Exotic Options under Levy Models: An Overview

Wim Schoutens∗

September 28, 2004

Abstract

In this paper we overview the pricing of several so-called exotic options

in the nowdays quite popular exponential Levy models.

∗K.U.Leuven, U.C.S., W. De Croylaan 54, B-3001 Leuven, Belgium. Email:

Wim.Schoutens@wis.kuleuven.ac.be

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1 Introduction

In recent years more and more attention has been given to stochastic models offinancial markets which depart from the traditional Black-Scholes model [18].Nowadays, a battery of models is available. Some of the most popular and stilltractable models are the Levy models. For an introduction on these modelsapplied to finance, we refer to [96]. These models are able to take into accountdifferent important stylised features of financial time series. However, since themodels are of a higher complexity, pricing financial derivatives and in particularthe so-called exotic derivatives is not easy. Recently a lot of papers on this topicappeared in the literature. For a state-of-the art we refer to [60] and the referencecited therein. Here, we give a brief overview of the literature and the differenttechniques to price vanilla and exotic options. We first focus on Europeanoptions. We depart by pricing the vanilla call options, using i.a. (Fast) Fouriertransforms. Next, we move to more complicated payoff function and will treatdigital, barrier, lookback, Asian and American options using i.a. Wiener-Hopffactorization theory, numerical simulation algortihms solving Partial Integro-Differential Equation/Inequalities (PIDE/PIDI) and comonotonicity theory.

This paper is organized as follows. First we give a brief introduction to Levyprocesses and the associated Levy market models. In Section 2, we overview thepricing of European-type options. Section 3 is devoted to American options.

1.1 Levy Processes

Suppose φ(u) is the characteristic function of a distribution. If for every positiveinteger n, φ(u) is also the nth power of a characteristic function, we say thatthe distribution is infinitely divisible.

One can define for every such an infinitely divisible distribution a stochasticprocess, X = {Xt, t ≥ 0}, called Levy process, which starts at zero, has indepen-dent and stationary increments and such that the distribution of an incrementover [s, s+ t], s, t ≥ 0, i.e. Xt+s −Xs, has (φ(u))t as characteristic function.

Every Levy process has a cadlag modification which is itself a Levy process.We always work with this cadlag version. So sample paths of a Levy processare a.e. continuous from the right and have limits from the left. Moreover wealways will work in the sequel with the natural filtration generated by the Levyprocess X .

The cumulant characteristic function function ψ(u) = logφ(u) is often calledthe characteristic exponent and it satisfies the following Levy-Khintchine for-mula:

ψ(u) = iγu−σ2

2u2 +

∫ +∞

−∞

(exp(iux) − 1 − iux1{|x|<1})ν(dx), (1)

where γ ∈ R, σ2 ≥ 0 and ν is a measure on R\{0} with

∫ +∞

−∞

inf{1, x2}ν(dx) =

∫ +∞

−∞

(1 ∧ x2)ν(dx) <∞.

2

We say that our infinitely divisible distribution has a triplet of Levy character-istics (or Levy triplet for short) [γ, σ2, ν(dx)]. The measure ν is called the Levymeasure of X .

General reference works on Levy processes are [16], [91] and [8].

1.2 The Levy Market Model

If one departs from the Black-Scholes world one typically enters into the so-called incomplete market models. Roughly speaking incompleteness means thata general contingent claim can not be perfectly hedged. Most models are notcomplete, and most practitioners believe the actual market is not complete. Thequestion of completeness is linked with the uniqueness of the martingale measure[40]. In incomplete markets, we have to choose an equivalent martingale measurein some way and this is not always clear. Actually, the market is choosing themartingale measure for us. We do not go into detail about the choice of thismeasure and assume an appropriate (risk-neutral) martingale measure Q hasbeen chosen. Throughout, we will work under this risk-neutral measure Q.Note that it is quite common in practice to estimate (calibrate) the risk-neutralmeasure directly from market data (the volatility surface).

We work under a market which consists of one riskless asset (the bond) withprice process given by Bt = exp(rt) and one non-dividend paying risky asset(the stock or index). The model for the risky asset is given by

St = S0 exp(Xt),

where X = {Xt, t ≥ 0} is (under Q) a Levy process. Since we immediately workin a risk-neutral setting, the discounted stock price is a martingale and hence:

EQ[exp(−r(t− s))St|Fs] = Ss, 0 ≤ s ≤ t,

where F = {Ft, 0 ≤ t ≤ T} is the natural filtration of X = {Xt, 0 ≤ t ≤ T}This market model is often called the exponential Levy market model. Thelog-returns log(St+s/St) of such a model follow the distribution of incrementsof length s of the Levy process X .

In the literature several particular choices for the Levy processes were studiedin detail. [71] and [72] have proposed a VG Levy process (see also [69] and [70].In [45] the Hyperbolic model was proposed, and in [13] the NIG model (seealso [14] and [89]). All three above mentioned models where brought togetheras special cases of the Generalized Hyperbolic Model, which was developed byEberlein and co-workers in a series of paper ([46], [47], [48], [90] and [83]). [31]introduced the CGMY model; this familiy of distributions is by some authorsalso called the KoBoL family refering to [59] and [23] (see also [20], [36], [73],[24], and [26]). Finally, the Meixner model was used in [94] (see also [52], [92],[93] and [95]).

An accesible introduction, together with theoretical motivations to this Levymarket, can be found for example [49]. Some theoretical motivation for consid-ering Levy processes in finance can also be found in [61]. For an overview of

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the theory and the applications of Levy processes in finance we refer to [96] and[37].

2 European Options

Given our market model, let F ({St, 0 ≤ t ≤ T}) denote the payoff of an Euro-pean derivative at its time of expiry T . In case of the European call with strikeprice K, we have F ({St, 0 ≤ t ≤ T}) = F (ST ) = (ST −K)+. According to thefundamental theorem of asset pricing (see [40]) the arbitrage free price Vt of thederivative at time t ∈ [0, T ] is given by

Vt = EQ[exp(−r(T − t))F ({Su, 0 ≤ u ≤ T})|Ft], (2)

where the expectation is taken with respect to an equivalent martingale measureQ. The factor exp(−r(T − t)) is called the discounting factor.

2.1 Monte-Carlo Techniques

European options can easily be priced by making use of Monte-Carlo simula-tion. The idea is simple and classical. In order to evaluate the expectation in(2), one simulates a hugh amount of paths of the underlying Levy process andhence the stock price process. For every path one calculates the payoff function,one discounts, and finally, averages over all the paths to obtain a Monte-Carloestimate of the value of the option.

The generation of the paths of the driving Levy process is thus crucial. Sim-ulation schemes for general and some particular processes can be found in [96].Very accurate and fast simulation schemes for the VG and NIG setting basedon the construction of Gamma and Inverse Gaussian bridges respectively, canbe found in [85] and [86] (see also [102]). An compound-poisson approximationiin order to simulate a general Levy process is described in [10].

2.2 European Call Options

If we know the density function of ST , we can just (numerically) calculate theprice of a vanilla option as the discounted expected value of the payoff. On theother hand, we do not have always the density function available. However inmost cases we have the characteristic function of our stock price process (or thelog of it) in the risk-neutral world at hand.

Let C = C(K,T ) be the price at time t = 0 of an European call option withstrike K and maturity T . Next, we overview some ways to calculate the optionprice.

2.2.1 Pricing through the Density Function

If we know the density function, fQ(s, T ) of our stock price at the expiry Tunder the risk-neutral measure Q, we can easily price European call and putoptions, by just calculating the expected value.

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For an European call option with strike price K and time to expiration T ,the value at time 0 is therefore given by the expectation of the payoff under themartingale measure:

C = C(K,T ) = EQ[exp(−rT ) max{ST −K, 0}]

= exp(−rT )

∫ ∞

0

fQ(s, T ) max{s−K, 0}ds

= exp(−rT )

∫ ∞

K

fQ(s, T )(s−K)ds

= exp(−rT )

∫ ∞

K

fQ(s, T )sds−K exp(−rT )Π2,

where Π2 is the probability (under Q) of finishing in the money. Note that wealready have assumed that fQ lives on the non-negative real numbers since thestock price is always bigger than zero.

2.2.2 Pricing through the Characteristic Function

More explicit pricing methods for the classical vanilla options which can beapplied in general when the characteristic function of the risk-neutral stockprice process is known, were developed in [12] and in [30].

Let as usual S = {St, 0 ≤ t ≤ T} denote the stock price process and denoteby φ(u) the characteristic function of the random variable logST , i.e. φ(u) =EQ[exp(iu log(ST ))] = EQ[exp(iu(logS0 +XT ))].

In [12], one shows very generally one may write

C = C(K,T ) = S0Π1 −K exp(−rT )Π2, (3)

where Π1 and Π2 are obtained by computing the integrals

Π1 =1

2+

1

π

∫ ∞

0

Re

(

exp(−iu logK)E[exp(i(u− i) logST )]

iuE[ST ]

)

du

=1

2+

1

π

∫ ∞

0

Re

(

exp(−iu logK)φ(u− i)

iuφ(−i)

)

du

Π2 =1

2+

1

π

∫ ∞

0

Re

(

exp(−iu logK)E[exp(iu logST )]

iu

)

du

=1

2+

1

π

∫ ∞

0

Re

(

exp(−iu logK)φ(u)

iu

)

du.

The probability of finishing in the money corresponds with Π2. Similarly,the delta (i.e. the change in the value of the option compared with the changein the value of the underlying asset) of the option corresponds with Π1.

Another method was developed by Carr and Madan in [30]. It can be appliedin general when the characteristic function of the risk-neutral stock price processis known.

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Let α be a positive constant such that the αth moment of the stock priceexists. For all stock price models encountered here, typically a value of α = 0.75will do fine. Carr and Madan then showed that the price C(K,T ) of an Europeancall option with strike K and time to maturity T is given by:

C = C(K,T ) =exp(−α log(K))

π

∫ +∞

0

exp(−iv log(K))%(v)dv, (4)

where

%(v) =exp(−rT )E[exp(i(v − (α+ 1)i) log(ST ))]

α2 + α− v2 + i(2α+ 1)v

=exp(−rT )φ(v − (α + 1)i)

α2 + α− v2 + i(2α+ 1)v.

Using Fast Fourier Transforms, one can compute within a second the com-plete option surface on an ordinary computer.

Related to the later formula is the work of Raible [84] in which bilateralLaplace transforms are used for the valuation of a series of European options.Pricing formulas were worked out not only for plain vanilla European call andput option, but also for more complex payoff structures, such as quantos andpower options. A similar method can be found in [65]; for extenssions, unifica-tion and error bounds we refer to [63].

2.3 European Options with Payoff only Depending on Stock

Price Value at Maturity

In this section we discuss how to price an European option of which the payoffonly depends on ST , i.e. F ({St, 0 ≤ t ≤ T}) = F (ST ). Examples are theEuropean call (F (ST ) = (ST −K)+), the European put (F (ST ) = (K −ST )+),the digital option (F (ST ) = 1(ST > K)) and many others.

2.3.1 The PIDE Approach

This approach is based on the numerical solution of some PIDE. Let us denote byG(x) = F (exp(x)). Then the price V (τ, x) of this option with time to maturityτ = T − t and current log stock price x = logSt, can be found by solving thePIDE

∂V (τ, x)

∂τ−LV (τ, x) + rV (τ, x) = 0 in (0, T ) × R, (5)

where

LV (τ, x) = (r −σ2

2)∂V (τ, x)

∂x+σ2

2

∂2V (τ, x)

∂x2(6)

+

∫ +∞

−∞

(

V (τ, x+ y) − V (τ, x) − (exp(y) − 1)∂V (τ, x)

∂x

)

ν(dy)

6

is the infinitesimal generator of the transition semigroup of the driven Levyprocess.

This PIDE must be solved subject to the initial condition V (0, x) = F (exp(x)).In [54] a numerical solution scheme for this PIDE was worked out under a VGmodel (see also [7] and [6]). Using a backward scheme, one solves iterativelythis system.

2.4 Barrier Options

The payoff of a barrier option depends on whether the price of the underlyingasset crosses a given threshold (the barrier) before maturity. The simplest bar-rier options are “knock in” options which come into existence when the price ofthe underlying asset touches the barrier and “knock-out” options which comeout of existence in that case. For example, an up-and-out call has the samepayoff as a regular plain vanilla call if the price of the underlying asset remainsbelow the barrier over the life of the option but becomes worthless as soon asthe price of the underlying asset crosses the barrier.

Let us consider contracts of duration T , and denote the maximum and min-imum process, resp., of a process X = {Xt, 0 ≤ t ≤ T} as

MXt = sup{Xu; 0 ≤ u ≤ t} and mX

t = inf{Xu; 0 ≤ u ≤ t}, 0 ≤ t ≤ T.

Let us denote with 1(A) the indicator function, which has a value 1 if A istrue and zero otherwise.

For single barrier options, we will focus on the following types of call options:

• The down-and-out barrier call is worthless unless its minimum remainsabove some low barrier H , in which case it retains the structure of anEuropean call with strike K. Its initial price is given by:

DOBC = exp(−rT )EQ[(ST −K)+1(mST > H)].

• The down-and-in barrier call is a standard European call with strike K,if its minimum went below some low barrier H . If this barrier was neverreached during the life-time of the option, the option is worthless. Itsinitial price is given by:

DIBC = exp(−rT )EQ[(ST −K)+1(mST ≤ H)].

• The up-and-in barrier call is worthless unless its maximum crossed somehigh barrier H , in which case it retains the structure of an European callwith strike K. Its price is given by:

UIBC = exp(−rT )EQ[(ST −K)+1(MST ≥ H)].

• The up-and-out barrier call is worthless unless its maximum remains belowsome high barrier H , in which case it retains the structure of an Europeancall with strike K. Its price is given by:

UOBC = exp(−rT )EQ[(ST −K)+1(MST < H)].

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The put-counterparts, replacing (ST − K)+ with (K − ST )+, can be definedalong the same lines.

We note that the value, DIBC, of the down-and-in barrier call option withbarrier H and strike K plus the value, DOBC, of the down-and-out barrieroption with same barrier H and same strike K, is equal to the value C of thevanilla call with strike K. The same is true for the up-and-out together withthe up-and-in:

DIBC +DOBC = C = UIBC + UOBC. (7)

The valuation of barrier options under our setting is a hard mathematicalproblem. The main problem is that the distribution of the minimum and maxi-mum process and the related overshoot distribution associated with the passageof the underlying Levy process across a barrier is not known explicitly.

The problem in case the driving process is a spectrally positive/negativeLevy process has been treated in i.a. [87], [98] and [11].

Making use of the memoryless property of the exponential distribution, [57]and [58] have derived explicit formulas for a jump diffusion model where thejumps are double-exponentially distributed. Under the same model [66] derivessimilar formulas using fluctuation theory.

2.4.1 The Wiener-Hopf Approach

Fluctuation theory and the Wiener-Hopf factorization of Levy processes play acrucial role in the analyses of the (distribution of the) minimum and miximumprocess. The results date back to [17] and [51]. See also the books [91, Chapter9] and [16, Chapter VI].

Let θ denote a random variable exponentially distributed with parameter q,independent of X . Then, the Wiener-Hopf factorization of the Levy process Xstates that

E[exp(izXθ)] = E[exp(izMXθ )] ·E[exp(izmX

θ )]

or equivalently

q(q − ψ(z))−1 = ϕ+q (z) · ϕ−

q (z), z ∈ R,

where ψ denotes the characteristic exponent of X . So the characteristic func-tion taken at an exponential distributed time factorizes into the characteristicfunction of the mimimum at an exponential time and the characteristic functionof the maximum at an exponential time.

The functions ϕ+q and ϕ−

q have the following representations

ϕ+q (z) = exp

[

∫ t=∞

t=0

∫ x=∞

x=0

t−1 exp(−qt)(exp(izx) − 1)dPQ(x; t)dt]

ϕ−q (z) = exp

[

∫ t=∞

t=0

∫ x=0

x=−∞

t−1 exp(−qt)(exp(izx) − 1)dPQ(x; t)dt]

8

where PQ(x; t) = PrQ(Xt ≤ x) is the (risk-neutral) cumulative distributionfunction of Xt.

Barrier options under a Levy market were considered by [27]. The resultsrely on the Wiener-Hopf decomposition and one uses analytic techniques; theyapply methods from potential theory and pseudodifferential operators to deriveformulas for barrier and touch options. Similar and totally general results byprobalistic methods are described by [80]. The numerical calculations neededare of high complexity: numerical integrals with dimension 3 or 4 are needed,together with numerical inversion methods.

2.4.2 The PIDE Approach

For down-and-out barrier options, with a payoff function of the form F (ST )1(mST >

H) or F (ST )1(MST < H), one can proceed along the same lines as the procedure

described in Section 2.3. The only difference is that at each time step on hasto check whether the barrier has been crossed or not. If the barrier has beencrossed, the value of the option at that point of the grid is set equal to zero.By this one forces that if the barrier H has been crossed, the option is and willremain worthless. Down-and-in barrier option price can be obtain using (7).

For more details see [38] and [39]. This technique has been used in [33] in acredit-risk setting.

2.5 Lookback Options

The lookback call (put) option with floating strike has the particular feature ofallowing its holder to buy (sell) the stock at the minimum (maximum) it hasachieved over the life of the option.

Using risk-neutral valuation and after choosing an equivalent martingalemeasure Q, we have that the initial, i.e. t = 0, price of a floating strike lookbackoption is given by

Lfloating(T ) = exp(−rT )EQ[MST − ST ].

Similmarly, the price of a fixed strike lookback option is given by

Lfix(K,T ) = exp(−rT )EQ[(MST −K)+].

The above lookbacks are so-called continously monitored options, since theinfimum and the supremum runs over a (continuous) time interval. Often, theterms of the contract are modified and there are only a discrete number ofobservations, for example at the close of each trading day. These discretelymonitored options have received much less attention in the literature. [19] useFourier methods and Spitzer’s identity to derive formulas for fixed strike look-back options. Under the Black-Scholes framework [28] provide a way of adjustingthe continuous prices formulas for the situation of periodical observations.

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2.5.1 The Wiener-Hopf Approach

The double Laplace transform of the price of the fixed strike lookback wasobtained in [79]. Choose γ > 1 and α > 0 such that EQ[exp(2X1)] < exp(r+α)and let k = log(K/S0). If k = log(K/S0) > 0, then for all v, u > 0 we have:

∫ T=+∞

T=0

∫ k=+∞

k=0

exp(−(v + α)T − (u+ γ)k)Lfix(S0 exp(k), T )dkdT = (8)

S01

v + r + α

1

(u+ γ)(u+ γ − 1)[ϕ+

v+r+α(i(u+γ−1))+(u+γ−1)ϕ+v+r+α(−i)−u−γ].

Using the symmetry results of [43] (see also [44]) the case of the floating strikelookback option can be extracted out of the fixed strike price.

However, in general the Wiener-Hopf factors are not known explicitly andnumerical computation are typically extremely time-consuming. In case of theso-called processes of exponential type (RLPE), which contains the popularclasses of the Generalized Hyperbolic and Variance Gamma processes, [27] pro-vide some more efficient formulas for the Wiener-Hopf factors.

2.6 Asian Options

In this section we basically consider the pricing of an European-style arithmeticaverage call option with strike price K, maturity T and n averaging days 0 =t0 ≤ t1 < . . . < tn = T .

Its price is according to a risk-neutral pricing measure Q at time t given by

AAt(K,T ) =exp(−r(T − t))

nEQ

(

n∑

k=1

Stk− nK

)+∣

∣

∣Ft

, (9)

where {Ft, 0 ≤ t ≤ T} denotes the natural filtration of S.

Pricing of (arithmetic) Asian options is even in the Black-Scholes worldnot straightforward. The main difficulty in evaluating (9) is to determine thedistribution of the dependent sum

∑n

k=1 Stk. In general no explicit analytical

expression for (9) is available. One can use Monte Carlo simulation techniques toobtain numerical estimates of the price (see [21, 22, 55]). Hartinger and Pedota[53] apply Quasi Monte-Carlo methods for the valuation of Asian options inthe Hyperbolic model. These approaches are rather time consuming and therelated hedging problem is even more difficult. In [100] one shows that thepricing function is satisfying a PIDE in the case of semimartingale models and inparticular for Levy models. For an approach based on Fast Fourier Transforms,see [15, 34].

An alternative is to use approximations of the distribution of the average.Next, we discuss briefly such a technique, which was first developed for theBlack-Scholes case ([62], [99], [101]) and later on adapted to Levy Models ([3],[4], [2]).

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An alternative route is to try to derive upper and lower bounds for the optionprice. This can nicely be done by the use of comonotonic theory as describedin [97], [41], [42], [1] and [5].

The results presented below deal with the fixed-strike arithmetic averagecall option. However, many of them translate immediately to put options andfloating-strike options (using put-call parity and symmetries of floating and fixedstrike Asian options recently established for exponential Levy models in [43] (seealso [44])).

2.6.1 The Moment Matching Approach

The idea is to first compute the first moments of the dependent sum∑n

k=1 Stk

in (9). Next, we will replace it by a more tractable distribution with identicalfirst moments.

Due to the independence and stationarity of increments of Levy processes,a simple and fast algorithm to compute the mth moment can be derived for thesum An =

∑n

k=1 Stk.

Denote by

Ri =Sti

Sti−1

, i = 1, . . . , n

and setLn = 1 and Li−1 = 1 +RiLi, i = 2, . . . , n.

Then we have

n∑

k=1

Stk= S0(R1 +R1R2 + · · · +R1R2 . . . Rn) = S0R1L1.

Due to the independent increments property of the underlying Levy process,one can write

EQ[Amn ] = Sm

0 EQ[(R1L1)m] = Sm

0 EQ[Rm1 ]EQ[Lm

1 ].

The calculation of the mth moment of the variables Ri and Li can be donein the following way: First note that

EQ[Rki ] = (EQ[exp(k X1)])

ti−ti−1 , for all i = 1, . . . , n (10)

so that one just has to evaluate the risk-neutral moment generating function ofX1 at k, given it exists. Furthermore we have

EQ[Lmi−1] = EQ[(1 + LiRi)

m] =

m∑

k=0

(

m

k

)

EQ[Lki ]EQ[Rk

i ]. (11)

Starting with EQ[Lkn] = 1, k = 0, . . . ,m, one can then apply recursion (11)

together with (10) to obtain EQ[Lm1 ] and hence EQ[(An)m].

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These moments can now be used to approximate the distribution of An =∑n

k=1 Stkby another more tractable distribution with identical first moments.

A natural and usual effective choice is to approximate An by a distribution ofthe same class as X . These kind of approximations have been worked out indetail for the Normal Inverse Gaussian Levy model in [3] and for the VarianceGamma Levy model in [4]. They turn out to be a quick and accurate alternativeto other numerical pricing techniques, the approximation error typically beingless than 0.5%.

2.6.2 The Comontonic Approach: Pricing and Static Hedging

An other approach is by using comonotonic theory. One derives an upper boundfor the price and at the same time the theory gives a static super-hedging interms of vanilla options. Assume for simplicity that t = 0 and that the averaginghas not yet started. First note, that for anyK1, . . . ,Kn ≥ 0 withK =

∑n

k=1 Kk,we have a.s.

(

n∑

k=1

Stk− nK

)+

=(

(St1 −nK1) + · · ·+ (Stn−nKn)

)+

≤

n∑

k=1

(Stk− nKk)

+.

Hence

AA0(K,T ) =exp(−rT )

nEQ

(

n∑

k=1

Stk− nK

)+∣

∣

∣F0

≤exp(−rT )

n

n∑

k=1

EQ

[

(Stk− nKk)

+∣

∣

∣F0

]

=exp(−rT )

n

n∑

k=1

exp(rtk)EC0(κk, tk), (12)

where EC0(κk, tk) denotes the price of an European call option at time 0 withstrike κk = nKk and maturity tk.

In terms of hedging, this means that we have the following static super-hedging strategy: for each averaging day tk, buy exp(−r(T − tk))/n Europeancall options at time t = 0 with strike κk and maturity tk and hold these untiltheir expiry. Then put their payoff on the bank account.

Since the upper bound (12) holds for all combinations of κk ≥ 0 that satisfy∑n

k=1 κk = nK, one still has the freedom to choose strike values.Note that, if 0 ≤ r, the choice κk = K (k = 1, . . . , n) immediately implies

thatAA0(K,T ) ≤ EC0(K,T ).

(See also [55] and [81]).

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However, one naturally looks for that combination of κk’s which minimizesthe right-hand side of (12). In the Black-Scholes setting this optimization prob-lem was solved in [81] by using Lagrange multipliers. In a Levy setting, the prob-lem was tackled in [2], [3] and [4]. In the general case of arbitrary arbitrage-freemarket models, this optimal combination can be determined by using stop-losstransforms and the theory of comonotonic risks. For a general introduction ofthe comontonic theory, see [41] and [42].

Let F (xk; tk) = PQ (Stk≤ xk |F0) (xk, tk > 0) denote the marginal (risk-

neutral) distribution function of Stk. Then the optimal choice (see [1]) of strike

prices is given by

nκk = F−1 (FSc(nK); tk) , k = 1, . . . , n,

where FSc is the distribution function of the comonotone sum of St1 , . . . , Stn

determined by F−1Sc (x) =

∑n

k=1 F−1(x; tk).

In [1] a numerical study of the performance of this superhedging strategyfor Normal Inverse Gaussian, Variance Gamma and Meixner Levy models wasperformed. It turns out that the strategy is quite effective, in particular forlow values of the strike price K. For an option with moneyness of 80%, thedifference between the hedging cost and the estimated option price is typicallyaround 1.5%, whereas the classical hedge with the European call leads to adifference of almost 10%. For options out of the money, the difference increases,but in view of the easy and cheap way in which this hedge can be implementedin practice, the comonotonic approach seems to be competitive also in thesecases. Furthermore, from a hedger’s point of view, the related static hedgingstrategy is very convenient and is not exposed to the risk inherent in dynamichedging, like transaction costs, liquidity problems, montinoring costs etc.

3 American Options

The valuation of American options under Levy process driven models is a quitehard task and no general analytical solutions are available. For the perpetualcase, i.e. with infinite time horizon, one can use the Wiener-Hopf factorizationtheory.

For the valuation of finite time horizon American options one can followtwo numerical methods: (1) numerically solving PIDI’s and (2) applying MonteCarlo methods adapted for optimal stopping problems.

3.1 The Perpetual American Option

An American perpetual option is a contract between two parties, in which thefirst one, the holder, buys the right to receive at a future time Θ, that he chooses,from the other party, the seller an amount F (SΘ). Call and put options havethe reward function F (x) = (x −K)+ and F (x) = (K − x)+ respectively.

The optimal Θ will dependent on the evolution of the stock prices and assuch is a random variable. For classical American options (see Section 3.2), the

13

contract includes an exercise time T <∞, the maturity, at or before the holdercan exercise: 0 ≤ Θ ≤ T . In the perpetual case there is no expiry. So T = ∞.

In [24] and [25] some expicite formulas in terms of the Wiener-Hopf factorswere derived using the theory of pseudo-differential operators.

Using probabilistic techniques, [77] studies perpetual American call and putoptions in terms of the overall supremum or infinimum of the Levy process, us-ing a random walk approximation to the process. Explicit formulae are obtainedin [77] under the assumption of mixed-exponentially distributed and arbitrarynegative jumps for the call options; and negative mixed-exponentially distrib-uted and arbitrary positive jumps for put options. These results generalize theclosed formulas of [68] and [76] for the Black-Scholes setting.

Asmussen, Avram and Pistorius [9] find explicit expressions if the drivenLevy process has two-sided phase-type jumps; the solution uses the Wiener-Hopf factorization and can also be applied to regime-switching Levy processeswith phase-type jumps.

Finally, we mention that in [35] one obtains formulas for the excercise bound-ary when jumps are either only positive or only negative (for a currency marketsetting driven by Levy processes).

3.2 The Standard American Option

In our Levy setting one can show, using the strong Markov property of the stockprice process, that the time t value, vt, of an American option with rewardfunction F (x) and time of maturity T , is a function of time to maturity τ =T − t and the current stock price St (or equivalently the log stock price xt =logSt) and is given by the highest value obtained by maximizing over all allowedexcercise strategies:

vt = v(τ, xt) = ess supΘ∈T (t,T )EQ[exp(−r(Θ − t))F (SΘ)|Ft]

where T (t, T ) denotes the set of nonanticipating exercise times Θ, satisfyingt ≤ Θ ≤ T .

We discuss two approaches to calculate the function v(τ, x) numerically.

3.2.1 The PIDI Approach

This approach is based on the numerical solution of some PIDI’s. Consider anAmerican type option with reward function F (x), e.g. in the put case we haveF = (K − x)+. Let us denote by G(x) = F (exp(x)). Then the price v(τ, x) ofthis American option with time to maturity τ and current log stock price x, canbe found by solving the PIDI

∂v(τ, x)

∂τ−Lv(τ, x) + rv(τ, x) ≥ 0 in (0, T )× R, (13)

14

subject to the boundary conditions

v(τ, x) −G(x) ≥ 0, a.e. in [0, T ]× R

(v(τ, x) −G(x))

(

∂v(τ, x)

∂τ−Lv(τ, x) + rv(τ, x)

)

= 0, in (0, T ) × R

v(0, x) = G(x),

where L is as in (6) the infinitesimal generator of the transition semigroupof the driven Levy process. Explicite numerical schemes, using wavelets, canbe found in [74] and [75]. They consider a variational inequality formulationcombined with a convenient wavelet basis for compression. Almendral solvesin [6] the problem numerically using implicite-explicite methods in case of theCGMY process. Here, one essentially works with a formulation of the problemas a Linear Complementarity Problem, and uses standard finite differences.To deal with the singularity of the jump measure at the origin, an compoundPoissson-Brownian approximation in line with [10] is used. Basically, one solvesthe problem iteratively (backwards). For each time step one needs to solvetridiagonal linear complementarity problems. A similar method was alreadyproposed in [54] in the special case of the VG process; see also [7].

Equation (13) is a backward equation in (log) spot and time to maturity.Carr and Hirsa [29] transforms these equations into forward equation in strikeand time of maturity and discuss the benifits of doing this under certain cir-cumstances.

3.2.2 The Monte-Carlo Approach

An other alternative is to use Monte-Carlo methods suitably adapted fo optimalstopping problems. This problem is tackled in [88] for spectrally one-sided Levyprocesses. The least squares Monte Carlo method ([67] and [32]) allows toapproximate conditional expectations. For an overview see [50]. Bermudianoptions were priced by Kellezi and Webber in [56] using a lattice method. Bytaking limits of the Bermudian option prices one can obtain in principle theprice of the corresponding American version. Finally, Levendorskii develops in[64] a method using Carr’s randomization and the method of lines to formulatean algoritm to obtain an approxiamtion for the price of an American option.

Acknowledgements

Wim Schoutens is a Postdoctoral Fellow of the Fund for Scientific Research -Flanders (Belgium) (F.W.O. - Vlaanderen).

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