Backward Stochastic Differential Equations Approach to Hedging, Option Pricing, and Insurance Problems

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Research Article

Backward Stochastic Differential Equations Approach to

Hedging, Option Pricing, and Insurance Problems

Francesco Cordoni

1

and Luca Di Persio

2

1_{Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Trento, Italy} 2_{Department of Computer Science, University of Verona, Strada le Grazie 15, 37134 Verona, Italy}

Correspondence should be addressed to Luca Di Persio; dipersioluca@gmail.com Received 30 May 2014; Accepted 25 August 2014; Published 11 September 2014 Academic Editor: Ciprian A. Tudor

Copyright © 2014 F. Cordoni and L. Di Persio. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the present work we give a self-contained introduction to financial mathematical models characterized by noise of L´evy type in the framework of the backward stochastic differential equations theory. Such techniques will be then used to analyse an innovative model related to insurance and death processes setting.

1. Introduction

The main goal of the work consists in the analysis of problems characterizing modern financial markets which can be efficiently studied using techniques coming from the theory of stochastic differential equations in backward form

(BSDEs); see, for example, [1–3]. We start giving an overview

of classical approaches, for example, the standard

Black-Scholes (B-S) model,Section 3.1, which are related to the use

of stochastic perturbations of Brownian type. Then we will enrich the description of the relevant stock’s dynamics by

considering first stochastic jumps,Section 3.3, and after full

L´evy type noises, also including related Girsanov theorem

and Esscher transform, Section 2.2. Such an analysis will

allow us,Section 3.4, to study some original applications in

particular with respect to life insurance’s problems and linked death processes.

A key ingredient of our analysis will be an extensive use of the theory of backward stochastic differential equations (BSDEs) introduced by Bismut (1973), for the linear case, and generalized by Pardoux and Peng (1990) in the general nonlinear case in the Brownian framework; see, for example

[4].

In [5] Pardoux and Peng provided also a

Feynman-Kac type theorem for solution of nonlinear parabolic partial differential equation (PDE). We would like to underline that

BSDEs techinques provide powerful instruments to analyse a heterogeneous class of concrete problems, spanning from biology to finance, from population dynamics to particle

theory, and so forth; see, for example, [2, 3, 6, 7] and

references therein.

In the mathematical finance framework BSDEs tech-niques gained a great attention by both practitioners and academics in particular with respect to problems which arise in option pricing, portfolio hedging, market utility maximization, risk measures, and so forth. Latter interests have been increasingly developed both in financial as well in insurance frameworks, particularly under the influence of the related European directives, namely, Basilea and Solvency;

see, for example, [3] for one of the first review on the

subject and [1,2] for a more extensive introduction to recent

developments.

First approaches to such kind of general quantitative

economic questions have been following the idea of Black

and Scholes, hence developed in a Brownian setting, namely, allowing the dynamics of the interested quantities to be driven by Brownian measure/noise. Nevertheless empirical evidence, for example, working with stock prices’ behaviours, has pointed out that the traditional setup was based on the geometric Brownian motion; it is not fully satisfactory since it lacks an accurate description of financial data; see, for

example, [8,9]. A typical example of such type of issues arise

Volume 2014, Article ID 152389, 11 pages http://dx.doi.org/10.1155/2014/152389

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in the study of the implied volatility surface; see, for example,

[8,10].

Moreover, by its own nature, Brownian models are not able to capture those phenomena which are characterized by abrupt changes in economic quantities of interest, changes which have become more and more frequently determining in current financial fields.

Latter issues have promoted the development of more flexible models causing an explosion of the related mathe-matical literature starting from the late 90s; see, for example,

[1,2,11,12] and references therein.

Following these impulses great improvements towards more realistic models to describe and forecast movements of relevant financial quantities have been achieved taking into account L´evy type noises, hence allowing for asset price dynamics influenced by random perturbation with both diffusive and jump components. The L´evy random jump part is a key ingredient to capture sudden variations of prices which could happen, for example, in presence of turbu-lent economics dynamics originated by unexpected political events, natural disasters, abrupt variations of commodities’ prices, and so forth.

Fundamental results related to BSDEs in presence of

L´evy type drivers are given by Rong in [13], where BSDEs

driven by a Brownian motion plus a Poisson point process

are studied, and in [14], where Ouknine exploited the integral

representation of a square-integrable random variable in terms of a Poisson random, to study the case of a BSDEs driven by a Poisson random measure.

Moreover, in [15], Nualart and Schoutens proved both a

martingale representation theorem for L´evy processes satisfy-ing some exponential moment condition and a Feynman-Kac formula using a Teugels type orthonormalization procedure. In what follows, we will consider the wealth processes dynamic of a portfolio, composed by a riskless asset, for example, a bond or a bank account, and a risky security whose dynamic is modelled using a BSDE driven by a L´evy process, hence generalizing the classic approach based on Brownian stochastic driver.

In particular the paper is organized as follows. In

Section 2basic definitions for the mathematical framework within to develop BSDEs’ theory characterized by L´evy type noise are stated and the role played by both the Girsanov

theo-rem and the Esscher transform is underlined with applications

in view. InSection 3the financial background will be set up

and we will provide related examples; see Sections3.1,3.2, and

3.3; moreover inSection 3.4a novel insurance application is

presented.

2. Mathematical Framework

In this section we will give basic definitions and results which allow to set up a suitable framework within hedging/pricing

problems which will be analysed inSection 3.

Let 𝑊_𝑡 = {𝑊_𝑡}_{𝑡∈[0,𝑇]}, 𝑇 ∈ R+, be a one dimensional

Brownian motion on a probability space (Ω, F, P) and let

𝑁(𝑑𝑥, 𝑑𝑡) be a Poisson random measure on R+× R∗, where

we have denotedR∗ := R \ {0}, independent of 𝑊_𝑡, with

compensator](𝑑𝑥)𝑑𝑡 and such that it is a 𝜎-finite L´evy

mea-sure on(R∗, B(R∗)), with ̃𝑁(𝑑𝑡, 𝑑𝑢) := 𝑁(𝑑𝑥, 𝑑𝑡) − ](𝑑𝑥)𝑑𝑡

representing its compensated measure. Let(F_𝑡)_{𝑡∈[0,𝑇]}be the

filtration generated (jointly) by𝑊_𝑡 and 𝑁_𝑡, whileP is the

predictable𝜎-algebra on [0, 𝑇] × Ω. Throughout the paper

the following notations will be used.

(i)𝐿2(F_𝑇) is the set of random variables which are F_𝑇

measurable and square integrable:

𝐿2(F_𝑇) := {𝑍 : Ω 󳨀→ R : ∫

Ω|𝑍 (𝜔)|

2_{P (𝑑𝜔) < ∞} , (1)}

endowed with the scalar product ⟨𝑌, 𝑍⟩ := ∫

Ω𝑌 (𝜔) 𝑍 (𝜔) P (𝑑𝜔) ; (2)

(ii)H2,𝑇is the set of real-valued progressively

measur-able processes: H2,𝑇:= {𝑋 = {𝑋_𝑡: 0 ≤ 𝑡 ≤ 𝑇} : E [∫𝑇 0 󵄨󵄨󵄨󵄨𝑋𝑡(𝜔)󵄨󵄨󵄨󵄨 2_{𝑑𝑡] < ∞} ,} (3)

endowed with the scalar product

⟨𝑋1, 𝑋2⟩_H := E [∫𝑇

0 𝑋 1

𝑡(𝜔) 𝑋2𝑡(𝜔) 𝑑𝑡] ; (4)

(iii)𝐿2(R∗, ]) is the set of square ]-measurable functions

𝐿2(R∗, ]) := {𝛾 : R∗ 󳨀→ R : ∫

R∗󵄨󵄨󵄨󵄨𝛾(𝑥)󵄨󵄨󵄨󵄨

2_{] (𝑑𝑥) < ∞} ,}

(5) endowed with the scalar product

⟨𝛾, 𝜂⟩_]:= ∫

R∗𝛾 (𝑥) 𝜂 (𝑥) ] (𝑑𝑥) ; (6)

(iv)H2,𝑇_] is the set of predictable progressively measurable

stochastic processes H2,𝑇_] := {𝛾 = {𝛾_𝑡(𝜔, 𝑥) : 0 ≤ 𝑡 ≤ 𝑇, 𝜔 ∈ Ω, 𝑥 ∈ R∗} : E [∫𝑇 0 ∫R∗󵄨󵄨󵄨󵄨𝛾𝑡(𝜔, 𝑥)󵄨󵄨󵄨󵄨 2_{] (𝑑𝑥) 𝑑𝑡] < ∞} ,} (7)

endowed with the scalar product

⟨𝛾1_{, 𝛾}2_⟩ H:= E [∫ 𝑇 0 ∫R∗𝛾 1 𝑡(𝜔, 𝑥) 𝛾𝑡2(𝜔, 𝑥) ] (𝑑𝑥) 𝑑𝑡] ; (8)

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(v)𝑆2,𝑇is the set of real-valued c´adl´ag adapted processes 𝑆2,𝑇:= {𝑋 = {𝑋𝑡(𝜔) : 0 ≤ 𝑡 ≤ 𝑇} : E [ sup 0≤𝑡≤𝑇󵄨󵄨󵄨󵄨𝑋𝑡(𝜔)󵄨󵄨󵄨󵄨 2_{] < ∞} ,} (9)

endowed with the norm

‖𝑋‖_𝑆:= E [ sup

0≤𝑡≤𝑇󵄨󵄨󵄨󵄨𝑋𝑡(𝜔)󵄨󵄨󵄨󵄨

2_{] ;} ₍₁₀₎

(vi)𝜏₀is the set of stopping times𝜏, such that 𝜏 ∈ [0, 𝑇],

a.s.

Moreover we will use the following simplified notations:

𝐿2:= 𝐿2(F𝑇), H := H2,𝑇,𝐿2]:= 𝐿2(R∗, ]), and H]:= H2,𝑇] ,

which are Hilbert spaces, with the respective scalar product,

whenever it makes sense. Furthermore we set𝑆 := 𝑆2,𝑇and

we will omit the explicit dependence on𝜔 whenever it does

not cause misinterpretations.

2.1. Linear BSDE. In what follows we introduce some

funda-mental results in the theory BSDEs; see, for example, [2,3], for

a deeper treatment on the topic. We start studying the cases of linear BSDEs for which a solution can be expressed as a conditional expectation of some specified known processes;

later, inSection 3, explicit formulae for such solutions will be

provided. Taking into account previous notations, we define a BSDE with jumps as follows.

Definition 1 (BSDE). A BSDE with jumps is an equation of

the form −𝑑𝑋𝑡= 𝑓 (𝑡, 𝑋𝑡−, 𝑌_𝑡, 𝑈_𝑡) 𝑑𝑡 − 𝑌_𝑡𝑑𝑊_𝑡 − ∫ R∗𝑈𝑡(𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥) , 𝑋_𝑇= 𝜉, (11)

where𝑊_𝑡is a one-dimensional Brownian motion, ̃𝑁(𝑑𝑡, 𝑑𝑥)

is the compensated Poisson random measure onR+ × R∗

defined above, and𝜉 is the so-called terminal condition.

Definition 2 (driver). A function𝑓 : [0, 𝑇]×Ω×R×R×𝐿2_] →

R is called a driver for the BSDE (11), if it holds the following

properties:

(i)𝑓 is P ⊗ B(R2) ⊗ B(𝐿2_])-measurable,

(ii)𝑓(⋅, 0, 0, 0) ∈ H.

Definition 3 (Lipschitz driver). Let𝑓 be a driver of a BSDE of

the form (11) in the sense ofDefinition 2; then𝑓 is said to be

a Lipschitz driver if there exists a constant𝐶 ≥ 0 such that,

for each(𝑥₁, 𝑦₁, 𝑢₁), (𝑥₂, 𝑦₂, 𝑢₂) ∈ R × R × 𝐿2_], the following

holds:

󵄨󵄨󵄨󵄨𝑓(𝑡,𝑤,𝑥1, 𝑦1, 𝑢1) − 𝑓 (𝑡, 𝑤, 𝑥2, 𝑦2, 𝑢2)󵄨󵄨󵄨󵄨

≤ 𝐶 (󵄨󵄨󵄨󵄨𝑥1− 𝑥2󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑦1− 𝑦2󵄨󵄨󵄨󵄨

+󵄨󵄨󵄨󵄨𝑢1− 𝑢2󵄨󵄨󵄨󵄨]) , 𝑑P ⊗ 𝑑𝑡 a.s.

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Definition 4. A solution to a BSDE is a triplet(𝑋_𝑡, 𝑌_𝑡, 𝑈_𝑡) ∈

𝑆 × H × H_]satisfying 𝑋_𝑡= 𝜉 + ∫𝑇 𝑡 𝑓 (𝑡, 𝑋𝑠−, 𝑌𝑠, 𝑈𝑠) 𝑑𝑠 − ∫ 𝑇 𝑡 𝑌𝑠𝑑𝑊𝑠 − ∫𝑇 𝑡 ∫R∗𝑈𝑠(𝑥) ̃𝑁 (𝑑𝑠, 𝑑𝑥) , (13)

where𝑋_𝑡is a c´adl´ag process,𝑌_𝑡is progressively measurable,

and𝑈_𝑡is integrable with respect to the compensated Poisson

measure ̃𝑁(𝑑𝑡, 𝑑𝑥).

Existence and uniqueness results for the problem (11)

can be established provided that suitable conditions on the

terminal condition𝜉 and the driver 𝑓 are satisfied; see, for

example, [16] and references therein for details. In particular

the following theorem holds.

Theorem 5. Let 𝑓 be as inDefinition 3and𝜉 ∈ 𝐿2; then there

exists a unique solution(𝑋_𝑡, 𝑌_𝑡, 𝑈_𝑡) ∈ 𝑆 × H × H_]of (11). With financial application in mind we will consider, as a special case of the previous result, BSDEs with linear

generator, hence taking𝑓 of the form

𝑓 (𝑡, 𝑋_𝑡, 𝑌_𝑡, 𝑈_𝑡) = 𝛼_𝑡+ 𝛽_𝑡𝑋_𝑡+ 𝛿_𝑡𝑌_𝑡+ ⟨𝑈_𝑡, 𝜁_𝑡⟩_], (14)

where 𝛽_𝑡 and 𝛿_𝑡 are a real-valued predictable processes,

supposed to be a.s. integrable with respect to𝑑𝑡 and 𝑑𝑊_𝑡,

(𝜁_𝑡(⋅))_{𝑡∈[0,𝑇]} is a real-valued predictable process defined on

[0, 𝑇]×Ω×R∗_{, that is,}_P⊗B(R∗_{)-measurable, and integrable}

with respect to ̃𝑁(𝑑𝑡, 𝑑𝑥) and 𝛼_𝑡∈ H.

The following fundamental result shows that the solution of a linear BSDE with jumps can be written as a conditional expectation via an exponential semimartingale.

Theorem 6. Let (𝑋_𝑡, 𝑌_𝑡, 𝑈_𝑡) be the solution in 𝑆 × H × H_]of

the following linear BSDE:

−𝑑𝑋_𝑡= (𝛼_𝑡+ 𝛽_𝑡𝑋_𝑡+ 𝛿_𝑡𝑌_𝑡+ ⟨𝑈_𝑡, 𝜁_𝑡⟩_]) 𝑑𝑡 − 𝑌_𝑡𝑑𝑊_𝑡

− ∫

R∗𝑈𝑡(𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥) ,

𝑋_𝑇= 𝜉;

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then, the process𝑋_𝑡satisfies

𝑋_𝑡= E [𝑆_𝑇𝜉 + ∫𝑇

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where the process𝑆_𝑡∈ 𝑆 satisfies

𝑑𝑆_𝑡= 𝑆_𝑡₋(𝛽_𝑡𝑑𝑡 + 𝛿_𝑡𝑑𝑊_𝑡+ ∫

R∗𝜁𝑡(𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥))

𝑆₀= 1.

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Proof. See, for example, [17].

2.2. Girsanov Theorem and the Change of Measure. In this

section we focus our attention on the Girsanov theorem, a

result which will be later used,Section 3, to obtain pricing

relation by a change of measure approach when the asset prices’ behaviour is represented by geometric Itˆo-L´evy pro-cesses, namely, by an analogue of the risk-neutral valuation technique used in the Black-Scholes framework. We would

like to underline that (see, for example, [3,8,18–20]) when

assets’ dynamics are driven by jump-diffusion processes perfect hedge does not exist; namely, it is not always possible to replicate the derivative payoff by a controlled portfolio of the basic securities. However there exist particular cases, for example, when the driving process is of geometric Itô-Lévy type, which can be successfully treated by mean of the so-called Esscher transformation which allow for the definition of a suitable risk-neutral density in the form of the Doléans-Dade exponential; see below for details. In particular the following particular case for the Girsanov theorem holds; see,

for example, [8,18,21].

Theorem 7 (Girsanov theorem). Let (Ω, F, (F_𝑡)_{𝑡∈[0,𝑇]}, P),

where(F_𝑡)_{𝑡∈[0,𝑇]}is the filtration generated (jointly) by𝑊_𝑡and

𝑁_𝑡, be a filtered probability space and let𝑆_𝑡be a Geometric Itˆo-L´evy process of the form

𝑑𝑆_𝑡= 𝑆_𝑡−[𝜇_𝑡𝑑𝑡 + 𝜎_𝑡𝑑𝑊_𝑡+ ∫

R∗𝛾𝑡(𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥)] . (18)

Let one further assume that, for any𝛼 ∈ R, the following holds:

∫

|𝑥|≥1𝑒

𝛼𝑥_{] (𝑑𝑥) < ∞,}

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and let𝑢_𝑡,𝑙_𝑡(𝑥) be two F_𝑡-predictable processes such that

(i)

𝜇_𝑡+ 𝑢_𝑡𝜎_𝑡+ ∫

R∗𝛾𝑡(𝑥) 𝑙𝑡(𝑥) ] (𝑑𝑥) = 0, (20)

(ii) the process𝑙_𝑡(𝑥) satisfies 𝑙_𝑡(𝑥) ≥ 0,

(iii) the process𝑍_𝑡defined by the solution of the following

SDE:

𝑑𝑍_𝑡= 𝑍_𝑡−[𝑢_𝑡𝑑𝑊_𝑡+ ∫

R∗𝑙𝑡(𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥)] = 𝑍𝑡−𝑑𝑌𝑡,

𝑍₀= 1,

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where the process𝑌_𝑡satisfies

𝑑𝑌_𝑡= 𝑢_𝑡𝑑𝑊_𝑡+ ∫

R∗𝑙𝑡(𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥) , (22)

is well defined and satisfies

E [𝑍_𝑇] = 1. (23)

Then there exists a probability measure Q on F_𝑇 which is equivalent toP and such that

𝑑Q = 𝑍𝑇𝑑P. (24)

Note that in financial applications the Girsanov theorem is particularly useful since it allows the discounted stock price

𝑒−𝑟𝑡_𝑆

𝑡, where𝑟 is the risk-free interest rate and 𝑆𝑡is the stock

price, to be a local martingale with respect to the measureQ.

Theorem 7straightforwardly implies the following result; see,

for example, [22].

Corollary 8. Let 𝑢_𝑡and𝑙_𝑡(𝑥) ≥ 0 be predictable processes such

that the process𝑍_𝑡satisfying

𝑑𝑍_𝑡= 𝑍_𝑡−[𝑢_𝑡𝑑𝑊_𝑡+ ∫

R∗𝑙𝑡(𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥)] , (25)

is well defined for0 ≤ 𝑡 ≤ 𝑇. Suppose that

E [𝑍_𝑇] = 1, (26)

and define the probability measureQ on F_𝑇by

𝑑Q = 𝑍_𝑇𝑑P; (27)

then

(i) the process𝑊_𝑡Qdefined by

𝑊_𝑡Q= 𝑊_𝑡− ∫𝑡

0𝑢𝑠𝑑𝑠, (28)

is aQ-Brownian motion and

(ii) the random measure ̃𝑁Q(𝑑𝑡, 𝑑𝑥) defined by

̃ 𝑁Q_{(𝑑𝑡, 𝑑𝑥) = ̃}_{𝑁 (𝑑𝑡, 𝑑𝑥) − 𝑙} 𝑡(𝑥) ] (𝑑𝑥) 𝑑𝑡, (29) such that ∫𝑡 0∫R∗𝑁̃ Q_{(𝑑𝑠, 𝑑𝑥)} = ∫𝑡 0∫R∗𝑁 (𝑑𝑠, 𝑑𝑥) − ∫̃ 𝑡 0∫R∗𝑙𝑠(𝑥) ] (𝑑𝑥) 𝑑𝑠, (30) is aQ-local martingale.

Remark 9. We would like to stress that 𝜎_𝑡, the diffusion

coefficient, and𝑁, the Poisson random measure of process

jumps, do not change passing from the original probabilityP

to the equivalent measureQ, since they are path properties

of the process. Heuristically speaking, while process’ paths do not change, the Girsanov transformation changes their probability to be realized.

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Taking into consideration L´evy process implies that

con-dition (20) is satisfied provided the existence of two suitable

processes𝑢_𝑡and𝑙_𝑡and this leads to the identification of more

than one equivalent martingale measureQ; therefore related

markets are incomplete; moreover only particular choices of

the processes𝑢_𝑡and𝑙_𝑡allow for an equivalent measureQ with

a physical meaning. In some particular cases, for example, when the driving process is an exponential Itˆo-L´evy type process, a meaningful choice is represented by the so-called

Esscher transformation which is defined taking𝑙_𝑡= (𝑒𝛼𝑥− 1),

𝛼 ∈ R, and it has the good physical property of minimizing

the relative entropy; see, for example [23].

Considering such a transformation in relation with (25),

the resulting density𝑍_𝑇, solution of𝑑𝑍_𝑡= 𝑍_𝑡₋𝑑𝐿_𝑡, where the

process𝐿_𝑡satisfies

𝑑𝐿_𝑡= 𝑢_𝑡𝑑𝑊_𝑡+ ∫

R∗𝑙𝑡(𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥) , (31)

is called Dol´eans-Dade exponential or stochastic exponential,

and it will be denoted by𝑍_𝑇 = E(𝐿)_𝑇. In particular if 𝑍

evolves according to (25), we have

𝑍_𝑇= E(𝐿)𝑇= exp {𝐿𝑡−1₂[𝐿𝑐𝑡, 𝐿𝑐𝑡]} ∏ 0≤𝑠≤𝑡[1 + Δ𝐿𝑠] 𝑒 −Δ𝐿𝑠 = exp {∫𝑇 0 𝑢𝑠𝑑𝑊𝑠− 1 2∫ 𝑇 0 𝑢 2 𝑠𝑑𝑠 − ∫𝑇 0 ∫R∗𝑙𝑡(𝑥) ] (𝑑𝑥) 𝑑𝑠} × ∏ 0≤𝑠≤𝑇 (1 + 𝑙_𝑠(Δ𝐿_𝑠)) , (32)

where we have denoted by𝐿𝑐_𝑡the continuous part of𝐿_𝑡with

[⋅, ⋅]being the quadratic variation and denoted by Δ𝑌𝑡:= 𝑌𝑡−

𝑌𝑡−the jump occurring at time𝑡. We refer to [18], in particular

to Sections5.4.3, 5.4.4, and 5.4.5 and references therein, for

further details about both the Esscher transformation and the Dol´eans-Dade exponentials. Let us note that (see, for

example, [18, Chapter 5])𝐿_𝑡is a local martingale since it can

be rewritten as a L´evy type stochastic integral thanks to the following result.

Theorem 10. Let one assume that for any 𝑡 ≥ 0√𝐺_𝑡 is a

predictable process such that

E ∫𝑡

0𝐺 (𝑠) 𝑑𝑠 < ∞, (33)

and that𝑡 ≥ 0 𝐾(𝑡, 𝑥) is a predictable process satisfying

E ∫𝑡 0∫|𝑥|>1|𝐾 (𝑠, 𝑥)| 2_{] (𝑑𝑥) 𝑑𝑠 < ∞,} P (∫𝑡 0∫|𝑥|>1|𝐾 (𝑠, 𝑥)| 2_{] (𝑑𝑥) 𝑑𝑠 < ∞) = 1;} (34)

then𝐿_𝑡is any L´evy type stochastic integral of the form

𝑑𝐿_𝑡= 𝐺_𝑡𝑑𝑡 + 𝐹_𝑡𝑑𝑊_𝑡+ ∫

|𝑥|<1𝐻 (𝑡, 𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥)

+ ∫

|𝑥|>1𝐾 (𝑡, 𝑥) 𝑁 (𝑑𝑡, 𝑑𝑥) ;

(35)

then𝐿_𝑡is a local martingale if and only if

𝐺_𝑡+ ∫

|𝑥|>1𝐾 (𝑡, 𝑥) ] (𝑑𝑥) = 0, a.s. (36)

3. Financial Framework

In what follows we will use results stated in Section 2 to

analyse the problem of pricing and hedging contingent claim written on underlyings subjected to a risk of both diffusive and jump type, hence allowing for underlyings’ dynamics with random discontinuities. Our financial framework will be defined as a market composed by two securities: a riskless asset, for example, a bond, a bank account, and so forth,

𝐵 = {𝐵_𝑡 : 0 ≤ 𝑡 ≤ 𝑇}, solution to the following deterministic

differential equation:

𝑑𝐵_𝑡= 𝑟_𝑡𝐵_𝑡𝑑𝑡, 0 ≤ 𝑡 ≤ 𝑇,

𝐵₀= 1, (37)

where𝑟_𝑡is the risk-free interest rate, and a risky security (or

stock)𝑆 := {𝑆_𝑡: 0 ≤ 𝑡 ≤ 𝑇} solution to the following:

𝑑𝑆_𝑡= 𝑆_𝑡−[𝜇_𝑡𝑑𝑡 + 𝜎_𝑡𝑑𝑊_𝑡+ ∫

R∗𝛾𝑡(𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥)] ,

0 ≤ 𝑡 ≤ 𝑇,

𝑆₀= 𝑠 ∈ R,

(38)

where𝜇_𝑡and𝜎_𝑡are predictable processes called respectively

drift and diffusion, while𝛾_𝑡(𝑥) ∈ H_]is the jump component.

We thus have that the stock price 𝑆_𝑡 is a c`adl`ag process

described by a geometric Itˆo-L´evy process. Thus the investor in the risky asset is exposed to a diffusion risk, caused by the Brownian component, as well as to a jump component risk

given by∫_R∗𝛾𝑡(𝑥) ̃𝑁(𝑑𝑡, 𝑑𝑥).

In what follows we will denote by𝑉 := {𝑉_𝑡: 0 ≤ 𝑡 ≤ 𝑇} the

wealth stochastic process representing the total value of the

investor’s portfolio at time𝑡, given an initial wealth 𝑉₀ > 0.

In particular the investor, at a given time𝑡, holds 𝜋_𝑡share of

the risky stock, whilst the remaining part of his total wealth,

𝑉_𝑡− 𝜋_𝑡, is invested in the riskless bond. Let us note that, in

order to avoid an arbitrage opportunity (see, for example,

[24]) we further assume the portfolio to be self-financing; that

is, there is no exogenous infusion or withdrawal of money. This implies that the instantaneous variation of the wealth value is caused uniquely by assets’ prices variations and not by injecting or withdrawing funds from outside; hence the self-financing condition reads as follows.

(6)

Definition 11 (self-financing strategy). A self-financing

strat-egy is a pair(𝑉_𝑡, 𝜋_𝑡), where 𝜋_𝑡is a predictable process such

that 𝑉_𝑡= 𝑉_𝑇+ ∫𝑇 𝑡 ((𝑉𝑡− 𝜋𝑡) 𝑑𝐵_𝑡 𝐵_𝑡 + 𝜋𝑡 𝑑𝑆_𝑡 𝑆_𝑡 ) , (39) with E [∫𝑇 0 󵄨󵄨󵄨󵄨𝜎𝑡𝜋𝑡󵄨󵄨󵄨󵄨 2_{𝑑𝑡] < +∞,} E [∫𝑇 0 𝜋 2 𝑡𝛾𝑡2(𝑥) ] (𝑑𝑥) 𝑑𝑡] < +∞. (40)

Within the framework defined inSection 3, we have that

a contingent claim, with payoff at maturity time𝑇 given by 𝜉,

is said to be hedgeable if there exists a self-financing strategy

(𝑉_𝑡, 𝜋_𝑡) of the form (39) that replicates its payoff, in particular

the following result holds.

Proposition 12. A self-financing strategy solves a linear

Lips-chitz BSDE of the form

−𝑑𝑉_𝑡= 𝑟_𝑡𝑉_𝑡𝑑𝑡 − 𝜋_𝑡𝜎_𝑡𝑢_𝑡𝑑𝑡 − ⟨𝜋_𝑡𝛾_𝑡, 𝑙_𝑡⟩_]𝑑𝑡 + 𝜋_𝑡𝜎_𝑡𝑑𝑊_𝑡

+ ∫

R∗𝜋𝑡𝛾𝑡(𝑥) ̃𝑁 (𝑑𝑡, 𝑑𝑥) ,

𝑉𝑇= 𝜉.

(41)

Proof. Let us consider an asset evolving according to (38)

together with a risk-less security as in (37). A straightforward

substitution of (38) and (37) into equation for the

self-financing strategy (39) leads to the conclusion that the

self-financing strategy solves the BSDE:

−𝑑𝑉_𝑡= 𝑟_𝑡𝑉_𝑡𝑑𝑡 + 𝜋_𝑡(𝜇_𝑡− 𝑟_𝑡) 𝑑𝑡 + 𝜋_𝑡𝜎_𝑡𝑑𝑊_𝑡

+ ∫

𝑉_𝑇= 𝜉.

(42)

Note that applying the GirsanovTheorem 7to the stock price

(39) with Radon-Nikodym density given as in (32) we have

that the martingale condition (20) inTheorem 7has to hold to

have the discounted price𝑒− ∫0𝑡𝑟𝑠𝑑𝑠𝑆

𝑡to be a martingale under

a risk-neautral measureQ equivalent to the real world measure

P; therefore substituting (20) into (42), we obtain that the

self-financing strategy𝑉_𝑡has to satisfy the following BSDE:

+ ∫

𝑉_𝑇= 𝜉,

(43)

which is an equation of the form (15) with

𝑋_𝑡= 𝑉_𝑡, 𝑌_𝑡= −𝜋_𝑡𝜎_𝑡, 𝑈_𝑡= −𝜋_𝑡𝛾_𝑡(𝑥) ,

𝛼_𝑡= 0, 𝛽_𝑡= 𝑟_𝑡, 𝛿_𝑡= 𝑢_𝑡, 𝜁_𝑡= 𝑙_𝑡. (44)

Proposition 12 implies that we can applyTheorem 6to obtain a formula for pricing a contingent claim in a mar-ket consisting of a risky asset driven by a jump-diffusion dynamic; indeed we have the following.

Theorem 13. Let (𝑉_𝑡, 𝜋_𝑡𝜎_𝑡, 𝜋_𝑡𝛾_𝑡) be the solution in 𝑆 × H × H_]

of the following linear BSDE:

+ ∫

𝑉_𝑇= 𝜉,

(45)

so that the assumptions ofTheorem 5are satisfied. Let one fur-ther suppose that the assumptions of the GirsanovTheorem 7

hold, so that there exists an equivalent measureQ with Radon-Nikodym density𝑍 as in (32); then a solution of the BSDE (45)

is given by 𝑉_𝑡= E [𝑒− ∫𝑡𝑇𝑟𝑠𝑑𝑠_𝑍 𝑡𝜉 | F𝑡] = EQ[𝑒− ∫ 𝑇 𝑡 𝑟𝑠𝑑𝑠_{𝜉 | F} 𝑡] , 0 ≤ 𝑡 ≤ 𝑇 a.s. (46)

Proof. The proof is a straightforward application of

Theorem 6 to the particular case of a BSDE of the form

(45).

From now on we will deal mainly with a BSDE of the form

of (41).

3.1. The Black-Scholes Model. Let us consider the standard

Black-Scholes (BS) model, where the driving process is a

diffusion without jumps; see, for example, [25,26] for details;

hence the riskless bond𝐵_𝑡solves

𝑑𝐵_𝑡= 𝑟𝐵_𝑡𝑑𝑡, 0 ≤ 𝑡 ≤ 𝑇,

𝐵₀= 1, (47)

while the risky asset𝑆_𝑡is the solution of

𝑑𝑆_𝑡= 𝜇𝑆_𝑡𝑑𝑡 + 𝜎𝑆_𝑡𝑑𝑊_𝑡, 0 ≤ 𝑡 ≤ 𝑇,

𝑆₀= 𝑠, (48)

for some constant parameters𝑟, 𝜇, and 𝜎. It is well known

that the BS model describes a complete and free of arbitrage market; hence there exists a unique equivalent risk-neutral

measureQ, and 𝑒−𝑟𝑡𝑆_𝑡is aQ-martingale. Applying Girsanov

Theorem 7to the Brownian case with𝑢_𝑡= (𝑟 − 𝜇)/𝜎, we have

that under the new measureQ := 𝑍_𝑇P, with

𝑍𝑇= exp {∫ 𝑇 0 𝑟 − 𝜇 𝜎 𝑑𝑊𝑠− 1 2∫ 𝑇 0 ( 𝑟 − 𝜇 𝜎 ) 2 𝑑𝑠} , (49)

the stock price evolves according to

𝑑𝑆_𝑡= 𝑟𝑆_𝑡𝑑𝑡 + 𝜎𝑆_𝑡𝑑𝑊Q

(7)

Suppose that we want to hedge a call option𝜉 written on an

underlying whose dynamic is described by (48), with payoff

given by Φ(𝑆_𝑇) = max(0, 𝑆_𝑇 − 𝐾) =: (𝑆_𝑇 − 𝐾)₊, where

𝐾 is the so-called strike price; then we have to construct

a replicating portfolio according toProposition 12, without

jump component. The latter implies that we have to solve the following BSDE:

𝑉_𝑡= (𝑆_𝑇− 𝐾)₊− ∫𝑇

𝑡 (𝑟𝑉𝑠+ 𝜋𝑠𝜎𝑢𝑠) 𝑑𝑠 − ∫

𝑇

𝑡 𝜋𝑠𝜎 𝑑𝑊𝑠. (51)

ByTheorem 6 together with GirsanovTheorem 7 we have that the initial value of the replicating portfolio, which coincides with the fair price of the claim, is given by

𝑋₀= E_Q[𝑒−𝑟𝑇(𝑆_𝑇− 𝐾)₊]

= 𝐾𝑒−𝑟𝑇Φ (−𝑑 (0, 𝑆₀− 𝜎√𝑇)) − 𝑆₀Φ (−𝑑 (0, 𝑆₀)) , (52)

a result obtained exploiting the fact that, for any given𝑡 ∈

[0, 𝑇], the random variables ln 𝑆_𝑡 are normally distributed;

therefore we explicitly know their density functions,Φ being

the corresponding cumulative distribution function and

𝑑 (𝑡, 𝑆_𝑡) := ln(𝑆𝑡/𝐾) + (𝑟 + (𝜎

2_{/2)) (𝑇 − 𝑡)}

𝜎√𝑇 − 𝑡 . (53)

It can be easily checked that the fair price given in (52)

coincides with the fair price obtained via standard methods;

see, for example, [26, Chapter 5].

3.2. Local Volatility Models. A first generalization of the

model proposed inSection 3.1is realized by the so-called local

volatility models (LVM), where both the drift and the volatility

parameters are no longer constant. Empirical evidences have shown that latter models, which in general do not have explicit solutions, fit better real data when compared to the standard BS setting, in particular with respect to the analysis

of the implied volatility surface; see, for example [27,28]. Let

us then consider a riskless bond𝐵_𝑡solution to

𝑑𝐵_𝑡= 𝑟_𝑡𝐵_𝑡𝑑𝑡, 0 ≤ 𝑡 ≤ 𝑇,

𝐵₀= 1, (54)

and a risky asset𝑆_𝑡

𝑑𝑆_𝑡= 𝜇 (𝑆_𝑡) 𝑑𝑡 + 𝜎 (𝑆_𝑡) 𝑑𝑊_𝑡, 0 ≤ 𝑡 ≤ 𝑇,

𝑆₀= 𝑠, (55)

where 𝑟_𝑡, 𝜇, and 𝜎 are F-predictable processes satisfying

standard Lipschitz continuity and linear growth assumptions,

so that there exists a unique solution to (48); see, for example,

[29]. Within this framework one of the most used models

is the constant elasticity of variance (CEV) model, where the

volatility term is of the form𝑆𝛽/2_𝑡 , for0 ≤ 𝛽 ≤ 2. Again

applying the Girsanov theorem, with𝑢_𝑡= (𝑟_𝑡− 𝜇(𝑆_𝑡))/𝜎(𝑆_𝑡),

the stock price under the risk-neutral measure Q evolves

according to

𝑑𝑆_𝑡= 𝑟_𝑡𝑑𝑡 + 𝜎 (𝑆_𝑡) 𝑑𝑊_𝑡Q, 0 ≤ 𝑡 ≤ 𝑇. (56)

If we want to construct a replicating portfolio to hedge

a contingent claim 𝜉, with terminal payoff given by 𝜉 =

(𝑆_𝑇− 𝐾)₊, then the portfolio dynamic has to solve the

following BSDE: 𝑉𝑡= (𝑆𝑇− 𝐾)+− ∫ 𝑇 𝑡 (𝑟𝑠𝑉𝑠+ 𝜋𝑠𝜎 (𝑆𝑠) 𝑢𝑠) 𝑑𝑠 − ∫𝑇 𝑡 𝜋𝑠𝜎 (𝑆𝑠) 𝑑𝑊𝑠; (57)

hence, byTheorem 13and (46), we have that the initial value

of the replicating portfolio is given by

𝑋₀= E_Q[𝑒−𝑟𝑇_(𝑆

𝑇− 𝐾)+] . (58)

Note that, as mentioned before, an analytic solution to (58),

namely, an explicit solution for the fair price problem, does not exist; nevertheless effective numerical approximations can be

given; see, for example, [8, Section 12].

3.3. The Black-Scholes Model with Jumps. A different

general-ization of the B-S model introduced inSection 3.1is obtained

considering a stock price driven by a general L´evy process instead of a standard Brownian motion. Latter generalization

is motivated by empirical results (see, for example, [8]) which

show the presence of leaps in the evolution of real stock mar-kets’ quantities; hence we are in presence of discontinuities that cannot be modelled using a Brownian type approach since the Brownian paths are continuous. Latter analysis implies the need for a more general type of random processes to be used, hence allowing for jumps in the stocks’ prices dynamics. Therefore a natural improvement is to consider L´evy type drivers. Nevertheless it is crucial to underline that even if L´evy processes fit better to empirical data than the Brownian counterpart does, they are mathematically more difficult to treat. In fact not only is an analytical solution to the pricing equation no longer available, but also the market is not anymore complete. Let us then consider a riskless bond

𝐵𝑡solution to

𝐵₀= 1, (59)

and a risky asset𝑆_𝑡

𝑑𝑆𝑡= 𝜇𝑆𝑡𝑑𝑡 + 𝜎𝑆𝑡𝑑𝑊𝑡+ ∫

R∗𝑆𝑡−𝑥 ̃𝑁 (𝑑𝑡, 𝑑𝑥) , 0 ≤ 𝑡 ≤ 𝑇,

𝑆₀= 𝑠,

(60)

where the Poisson component has L´evy measure ]. The

solution to (60) can be found exploiting the stochastic

exponential introduced in (32); in particular we have

𝑆_𝑡= 𝑠𝑒(𝜇−(𝜎2/2))𝑡+𝜎𝑊𝑡

𝑁𝑡 ∏

𝑖=1

(8)

where𝑁_𝑡is a Poisson process independent of the Brownian

motion𝑊_𝑡. Since we are now dealing with a L´evy process, a

unique equivalent measureQ does not exist; see, for

exam-ple, [18]; namely, it is not possible to uniquely determine

the solution couple (𝑢_𝑡, 𝑙_𝑡). Nevertheless, in the case of a

geometric L´evy process, a suitable choice can be obtained by

the so-called Esscher transform and the density𝑍_𝑇takes the

form of a Dol´eans-Dade exponential (32).

In particular if we define𝑢_𝑡:= (𝑟−𝜇)/𝜎 and 𝑙_𝑡= 𝑒𝑥−1 (see

Section 2.2) we have that the asset has mean return𝑟 under

the risk-neutral measureQ; hence the discounted price is a

Q-martingale; namely, it satisfies the following equation:

𝑑𝑆_𝑡= 𝑟𝑆_𝑡𝑑𝑡 + 𝜎𝑆_𝑡𝑑𝑊_𝑡Q+ ∫

R∗𝑆𝑡−𝑥 ̃𝑁

Q_{(𝑑𝑡, 𝑑𝑥) , 0 ≤ 𝑡 ≤ 𝑇,}

𝑆₀= 𝑠;

(62) moreover hedging the contingent claim with terminal payoff

given by𝜉 = (𝑆_𝑇− 𝐾)₊, we have that the portfolio satisfies the

following BSDE: 𝑉_𝑡= (𝑆_𝑇− 𝐾)₊− ∫𝑇 𝑡 (𝑟𝑉𝑠+ 𝜋𝑠𝜎𝑢𝑠+ ⟨𝑙𝑠, 𝜋𝑠𝑥⟩]) 𝑑𝑠 + ∫𝑇 𝑡 𝜋𝑠𝜎 𝑑𝑊𝑠+ ∫ 𝑇 𝑡 ∫R∗𝜋𝑠𝑥 ̃𝑁 (𝑑𝑠, 𝑑𝑥) , (63)

that is, a linear BSDE of the form (43); hence, byTheorem 13,

we have that the value𝑉₀of the portfolio at initial time is given

by

𝑋₀= E_Q[𝑒−𝑟𝑇(𝑆_𝑇− 𝐾)₊] . (64)

3.4. Life Insurance Portfolio. Backward stochastic differential

equations have a wide range of applications, from standard

examples, as stated in Sections3.1,3.2, and3.3, to less typical

scenario as the ones characterizing actuarial applications; see,

for example [2] for details. In what follows we solve a novel life

insurance pricing problem. Let us assume that we are dealing

with the B-S financial model described inSection 3.1; namely,

we have a riskless bond𝐵_𝑡which evolves according to

𝐵₀= 1, (65)

and a risky asset𝑆_𝑡satisfying

𝑑𝑆_𝑡= 𝜇𝑆_𝑡𝑑𝑡 + 𝜎𝑆_𝑡𝑑𝑊_𝑡1, 0 ≤ 𝑡 ≤ 𝑇

𝑆₀= 𝑠. (66)

Let us consider a stochastic mortality intensity𝐽 := {𝐽_𝑡}_{𝑡∈[0,𝑇]},

𝑇 < +∞, which is modelled by the following SDE:

𝑑𝐽_𝑡= 𝜇𝐽(𝐽_𝑡) 𝑑𝑡 + 𝜎𝐽(𝐽_𝑡) 𝑑𝑊_𝑡2,

𝐽₀= 𝑗, (67)

where the process𝐽 is independent of the financial market

described by the couple(𝐵, 𝑆). Since we aim at considering

a life insurance liability problem, then we will analyse the dynamics of life insurance equity-linked claims. The uncer-tainty in this type of investments comes mainly from the fact that we cannot know how mortality rates will evolve.

Let us now assume that we are willing to construct a

portfolio consisting of an arbitrary, but finite, number 𝑛

policies, each of which relates to a unique insured person

whose lifetime is modelled via 𝑛 inhomogeneous Poisson

process𝜏_𝑖,𝑖 = 1, . . . , 𝑛; that is,

P (𝜏_𝑖> 𝑡) = 𝑒∫0𝑡𝜆(𝑠) 𝑑𝑠, 𝑖 = 1, . . . , 𝑛, (68)

where𝜆 : [0, 𝑇] → R₊ is a deterministic function

repre-senting the mortality intensity; then the insurance payment

process𝑃 := {𝑃_𝑡}_{𝑡∈[0,𝑇]}can be modelled as follows:

𝑃𝑡= ∫ 𝑡 0(𝑛 − 𝑁𝑠) 𝐻𝑠𝑑𝑠 + ∫ 𝑡 0𝐺𝑠𝑑𝑁𝑠+ (𝑛 − 𝑁𝑇) 𝜉1{𝑡=𝑇}, 0 ≤ 𝑡 ≤ 𝑇, (69)

where the claims 𝐻, 𝐺, and 𝜉 : [0, 𝑇] × R₊ → R₊

are measurable functions;1_{𝑡=𝑇}stands for the characteristic

function of the set{𝑡 = 𝑇}, while 𝑁 is a counting measure

representing a death counting process for a life insurance

portfolio with𝑛 policies linked to 𝑛 different insured persons.

A classical credit default swap is modelled setting 𝑛 = 1

and𝐹 = 0, while a collective credit risk setting is obtained

taking𝜉 = 𝐻 = 0 and 𝐺 = 𝑔(𝑡)1_{𝑡=𝑇}so that it represents

the credit loss in case of default; see, for example, [2] and

references therein for more details. Heuristically the process 𝑃 can be seen as a stream of liabilities since it is defined by a

continuous payments stream𝐻, a random payment process

𝐺, which is financially defined as the death benefits process,

and a claim𝜉, the so-called survival benefit, which pays off

at maturity time. Latter defined claims, namely,𝐻, 𝜉, and 𝐺,

depend upon the financial market defined by (65) and (66).

During last years growing attention has been attracted by the so-called mortality derivatives. Such type of contracts are mainly stipulated in order to hedge mortality risk. We will

show that the BSDE approach introduced inSection 2can be

usefully exploited to explicitly find a strategy which replicates a mortality bond by mean of some specific investments.

Let us assume that the mortality bond is already priced; hence we already are within the risk-neutral framework with

respect to a certain risk-neutral measureQ; therefore, taking

into account definitions given in Section 2.2, the density

𝑍 is given as in (32) and the processes 𝑢_𝑡 and 𝑙_𝑡 are as

in Section 3.3. Our goal is to solve a hedging problem; namely, we aim at finding a replicating strategy for the

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we obtain that the replicating portfolio evolves according to the following BSDE:

𝑉_𝑡= (𝑛 − 𝐽_𝑇) 𝜉 + ∫𝑇 𝑡 (−𝑟𝑉𝑠− 𝑢𝑠𝑍𝑠) + 𝜆 (𝑠) (𝑛 − 𝑁𝑠−) 𝑍𝑠𝑙𝑠 + 𝐻_𝑠(𝑛 − 𝑁_𝑠₋) + 𝜆 (𝑠) 𝐺𝑠(1 + 𝑙𝑠) (𝑛 − 𝑁𝑠−) 𝑑𝑠 − ∫𝑇 𝑡 𝑍𝑠𝑑𝑊𝑠− ∫ 𝑇 𝑡 𝑈𝑠𝑁 (𝑑𝑠) , 0 ≤ 𝑡 ≤ 𝑇,̃ (70)

and we are in position to applyTheorem 13 to find that a

solution of (70) is given by

𝑉_𝑡= EQ[∫𝑇

𝑡 𝑒

−𝑟(𝑠−𝑡)_𝑑𝑃

𝑠| F𝑡] . (71)

Proposition 14. In a market such that the unsystematic

mortality risk can be diversified, a solution(𝑉_𝑡, 𝑌_𝑡, 𝑍_𝑡) ∈ 𝑆 ×

H × H_]to the BSDE 𝑉_𝑡= 𝜉 + ∫𝑇 𝑡 (−𝑟𝑉𝑠− 𝑢𝑠𝑍𝑠) + 𝜆 (𝑠) (𝑛 − 𝑁𝑠−𝑍𝑠𝑙𝑠) + 𝐻_𝑠(𝑛 − 𝑁_𝑠₋) + 𝜆 (𝑠) 𝐺𝑠(1 + 𝑙𝑠) (𝑛 − 𝑁𝑠−) 𝑑𝑠 − ∫𝑇 𝑡 𝑌𝑠𝑑𝑊𝑠− ∫ 𝑇 𝑡 𝑍𝑠𝑁 (𝑑𝑠) , 0 ≤ 𝑡 ≤ 𝑇,̃ (72)

when the claim to hedge is given by𝜉 = (𝑛 − 𝑁_𝑇)(𝑆_𝑇− 𝐾)₊and the mortality rate is constant, is given by

𝑉𝑡= (𝑛 − 𝑁𝑡) 𝑒−𝜆(𝑇−𝑡)[𝑒−𝑟(𝑇−𝑡)Φ (𝑑 (𝑡, 𝑆𝑡) − 𝜎√𝑇 − 𝑡) − 𝑆_𝑡Φ (−𝑑 (𝑡, 𝑆_𝑡))] , 𝑌_𝑡= (𝑛 − 𝑁_𝑡) 𝑒−𝜆(𝑇−𝑡)𝜎𝑆_𝑡Φ (−𝑑 (𝑡, 𝑆_𝑡)) , 𝑍_𝑡= 𝑒−𝜆(𝑇−𝑡)[𝐾𝑒−𝑟(𝑇−𝑡)Φ (𝑑 (𝑡, 𝑆_𝑡) − 𝜎√𝑇 − 𝑡) − 𝑆_𝑡Φ (−𝑑 (𝑡, 𝑆_𝑡))] , (73)

whereΦ is the cumulative distribution of the standard Gaus-sian random variable and𝑑 is given as in (53).

Proof. We are interested in hedging the claim 𝜉 = (𝑛 −

𝑁𝑇)(𝑆𝑇 − 𝐾)+, which is a call option, with strike price𝐾,

written on a given fund (seeSection 3.1) which collects 𝑛

insured persons. In the present case𝐻 = 𝐺 = 0, moreover

since we have assumed that the market believes that the

unsystematic mortality risk can be diversified, then𝑙_𝑡 = 0;

see, for example, [2, Section 9.4], for details. It follows that

the replicating portfolio for the claim𝜉 (see (70)) turns out to

be described by the following BSDE:

𝑉_𝑡= 𝜉 + ∫𝑇 𝑡 (−𝑟𝑉𝑠− 𝑢𝑠𝑍𝑠) 𝑑𝑠 − ∫ 𝑇 𝑡 𝑌𝑠𝑑𝑊𝑠 − ∫𝑇 𝑡 𝑍𝑠 ̃ 𝑁 (𝑑𝑠) , 0 ≤ 𝑡 ≤ 𝑇, (74)

or equivalently, under the risk-neutral measure Q, by the

following: 𝑉_𝑡= 𝜉 − ∫𝑇 𝑡 𝑟𝑉𝑠𝑑𝑠 − ∫ 𝑇 𝑡 𝑌𝑠𝑑𝑊 Q 𝑠 − ∫ 𝑇 𝑡 𝑍𝑠 ̃ 𝑁Q(𝑑𝑠) , 0 ≤ 𝑡 ≤ 𝑇, (75)

which is an equation of the form (41) and since assumptions

of Theorem 5 are satisfied, it admits a unique solution.

Moreover, exploiting Theorem 13, we can compute such a

solution𝑉_𝑡from the standard B-S model stated inSection 3.1.

In particular by the B-S pricing formula (51), the BS price

denoted by𝑃_𝐵𝑆reads as follows:

𝑉_𝑡= E [𝑒−𝜆(𝑇−𝑡)𝜉 | F_𝑡] = (𝑛 − 𝑁_𝑡) 𝑒−𝜆(𝑇−𝑡)𝑃_𝐵𝑆; (76)

therefore a solution to (75) is given by

𝑉_𝑡= (𝑛 − 𝑁_𝑡) 𝑒−𝜆(𝑇−𝑡)[𝐾𝑒−𝑟(𝑇−𝑡)Φ (𝑑 (𝑡, 𝑆_𝑡) − 𝜎√𝑇 − 𝑡) − 𝑆𝑡Φ (−𝑑 (𝑡, 𝑆𝑡))] , 𝑌_𝑡= (𝑛 − 𝑁_𝑡) 𝑒−𝜆(𝑇−𝑡)𝜎𝑆_𝑡Φ (−𝑑 (𝑡, 𝑆_𝑡)) , 𝑍_𝑡= 𝑒−𝜆(𝑇−𝑡)_[𝐾𝑒−𝑟(𝑇−𝑡)_{Φ (𝑑 (𝑡, 𝑆} 𝑡) − 𝜎√𝑇 − 𝑡) − 𝑆_𝑡Φ (−𝑑 (𝑡, 𝑆_𝑡))] . (77)

In particular the fair price of the claim𝜉, regardless of the

initial wealth𝑉₀of the portfolio, is given by

𝑉₀= 𝑛𝑒−𝜆𝑇[𝐾𝑒−𝑟𝑇Φ (𝑑 (0, 𝑆₀) − 𝜎√𝑇)

− 𝑆₀Φ (−𝑑 (0, 𝑆₀))] . (78)

4. Conclusion

The theory of backward stochastic differential equations has been characterized by an impressive growth of interest from the applicative point of view under the pushing of a wide variety of different scientific communities, spanning from pure mathematicians to engineers and from biologists to risk management’s practitioners. A key ingredient behind such an increasing interest relies in the flexibility of the BSDEs approach which can be further improved allowing to include a rather general type of stochastic perturbations.

In the present paper we have showed how BSDEs

set-ting (see Section 2) and related results (see, for example,

the predictable representation theorem,Theorem 13, and the

Girsanov theorem,Theorem 7) can be fruitfully exploited to obtain concrete results in a wide set of problems which arise in modern theory of quantitative finance, with a focus

on those related to option pricing/hedging (seeSection 3).

In particular, after having recalled standard results about

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we moved to the more recent settings characterized by the BSDEs’ approach with perturbations of L´evy type (see

Section 2) whose related techniques are then exploited (see

Section 3.4) to develop an innovative model focusing our attention on a less standard life insurance scenario. We would like to note that, despite the fact that the mathematical literature about such kind of problems is not so developed, insurance contracts actually constitute a large market share within the set of all the traded contract nowadays. Let us note that even if the generalizations of the Black-Scholes model to nonconstant volatility models plus jump perturbations (see

Sections 3.2 and 3.3) better fit empirical time series, they

miss analytical solutions: therefore numerical evaluation is

necessary; see, for example [8,18].

We would like to underline that BSDEs’ field of potential applications is much wider than it has been stated in the present paper. As examples there are recent applications con-cerning the so-called dynamic risk measures which are based upon nonlinear expectations (g-expectations) developed by

Peng (see, for example [2,30,31]) BSDEs with time-delayed

generator (see, for example, [2,32]) where the problem of a

big investor who can influence the market prices is treated,

and BSDE with reflecting barriers (see, for example, [2]) also

in connection with applications to exotic American options pricing and portfolio with consumption problems (see, for

example, [33]).

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the anonymous referee for his comments and suggestions.

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