Pricing insurance drawdown-type contracts with underlying L\'evy assets

0.76 MB·

by  Zbigniew Palmowski

PRICING INSURANCE DRAWDOWN-TYPE CONTRACTS
WITH UNDERLYING LE´VY ASSETS
ZBIGNIEWPALMOWSKIANDJOANNATUMILEWICZ
Abstract. In this paper we consider some insurance policies related with drawdown and drawup
eventsoflog-returnsforanunderlyingassetmodeledbyaspectrallynegativegeometricL´evyprocess.
We consider four contracts among which three were introduced in [16] for a geometric Brownian
motion. Thefirstoneisaninsurancecontractwhereprotectionbuyerpaysaconstantpremiumuntil
thedrawdownoffixedsizeoflog-returnsoccurs. Inreturnhe/shereceivescertaininsuredamountat
7 thedrawdownepoch. Nextinsurancecontractprovidesprotectionfromanyspecifieddrawdownwith
1 adrawupcontingency. Thiscontractexpiresearlyifcertainfixeddrawupeventoccurspriortofixed
0 drawdown. Thelasttwocontractsareextensionsofthepreviousonesbyadditionalcancellablefeature
2 whichallowsaninvestortoterminatethecontractearlier. Wefocusontwoproblems: calculatingthe
fairpremiumpforthebasiccontractsandidentifyingtheoptimalstoppingruleforthepoliceswith
n cancellablefeature. Todothiswesolvesometwo-sidedexitproblemsrelatedwiththedrawdownand
a thedrawupofspectrallynegativeL´evyprocesseswhichisofownscientificinterest. Wealsoheavily
J
relyonatheoryofoptimalstopping.

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Keywords. insurance contract (cid:63) fair valuation (cid:63) drawdown (cid:63) drawup (cid:63) L´evy process (cid:63) optimal

] stopping.
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Contents
v
1
1. Introduction 2
9
2. Preliminaries 3
8
1 3. Drawdown insurance contract 4
0 3.1. Fair premium 4
. 3.2. Cancellable feature 6
1
0 4. Incorporating drawup cotingency 8
7 4.1. Fair premium 8
1 4.2. Cancellable feature 10
:
v 4.3. Striking simple case when a=b 12
i 5. Numerical analysis 13
X
5.1. Fair premium for drawdown insurance 14
r
a 5.2. Cancellable drawdown insurance 15
5.3. Fair premium for drawup contingency when a>b 16
5.4. Fair premium for drawup contingency when a=b 18
5.5. Cancellable drawup contingency for a>b 19
5.6. Cancellable drawup contingency for a=b 19
6. Conclusions 19
References 20
Date:January10,2017.
1991 Mathematics Subject Classification. C61,G01,G13,G22.
ThisworkispartiallysupportedbyNationalScienceCentreGrantNo. 2015/17/B/ST1/01102(2016-2019).
1
2 Z.Palmowski—J.Tumilewicz
1. Introduction
The drawdown of a given process is the distance of the current value away from the maximum
value it has attained to date. Similarly, the drawup of a given process is defined as the current rise of
the present value over the running minimum. Both of them have been customarily used as dynamic
risk measures. In fact, drawdown process does not only provide dynamic measures of risk, but can
also be viewed as measures of relative regret. Similarly drawup process can be viewed as measures of
relative satisfaction. Thus, a drawdown or a drawup of a certain number may signal the time when
investor may choose to change his/her investment position, which depends on his/her perception of
future moves of the market and his/her risk aversion.
The interest in drawdown asset has been strongly raised by recent financial crisis. Large market
drawdown may bring portfolio losses, liquidity shocks and even future recessions. Therefore risk
management of drawdown has become so important among practitioners; see e.g. [2] for portfolio
optimization under constraints on the drawdown process, [1, 5] for the distribution of the maximum
drawdownofdriftedBrownianmotionandtime-adjustedmeasureofperformanceknownastheCalmar
ratioand[11,13,14]fordrawdownprocessasdynamicmeasureofrisk. Foranoverviewoftheexisting
techniques for analysis of market crashes as well as a collection of empirical studies of the drawdown
process and the maximum drawdown process, please refer to Sornette [12].
Itisthennaturalthatfundmanagershavestrongincentivetoseekinsuranceagainstdrawdown. In
fact, as works of [1, 13, 14] argue, some market-traded contracts, such as vanilla and look-back puts,
have only limited ability to insure the market drawdown. Therefore the drawdown protection can be
useful also for individual investors.
In this paper we follow Zhang et al. [16] pricing some insurance contracts against drawdown (and
drawup) events of log-returns of stock price modeled by exponential L´evy process and identifying the
optimal stopping rules. We also identify for these contracts so-called fair premium rates for which
contracts prices equal zero.
Initssimplestform,thefirstdrawdowninsurancecontractinvolvesacontinuouspremiumpayment
by the investor (protection buyer) to insure a drawdown of log-returns of the underlying asset over
a pre-specified level. Possible buyer of this insurance contract might think that it is unlikely to get
largedrawdownandhe/shemightwanttostoppayingthepremium. Thereforeweexpandthesimplest
contractbyaddingcancellablefeature. Inthiscase,theinvestorreceivesrighttoterminatethecontract
earlierandinthiscasehe/shepaypenaltyfordoingso. Weshowthattheinvestor’soptimalcancellable
time is based on the first passage time of the drawdown of log-return process.
Moreover,wealsoconsiderarelatedinsurancecontractthatprotectstheinvestorfromadrawdown
of log-return of the asset price preceding a drawup related with it. In other words, the insurance
contract expires early if a drawup event occurs prior to a drawdown. From the investors perspective,
when a drawup is realized, there is little need to insure against a drawdown. Therefore, this drawup
contingency automatically stops the premium payment and it is an attractive feature that could po-
tentially reduce the cost of the drawdown insurance. Finally, we also added cancellable feature to this
contract.
Zhangetal. [16]consideredonlyariskyassetmodeledbythegeometricBrownianmotion. However,
inrecentyears,empiricalstudyoffinancialdatarevealsthefactthatthedistributionofthelog-return
of stock price exhibits features which cannot be captured by the normal distribution such as heavy
tailsandasymmetry. Forthepurposeofreplicatingmoreeffectivelythesefeaturesandforreproducing
a wide variety of implied volatility skews and smiles, there has been a general shift in the literature to
modeling a risky asset with an exponential L´evy process as an alternative to exponential of a linear
Brownianmotion;seeKyprianou[3]andØksendalandSulem[8]foroverviews. Thereforelookingfora
betterfittingoftheevolutionofthestockpriceprocesstotherealdata,inthispaperwepricederivative
securities in market by a general geometric spectrally negative L´evy process. That is, logarithm of a
risky asset in our case will be a process with stationary and independent increments with no positive
jumps.
Thelastcontractanalyzedinthispapertakingintoaccountdrawdownanddrawupwithcancellable
feature is considered for the first time in literature. At the same, although it is the most complex one,
it produces very interesting and surprising results. In particular, we discover new phenomenon for
optimal stopping contract rule in this contract. In the phenomenon, the investor’s stopping rule is
also at a first passage time of the drawdown of log-returns process, similarly like it is for the second
PricinginsuranceLe´vy-drawdown-typecontracts 3
contract without drawup contingency. Still, the level of termination is different taking into account
drawup event.
Our approach is based on the classical fluctuation theory for the spectrally negative L´evy processes
(related with so-called scale functions) and some new exit identities for reflected L´evy processes. The
latter ones identify two-sided exit problems for drawup and drawdown first passage times. A key
elementofourapproachispathanalysisandusingsomeresultsofMijatovi´candPistorius[6]. Wealso
heavily use optimal stopping theory. In a market where the underlying dynamics for the stock price
process is driven by the exponential of a linear Brownian motion the valuation is transformed into a
free boundary problem. However, by allowing jumps to appear in the sample paths of the underlying
dynamics of the stock price process, this idea breaks down. To tackle these infinite horizon problem
we use the so-called ,,guess and verify” method. For this method, one guesses what the optimal value
functionandoptimalstoppingshouldbeand thentriestoverifythat this candidatesolutionisindeed
the optimal one by putting it through a verification theorem. What is meant by the latter is that the
value function identified by guessed stopping rule applied to the log-return price process constructs a
smallest, in some sense, discounted supermartingale.
In this paper we also analyze many particular examples and give extensive numerical analysis
showing the dependence of the contract and stopping time on the model’s parameters. We focus
mainly on the case when logarithm of the asset price is a linear Brownian motion or drift minus
compound Poisson process (so-called Cram´er-Lundberg risk process).
The paper is organized as follows. In Section 2 we introduce main definitions, notations and main
fluctuation identities. We analyze insurance contracts based on drawdown and additional drawup in
Sections 3 and 4, respectively. We finish our paper by the numerical analysis performed in Section 5
and Conclusions in Section 6.
2. Preliminaries
We work on complete filtered probability space (Ω,F,P) satisfying the usual conditions. We model
a logarithm of the underlying risky asset price logS by a spectrally negative L´evy process X , that is
t t
S =exp{X } is a geometric L´evy process. This means that X is a stationary stochastic process with
t t t
independent increments, right-continuous paths with left limits and has only negative jumps.
Manyidentitieswillbegivenintermsofso-calledscalefunctionswhicharedefinedinfollowingway.
We start from defining so-called Laplace exponent of X :
t
(1) ψ(φ)=logE[eφX1]
which is well defined for φ≥0 due to the absence of positive jumps. Recall that by L´evy-Khintchine
theorem:
(2) ψ(φ)=µφ+ 1σ2φ2+(cid:90) (cid:0)e−φu−1+φu1 (cid:1)Π(du),
2 (u<1)
(0,∞)
whichisanalyticforIm(φ)≤0,whereµandσ ≥0arerealandΠisso-calledL´evymeasure. Itiseasy
to observe that ψ is zero at the origin, tends to infinity at infinity and is strictly convex. We denote
by Φ:[0,∞)→[0,∞) the right continuous inverse of ψ so that it satisfies the following:
Φ(r)=sup{φ>0:ψ(φ)=r} and ψ(Φ(r))=r for all r ≥0.
For r ≥ 0 we define a continuous and strictly increasing function W(r) on [0,∞) with the Laplace
transform given by:
(cid:90) ∞ 1
(3) e−φuW(r)(u)du= ,
ψ(φ)−r
0
where ψ is a Laplace exponent of X given in (1). It is so-called the first scale function. The second
t
one is related with the first one via the following relationship:
(cid:90) u
(4) Z(r)(u)=1+r W(r)(φ)dφ.
0
In this paper we will assume that
(5) W(r) ∈C1(R )
+
4 Z.Palmowski—J.Tumilewicz
forR =[0,∞). ThisassumptionissatisfiedwhentheprocessX hasnon-trivialgaussiancomponent
+ t
or it is of unbounded variation or the jumps have the density; see [4, Lem. 2.4]. The scale functions
are used in two-sided exit formulas:
(cid:104) (cid:105) W(r)(x)
(6) Ex e−rτa+;τa+ <τ0− = W(r)(a),
(cid:104) (cid:105) W(r)(x)
(7) Ex e−rτ0−;τ0− <τa+ =Z(r)(x)−Z(r)(a)W(r)(a),
where x≤a, r ≥0 and
(8) τ+ =inf{t≥0:X ≥a}, τ− =inf{t≥0:X ≤a}
a t a t
are the first passage times. We also used the following notational convention: E[· 1 ] = E[·;A] for
{A}
any event A.
Let us denote:
X =supX , X = infX .
t s t s
s≤t s≤t
Inthispaper,weanalyzesomeinsurancecontractsrelatedwiththedrawdownanddrawupprocessesof
log-returnoftheassetpriceS ,thatis,withthedrawdownanddrawupprocessesofX . Thedrawdown
t t
is the difference between running maximum of the underlying process and its current value and the
drawup is difference between process current value and its running minimum. Here, we additionally
assumethatthedrawdownanddrawupprocessesstartfromsomepointsy >0andz >0,respectively.
That is,
(9) D =X ∨y−X , U =X −X ∧(−z).
t t t t t t
Above the values y and −z can be interpreted as historical maximum and historical minimum of
process X. The crucial for further work are the following first passage times of the drawdown process
and the drawup process, respectively:
(10) τ+(a)=inf{t≥0:D ≥a}, τ−(a)=inf{t≥0:D ≤a},
D t D t
(11) τ+(a)=inf{t≥0:U ≥a}, τ−(a)=inf{t≥0:U ≤a}.
U t U t
Later, we will use the following notational convention:
P [·]:=P[·|D =y], P [·]:=P[·|D =y,U =z], P [·]:=P[·|X =x,D =y,U =z].
|y 0 |y|z 0 0 x|y|z 0 0 0
Finally, we denote P [·]:=P[·|X =x] with P=P and E ,E ,E ,E ,E are the corresponding
x 0 0 |y |y|z x|y|z x
expectations to above measures. We finish this section with two main formulas (the fist one is given
in Mijatovi´c and Pistorius [6, Thm.4] and the second one follows from (6)) that identify the joint laws
of {τ+,X , X } and {τ−(θ), X }:
U τ+ τ+ D τ−(θ)
U U D
E(cid:20)e−rτU+(b)+uXτU+(b);XτU+(b) <v(cid:21)=eub11++(r(r−−ψψ(u(u))))(cid:82)0(cid:82)b−bve−e−uyuWyW(r()r()y(y)d)ydy
0
W(r)(b−v)
(12) −e−u(b−v) ,
W(r)(b)
(cid:104) (cid:105) (cid:104) (cid:105) W(r)(x)
(13) E|y e−rτD−(θ);XτD−(θ) >−x =Ex e−rτy+−θ+x;τy+−θ+x <τ0− = W(r)(y−θ+x).
3. Drawdown insurance contract
3.1. Fair premium. In this section, we consider the insurance contract in which protection buyer
pays a constant premium p ≥ 0 continuously until the drawdown of log-returns of the asset price of
size a > 0 occurs. In return she/he receives the insured amount α ≥ 0 at the drawdown epoch. Let
r ≥ 0 be the risk-free interest rate. The contract price is equal to the discounted value of the future
cash-flows:
(cid:34) (cid:90) τ+(a) (cid:35)
(14) f(y,p)=E|y − D e−rtpdt+αe−rτD+(a) .
0
PricinginsuranceLe´vy-drawdown-typecontracts 5
Note that in this contract the investor wants to protect herself/himself from the asset price St =eXt
falling down from the previous maximum more than fixed level ea for some a > 0. In other words,
she/he believes that even if the price will go up again after the first drawdown of size ea it will not
bring her/him sufficient profit. Therefore, she/he is ready to take this type of contract to reduce loss
by getting α>0 at the drawdown epoch.
Note that
(cid:16)p (cid:17) p
(15) f(y,p)= +α ξ(y)− ,
r r
where
(cid:104) (cid:105)
(16) ξ(y):=E|y e−rτD+(a)
is the conditional Laplace transform of τ+(a) given that D = y ∈ (0,a). To price the contract (14)
D 0
we start from identifying the crucial function ξ.
Proposition 1. The conditional Laplace transform ξ(·) is given by
W(r)(a)
(17) ξ(y)=Z(r)(a−y)−rW(r)(a−y) .
W(cid:48)(r)(a)
Proof. Note that τ−(0) is the first time that the drawdown process D pass the level 0, which means
D t
that the the process X attains its historical maximum. It is done in a continuous way by assumed
t
spectralnegativityofL´evyprocessX. BythestrongMarkovpropertyofprocessD atτ−(0)wehave
t D
that
(cid:104) (cid:105)
ξ(y)=E|y e−rτD+(a)
(cid:104) (cid:105) (cid:104) (cid:105)
=E|y e−rτD+(a);τD+(a)<τD−(0) +E|y| e−rτD−(0);τD−(0)<τD+(a) ξ(0)
(cid:104) (cid:105) (cid:104) (cid:105)
=Ea−y e−rτ0−;τ0− <τa+ +Ea−y e−rτa+;τa+ <τ0− ξ(0)
W(r)(a−y) W(r)(a−y)
(18) =Z(r)(a−y)−Z(r)(a) + ξ(0),
W(r)(a) W(r)(a)
where the third equation follows from the two-sided exit formulas given in (6) - (7). Therefore the
problem of finding ξ is reduced to identifying ξ(0). The latter one can be obtained from [10, Prop.
2(ii), p. 191]:
W(r)(a)
(19) ξ(0)=Z(r)(a)−rW(r)(a) .
W(cid:48)(r)(a)
This completes the proof. (cid:3)
Thus we have the following theorem.
Theorem 1. The value of the contract (14) is given in (15) for ξ identified in (17).
The fair situation for both sides, the insurance company and investor, is when contract price at
conclusion moment equals 0. We say then that the premium p∗ is fair when
f(y,p∗)=0.
From (15) using Proposition 1 we derive the following theorem.
Theorem 2. For the contract (14) the fair premium equals:
rαξ(y)
(20) p∗ = .
1−ξ(y)
6 Z.Palmowski—J.Tumilewicz
3.2. Cancellable feature. We now extend the previous contract by cancellable feature. In other
words, we give the investor right to terminate the contract by paying fixed fee c≥0 at any time prior
to a pre-specifies drawdown of log-return of the asset price of size a > 0. This contract is addressed
to the investors who are not willing to pay premium any longer after they stopped to believe that a
large drawdown of the asset price may happen. The contract value equals then:
(cid:34) (cid:90) τ+(a)∧τ (cid:35)
(21) F(y,p)= supE|y − D e−rtpdt−ce−rτ1(τ<τ+(a))+αe−rτD+(a)1(τ+(a)≤τ) ,
τ∈T 0 D D
where T is a family of all F -stopping times.
t
Oneofthemaingoalsofthispaperistoidentifytheoptimalstoppingruleτ∗ thatrealizestheprice
F(y,p). We start from the simple observation.
Proposition 2. The cancellable drawdown insurance value admits the following decomposition:
(22) F(y,p)=f(y,p)+G(y,p),
where
(23) G(y,p):= supg (y,p),
τ
τ∈T
(24) f˜(y,p):=−f(y,p)−c
for
(cid:104) (cid:105)
(25) g (y,p):=E e−rτf˜(D ,p);τ <τ+(a)
τ |y τ D
and f(·,·) is defined in (14).
Proof. Using 1 =1−1 in (21) we obtain:
(τ≥τ+(a)) (τ<τ+(a))
D D
(cid:34) (cid:90) τ+(a) (cid:35)
F(y,p)=E|y − D e−rtpdt+αe−rτD+(a)
0
(cid:34)(cid:90) τ+(a) (cid:35)
+supE|y D e−rtpdt−αe−rτD+(a)1(τ<τ+(a))−ce−rτ1(τ<τ+(a)) .
τ∈T τ∧τ+(a) D D
D
Notethatthefirsttermdoesnotdependonτ. Thesecondtermdependsonτ onlythroughτ <τ+(a).
D
Then by strong Markov property we get:
F(y,p)=f(y,p)
(cid:34)(cid:90) τ+(a) (cid:35)
+supE|y D e−rtpdt−αe−rτD+(a)1(τ<τ+(a))−ce−rτ1(τ<τ+(a))
τ∈T τ D D
(cid:34) (cid:32)(cid:90) τ+(a) (cid:33) (cid:35)
=f(y,p)+supE|y e−rτE|Dτ D e−rtpdt−αe−rτD+(a)−c ; τ <τD+(a) .
τ∈T 0
This completes the proof. (cid:3)
Observe now that f˜(y,p) in (24) is a decreasing function with respect to y. Thus, if f˜(0+,p) < 0,
then the optimal stopping strategy for the investor is to never terminate the contract, that is τ =∞.
To eliminate this trivial case we assume from now that
(26) f˜(0+,p)>0
which is equivalent to saying that
p (cid:16)p (cid:17)
(27) −c> +α ξ(0+)≥0.
r r
In order to determine the optimal cancellation strategy for our contract it is sufficient to solve the
optimal stopping problem represented by second term in (22), that is to identify G(y,p). We will use
the ,,guess and verify” approach. This means that we first guess the candidate stopping rule and then
verify if this is truly the optimal stopping rule using the Verification Lemma given below.
PricinginsuranceLe´vy-drawdown-typecontracts 7
Lemma 3.1. Let Υ be a right-continuous process living in some Borel state space B killed at some
t
FΥ-stopping time τ , where FΥ is a right-continuous natural filtration of Υ. Consider the following
t 0 t
stopping problem:
(28) v(φ)= sup E(cid:2)e−rτV(Υ )|Υ =φ(cid:3)
τ 0
τ∈TΥ
for some function V and the family FΥ-stopping times TΥ. Assume that
t
(29) P(lim e−rtV(Υ )<∞|Υ =φ)=1.
t 0
t→∞
The pair (v∗,τ∗) is a solution of stopping problem (28), that is
(cid:104) (cid:105)
v∗(φ):=E e−rτ∗V(Υ )|Υ =φ ,
τ∗ 0
if the following conditions are satisfied:
i) v∗(φ)≥V(φ) for all φ∈B,
ii) the process e−rtv∗(Υ ) is right continuous supermartingale.
t
Proof. The proof follows the same arguments as the proof of [3, Lem. 9.1, p. 240]; see also [9, Th.
2.2, p. 29]. (cid:3)
Using above Verification Lemma 3.1 we will prove that the first passage time of the drawdown
process D below some level θ is the optimal stopping time for (23) (hence also for (22)). That is, we
t
will prove that
(30) τ∗ =τ−(θ)∈T
D
for an appropriate chosen θ ∈[0,a).
For the stopping rule (30) and for y >θ we will compute explicitly g (y,p) given in (25). Note
τ−(θ)
D
that, if y >θ, then
(cid:104) (cid:105) W(r)(a−y)
(31) g(y,p,θ):=gτD−(θ)(y,p)=f˜(θ,p)E|y e−rτD−(θ);τD−(θ)<τD+(a) =f˜(θ,p)W(r)(a−θ).
Furthermore, if y ≤θ then the investor will terminate contract immediately:
(cid:104) (cid:105)
(32) g(y,p,θ)=E|y e−rτD−(θ)f˜(Dτ−(θ),p);τD−(θ)<τD+(a) =f˜(y,p).
D
Thus, for θ ∈[0,y] we have:
F(y,p,θ)=f(y,p)+g(y,p,θ)
(cid:16)p (cid:17) (cid:16)p (cid:17)W(r)(a−y) (cid:16)p (cid:17) W(r)(a−y) p
(33) = +α Z(r)(a−y)+ −c − +α Z(r)(a−θ) − .
r r W(r)(a−θ) r W(r)(a−θ) r
Recall that by (22) the optimal level θ∗ maximizes both value functions F(y,p) and G(y,p). Thus
(cid:26) (cid:27)
∂
(34) θ∗ =inf θ ∈[0,a): g(y,p,θ)=0 and for all ς ≥0 g(y,p,ς)≤g(y,p,θ) .
∂θ
Note that θ∗ >0 because g(y,p,θ) increases at θ =0. Indeed,
(cid:20) ∂ (cid:21) W(r)(a−y) W(r)(a−y)W(cid:48)(r)(a)
g(y,p,θ) =f˜(cid:48)(0) +f˜(0) >0,
∂θ W(r)(a) (W(r)(a))2
|θ=0
where the last inequality follows from assumption (26) and fact that f˜(cid:48)(0) = −(p +α)ξ(cid:48)(0) = 0. We
r
will verify now that (30) indeed holds true, that is, τ−(θ∗) is an optimal stopping rule.
D
Theorem 3. Assume that (27) holds. The stopping time τ−(θ∗), with θ∗ defined in (34), is the
D
optimal stopping rule for the stopping problems (23) and (21). Moreover, the price of the drawdown
insurance contract with cancellable feature equals F(y,p)=f(y,p)+g(y,p,θ∗).
Proof. Basedontheoptimalstoppingproblem(23)itissufficienttocheckthatτ∗ =τ−(θ∗)fulfillstwo
D
conditions presented in Verification Lemma 3.1 with Υ = D , B = R , τ = τ+(a), V(x) = f˜(x,p).
t t + 0 D
Note that in this case the assumption (29) is clearly satisfied. In order to prove (i) of Verification
8 Z.Palmowski—J.Tumilewicz
Lemma 3.1 it suffices to show that g(y,p,θ)−f˜(y,p) ≥ 0 for some θ. Observe that taking θ = y for
y ∈(0,a) produces:
(cid:104) (cid:105)
(35) g(y,p,a)−f˜(y,p)=E|y e−rτD−(y)f˜(Dτ−(y),p);τD−(y)<τD+(a) −f˜(y,p)=0,
D
where f˜(y,p) is given in (24). Thus condition (i) of Verification Lemma 3.1 follows from the fact that
θ∗ maximizes the function g(y,p,·).
To prove condition (ii) of Verification Lemma 3.1 note that by the strong Markov property the
process
e−r(t∧τD+(a)∧τD−(θ∗))g(Dt∧τ+(a)∧τ−(θ∗),p,θ∗)=E|y[e−rsg(Ds,p,θ∗)|Ft∧τ+(a)∧τ−(θ∗)]
D D D D
is a martingale. Hence
A g(y,p,θ∗)−rg(y,p,θ∗)=0
D
for y ∈(θ∗,a) and for the full generator A of the process D. Moreover, from (32) we know that for
D
y ∈(0,θ∗) we have g(y,p,θ∗)=f˜(y,p). Therefore, for y ∈(0,θ∗),
A g(y,p,θ∗)−rg(y,p,θ∗)=A f˜(y,p)−rf˜(y,p)
D D
=−A f(y,p)+rf(y,p)+rc
D
(cid:16)p (cid:17) (cid:16)p (cid:17)
=− +α [A ξ(y)−rξ(y)]−r −c .
r D r
Now, the strong Markov property of the process Dt implies that process e−rt∧τD+(a)ξ(Dt∧τ+(a)) =
E [e−rsξ(D )|F ] is a martingale. Hence A ξ(y)−rξ(y)=0 for y ∈(0,θ∗) since θ∗ <aD.
|y s t∧τ+(a) D
D
Thus the process e−rt∧τD+(a)g(Dt∧τ+(a),p,θ∗) is a supermartingale because for y ∈(0,θ∗) we have:
D
(cid:16)p (cid:17)
A g(y,p,θ∗)−rg(y,p,θ∗)=−r −c ≤0,
D r
where the last inequality follows from the assumption (27). This completes the proof. (cid:3)
4. Incorporating drawup cotingency
4.1. Fair premium. The investor might like to buy a contract which has some maturity conditions.
This means that this contract would end when these conditions are fulfilled. Therefore in this paper
we also consider the insurance contract which provides protection from any specified drawdown of
log-return of the asset price with certain drawup contingency. In particular, this contract may expire
earlier if a fixed drawup event of log-return of the stock price occurs prior to some fixed drawdown of
it. Choosing the drawup event is very natural since it corresponds to some market upward trend and
thereforetheinvestormaywanttostoppayingpremiumwhenthiseventhappens. Underarisk-neutral
measure the value of this contract equals:
(cid:34) (cid:90) τ+(a)∧τ+(b) (cid:35)
(36) k(y,z,p):=E|y|z − D U e−rtpdt+αe−rτD+(a)1(τ+(a)≤τ+(b)) ,
0 D U
for some fixed a > b > 0. At the beginning we will find this value function and then we identify the
fair premium p∗ under which
(37) k(y,z,p∗)=0.
Note that
(cid:16)p (cid:17) p p
(38) k(y,z,p)= +α ν(y,z)+ λ(y,z)− ,
r r r
where
(cid:104) (cid:105)
ν(y,z):=E|y|z e−rτD+(a); τD+(a)≤τU+(b) ,
(cid:104) (cid:105)
λ(y,z):=E|y|z e−rτU+(b); τU+(b)<τD+(a) .
To get the formulas for ν and λ we have to make some additional observations.
PricinginsuranceLe´vy-drawdown-typecontracts 9
Proposition 3. Let y and z denote starting position for drawdown and drawup processes, respectively.
For a>b≥0 the following events are equivalent:
(cid:8)τ+(b)<τ+(a), D =y, U =z(cid:9)={τ+ <τ− }
U D 0 0 b−z (y−a)∨(−z)
(cid:110) (cid:111)
(39) ∪ X ∨y−X <a, X ≤−z ,
τ+(b) τ+(b) τ+(b)
U U U
(cid:8)τ+(a)<τ+(b), D =y, U =z(cid:9)={τ− <τ+ , y−a≥−z}
D U 0 0 y−a (b−z)
(cid:110) (cid:111)
(40) ∪ X ∨y−X ≥a, X ≤−z, X ≤b−z, y−a<−z .
τ+(b) τ+(b) τ+(b) τ+(b)
U U U U
Proof. The proof of this proposition follows from the geometric path arguments. To prove (39) note
that
(cid:8)τ+(b)<τ+(a), D =y, U =z(cid:9)=(cid:110)τ+(b)<τ+(a), X >−z, D =y, U =z(cid:111)
U D 0 0 U D τ+(b) 0 0
U
(cid:110) (cid:111)
∪ τ+(b)<τ+(a), X ≤−z, D =y .
U D τ+(b) 0
U
The event {X > −z, U = z} is equivalent to the requirement that X = b−z and that the
τ+(b) 0 τ+(b)
U U
underlying process X cannot cross y−a level before τ+(b). Therefore,
t U
(cid:110) (cid:111) (cid:110) (cid:111)
(41) τ+(b)<τ+(a), X >−z, D =y, U =y = τ+ <τ− .
U D τ+(b) 0 0 b−z (y−a)∨(−z)
U
Moreover, when X ≤ −z we have X = b−X . Using now the fact that the drawdown
τ+(b) τ+(b) τ+(b)
U U U
occurs after drawup, we can conclude that D <a−b. Thus
τ+(b)
U
(cid:110) (cid:111) (cid:110) (cid:111)
(42) τ+(b)<τ+(a), X ≤−z, D =y = X ∨y−X <a, X ≤−z .
U D τ+(b) 0 τ+(b) τ+(b) τ+(b)
U U U U
Observations (41) and (42) complete the proof of (39). To prove (40) we again consider two scenarios:
(cid:8)τ+(a)<τ+(b), D =y, U =z(cid:9)=(cid:8)τ+(a)<τ+(b), D =y, U =z, y−a<−z(cid:9)
D U 0 0 D U 0 0
∪(cid:8)τ+(a)<τ+(b), D =y, U =z, y−a≥−z(cid:9).
D U 0 0
The case y −a > −z together with the assumption b ≤ a imply that b−a < y. This means that
the event when drawdown occurs before drawup is the same as the one when process X crosses y−a
before it hits b−z. That is,
(cid:8)τ+(a)<τ+(b), D =y, U =z, y−a≥−z(cid:9)=(cid:110)τ− <τ+ , y−a≥−z(cid:111).
D U 0 0 y−a (b−z)
If y−a≤−z then the process X is crossing −z before the drawdown event occurs. Additionally, the
underlying process X can cross level y but it cannot cross level b−z, because otherwise the drawup
would occur. Thus,
(cid:8)τ+(a)<τ+(b), D =y, U =z, y−a<−z(cid:9)
D U 0 0
(cid:110) (cid:111)
= X ∨y−X >a, X ≤−z, X ≤b−z, y−a<−z .
τ+(b) τ+(b) τ+(b) τ+(b)
U U U U
This completes the proof of (40). (cid:3)
Note that, for b<a, we have
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
E|y|z e−rτU+(b); τD+(a)<τU+(b) =E|y|z e−rτD+(a); τD+(a)<τU+(b) E e−rτU+(b)
Proposition 3 and above observation produce the following crucial corollary.
Corollary 4. For a>b we have:
(cid:104) (cid:105)
ν(y,z)=E e−rτy−−a;τy−−a <τ(+b−z) 1(y+z≥a)
(cid:104) (cid:105) 1
+E e−rτU+(b);XτU+(b)∨y−XτU+(b) ≥a,XτU+(b) ≤−z, XτU+(b) ≤b−z E(cid:104)e(y−+rzτ<U+a(b))(cid:105)
and
(cid:104) (cid:105)
λ(y,z)=E e−rτb+−z;τ+ <τ−
b−z (y−a)∨(−z)
(cid:104) (cid:105)
+E e−rτU+(b);Xτ+(b)∨y−Xτ+(b) <a, Xτ+(b) ≤−z .
U U U
10 Z.Palmowski—J.Tumilewicz
Both functions λ and ν can be now calculated by taking inverse Laplace transform of (12).
Theorem 5. The price of the contract (36) is given in (38) with λ and ν identified in Corollary 4.
From (38) if follows the following theorem.
Theorem 6. For the contract (36) the fair premium defined in (37) equals:
rαν(y,z)
(43) p∗ = ,
1−λ(y,z)−ν(y,z)
where functions λ and ν are given in Corollary 4.
4.2. Cancellable feature. We will also consider additional possibility of terminating the previous
contract. Now, the protection buyer can terminate the position by paying fee c≥0 for doing it. The
value of this contract equals then
(cid:34) (cid:90) τ+(a)∧τ+(b)∧τ
K(y,z,p):=supE − D U e−rtpdt
|y|z
τ∈T 0
(cid:35)
(44) +αe−rτD+(a)1(τ+(a)<τ+(b)∧τ)−ce−rτ1(τ<τ+(a)∧τ+(b)) .
D U D U
Similarlylikeinthecaseofcancellabledrawdowncontractwecanrepresentthecontractvaluefunction
as the sum of two parts: one without cancellable feature and one that depends on a stopping time τ.
Proposition 4. The cancellable drawup insurance value admits the decomposition:
(45) K(y,z,p)=k(y,z,p)+H(y,z,p),
where
(46) H(y,z,p):= suph (y,z,p),
τ
τ∈T
(cid:104) (cid:105)
(47) h (y,z,p):=E e−rτk˜(D ,U ,p); τ <τ+(a)∧τ+(b) ,
τ |y|z τ τ D U
(48) k˜(y,z,p):=−k(y,z,p)−c
and k is given in (38).
Proof. The proof follows from the following equality:
(cid:34) (cid:90) τ+(a)∧τ+(b) (cid:35)
K(y,z,p)=E|y|z − D U e−rtpdt+αe−rτD+(a)1(τ+(a)<τ+(b))
0 D U
+supE (cid:34)e−rτ1 E (cid:104)(cid:90) τD+(a)∧τU+(b)e−rtpdt(cid:105)
τ∈T |y|z (τ<τD+(a)∧τU+(b)) |Dτ|Uτ 0
(cid:35)
(49) −αe−rτD+(a)1(τ<τ+(a)<τ+(b))−ce−rτ1(τ<τ+(a)∧τ+(b)) .
D U D U
(cid:3)
At the beginning note that, if k˜(D ,U ) < 0 for all θ, then it is not optimal to terminate
τ−(θ) τ−(θ)
D D
the contract and hence τ = ∞. For avoiding this case we assume from this point, that there exist θ
0
for which k˜(D ,U )>0. We can rewrite this assumption as follows:
τD−(θ0) τD−(θ0)
p (cid:16)p (cid:17) p
(50) −c> +α ν(θ ,y+z−θ )+ λ(θ ,y+z−θ )≥0
r r 0 0 r 0 0
for y+z ≥a and
p (cid:16)p (cid:17) p
(51) −c> +α ν(θ ,y−x −θ )+ λ(θ ,y−x −θ )≥0
r r 0 0 0 r 0 0 0
for y+z <a, where x satisfies k˜(θ ,y−x −θ )= min k˜(θ ,y−x−θ ). Additionally, because
0 0 0 0 0 0
x∈(y−a,−z)
of the presence of indicator in (47) without loss of generality we can assume that
b−z >y−θ.