e−iux(−x − K2)dx

+

∫ K1

−K1

e−iux(K1 − K2)dx +

∫ K2

K1

e−iux(x − K2)dx]

=1π

1u2

(cos uK2 − cos uK1) ∈ L1(R),


References

• In particular since LT2 ≥ 0 a.s., we can write

πCSt = E [h(LT2)− k |Ft ] + k = E

[h(LT2)− k

∣∣Gt]

+ k

= E[∫ ∞−∞

e iuLT2 ˆh(u)du∣∣∣∣Ft

]+ k

=

∫ ∞−∞

E[e iuLT2

∣∣∣Ft

]ˆh(u)du + k

=

∫ ∞−∞

ct(u)ˆh(u)du + k

=1π

∫ ∞−∞

ct(u)

u2(cos uK2 − cos uK1)du + K2 − K1, (25)

where ct(u) is the conditional characteristic function of the indexprocess.


References

Re-estimation factors

• We now suppose that the re-estimation factors are given by positiveaffine martingales.

DefinitionA Markov process A = (At ,Px) on [0,∞] is called an affine process if thereexist C-valued functions φ(t, u) and ψ(t, u), defined on R+ × R, such that

E[e iuAT2

∣∣∣Ft

]= eφ(T2−t,u)+ψ(T2−t,u)At . (26)

for t ≥ 0.• By Theorem 4.2 of [6] we have that for positive affine martingales

φ(t, u) ≡ 0.


References

ExampleIf A has no jump part then A is called Feller diffusion. In that case thepositive affine martingale dynamics is given by

dAt =√αAtdWt ,

where Wt is a standard Brownian motion. In this case ψ is the solution of

ψ′t =12αψ2

t . (27)

given by

ψ(t, u) ≡ 0 or ψ(t, u) = − 112αt + i

u

, u ∈ R.


References

ExampleIn order to give an example of a positive affine martingale including jumpswhere we can solve for ψ explicitly, we consider the jump density µ(dy) forthe re-estimation process as

µ(dy) =3

4√π

dyy5/2

.

In this case we obtain

ψ(t, u) ≡ 0 or ψ(t, u) = − 4α2

(1 + W ((−1 +

2α

√iu

)e−tα−1+ 2

α

√iu )

)−2

,

where W (·) is the Lambert W function. The Lambert W function W (z) isdefined to be the function satisfying W (z)eW (z) = z , z ∈ C.


References

(Local) Risk Minimization

• We focus on the case where the asset prices discounted with themoney market account are given by local martingales (see [9] , [18]).We denote

Sδt :=SδtS0

t, t ≥ 0.

• If we choose the P-numeraire portfolio as discounting factor, thebenchmarked underlyings are always local martingale, when S is givenby a continuous semimartingale and also for a wide class ofjump-diffusion market models, see [12]. For benchmarked local riskminimization, we refer to [2].

• Let T > 0 be some fixed maturity.


References

DefinitionAn L2-admissible strategy is any Rd+1-valued predictable vector processδ = δt = (δ0t , δ

1t , . . . , δ

dt )>, t ∈ [0,T ] with(E

[∫ T

0δ>u d[S ]uδu

]) 12

<∞, (28)

such that the discounted value process Sδt =∑d

j=0 δjt S

jt is right-continuous and

square-integrable (i.e. Sδt ∈ L2(Ft ,P), for each t ∈ [0,T ]). We denote by L2(S)the space of all L2-admissible strategies.


References

DefinitionFor any strategy δ ∈ L2(S), the profit & loss or cost process C δ is definedby

C δt := Sδt −∫ t

0δu · dSu − Sδ0 , t ∈ [0,T ]. (29)

Here C δt describes the total costs incurred by δ over the interval [0, t].

DefinitionFor a strategy δ ∈ L2(S), the corresponding risk at time t is defined by

Rδt := E[(

C δT − C δt)2∣∣∣∣Ft

], t ∈ [0,T ],

where the profit & loss process C δ, given in (29), is assumed to besquare-integrable.


References

DefinitionGiven a discounted contingent claim HT ∈ L2(FT ,P), an L2-admissiblestrategy δ is said to be risk-minimizing if the following conditions hold:(i) SδT = HT , P-a.s.;

(ii) for any admissible strategy δ such that S δT = SδT P-a.s., we have

Rt(δ) ≤ Rt(δ) P− a.s.,∀t ∈ [0,T ].

LemmaAny risk-minimizing strategy is mean-self-financing, i.e. the associatedprofit & loss process C δ is a P-martingale for all t ∈ [0,T ].


References

PropositionSuppose that S is a local martingale. Then every discounted contingentclaim HT ∈ L2(FT ,P) admits a risk-minimizing strategy δ with SδT = HT .

Proof.Since the space I2(S) :=

∫φdS | φ ∈ L2(S)

is stable, any

HT ∈ L2(FT ,P) admits a martingale representation (Galtchouk-Kunita-Watanabe decomposition, in short GKW-decomposition) of the form

HT = H0 +

∫ T

0ξHu · dSu + LH

T , P− a.s., (30)

where ξH ∈ L2(S) and LH = LHt , t ∈ [0,T ] is a martingale in M2

0(P), stronglyorthogonal to each component of S .


References

• The associated risk-minimizing strategy δ is given by

δt = ξHt ,

• its discounted value process is the martingale

Sδt = Ht = E[HT∣∣Ft]

• and the profit & loss equals the martingale

C δt = H0 − Sδ0 + LHt = LH

t , t ∈ [0,T ].


References

• In particular (local) risk-minimization naturally appears as suitablehedging method for the new financial instruments recently introducedto hedge against systematic mortality risk in life insurance contracts,where the market incompleteness is due to the presence of anadditional source of randomness, that is “orthogonal” to the asset pricedynamics, but not necessarily independent of them and vice versa.

• This is the case of the so-called mortality derivatives (survival swaps,longevity bonds) and of the unit-linked life insurance contracts, i.e.contracts where the insurance benefits depend on the price of somespecific traded stock, see for example [3], [4], [7], [10], [11], [15]and [16].


References

Mortality risk

• A large number of life insurance and pensions products have mortalityand longevity as a primary source of risk.

• (Systematic) mortality risk denotes here all forms of deviations inaggregate mortality rates from those anticipated at different times andover different times horizons.

• Longevity risk refers to the risk that aggregate survival rates for givencohorts are higher than anticipated.

• Short-term, catastrophic mortality risk is the risk, that over shortperiod of time, mortality rates are very much higher than would benormally experienced (such as for example in the case of a pandemicinfluenza or a natural catastrophe).


References

• Although mortality and longevity risk can be re-insured, there isinadequate reinsurance capacity on a global basis to address effectivelythese risks.

• In addition systematic mortality risk cannot be diversified away bypooling, but on the contrary its impact increases for larger portfoliosof insurers.

• Hence several new instruments having mortality and longevity indexesas basis factors have been introduced on the financial markets asalternative source of risk diversification.


References

Mortality derivatives

• Longevity bonds, where coupon payments are linked to the number ofsurvivors in a given cohort.

• Short-dated, mortality securities: market traded securities, whosepayments are linked to a mortality index. They allow the issuer toreduce its exposure to short-term catastrophic mortality risk.

• Survivor swaps, where counterparties swap a fixed series of paymentsfor a series of payments linked to the number of survivors in a givencohort.

• Mortality options: financial contracts with mortality rate as underlying.


References

Application III: Dynamic hedging with longevity bonds

• We follow the approach of [4].• Let T > 0 be some fixed maturity. The time of death τ > 0 of aperson, who is x-years old at time 0 (commonly referred to as (x)), ismodeled as a random variable with P(τ > t) > 0 for any t ∈ [0,T ],and we denote by Ht = Iτ≤t the counting process of death. Let (Ht)be the filtration, generated by H.

• We assume that the mortality intensity is given by a positivestochastic process µ, progressively measurable with respect to theaugmented natural filtration (Ft) of some Brownian motion W .


References

• Let (Gt) = (Ht) ∨ (Ft).• Martingale invariance property: every (Ft)-martingale remains amartingale in the larger filtration (Gt).

• In particular, W is a martingale in (Gt), and then by Lévy’scharacterization a Brownian motion.


References

• The survival probability process G associated to τ is given by

Gt := P (τ > t|Ft) = exp(−∫ t

0µu du

)=: exp (−Γt)

• The martingale M associated with the one-jump process H is given as

Mt = Ht −∫ t

0(1− Hu)µudu. (31)


References

• We now assume that it is possible to trade on the financial market inan instrument called a longevity bond.

• For the sake of simplicity, we assume a constant interest rate r .• The discounted value process associated with the longevity bond isthus given by the conditional expectation

Vt = E[∫ T

0e−ruGudu

∣∣∣∣Ft

]. (32)


References

• LeteΓT ∈ L2(P). (33)

• The spaces L2(W ), L2(M) consist of all predictable θ, ψ such that

E[∫ T

0θ2s ds

]<∞, E

[∫ T

0ψ2

s dΓs

]<∞.

• The space Θ of admissible strategies consists of all predictable ϑ suchthat

E[∫ T

0ϑ2s d 〈V 〉s

]<∞.

If ϑ ∈ Θ, then∫ϑdV is a square-integrable martingale.


References

• Assume that we have sold at time 0 a unit of a pure endowment withpresent value

Cpe = e−rT Iτ>T = e−rT (1− HT ).

Here one unit of cash will be paid to the policyholder given that (x) isstill alive at maturity T .


References

TheoremUnder the hypotheses above, the GKW- decomposition of the pureendowment Cpe is given by

Cpe = E[e−rT e−ΓT

]+

∫ T

0αW

s dWs +

∫ T

0+αM

s dMs , (34)

where the predictable integrands αW and αM are given as

αWs = ξs(1− Hs−)e−Γs (35)

αMs = −Us(1− Hs−)e−Γs , (36)

where Ut = E[e−rT e−ΓT

∣∣Ft]

= E[e−rT e−ΓT

]+∫ t0 ξsdWs .


References

PropositionThe GKW-decomposition of the discounted value of the longevity bond is

Vt = E[

∫ T

0e−ruGu du|Ft ] = V0 +

∫ t

0ξs dWs ,

for ξ ∈ L2(W ).


References

• Our goal is now to find the GKW-decomposition for the pureendowment Cpe in terms of V :

E [Cpe |Gt ] = c +

∫ t

0ϑ∗u dVu + V⊥t , (37)

where V⊥ is a square-integrable martingale, strongly orthogonal to Vwith decomposition

V⊥t =

∫ t

0+δMu dMu.

• Here 1 +∫ t0 ϑ∗u dVu can be interpreted as the part of the risk that can

be perfectly replicated by means of our optimal hedging strategy ϑ∗,and V⊥t as the part of the risk that is totally unhedgeable.


References

• The integrand ϑ∗ in the GKW- decomposition (37) is determineduniquely by the equation

ϑ∗ξ = δW . (38)

• Here uniqueness is understood modulo the following equivalencerelation: if ϑ, ψ ∈ Θ, then

ϑ ∼ ψ if∫ T

0ϑ2t d [V ]t =

∫ T

0ψ2

t d [V ]t . (39)

• In particular, the predictable process ϑ∗ ∈ Θ gives the uniquerisk-minimizing strategy of the claim by trading in the underlyinglongevity bond.


References

[1] D. Becherer. The numeraire portfolio for unbounded semimartingales.Finance and Stochastics, 5(3):327–341, 2001.

[2] F. Biagini, A. Cretarola, and E. Platen. Local risk-minimization underthe benchmark approach. Preprint, 2011.

[3] F. Biagini and I. Schreiber. Local risk-minimization for mortalityderivatives. Preprint, 2011.

[4] F. Biagini, Rheinländer T, and J. Widenmann. Hedging mortalityclaims with longevity bonds. Preprint, 2010.

[5] F. Biagini, Bregman Y., and Meyer-Brandis T. Pricing of catastropheinsurance options under immediate loss reestimation. Journal ofApplied Probability, 45:831–845, 2008.

[6] F. Biagini, Bregman Y., and Meyer-Brandis T. Pricing of catastropheinsurance options written on a loss index with reestimation. Insurance:Mathematics and Economics, 43:214–232, 2008.

[7] M. Dahl and T. Møller. Valuation and hedging of life insuranceliabilities with systematic mortaliy risk. Preprint, 2006.


References

[8] F. Delbaen and J. Haezendonck. A martingale approach to premiumcalculation principles in an arbitrage free market. Insurance:Mathematics and Economics, 8(4):269–277, 1989.

[9] H. Föllmer and D. Sondermann. Hedging of non-redundant contingentclaims. In W. Hildenbrand and A. Mas-Colell, editors, Contributions toMathematical Economics, pages 205–223. North Holland, 1986.

[10] T. Møller. Risk-minimizing hedging strategies for unit-linked lifeinsurance contracts. ASTIN Bulletin, 28:17–47, 1998.

[11] T. Møller. Risk-minimizing hedging strategies for insurance paymentprocesses. Finance and Stochastics, 5:419–446, 2001.

[12] E. Platen. Arbitrage in continuous complete markets. Adv. in Appl.Probab., 34(3):540–558, 2002.

[13] E. Platen. A benchmark framework for risk management. InStochastic Processes and Applications to Mathematical Finance,Proceedings of the Ritsumeikan Intern. Symposium,: 305-335, 2004.


References

[14] E. Platen and D. Heath. A Benchmark Approach to QuantitativeFinance. Springer Finance, Springer-Verlag Berlin Heidelberg, 2006.

[15] M. Riesner. Hedging life insurance contracts in a lévy process financialmarket. Insurance: Mathematics and Economics, 38:599–608, 2006.

[16] M. Riesner. Locally risk-minimizing hedging of insurance paymentstreams. ASTIN Bulletin, 32:67–92, 2007.

[17] T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels. StochasticProcesses for Insurance and Finance. John Wiley & Sons, 1999.

[18] M. Schweizer. A guided tour through quadratic hedging approaches.In J. Cvitanic E. Jouini and M. Musiela, editors, Option Pricing,Interest Rates and Risk Management, pages 538–574. CambridgeUniversity Press, Cambridge, 2001.

[19] D. Sondermann. Reinsurance in arbitrage-free markets. Insurance:Mathematics and Economics, 10:191–202, 1991.


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