Exotic options under Lévy models: An overview

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Exotic options under Lévy models: An overview

Wim Schoutens

∗

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

Abstract

Keywords:Lèvy processes; Financial derivatives; Exotic options

1. Introduction

In recent years more and more attention has been given to stochastic models of ﬁnancial markets which depart from the traditional Black–Scholes model[18]. Nowadays, a battery of models is available. Some of the most popular and still tractable models are the Lévy models. For an introduction on these models applied to ﬁnance, we refer to

[57]and[94]. These models are able to take into account different important stylized features of financial time series. However, since the models are of a higher complexity, pricing financial derivatives and in particular the so-called exotic derivatives is not easy. Recently, a lot of papers on this topic appeared in the literature. For a state of the art, we refer to[61]and the reference cited therein. Here, we give a brief overview of the literature and the different techniques to price vanilla and exotic options. We first focus on European options. We depart by pricing the vanilla call options, using i.e. (fast) Fourier transforms. Next, we move to more complicated payoff functions and will treat digital, barrier, lookback, Asian and American options using i.e. Wiener–Hopf factorization theory, numerical simulation algorithms solving partial integro-differential equation/inequalities (PIDE/PIDI) and comonotonicity theory.

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

1.1. Lévy processes

Suppose(u)is the characteristic function of a distribution. If for every positive integern,(u)is also thenth power of a characteristic function, we say that the distribution is inﬁnitely divisible.

One can deﬁne for every such an inﬁnitely divisible distribution a stochastic process,X= {X_t, t0}, called Lévy process, which starts at zero, has independent and stationary increments and such that the distribution of an increment over[s, s+t],s, t0, i.e.X_t₊_s−X_s, has((u))t as characteristic function.

∗_{Tel.: +32 16 32 20 27; fax: +32 16 32 28 31.}

E-mail address:[email protected].

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Every Lévy process has a càdlàg modiﬁcation which is itself a Lévy process. We always work with this càdlàg version. So sample paths of a Lévy process are a.e. continuous from the right and have limits from the left. Moreover we always will work in the sequel with the natural ﬁltration generated by the Lévy processX.

The cumulant characteristic function(u)=log(u)is often called thecharacteristic exponentand it satisﬁes the followingLévy–Khintchine formula:

(u)=iu−2 2 u

2₊

_+∞

−∞ (exp(iux)−1−iux1{|x|<1})(dx), (1) where∈R,20 andis a measure onR\{0}with

_+∞ −∞ inf{1, x 2_}₍_d_x)₌ _+∞ −∞ (1∧x 2₎₍_d_{x) <}_∞.

We say that our inﬁnitely divisible distribution has a triplet of Lévy characteristics (or Lévy triplet for short)[,2,(dx)]. The measureis called theLévy measureofX.

General reference works on Lévy processes are[16,90,8].

1.2. The Lévy market model

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

We work under a market which consists of one riskless asset (the bond) with price process given byB_t =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),

whereX= {X_t, t0}is (underQ) a Lévy process. Since we immediately work in a risk-neutral setting, the discounted stock price is a martingale and hence:

EQ[exp(−r(t−s))St|Fs] =Ss, 0st,

whereF={F_t,0tT}is the natural ﬁltration ofX={X_t,0tT}. This market model is often called the exponential Lévy market model. The log-returns log(S_t₊_s/S_t)of such a model follow the distribution of increments of lengthsof the Lévy processX.

In the literature, several particular choices for the Lévy processes were studied in detail. Madan and Seneta et al.

[71,72]have proposed a VG Lévy process (see also[69,70]). In[44]the Hyperbolic model was proposed, and in[13]

the NIG model (see also[14,88]). All three above-mentioned models were brought together as special cases of the generalized hyperbolic model, which was developed by Eberlein and co-workers in a series of paper[45,48,43,89,82]. Carr et al.[30]introduced the CGMY model; this family of distributions is by some authors also called the KoBoL family referring to[58]and[21](see also[20,36,75,22,24]). Finally, the Meixner model was used in[92](see also

[52,95,91,93]).

An accessible introduction, together with theoretical motivations to this Lévy market, can be found for example in[49,57]. Some theoretical motivation for considering Lévy processes in ﬁnance can also be found in[62]. For an overview of the theory and the applications of Lévy processes in ﬁnance we refer to[94]and[37].

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2. European options

Given our market model, letF ({S_t,0tT})denote the payoff of an European derivative at its time of expiry

T. In case of the European call with strike priceK, we haveF ({S_t,0tT})=F (S_T)=(S_T −K)+. According to the fundamental theorem of asset pricing (see[40]) the arbitrage free priceVt of the derivative at timet ∈ [0, T]is given by

Vt=EQ[exp(−r(T−t))F ({Su,0uT})|F_t], (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 simulation. The idea is simple and classical. In order to evaluate the expectation in (2), one simulates a huge amount of paths of the underlying Lévy process and hence 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-Carlo estimate of the value of the option.

The generation of the paths of the driving Lévy process is thus crucial. Simulation schemes for general and some particular processes can be found in[94]. Very accurate and fast simulation schemes for the VG and NIG setting based on the construction of Gamma and Inverse Gaussian bridges respectively, can be found in[84,85](see also[101]). An compound-Poisson approximation in order to simulate a general Lévy process is described in[10].

2.2. European call options

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

LetC =C(K, T )be the price at timet =0 of an European call option with strikeKand maturityT. Next, we overview some ways to calculate the option price.

2.2.1. Pricing through the density function

If we know the density function,f_Q(s, T )of our stock price at the expiryTunder the risk-neutral measureQ, we can easily price European call and put options, by just calculating the expected value.

For an European call option with strike priceKand time to expirationT, the value at time 0 is therefore given by the expectation of the payoff under the martingale 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−Kexp(−rT )2,

where2is the probability (underQ) of ﬁnishing in the money. Note that we already have assumed thatf_Qlives on the non-negative real numbers since the stock 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 be applied in general when the characteristic function of the risk-neutral stock price process is known, were developed in[12]and in[32].

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Let as usualS= {S_t,0tT}denote the stock price process and denote by(u)the characteristic function of the random variable logST, i.e.(u)=EQ[exp(iulog(ST))] =EQ[exp(iu(logS0+XT))].

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

C=C(K, T )=S01−Kexp(−rT )2, (3)

where1and2are obtained by computing the integrals 1=1 2+ 1 _∞ 0 Re

exp(−iulogK)E[exp(i(u−i)logS_T)] iuE[S_T] du =1 2+ 1 _∞ 0 Re

exp(−iulogK)(u−i) iu(−i) du 2=1 2+ 1 _∞ 0 Re

exp(−iulogK)E[exp(iulogS_T)] iu du =1 2+ 1 _∞ 0 Re

exp(−iulogK)(u) iu

du.

The probability of ﬁnishing in the money corresponds with2. Similarly, the delta (i.e. the change in the value of the option compared with the change in the value of the underlying asset) of the option corresponds with1.

Another method was developed by Carr and Madan in[32]. It can be applied in general when the characteristic function of the risk-neutral stock price process is known.

Letbe a positive constant such that theth moment of the stock price exists. For all stock price models encountered here, typically a value of=0.75 will do ﬁne. Carr and Madan then showed that the priceC(K, T )of an European call option with strikeKand time to maturityTis given by

C=C(K, T )=exp(−log(K))

_+∞

0

exp(−ivlog(K)) (v)dv, (4)

where

(v)=exp(−rT )E[exp(i(v−(+1)i)log(ST))] 2₊₋_v2₊_i₍₂₊₁_)v =

exp(−rT )(v−(+1)i) 2₊₋_v2₊_i₍₂₊₁_)v.

Using Fast Fourier Transforms, one can compute within a second the complete option surface on an ordinary computer. Related to the later formula is the work of Raible[83]in which bilateral Laplace transforms are used for the valuation of a series of European options. Pricing formulas were worked out not only for plain vanilla European call and put option, but also for more complex payoff structures, such as quantos and power options. A similar method can be found in[66]; for extensions, uniﬁcation 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 payoff only depends onST, i.e.F ({St,0t T})=F (ST). Examples are the European call(F (ST)=(ST −K)+), the European put(F (ST)=(K−ST)+), the digital option(F (S_T)=1(S_T > 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 (, x)of this option with time to maturity=T −t and current log stock pricex=logS_t, can be found by solving the PIDE

jV (, x)

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where LV (, x)= r−2 2 jV (, x) jx + 2 2 j2_{V (}_{, x)} jx2 + _+∞ −∞ V (, x+y)−V (, x)−(exp(y)−1)jV (, x) jx (dy) (6) is the inﬁnitesimal generator of the transition semigroup of the driven Lévy process.

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

2.4. Barrier options

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

Let us consider contracts of durationT, and denote the maximum and minimum process, resp., of a processX= {Xt,0tT}as

MX

t =sup{Xu;0ut} and mX_t =inf{X_u;0ut}, 0tT.

Let us denote with 1(A)the indicator function, which has a value 1 ifAis true 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 remains above some low barrierH, in which case it retains the structure of an European call with strikeK. Its initial price is given by

DOBC=exp(−rT )E_Q[(S_T −K)+1(mS_T> H )].

• The down-and-in barrier call is a standard European call with strike K, if its minimum went below some low barrierH. If this barrier was never reached during the life-time of the option, the option is worthless. Its initial price is given by:

DIBC=exp(−rT )E_Q[(S_T −K)+1(mS_TH )].

• The up-and-in barrier call is worthless unless its maximum crossed some high barrierH, in which case it retains the structure of an European call with strikeK. Its price is given by:

UIBC=exp(−rT )E_Q[(S_T −K)+1(M_TSH )].

• The up-and-out barrier call is worthless unless its maximum remains below some high barrierH, in which case it retains the structure of an European call with strikeK. Its price is given by:

UOBC=exp(−rT )E_Q[(S_T −K)+1(M_TS< H )].

The put-counterparts, replacing(S_T −K)+with(K−S_T)+, can be deﬁned along the same lines.

We note that the value, DIBC, of the down-and-in barrier call option with barrierHand strikeKplus the value, DOBC, of the down-and-out barrier option with same barrierHand same strikeK, is equal to the valueCof the vanilla call with strikeK. The same is true for the up-and-out together with the up-and-in:

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The valuation of barrier options under our setting is a hard mathematical problem. The main problem is that the distribution of the minimum and maximum process and the related overshoot distribution associated with the passage of the underlying Lévy process across a barrier is not known explicitly.

The problem in case the driving process is a spectrally positive/negative Lévy process has been treated in i.e.

[86,96,11].

Making use of the memoryless property of the exponential distribution,[59,60]have derived explicit formulas for a jump diffusion model where the jumps are double-exponentially distributed. Under the same model[67]derives similar formulas using ﬂuctuation theory.

2.4.1. The Wiener–Hopf approach

Fluctuation theory and the Wiener–Hopf factorization of Lévy processes play a crucial role in the analysis of the (distribution of the) minimum and maximum process. The results date back to[17]and[51]. See also the books[90, Chapter 9]and[16, Chapter VI].

Letdenote a random variable exponentially distributed with parameterq, independent ofX. Then, theWiener–Hopf factorizationof the Lévy processXstates that

E[exp(izX)] =E[exp(izMX)] ·E[exp(izmX)] or equivalently

q(q−(z))−1₌+

q(z)·−q(z), z∈R,

wheredenotes the characteristic exponent ofX. So the characteristic function taken at an exponential distributed time factorizes into the characteristic function of the minimum at an exponential time and the characteristic function of the maximum at an exponential time.

The functions+_q and−_q have the following representations: +_q(z)=exp _t_=∞ t=0 _x_=∞ x=0 t−1 exp(−qt)(exp(izx)−1)dP_Q(x;t)dt , − q(z)=exp _t_=∞ t=0 _x₌0 x=−∞t −1 exp(−qt)(exp(izx)−1)dP_Q(x;t)dt ,

whereP_Q(x;t)=Pr_Q(X_tx)is the (risk-neutral) cumulative distribution function ofX_t.

Barrier options under a Lévy market were considered by[25]. The results rely on the Wiener–Hopf decomposition and one uses analytic techniques; they apply methods from potential theory and pseudodifferential operators to derive formulas for barrier and touch options. Similar and totally general results by probalistic methods are described by

[80]. The numerical calculations needed are 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 formF (S_T)1(mS_T > H )orF (S_T)1(M_TS< 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 has to check whether the barrier has been crossed or not. If the barrier has been crossed, the value of the option at that point of the grid is set equal to zero. By this one forces that if the barrierHhas been crossed, the option is and will remain 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[29]in a credit-risk setting. 2.5. Lookback options

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

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Using risk-neutral valuation and after choosing an equivalent martingale measureQ, we have that the initial, i.e.

t=0, price of a ﬂoating strike lookback option is given by

Lﬂoating(T )=exp(−rT )EQ[M_TS−ST].

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

Lﬁx(K, T )=exp(−rT )E_Q[(M_TS −K)+].

The above lookbacks are so-called continuously monitored options, since the infimum and the supremum runs over a (continuous) time interval. Often, the terms of the contract are modified and there are only a discrete number of observations, for example at the close of each trading day. These discretely monitored options have received much less attention in the literature.[19]use Fourier methods and Spitzer’s identity to derive formulas for fixed strike lookback options. Under the Black–Scholes framework[27]provide a way of adjusting the continuous prices formulas for the situation of periodical observations.

2.5.1. The Wiener–Hopf approach

The double Laplace transform of the price of the ﬁxed strike lookback was obtained in[79]. Choose>1 and>0 such thatEQ[exp(2X1)]<exp(r+)and letk=log(K/S0). Ifk=log(K/S0) >0, then for allv, u >0 we have

_T_=+∞ T=0 _k_=+∞ k=0 exp(−(v+)T −(u+)k)L_{f ix}(S0exp(k), T )dkdT =S0 1 v+r+ 1 (u+)(u+−1)[ + v+r+(i(u+−1))+(u+−1)+_v₊_r₊(−i)−u−]. (8) Using the symmetry results of[46](see also[47]) the case of the ﬂoating strike lookback option can be extracted out of the ﬁxed strike price.

However, in general the Wiener–Hopf factors are not known explicitly and numerical computation are typically extremely time-consuming. In case of the so-called processes of exponential type (RLPE), which contains the popular classes of the generalized hyperbolic and variance gamma processes,[25]provide 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 arithmetic average call option with strike price K, maturityTandnaveraging days 0=t0t1<· · ·< tn=T.

Its price is according to a risk-neutral pricing measureQat timetgiven by

AAt(K, T )=exp(−r(T_n −t))EQ _n k=1 Stk−nK + Ft , (9)

where{F_t,0tT}denotes the natural ﬁltration ofS.

Pricing of (arithmetic) Asian options is even in the Black–Scholes world not straightforward. The main difﬁculty in evaluating (9) is to determine the distribution of the dependent sum n_k₌₁S_t_k. In general no explicit analytical expression for (9) is available. One can use Monte Carlo simulation techniques to obtain numerical estimates of the price (see[26,28,55]). Hartinger and Pedota[53]apply Quasi Monte-Carlo methods for the valuation of Asian options in the Hyperbolic model. These approaches are rather time consuming and the related hedging problem is even more difﬁcult. In[99]one shows that the pricing function is satisfying a PIDE in the case of semimartingale models and in particular for Lévy 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 the Black–Scholes case ([65,98,100]) and later on adapted to Lévy Models ([1,3,4]).

An alternative route is to try to derive upper and lower bounds for the option price. This can nicely be done by the use of comonotonic theory as described in[97,41,42,2,5].

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The results presented below deal with the fixed-strike arithmetic average call option. However, many of them translate immediately to put options and floating-strike options (using put-call parity and symmetries of floating and fixed strike Asian options recently established for exponential Lévy models in[46](see also[47])).

2.6.1. The moment matching approach

The idea is to first compute the first moments of the dependent sumn_k₌₁Stk in (9). Next, we will replace it by a more tractable distribution with identical first moments.

Due to the independence and stationarity of increments of Lévy processes, a simple and fast algorithm to compute themth moment can be derived for the sumA_n=n_k₌₁S_tk.

Denote by Ri=_SSti ti−1 , i=1, . . . , n and set Ln=1 and L_i₋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 Lévy process, one can write

EQ[Amn] =S0mEQ[(R1L1)m] =S0mEQ[R1m]EQ[Lm1].

The calculation of themth moment of the variablesR_iandL_i can be done in the following way: ﬁrst note that

EQ[Rki] =(EQ[exp(kX1)])ti−ti−1 for alli=1, . . . , n (10) so that one just has to evaluate the risk-neutral moment generating function ofX1atk, given it exists. Furthermore, we have EQ[Lmi−1] =EQ[(1+LiRi)m] = m k=0 _m k EQ[Lki]EQ[Rik]. (11)

Starting withE_Q[Lk_n] =1,k=0, . . . , m, one can then apply recursion (11) together with (10) to obtainE_Q[Lm₁]and henceE_Q[(A_n)m].

These moments can now be used to approximate the distribution of A_n=n_k₌₁S_t_k by another more tractable distribution with identical ﬁrst moments. A natural and usual effective choice is to approximateA_nby a distribution of the same class asX. These kind of approximations have been worked out in detail for the normal inverse Gaussian Lévy model in[4]and for the variance gamma Lévy model in[3]. They turn out to be a quick and accurate alternative to other numerical pricing techniques, the approximation error typically being less 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 bound for the price and at the same time the theory gives a static super-hedging in terms of vanilla options. Assume for simplicity thatt=0 and that the averaging has not yet started. First note, that for anyK1, . . . , Kn0 withK=

_n k=1Kk, we have a.s. _n k=1 Stk−nK + =((St1−nK1)+ · · · +(S_t_n−nK_n))+ n k=1 (Stk−nKk)+.

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Hence AA0(K, T )= exp(−rT ) n EQ _n k=1 Stk−nK + F0 exp(−_nrT ) n k=1 EQ[(Stk −nKk)+|F0] =exp(−rT ) n n k=1 exp(rt_k)EC0(k, tk), (12)

whereEC0(_k, t_k)denotes the price of an European call option at time 0 with strike_k=nK_kand maturityt_k. In terms of hedging, this means that we have the following static super-hedging strategy: for each averaging daytk, buy exp(−r(T −tk))/nEuropean call options at timet =0 with strikek and maturitytk and hold these until their expiry. Then put their payoff on the bank account.

Since the upper bound (12) holds for all combinations of_k0 that satisfyn_k₌1_k=nK, one still has the freedom to choose strike values.

Note that, if 0r, the choice_k=K (k=1, . . . , n)immediately implies that

AA0(K, T )EC0(K, T ). (See also[55,81]).

However, one naturally looks for that combination of _k’s which minimizes the right-hand side of (12). In the Black–Scholes setting this optimization problem was solved in[81]by using Lagrange multipliers. In a Lévy setting, the problem was tackled in[1,3,4]. In the general case of arbitrary arbitrage-free market models, this optimal combination can be determined by using stop-loss transforms and the theory of comonotonic risks. For a general introduction of the comontonic theory, see[41,42].

LetF (x_k;t_k)=P_Q(S_tkx_k|F0) (xk, tk>0)denote the marginal (risk-neutral) distribution function ofStk. Then the optimal choice (see[1]) of strike prices is given by

nk=F−1(FSc(n K);t_k), k=1, . . . , n,

whereF_Scis the distribution function of the comonotone sum ofS_t1, . . . , S_t_ndetermined byF_S−c1(x)=n_k₌1F−1(x;tk). In [2]a numerical study of the performance of this superhedging strategy for normal inverse Gaussian, variance gamma and Meixner Lévy models was performed. It turns out that the strategy is quite effective, in particular for low values of the strike priceK. For an option with moneyness of 80%, the difference between the hedging cost and the estimated option price is typically around 1.5%, whereas the classical hedge with the European call leads to a difference 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 implemented in practice, the comonotonic approach seems to be competitive also in these cases. Furthermore, from a hedger’s point of view, the related static hedging strategy is very convenient and is not exposed to the risk inherent in dynamic hedging, like transaction costs, liquidity problems, monitoring costs etc.

3. American options

The valuation of American options under Lévy process driven models is a quite hard task and no general analytical solutions are available. For the perpetual case, i.e. with inﬁnite time horizon, one can use the Wiener–Hopf factorization theory.

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

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3.1. The perpetual American option

An American perpetual option is a contract between two parties, in which the ﬁrst one, the holder, buys the right to receive at a future time, that he chooses, from the other party, the seller an amountF (S). Call and put options have the reward functionF (x)=(x−K)+andF (x)=(K−x)+, respectively.

The optimalwill dependent on the evolution of the stock prices and as such is a random variable. For classical American options (see Section 3.2), the contract includes an exercise timeT <∞, the maturity, at or before the holder can exercise: 0T. In the perpetual case there is no expiry. SoT = ∞.

In[22,23]some explicit formulas in terms of the Wiener–Hopf factors were derived using the theory of pseudo-differential operators.

Using probabilistic techniques, Mordecki[78]studies perpetual American call and put options in terms of the overall supremum or inﬁmum of the Lévy process, using a random walk approximation to the process. Explicit formulae are obtained in[78]under the assumption of mixed-exponentially distributed and arbitrary negative jumps for the call options; and negative mixed-exponentially distributed and arbitrary positive jumps for put options. These results generalize the closed formulas of[76,77]for the Black–Scholes setting.

Asmussen et al.[9]ﬁnd explicit expressions if the driven Lévy process has two-sided phase-type jumps; the solution uses the Wiener–Hopf factorization and can also be applied to regime-switching Lévy processes with phase-type jumps. Finally, we mention that in[35]one obtains formulas for the exercise boundary when jumps are either only positive or only negative (for a currency market setting driven by Lévy processes).

3.2. The standard American option

In our Lévy setting one can show, using the strong Markov property of the stock price process, that the timetvalue,

vt, of an American option with reward functionF (x)and time of maturityT, is a function of time to maturity=T−t and the current stock priceS_t(or equivalently the log stock pricex_t=logS_t) and is given by the highest value obtained by maximizing over all allowed exercise strategies:

vt =v(, xt)=ess sup_∈T_{(t,T )}E_Q[exp(−r(−t))F (S)|F_t]

whereT(t, T )denotes the set of nonanticipating exercise times, satisfyingtT. We discuss two approaches to calculate the functionv(, x)numerically.

3.2.1. The PIDI approach

This approach is based on the numerical solution of some PIDIs. Consider an American type option with reward functionF (x), e.g. in the put case we haveF=(K−x)+. Let us denote byG(x)=F (exp(x)). Then the pricev(, x) of this American option with time to maturityand current log stock pricex, can be found by solving the PIDI

jv(, x)

j −Lv(, x)+rv(, x)0 in(0, T )×R, (13) subject to the boundary conditions

v(, x)−G(x)0, a.e. in[0, T] ×R, (v(, x)−G(x)) jv(, x) j −Lv(, x)+rv(, x) =0, in(0, T )×R, v(0, x)=G(x),

whereLis as in (6) the inﬁnitesimal generator of the transition semigroup of the driven Lévy process. Explicit numerical schemes, using wavelets, can be found in[73,74]. They consider a variational inequality formulation combined with a convenient wavelet basis for compression. Almendral solves in[6]the problem numerically using implicit–explicit methods in case of the CGMY process. Here, one essentially works with a formulation of the problem as a Linear Complementarity Problem, and uses standard ﬁnite differences. To deal with the singularity of the jump measure at the origin, an compound Poisson–Brownian approximation in line with[10]is used. Basically, one solves the problem

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iteratively (backwards). For each time step one needs to solve tridiagonal linear complementarity problems. A similar method was already proposed in[54]in the special case of the VG process; see also[7].

Eq. (13) is a backward equation in (log) spot and time to maturity. Carr and Hirsa[31]transforms these into forward equation in strike and time of maturity and discuss the beneﬁts of doing this under certain circumstances.

3.2.2. The Monte-Carlo approach

An other alternative is to use Monte-Carlo methods suitably adapted for optimal stopping problems. This problem is tackled in[87]for spectrally one-sided Lévy processes. The least squares Monte Carlo method[68,33]allows to approximate conditional expectations. For an overview see[50]. Bermudian options were priced by Këllezi and Webber in[56]using a lattice method. By taking limits of the Bermudian option prices one can obtain in principle the price of the corresponding American version. Finally, Levendorskii develops in[64]a method using Carr’s randomization and the method of lines to formulate an algorithm to obtain an approximation 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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