Prices and Sensitivities of Barrier and First-Touch Digital Options in Levy-Driven Models

Abstract

We present a fast and accurate FFT-based method of computing the prices and sensitivities of barrier options and first-touch digital options on stocks whose log-price follows a Lévy process. The numerical results obtained via our approach are demonstrated to be in good agreement with the results obtained using other (sometimes fundamentally different) approaches that exist in the literature. However, our method is computationally much faster (often, dozens of times faster). Moreover, our technique has the advantage that its application does not entail a detailed analysis of the underlying Lévy process: one only needs an explicit analytic formula for the characteristic exponent of the process. Thus our algorithm is very easy to implement in practice. Finally, our method yields accurate results for a wide range of values of the spot price, including those that are very close to the barrier, regardless of whether the maturity period of the option is long or short.

Full text

Electronic copy available at: http://ssrn.com/abstract=1155149
PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH
DIGITAL OPTIONS IN L´
EVY-DRIVEN MODELS
MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
Abstract. We present a fast and accurate FFT-based method of computing the
prices and sensitivities of barrier options and first-touch digital options on stocks
whose log-price follows a L´evy process. The numerical results obtained via our
approach are demonstrated to be in good agreement with the results obtained using
other (sometimes fundamentally different) approaches that exist in the literature.
However, our method is computationally much faster (often, dozens of times faster).
Moreover, our technique has the advantage that its application does not entail a
detailed analysis of the underlying L´evy process: one only needs an explicit analytic
formula for the characteristic exponent of the process. Thus our algorithm is very
easy to implement in practice. Finally, our method yields accurate results for a
wide range of values of the spot price, including those that are very close to the
barrier, regardless of whether the maturity period of the option is long or short.
Key words and phrases: Option pricing, greeks, barrier options, first-touch digitals,
L´evy processes, KoBoL processes, CGMY model, Normal Inverse Gaussian pro-
cesses, Variance Gamma processes, Fast Fourier transform, Carr’s randomization,
Wiener-Hopf factorization.
Acknowledgement: The first author would like to thank Marc Jeannin for very
helpful email correspondence.
Contents
Introduction 2
1. Examples of options and underlying processes 6
2. Calculation of the prices and the Greeks 10
3. Expected present value operators and FFT 14
4. Practical implementation of the algorithms 22
5. Numerical examples for barrier options 26
6. Numerical examples for first-touch digitals 28
Appendix A. Wiener-Hopf factorization 29
Appendix B. Applications of FFT 35
References 36
M.B.: Department of Mathematics, University of Chicago, 5734 S. University Ave., Rm. 208C,
Chicago, IL 60637. Email address: mitya@math.uchicago.edu (corresponding author).
S.L.: Department of Economics, The University of Texas at Austin, 1 University Station C3100,
Austin, TX 78712–0301. Email address: leven@eco.utexas.edu.
1

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2 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
Introduction
The problem of pricing and hedging barrier options has attracted much attention
in the recent years, both from the theoretical finance side and from the practitioners’
side. For instance, a rather comprehensive review of the 1965–1995 literature on the
pricing of barrier options given in [11] lists about 30 articles, while hundreds of new
works devoted to the same topic have appeared since 1995.
In the present paper, for the sake of brevity, we only present detailed algorithms
for computing the prices and sensitivities of two types of options: down-and-out
barrier put options and down-and-in first-touch digital options. Simple modifications
of these algorithms can be applied to other types of knock-out barrier options, as
well as to up-and-in first-touch digital options; moreover, considering combinations
of these options allows one to calculate the prices and sensitivities of various kinds
of single-barrier options with rebates. The list of situations where our techniques are
applicable can easily be extended.
The method we will describe has two important virtues. On the one hand, it is very
easy to implement1and works significantly faster than the other methods that are
available in the literature. On the other hand, it accurately reproduces the qualitative
behavior of the prices of barrier and first-touch digital options when the spot price
of the underlying approaches the barrier. As we will see momentarily, several other
commonly used methods do not share this quality.
Let V=Vbarr(S0, T ) denote the no-arbitrage price of a down-and-out barrier put
option on a non-dividend paying stock, where S0is the current (t= 0) spot price of
the stock, and Tis the maturity date of the option. Let Hdenote the barrier for
the option2. For many of the models of the spot price process S={St}t≥0that were
empirically shown to provide a very good fit to the observed prices of stocks and
European options, the delta, ∂V/∂S0, of the option, tends to +∞as S0approaches
Hfrom the right. Among such processes are the processes of the extended Koponen
family introduced in [5] (they were later used in [15] under the name “CGMY model”,
and in [7] under the name “KoBoL processes”; we adopt the latter terminology), of
order ν > 1, as well as the Normal Inverse Gaussian processes [2].
A graphical illustration of the irregular behavior of the option price under a NIG
process is provided by the solid line in Figure 1. We refer the reader to [7, Ch. 7]
for an asymptotic analysis of the value function of a first-touch digital option as the
current spot price of the underlying approaches the barrier; qualitatively, its behavior
is similar to the behavior of 1 −Vbarr(S0, T ), where Vbarr(S0, T ) is as above.
1This is the case even for a person who is not familiar with general option pricing theory. In fact,
the reader who has access to a library for computing fast Fourier transforms in her programming
language of choice will be able to execute programs based on our algorithm without any difficulty.
2The dependence of Vbarr (S0, T ) on Hand on the strike price of the option is suppressed from
our notation.

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PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 3
2000 2500 3000 3500
0
100
200
300
400
500
600
Current spot price, S0, of the underlying
Value of the option
NIG model
Black−Scholes model
Figure 1. The value function of a down-and-out barrier put option in the NIG
and Black-Scholes models. The strike price is K= 3500, the barrier is H= 2100,
the time to maturity is T= 1 year, the riskless rate is 3%, and the underlying stock
pays no dividends. (The example is taken from [18].)
Solid line: the graph of the value function calculated assuming that under a risk-
neutral measure chosen by the market, the log-price process {Xt= ln St}of the
underlying is a NIG process with parameters α= 8.858, β=−5.808, δ= 0.174.
Dashed line: same as above, except that Xis assumed to be a Brownian motion
with volatility σ≈0.2136, chosen so that the second (instantaneous) moment, σ2,
of X={Xt}is the same as the second (instantaneous) moment, δα2(α2−β2)−3/2,
of the NIG process in the first example.
We will now briefly explain where our work fits in the currently available literature
on the valuation of path-dependent options. In the framework of the Black-Scholes
market model [3], an explicit formula for the price of a barrier call option was obtained
by Merton [31]. Many subsequent works on barrier and first-touch digital options also
remained in the Black-Scholes framework (the interested reader may wish to consult,
for example, the bibliography lists in the papers [11] and [10]).
However, it is a known fact that the Black-Scholes model yields rather inaccurate
prices of barrier options near expiry, especially when the spot price is close to the
barrier. The reason is that the Black-Scholes value function VBS =Vbarr,BS(S0, T ) for
(say) a down-and-out barrier put option is continuously differentiable with respect to
S0within the closed interval [H, +∞). The dashed line in Figure 1 is an example of
such behavior (we can clearly see how the delta, ∂VBS /∂S0, of the option has a finite
limit as S0approaches Hfrom the right).
As a result, the relative errors of prices of barrier options computed using the
Black-Scholes model versus the more realistic stock pricing models can sometimes
reach several dozen percent near the barrier. We refer the reader to [20], where this
phenomenon was demonstrated using the example of the NIG model.

4 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
A simple extension of the Black-Scholes model is the double-exponential jump-
diffusion model introduced by Kou [23]. Closed-form expressions (that are amenable
to practical computations) for the value function VKou(S0, T ) of a barrier option in
Kou’s model are available [24, 25], but they are less explicit than the known formulas
in the Black-Scholes framework: what one has is a formula for the Laplace transform
of VKou (St, τ ) with respect to t(where τ=T−tis the time to expiry), and a numerical
Laplace inversion must be performed in order to calculate the values of VKou.
Recently, barrier options and first-touch digital options in a hyper-exponential
jump-diffusion model (which generalizes Kou’s model by allowing several jumps with
different means in the positive and negative directions) were considered by Jeannin
and Pistorius in [18], who derived explicit formulas for the Laplace transforms with
respect to the time variable of the value functions, deltas, gammas and thetas of the
options. They further showed that, approximating other L´evy-driven models (such
as Variance Gamma [30, 29, 28] and NIG) with suitable hyper-exponential models,
one can obtain accurate approximations to the prices and sensitivities of barrier and
first-touch digital options in the regions not too close to the barrier.
Nevertheless, hyper-exponential models (with nonzero Gaussian component) also
have the disadvantage that the value functions of barrier options in these models
are continuously differentiable up to the barrier. In other words, qualitatively these
functions exhibit behavior similar to that of the dashed line in Figure 1, whereas
value functions obtained from more realistic models of stock prices, such as NIG and
KoBoL, exhibit behavior similar to that of the solid line in Figure 1.
Other methods of pricing barrier options that use models with a tangible diffusion
component to approximate models with zero diffusion component (such as the method
of Cont and Voltchkova [16]) suffer from the same problem (see [27, 20] for an analysis
of the errors that result from applying these methods). This issue is important
because empirical studies of financial markets (see, e.g., [15]) show that, typically,
the dynamics of a stock has zero diffusion component.
Some of the numerical examples appearing in the present article are devoted to
comparing the performance of our method with the method of [18]. In particular, we
show that for down-and-out barrier put options under a NIG process, the agreement
of the results produced by the two methods is quite good, but, closer to the barrier,
some discrepancy can be observed, which agrees with the discrepancy one expects on
theoretical grounds, due to the difference in asymptotic behavior of the prices near
the barrier3. For down-and-in first-touch digitals under a V.G. process, the agreement
between the results produced by the two methods is even better (see §6.1).
3For instance, 7 −8% away from the barrier, the option prices computed using our algorithm are
about 5% higher than the prices computed using the method of [18]. More generally, we will see in
§5.1 that, qualitatively, the discrepancy between our results and the results of [18] is similar to the
discrepancy between the solid line and the dashed line in Figure 1.

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 5
Moreover, our algorithm works significantly faster than the algorithms based on
Laplace inversion with respect to the time variable (see §A.9 for an explanation). This
should not be surprising, since a number of works ([14, 7, 20, 17, 4], to name but a
few) demonstrated the computational superiority of option pricing methods based on
taking Fourier transforms with respect to the log-spot price of the underlying asset,
which is the strategy employed in the present article.
Our approach does not require changing the L´evy process that one uses to model
the log-price of the underlying stock, hence one is guaranteed that the qualitative
behavior of the approximations to the option prices obtained using our approach
is the same as that of the “exact” option prices. For the same reason, we expect
our algorithm to yield accurate results even for options with very short maturity
periods (where the Black-Scholes model is known to provide an especially poor fit
of the observed option prices). Furthermore, our method is fairly universal, in that
it applies to a very wide class of L´evy-driven models, including the Black-Scholes
model, Merton’s model [32], hyper-exponential jump-diffusion models, V.G. model,
NIG model and the KoBoL model. It is also fast and easy to use: as long as an
explicit formula for the L´evy exponent of the underlying process is available and
satisfies a certain regularity assumption (see §1.4 for the details), which is the case in
all the aforementioned examples, the general scheme of implementation of our method
always remains the same. All these features make our method a clear favorite among
all the computational methods of pricing barrier options that are known to us.
Finally, we remark that in [4] we developed a method for pricing barrier options
that is the same as the one employed in the present article. The contributions of
the current work consist of an extension of this method to computing sensitivities
for barrier and first-touch digital options, as well as of a number of new numerical
examples demonstrating the speed and accuracy of our techniques.
Organization of the paper. The rest of this text is organized as follows. In §1 we
recall some basic facts about L´evy processes, describe the class of L´evy processes to
which our method can be applied, and present the well-known formulas that express
the prices of barrier and first-touch digital options in probabilistic terms.
These probabilistic formulas are unsuitable for fast numerical realization, so in §2
we explain how Carr’s randomization approximation [12] can be used to replace the
original problem with a sequence of simpler option pricing problems. Each problem
in the sequence can then be solved using the Wiener-Hopf factorization method,
developed in option pricing by S.I. Boyarchenko and Levendorski˘
i [6, 7, 8, 27, 9].
The numerical realization of the key ingredients of the Wiener-Hopf factorization
method (the “expected present value operators” of a L´evy process) is described in
detail in §3, and is followed by explicit algorithms for computing the prices and
sensitivities of barrier and first-touch digital options in §4.

6 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
A selection of numerical examples demonstrating the speed and accuracy of our
techniques is presented and discussed in §§5–6. The two appendices provide back-
ground information of Wiener-Hopf factorization, as well as a number of concrete
recommendations on using FFT techniques in practice, making our paper essentially
self-contained and allowing the reader to implement all our algorithms very easily.
1. Examples of options and underlying processes
1.1. Some types of options. We begin by recalling the exercise rules for a few
barrier and first-touch digital options that are commonly traded in financial markets.
Adown-and-out barrier call (respectively, put ) on a given asset is determined by
three parameters: maturity date T, strike price K, and barrier H. The option expires
worthless if, at any moment in time t≤T, the price, St, of the underlying reaches
or falls below H. Otherwise, at time t=T, the owner of the option receives payoff
equal to (ST−K)+= max{ST−K, 0}(respectively, (K−ST)+).
An up-and-out barrier call (respectively, put) option is defined similarly, the only
difference being that it expires worthless if Streaches or exceeds the barrier Hfor
some t≤T. In principle, we do not have to assume a relationship between Hand
K, but we note that a down-and-out barrier put option (respectively, an up-and-out
barrier call option) is worthless unless H < K (respectively, H > K).
Adown-and-in (respectively, up-and-in)first-touch digital4option is determined
by its maturity date Tand barrier H. The option pays its owner $1 at the first
moment in time t≤Twhen the price, St, of the underlying reaches or falls below
(respectively, reaches or exceeds) H. If no such event occurs, the option expires
worthless at time T.
Remarks 1.1.(1) Down/up-and-out call/put barrier options are collectively referred
to as knock-out barrier options. There are four other types of (single-)barrier
options, namely, the knock-in ones, which become activated (as opposed to de-
activated) when the price of the underlying crosses the barrier in the specified
way. However, a package consisting of a knock-out barrier option and a knock-in
barrier option with the same parameters clearly has value equal to that of a Euro-
pean option of the corresponding type. Thus, knowing the prices and sensitivities
of knock-out barrier options and European options, one can calculate the prices
and sensitivities of knock-in barrier options. Hence it is unnecessary to consider
the latter type of options separately.
(2) It will be clear that our approach to calculating prices and sensitivities of down-
and-out barrier put options (respectively, down-and-in digital options) can be
4Sometimes (see, e.g., [18]) one refers to these options as American down-and-in (respectively, up-
and-in)digital options. The reason is that an American-style option with a binary payoff function
is always optimal to exercise at the first moment in time when the payoff becomes positive.

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 7
easily modified to cover the other three types of knock-out barrier options (re-
spectively, up-and-in digital options).
(3) A package consisting of a knock-out barrier option and a certain quantity of
knock-in digital options with the same barrier is equivalent to a knock-out barrier
option with constant rebate. Thus the latter options can also be easily priced
using our techniques.
1.2. L´evy-driven models. Our next goal is to recall the probabilistic formulas for
the values of the options listed in §1.1. First, however, we must specify the type of
option pricing models we will use. We consider a model frictionless market consisting
of a riskless bond and a stock. We assume that the riskless rate, r > 0, is constant,
and that Xt= ln St, the logarithm of the spot price of the stock, follows a L´evy
process under a chosen equivalent martingale measure (EMM).
The aforementioned EMM-condition means that the discounted price process of
the stock, e−rtStt≥0, is a martingale:
Ee−rt2St2St1=S=e−rt1S∀t2> t1≥0.
We remark that in general, an EMM (also called a “risk-neutral measure”) is not
unique. We assume that an EMM has been fixed once and for all, and all expectation
operators appearing in this text will be with respect to this measure.
For an exposition of the general theory of L´evy processes and their applications
to pricing derivative securities, we refer the reader to the monographs [34] and [7],
respectively. We recall that every L´evy process X={Xt}t≥0has a characteristic
exponent, which is a continuous function ψ:R−→ Csatisfying ψ(0) = 0 and
EeiξXt=e−tψ(ξ)∀ξ∈R, t ≥0;
and, conversely, the law of a L´evy process is uniquely determined by its characteristic
exponent [34, Thm. 7.10]. Some examples of L´evy processes that are commonly used
in empirical studies of financial markets are listed in §1.4 below.
In terms of the characteristic exponent of the log-price process {Xt}, the EMM-
condition can be written as follows:
(1.1) r+ψ(−i) = 0,
where we are implicitly assuming that ψ(ξ) admits analytic continuation into the
closed strip −1≤Im ξ≤0 (if this is not the case, then E[St] = ∞for all t > 0, i.e.,
the process {St}cannot be priced; we exclude this situation from our consideration).
1.3. No-arbitrage pricing formulas. Consider a down-and-out barrier put option
with maturity date T, strike price Kand barrier H, where the underlying asset
satisfies the assumptions of §1.2. Let τ(0,H]denote the first hitting time of the interval
(0, H] by the price process {St}t≥0; by definition, this means that
τ(0,H](ω) = inft≥0St(ω)∈(0, H]∀ω∈Ω

8 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
(by convention, inf ∅=∞). Since almost every trajectory of {St}is right-continuous
(as {Xt}is a L´evy process), it follows that for almost every ω∈Ω with τ(0,H](ω)<∞,
we have Sτ(0,H](ω)∈(0, H ], which justifies the use of the term “first hitting time.”
From the definition given in §1.1, it follows that the value function, vd.o.b.p.(s0, T ),
of the option under consideration, is given by the following stochastic expression:
(1.2) vd.o.b.p.(s0, T ) = Eln s0e−rT (K−ST)+·{τ(0,H]>T },
where s0is the current spot price of the underlying, {τ(0,H ]>T }denotes the indicator
function of the subset of Ω on which τ(0,H]> T (this subset corresponds to those
trajectories of {St}that do not reach or fall below the barrier Hprior to maturity),
and Eln s0denotes the expectation under the law of the process {ln s0+Xt}t≥0(we
recall that, by convention, X0= 0 a.e.).
With the same notation, consider a down-and-in first-touch digital option with
barrier Hand maturity date T. Its value function, vd.i.f.t.d. (s0, T ), is given by the
stochastic expression
(1.3) vd.i.f.t.d.(s0, T ) = Eln s0e−rτ(0,H ]·{τ(0,H]≤T}.
In §2 we will explain how the expressions that appear in formulas (1.2) and (1.3)
can be approximated with ones that are amenable to fast numerical calculations.
We also recall that in the same setup, if v=v(s0, T ) is the value function of one of
the options under consideration (where s0is the current spot price of the underlying
and Tis the maturity date), then the delta (respectively, gamma; respectively, theta)
of the option is defined as the derivative ∂v/∂s0(respectively, ∂2v/∂s2
0; respectively,
∂v /∂T ), assuming that the derivative exists.
1.4. L´evy processes of exponential type. From now on we will assume that there
exist λ−<−1<0< λ+such that the underlying L´evy process Xis of exponential
type (λ−, λ+). Roughly speaking, this means that the characteristic exponent, ψ(ξ),
of X, admits analytic continuation into the open strip Im ξ∈(λ−, λ+). Moreover,
ψ(ξ) grows at most polynomially as Re ξ→ ±∞ within every closed strip Im ξ∈
[ω−, ω+]⊂(λ−, λ+). A precise formulation (in terms of the L´evy density of X) is
given in [7, Definition 3.2]. The details are not important to us: for the applications
we have in mind, it will suffice to know that the examples below satisfy the definition.
(1) A Brownian motion (used in the classical Black-Scholes model [3]) is a L´evy
process of exponential type (−∞,∞). Its characteristic exponent is given by
σ2
2ξ2−iµξ, where σ > 0 is the volatility and µ∈Ris the drift of the process.
(2) In Merton’s model [32], the underlying log-price process is a L´evy process with
characteristic exponent ψ(ξ) = σ2
2ξ2−iµξ +λ·1−eimξ−s2
2ξ2, where σ, s, λ > 0
and µ, m ∈R. A process of this kind also has exponential type (−∞,∞).

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 9
(3) A hyper-exponential jump-diffusion process [1, 26, 18] has characteristic exponent
(1.4) ψ(ξ) = σ2
2ξ2−iµξ +λ+·
n+
X
j=1
ip+
jα+
jξ
iξ −α+
j
+λ−·
n−
X
k=1
ip−
kα−
kξ
iξ +α−
k
,
where n±are positive integers and α±
j, λ±, p±
j>0 satisfy Pn±
j=1 p±
j= 1. Kou’s
model [23], which was discovered earlier, can be obtained as a special case of
hyper-exponential jump-diffusion models by taking n+=n−= 1. A L´evy process
with characteristic exponent (1.4) is of exponential type max{−α−
k},min{α+
j}.
(4) L´evy processes of the extended Koponen family (generalizing the class of processes
introduced by Koponen [22]) were defined by Boyarchenko and Levendorski˘
i in
[5]. Later the same family of L´evy processes was used in [15] under the name
“CGMY-model,” and in [7] under the name “KoBoL processes.” We adopt the
latter terminology. The characteristic exponent of a KoBoL process of order
ν∈(0,2), ν6= 1, has the form5
(1.5) ψ(ξ) = −iµξ +c·Γ(−ν)·(−λ−)ν−(−λ−−iξ)ν+λν
+−(λ++iξ)ν,
where λ−<0< λ+are called the steepness parameters of the process, c > 0
is its intensity, and µ∈R. A KoBoL process with parameters as above has
exponential type (λ−, λ+), so there is no conflict of notation.
(5) Variance Gamma (V.G.) processes were first used in empirical studies of financial
markets by Madan and collaborators [30, 29, 28]. The characteristic exponent of
a V.G. process has the form6:
(1.6) ψ(ξ) = −iµξ +c·ln(−λ−−iξ)−ln(−λ−) + ln(λ++iξ)−ln(λ+),
where λ−<0< λ+,c > 0 and µ∈R. A V.G. process with these parameters is
also a L´evy process of exponential type (λ−, λ+).
(6) Normal Inverse Gaussian (NIG) processes were constructed by Barndorff-Nielsen,
and were applied to empirical studies of financial markets in [2]. The character-
istic exponent of a NIG process is of the form
(1.7) ψ(ξ) = −iµξ +δ·hα2−(β+iξ)21/2−(α2−β2)1/2i,
where α > |β|>0, δ > 0 and µ∈R. A NIG process with these parameters has
exponential type (β−α, β +α).
Other examples of L´evy processes of exponential type can be found in [7, Ch. 3].
5In the formulas below, and elsewhere in the text, we use the standard convention that zν=eν·ln z
for any ν∈Cand any z∈Csuch that z6∈ (−∞,0]. In turn, ln zdenotes the unique branch of
the natural logarithm function defined on the complex plane with the negative real axis (−∞,0]
removed, determined by the requirement that ln(1) = 0.
6What we present is not the most common way of writing the formula. Rather, we chose an
expression that is equivalent to the standard one and makes the analogy with (1.5) transparent.

10 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
2. Calculation of the prices and the Greeks
2.1. Carr’s randomization for barrier options. Carr’s randomization [12] is an
approximation procedure that replaces the problem of pricing a finite-lived option
with a sequence of option pricing problems for perpetual options. These problems can
be solved in closed form that is amenable to very fast numerical calculations using the
Wiener-Hopf factorization method developed in a series of works by S.I. Boyarchenko
and Levendorski˘
i [6, 7, 8, 27, 9]. The numerical efficiency of Carr’s randomization
approximation, as well as its empirically demonstrated accuracy, are the reasons we
prefer it to the currently available closed form solutions of pricing problems for barrier
and first-touch digital options (see, e.g., [7, Ch. 7–8], and also [24, 25, 18]).
In this subsection and the next one we will explain how Carr’s method can be
applied to barrier options, following the original reference, [12], rather closely7.
Throughout this section we work with a market satisfying the assumptions of §1.
Let us consider a down-and-out barrier contingent claim with barrier H, expiry date
T, and terminal payoff function G(x), which is a nonnegative measurable function
on R. If, at any time t≤Tprior to expiry, the price, St=eXt, of the underlying,
reaches or falls below H, the claim expires worthless. Otherwise, at expiry, the claim
yields payoff equal to G(XT). Contingent claims of this type provide a common
generalization of down-and-out barrier call and put options8considered in §1.1.
The formula for the value of this claim at time t= 0 can be written as follows:
(2.1) V(x, T ) = Exe−rT G(XT){τh>T },
where xis the current log-spot price of the underlying, h= ln Hand τhis the first
hitting time of the interval (−∞, h] by the process {Xt}t≥0. Carr’s insight [12] was
to allow the expiry date of the claim to be random, rather than deterministic. As
a first step, let T0∼Exp T−1be an exponentially distributed random variable with
mean T. With a slight abuse of notation, we define V(x, T 0) by formally replacing T
with T0in (2.1) everywhere (note that V(x, T 0) is still a deterministic quantity). It
turns out that V(x, T 0) can be calculated more explicitly.
Before giving a formula for V(x, T 0), let us introduce some notation. First we define
the supremum process Xand the infimum process Xof Xby
(2.2) Xt= sup
0≤s≤t
Xs, Xt= inf
0≤s≤tXs.
7Carr’s randomization is equivalent to the analytic method of lines due to Carr and Faguet [13].
The latter method, applied to barrier options, amounts to the time discretization of the partial
integro-differential equation (PIDE) satisfied by the value function of the option. The reader with
a more analytic background may prefer this viewpoint; it is explained and used in [20, 4].
8In particular, G(x) = (ex−K)+(respectively, G(x) = (K−ex)+) for a down-and-out barrier
call (respectively, put) option with strike price K. The reason behind allowing for more general
payoff functions will soon become apparent.

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 11
Given any q > 0, we let Tq∼Exp qdenote an exponentially distributed random
variable with mean q−1, and we define operators E+
qand E−
qacting on a nonnegative
measurable (or an arbitrary bounded measurable) function fon Ras follows:
(2.3) (E+
qf)(x) = Exf(XTq),(E−
qf)(x) = Exf(XTq).
With this notation, we have the following result.
Lemma 2.1. Let Tand T0be as above, and let q=r+ 1/T . Then
(2.4) V(x, T 0) = (1 + rT )−1· E −
q[h,+∞)(x)·(E+
qG)(x).
Proof. See §A.7. 
The right hand side of (2.4) is computable in practice, provided an explicit analytic
formula for the characteristic exponent of the underlying process {Xt}is available.
Example 2.2.If X={Xt}is a Brownian motion with volatility σand drift µ, then
E+
q(respectively, E−
q) is a convolution operator with exponentially decaying kernel
β+e−β+y[0,+∞)(y) (respectively, −β−e−β−y(−∞,0](y)), where β−<0< β+are the
roots of the quadratic equation σ2
2β2+µβ −q= 0. For Kou’s model, or, more
generally, for a hyper-exponential jump-diffusion (§1.4(3)), E±
qare linear combinations
of convolution operators of similar form. Thus very accurate and efficient numerical
realizations of the operators E±
qcan be designed in this case.
For other examples of L´evy processes listed in §1.4, efficient numerical realizations
of the operators E±
qare developed in [20, 4]; we recall one of them in §3 below. The
operators E±
qare called the expected present value (EPV) operators of the supremum
and infimum processes of X. We defer a more detailed discussion of these operators
to Appendix A, so as not to disrupt the flow of the present section.
2.2. Carr’s randomization and backward induction. In general, of course, one
cannot expect V(x, T 0) to be a good approximation to V(x, T ), since an exponentially
distributed maturity date is not a good approximation to a deterministic maturity
date. As suggested by Carr [12], we next divide the maturity period of the claim
into Nsubintervals, using points 0 = t0< t1<· · · < tN=T, and we replace each
sub-period [ts, ts+1 ] with an exponentially distributed random maturity period with
mean ∆s=ts+1 −ts. Moreover, these Nrandom maturity sub-periods are assumed
to be independent of each other and of the process X. (In [12], it is assumed that
∆s=T/N for all s, but, in principle, we do not have to impose this requirement.)
As in [12], we can calculate the value function of the claim with this new maturity
period using backward induction. Namely, let Vs(x) denote the value function of the
claim after the first smaturity sub-periods. Then, by definition, VN(x) = G(x), the
terminal payoff function. Moreover, for all 0 ≤s≤N−1, the function Vs(x) can
be interpreted as the value function of a down-and-out barrier contingent claim with

12 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
barrier H, terminal payoff function Vs+1(x), and exponentially distributed maturity
date with mean ∆s. We define qs=r+ ∆−1
sand obtain, by Lemma 2.1,
(2.5) Vs(x) = (1 + r∆s)−1· E−
qs[h,+∞)(x)·(E+
qsVs+1)(x).
The function V0(x) obtained at the last step of this algorithm is the desired Carr’s
randomization approximation to the value function V(x, T ) of the original claim with
deterministic maturity date T. As the mesh, maxs∆s, of the partition of the maturity
period of the claim approaches 0, we expect that V0(x) converges to V(x, T ).
2.3. Carr’s randomization for first-touch digitals. In the same situation as
before, let us now consider a down-and-in first-touch digital option with maturity
date Tand barrier H. Let V(x, T ) denote the value of this option at time t= 0,
where xis the current log-spot price of the underlying. As in §2.1, let h= ln H, and
let τhdenote the first hitting time of the interval (−∞, h] by the process {Xt}t≥0.
Formula (1.3) can then be rewritten as follows:
(2.6) V(x, T ) = Exe−rτh{τh≤T}.
The idea of Carr’s randomization remains the same as before. As a first step, we
replace Twith an exponentially distributed random variable T0with mean T, which
is independent of the process {Xt}. We will use the notation introduced in §2.1.
Lemma 2.3. In this situation, let q=r+ 1/T . Then V(x, T 0) = E−
q(−∞,h](x).
Proof. See §A.8. 
To obtain a more accurate randomization approximation to V(x, T ), we divide the
maturity period of the option into Nsubintervals, using points 0 = t0< t1<· · · <
tN=T, and we replace each sub-period [ts, ts+1] with an exponentially distributed
random maturity period with mean ∆s=ts+1 −ts. As before, these Nrandom
maturity sub-periods are assumed to be independent of each other and of the process
X, and we let Vs(x) denote the value function of the claim after the first smaturity
sub-periods. We also let qs=r+ ∆−1
sfor each 0 ≤s≤N−1.
Now, by definition, VN−1(x) can be interpreted as the value function of the down-
and-in first-touch digital option with barrier Hand random exponentially distributed
maturity date with mean ∆N−1. Using Lemma 2.3, we obtain
(2.7) VN−1(x) = E−
qN−1(−∞,h](x).
Next, for every s=N−2, N −3, . . . , 0, we calculate Vs(x) by noting that it can be
interpreted as the value of the following package consisting of two derivatives. The
first one is a down-and-in first-touch digital option with barrier Hand exponentially
distributed maturity date with mean ∆s. The second derivative is a down-and-out
barrier contingent claim with the same barrier and maturity, and with terminal payoff
function equal to Vs+1(x). Combining Lemmas 2.3 and 2.1, we obtain
(2.8) Vs(x) = E−
qs(−∞,h](x)+ (1 + r∆s)−1· E−
qs[h,+∞)(x)·(E+
qsVs+1)(x).

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 13
The function V0(x) obtained at the last step of this inductive procedure is Carr’s
randomization approximation to the value function V(x, T ) of the down-and-in first-
touch digital option with barrier Hand deterministic maturity date T. As the mesh,
maxs∆s, of the partition of the maturity period of the claim approaches 0, we expect
that V0(x) converges to V(x, T ).
Of course, in order to implement the inductive procedures described in §§2.2–2.3,
one needs to have efficient procedures for numerical realization of the action of the
EPV operators E±
q. Such procedures are described in §3 below. Armed with them,
we present explicit algorithms for the calculation of prices and sensitivities of barrier
and first-touch digital options in §4.
2.4. Calculation of the deltas and gammas. We end this section with a few
words on the calculation of the sensitivities of the two types of options considered
in §§2.1–2.3. Let Hbe a fixed barrier, let h= ln H, and let V(x, T ) denote the
value function of either a down-and-out barrier contingent claim, or a down-and-in
first-touch digital option, with barrier Hand (deterministic) maturity date T, where
xis the current log-spot price of the underlying.
We note that V(x, T ) = 0 (respectively, V(x, T ) = 1) for all x≤h, so we are only
interested in calculating the sensitivities in the region where the claim is alive (i.e.,
x>hand T > 0). Since the current spot price, s0, of the underlying, equals ex, the
chain rule yields the following formulas for the delta and the gamma of the claim:
(2.9) ∂V
∂s0
=e−x·∂V
∂x ,∂2V
∂s2
0
=e−x·∂
∂x e−x·∂V
∂x .
There remains, of course, the question of calculating the derivatives with respect to
xthat appear in formulas (2.9). In practice, we found that numerical differentiation
yields sufficiently accurate approximations; see §4.4 for the details.
2.5. Calculation of the thetas. We keep the notation of §2.4. In order to calculate
the theta, ∂V/∂T of the claim, we use the fact that in the region {x > h, T > 0}where
the claim is alive, its value function satisfies the following partial integro-differential
equation (a special case of formula (2.51) in [7, Thm. 2.13]):
(2.10) ∂
∂T +r−LV(x, T ) = 0.
In this PIDE, Ldenotes the infinitesimal generator [34, §31] of the L´evy process X.
Thus we obtain the following formula for the theta of the claim:
(2.11) ∂V
∂T =−(r−L)V(x, T ).
Again, there remains the question of calculating the right hand side of (2.11). Let ψ(ξ)
be the characteristic exponent of X(cf. §1.2). Then Lacts on oscillating exponents

14 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
via Leiξx =−ψ(ξ)eiξx , where ξ∈R. Expanding a more general function f(x) as a
Fourier integral, one deduces the formula [7, Eq. (2.37)]
Fx→ξ(Lf)(ξ) = −ψ(ξ)·(Fx→ξf)(ξ)∀ξ∈R,
where Fx→ξis the Fourier transform, normalized as follows:
(2.12) (Fx→ξf)(ξ) = Z∞
−∞
e−iξxf(x)dx.
The inverse Fourier transform, F−1
ξ→x, is then given by
(2.13) (F−1
ξ→xg)(x) = 1
2πZ∞
−∞
eiξxg(ξ)dξ,
and we deduce the following formula for the theta of the claim:
(2.14) ∂V
∂T =−1
2πZ∞
−∞
eiξx(r+ψ(ξ)) ·Fx→ξV(x, T )dξ
In practice, assuming that an approximation to the function V(x, T ) was found, we
use (2.14) and the enhanced and refined FFT techniques of §§3.4, 3.7 to calculate the
corresponding approximation to the theta of the claim (see §4.5 for the details).
3. Expected present value operators and FFT
3.1. Overview. In §2 we described general methods of approximate calculation of
the prices and sensitivities of down-and-out barrier put options and down-and-in
first-touch digital options. The main ingredients of these methods were forward and
inverse Fourier transforms, defined by (2.12)–(2.13), as well as the expected present
value operators E±
q, defined by (2.3). In the present section we explain an accurate
and efficient approach to realizing these ingredients numerically in practice. All the
material to be discussed in this section has already appeared in [4], so we will be as
concise as possible; however, the present article can be read independently of [4].
Our approach to the calculation of the action of the EPV operators is based on
realizing the latter as convolution operators; we explain this idea in more detail in §3.2
and §§3.5–3.6. The background for §§3.2, 3.5 is contained in Appendix A. Accurate
calculation of the action of convolution operators and of the forward Fourier transform
is based on the idea of enhancement (using piecewise-linear functions to approximate
more general ones), explained in §§3.3–3.4. Finally, the idea of refinement of the dual
grid, which is used to increase the accuracy of calculations involving inverse Fourier
transforms and to disentangle the interactions between the x-grid and the ξ-grid that
is imposed by the ordinary FFT techniques, is recalled in §3.7.

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 15
3.2. EPV operators via convolution. Let X={Xt}t≥0be a 1-dimensional L´evy
process, and let q > 0 be fixed. The EPV operators E±
qwere defined in (2.3). We
also define the EPV operator Eqof the process Xitself by
(3.1) (Eqf)(x) = Exf(XTq),
where Tq∼Exp qis an exponentially distributed random variable with mean q−1.
The justification of the terminology “expected present value operators” is recalled
in §A.2. In §§A.3–A.4 we derive formulas that express the action of Eqand E±
qin
terms of Fourier transforms, as well as in terms of convolution. To avoid unnecessary
repetition, we merely reproduce the latter formulas here, and refer the reader to
Appendix A for the background. The formulas (A.9), (A.11) are as follows:
(3.2) (Eqf)(x) = Z∞
−∞
f(x+y)gq(y)dy, (E±
qf)(x) = Z∞
−∞
f(x+y)g±
q(y)dy,
with the convolution kernels being given by
(3.3) gq(y) = 1
2πZ∞
−∞
qe−iyξ dξ
q+ψ(ξ), g±
q(y) = 1
2πZ∞
−∞
φ±
q(ξ)e−iyξ dξ,
where ψ(ξ) is the characteristic exponent of X(cf. §1.2) and φ±
q(ξ) are the Wiener-
Hopf factors of q(q+ψ(ξ))−1, defined by E±
q(eiξx) = φ±
q(ξ)eiξx. We explain how to
calculate φ±
qin practice in §3.5 below, and refer the reader to §A.5 for background
on the Wiener-Hopf factorization.
3.3. Enhanced realization of convolution operators. Let us consider one of the
formulas (3.2), or the definition of the Fourier transform (2.12). With the standard
approach to the numerical realization of these formulas, one truncates the improper
integral on the right hand side, replacing it with an integral over a bounded interval,
and uses a suitable quadrature rule to approximate the latter integral with a finite
sum. However, we demonstrated in [4] that sometimes this approach leads to very
large computational errors, and suggested an alternative: instead of discretizing the
integral on the right hand side of (2.12) or (3.2), one should first approximate the
function f(x) with a piecewise linear function, and then evaluate the resulting integral
involving this piecewise linear approximation explicitly (in a suitable sense).
In this subsection we recall, following [4], how this idea leads to what we call the
enhanced realization of convolution operators. The enhanced numerical realization
of the Fourier transform is discussed in §3.4 below.
Given functions f(x), g(x) defined on R, we would like to approximately calculate
(3.4) F(x) = Z∞
−∞
f(x+y)g(y)dy.
In order to discretize the problem, we assume given a uniformly spaced grid of points
~x = (xj)M
j=1, where xj=x1+ (j−1)∆ ∈R, and ∆ >0 is fixed. Let us write

16 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
fj=f(xj) for all j. Following the strategy we just described, we approximate f(x)
with a linear function on each of the intervals [xj, xj+1]:
(3.5) f(x)≈fj+ ∆−1·(fj+1 −fj)·(x−xj), xj≤x≤xj+1.
Of course, (3.5) is an exact equality for x=xjand for x=xj+1; inside the interval
(xj, xj+1), the error of this approximation is controlled by the size of the second
derivative f00(x) (assuming that the latter exists).
We now quote [4, Proposition 3.2]:
Proposition 3.1. Let us approximate f(x)by a piecewise linear function on the
interval [x1, xM]using (3.5), and let us approximate f(x)by 0outside of [x1, xM].
This leads to the following approximation of the values of the function F(x):
(3.6) F(xk)≈ −c1
k·f1−(c0
k−M−c1
k−M)·fM+
M
X
j=1
c∆
k−j·fj,
where fj=f(xj), and the coefficients c0
`,c1
`,c∆
`(for `∈Z)are defined by
c0
`=Z∆
0
g(y−`∆) dy, c1
`= ∆−1Z∆
0
y·g(y−`∆) dy,
and c∆
`=c0
`−c1
`+c1
`+1.
Remark 3.2.The values of the sum on the right hand side of (3.6) can be computed
efficiently for all 1 ≤k≤Musing the algorithm presented in §B.3 below.
3.4. FFT, iFFT and enhanced FFT. Even though, for the purposes of the present
article, the enhanced realization of convolution operators is the main computational
tool, the use of fast Fourier transforms cannot be avoided altogether. Indeed, iFFT
is needed in order to calculate the “convolution coefficients” c0
`,c1
`,c∆
`appearing
in Proposition 3.1 when g(y) is one of the functions gq(y) or g±
q(y) given by (3.3).
Furthermore, enhanced FFT is used in the calculation of the thetas of the options
discussed in §§1–2 (cf. §4.5 below).
We begin by recalling the most common approach to the numerical realization of
Fourier and inverse Fourier transforms. Let us consider two uniformly spaced grids,
~x = (xj)M
j=1 and ~
ξ= (ξk)M
k=1, so that
xj=x1+ (j−1)∆,1≤j≤Mand ξk=ξ1+ (k−1)ζ, 1≤k≤M,
where ∆, ζ > 0 are fixed. Using a simplified version of the trapezoid rule, we replace
(Fx→ξf)(ξ) and (F−1
ξ→xg)(x) (see (2.12), (2.13)) with the following two functions:
(3.7) (Ffastf)(ξ) = ∆ ·
M
X
j=1
f(xj)e−iξxj

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 17
and
(3.8) (F−1
fastg)(x) = ζ
2π·
M
X
k=1
g(ξk)eiξkx.
We refer to Ff ast and F−1
fast as the fast Fourier transform (FFT) and the inverse fast
Fourier transform (iFFT), respectively. In §B.2 we recall how the values of Ffastf
(respectively, F−1
fastg) on the grid ~
ξ(respectively, ~x) can be calculated efficiently under
the assumption that M∆ζ= 2π.
In order to obtain the enhanced numerical realization of the Fourier transform, we
truncate the integral on the right hand side of (2.12), replacing it with the integral
over [x1, xM], and replace the function f(x) that appears in the integrand with its
piecewise linear approximation defined by (3.5). As we explained in [4, §3], this leads
to the approximation (Fx→ξf)(ξ)≈(Fenhf)(ξ), where (Fenhf)(ξ) is the enhanced
fast Fourier transform of f, defined by
(Fenhf)(ξ) = eiξ∆+e−iξ∆−2
(iξ∆)2·(Ff astf)(ξ)
+1 + iξ∆−eiξ∆
(iξ∆)2·∆·f(x1)·e−iξx1
+1−iξ∆−e−iξ∆
(iξ∆)2·∆·f(xM)·e−iξxM.
(3.9)
Clearly, the calculation of (Fenhf)(ξ) easily reduces to that of (Ff ast f)(ξ).
3.5. Calculation of the Wiener-Hopf factors. In order to be able to implement
the enhanced realization of the operators E±
q, we must first know how to calculate
the values of the functions φ±
q(ξ) that appear in (3.3). Unfortunately, apart from a
few special cases (such as the hyper-exponential L´evy processes [1, 18]), no explicit
formulas for φ±
q(ξ) are known. Instead, one must use the integral formulas recalled
in §A.6. We will not repeat these formulas here, but will only give the discretized
versions thereof, after introducing some auxiliary notation.
As before, we consider a uniformly spaced grid of points ~
ξ= (ξk)M
k=1 in R, where
ξk=ξ1+ (k−1)ζfor all 1 ≤k≤Mand ζ > 0 is fixed. We assume that there exist
λ−<0< λ+such that the L´evy process X={Xt}t≥0is of exponential type (λ−, λ+)
(see §1.4), and recall that ψ(ξ) denotes the characteristic exponent of X(cf. §1.2).
We will obtain approximate formulas for φ±
q(ξk) by using a simplified version of
the trapezoidal rule to discretize (A.16). Due to the fact that the integrand in (A.16)
decays somewhat slower than |η|−2as Re η→ ±∞, it is sometimes necessary to use
an η-grid that is longer than the ξ-grid for this discretization, in order to guarantee
the desired precision of the calculation of φ±
q(ξk).

18 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
With these remarks in mind, and with the notation above, we present an algorithm
for the approximate calculation of the values (φ±
q(ξk))M
k=1.
•Select a positive integer mthat controls the length of the η-grid.
•Choose ω−∈(λ−,0) and ω+∈(0, λ+) such that there exists δ > 0 with
Re(q+ψ(η)) ≥δwhenever Im η∈[ω−, ω+]. As a rule of thumb, we recommend
taking ω±=λ±/3 in the algorithms that are based on Carr’s randomization
method (§2), since in these examples, qis rather large9.
•Define the η-grid as follows:
~η±= (η±
`)mM
`=1 , η±
`=−mMζ/2+(`−1)ζ+iω∓,
where ~η+(resp., ~η−) is used for the calculation of φ+
q(resp., φ−
q).
•Using the simplified trapezoid rule to discretize (A.16) leads to the following
approximation:
φ±
q(ξk)≈exp "±ζ·ξk
2πi
mM
X
`=1
ln(1 + q−1ψ(η±
`))
η±
`(ξk−η±
`)#.
•The last formula can be rewritten as follows:
(3.10) φ±
q(ξk)≈exp "±ζ·ξk
2πi
m
X
j=1
Ij(ξk)#,
where
Ij(ξk) =
jM
X
`=1+(j−1)M
ln(1 + q−1ψ(η±
`))
η±
`(ξk−η±
`),1≤j≤M.
•Noting that ξk−η±
`=ξ1+mMζ/2−iω∓−(k−`)ζdepends only on the
difference k−`, calculate each of the arrays (Ij(ξk))M
k=1 for j= 1,2, . . . , m
using the “fast convolution” algorithm of §B.3.
•Using the results of the previous step, calculate the right hand side of (3.10).
Remark 3.3.In practice, it is clearly computationally more efficient to calculate either
the values (φ+
q(ξk)) or the values (φ−
q(ξk)), and then use the analytic form (A.14)
of the Wiener-Hopf factorization formula, namely, φ+
q(ξ)φ−
q(ξ) = q(q+ψ(ξ))−1, to
calculate the values (φ−
q(ξk)) (respectively, (φ+
q(ξk))).
3.6. Enhanced realization of the EPV operators. In this subsection we recall
the formulas for the enhanced convolution realization of the operators Eqand E±
qthat
were obtained in [4, §4]. We keep the notation and assumptions of §3.2.
9When q > 0 is small, one has to use a somewhat different approach to the calculation of the
factors φ±
q. A discussion of this approach is beyond the scope of the present article.

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 19
3.6.1. Numerical realization of Eq.We quote [4, Prop. 4.1]:
Proposition 3.4. In the situation of Proposition 3.1, let g(y) = gq(y)be given by
(3.3). Then the coefficients c0
`,c1
`and c∆
`can be found from the following formulas:
c0
`=∆
2πZ∞
−∞
ei`∆ξ·qc0(ξ)dξ
q+ψ(ξ),
c1
`=∆
2πZ∞
−∞
ei`∆ξ·qc1(ξ)dξ
q+ψ(ξ),
c∆
`=∆
2πZ∞
−∞
ei`∆ξ·qc∆(ξ)dξ
q+ψ(ξ),
where
c0(ξ) = 1−e−iξ ∆
iξ∆, c1(ξ) = 1−e−iξ∆−iξ∆e−iξ∆
(iξ∆)2,
and
c∆(ξ) = eiξ ∆+e−iξ∆−2
(iξ∆)2.
Remark 3.5.The apparent singularities of the functions c0(ξ), c1(ξ) and c∆(ξ) at ξ= 0
are, of course, removable, as one can easily verify using the power series expansion of
the exponential function.
Remark 3.6.Let us comment on the question of calculating the coefficients c0
`, c1
`, c∆
`
appearing in Proposition 3.4. We restrict attention to c∆
`(the other ones can be
treated similarly). Analyzing formula (3.6) of Proposition 3.1, we see that, in practice,
we need to compute (c∆
`)M−1
`=1−M, which can be interpreted as the array of values of
the inverse Fourier transform of the function
g∆
q(ξ) = q∆·c∆(ξ)
q+ψ(ξ)=q∆·eiξ∆+e−iξ∆−2
(q+ψ(ξ)) ·(iξ∆)2
on the grid (`∆)M−1
`=1−M. Hence this array can be calculated approximately using the
inverse fast Fourier transform introduced in §3.4 and the algorithm of §B.2. We also
note that, in practice, the refined approach to iFFT (recalled in §3.7 below) must be
used in order to guarantee the desired precision of the calculations.
3.6.2. Numerical realization of E+
q.The enhanced convolution realization of E+
qis
slightly easier, because the kernel of E+
qis supported on [0,+∞) (this is immediate
from the interpretation of E+
qas a normalized EPV operator for the supremum process
of X; see §A.2). Thus, with the notation of Proposition 3.1, we have c0
`=0=c1
`for
` > 0, which leads to the following approximation:
(E+
qf)(xk)≈ −(c0
+,k−M−c1
+,k−M)·fM+
M
X
j=k
c+
k−j·fj(1 ≤k≤M),

20 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
where fj=f(xj) for 1 ≤j≤M, and
c0
+,` −c1
+,` =∆
2πZ∞
−∞
ei`∆ξ·φ+
q(ξ)·e−iξ∆+iξ∆−1
(iξ∆)2dξ
for 1 −M≤`≤0, and
c+
`=c0
+,` −c1
+,` +c1
+,`+1 =∆
2πZ∞
−∞
ei`∆ξ·φ+
q(ξ)·eiξ∆+e−iξ∆−2
(iξ∆)2dξ
for 1 −M≤`≤0.
Remark 3.7.The obvious analogue of Remark 3.6 is valid here as well, the only
difference being that in order to calculate the convolution coefficients c+
`and c0
+,`−c1
+,`,
we must first calculate the array of values of φ+
q(ξ) on a suitable grid in the ξ-space
using the algorithm of §3.5 (where “suitable” means “suitable for the application of
the refined iFFT technique of §3.7”).
3.6.3. Numerical realization of E−
q.The situation is similar to that of §3.6.2, except
that if we view E−
qas a convolution operator, then its kernel is supported on (−∞,0].
Thus, with the notation of Proposition 3.1, we have c0
`= 0 = c1
`for `≤0, which
leads to the following approximation:
(E+
qf)(xk)≈ −c1
−,k ·f1+
k
X
j=1
c−
k−j·fj(1 ≤k≤M),
where fj=f(xj) and
c1
−,` =∆
2πZ∞
−∞
ei(`−1)∆ξ·φ−
q(ξ)·eiξ∆−iξ∆−1
(iξ∆)2dξ
for 1 ≤`≤M, and
c−
`=c0
−,` −c1
−,` +c1
−,`+1 =∆
2πZ∞
−∞
ei`∆ξ·φ−
q(ξ)·eiξ∆+e−iξ∆−2
(iξ∆)2dξ
for 0 ≤`≤M−1. Note that we have an obvious analogue of Remark 3.7 here.
3.7. Refinement of the dual grid. We conclude the section by recalling another
important idea that appeared in [4]. From the viewpoint of the present article, its
main application is to increasing the precision of the iFFT-based calculation of the
convolution coefficients c0
`, c1
`, c∆
`, c0
+,` −c1
+,`, c+
`, c1
−,`, c−
`that appear in the formulas
of §3.6. A detailed discussion of why refinement of the ξ-grid is often indispensable
appears in op. cit.; here we will simply recall the framework introduced in op. cit.
and the realization of FFT and iFFT in this framework.

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 21
3.7.1. An improved setup for FFT. Let a uniformly spaced grid ~x = (xj)M
j=1 of points
in Ris given, where xj=x1+ (j−1)∆, and both Mand ∆ >0 are fixed. One
should choose two positive integers, M2and M3, that will be responsible, respectively,
for refining and stretching the ξ-grid, in the sense made precise below. Finally, one
should choose ξ1∈C, the desired initial point of the ξ-grid. The parameters M2,M3
and ξ1can be varied independently of Mand ∆.
The total number of points in the ξ-grid equals M1=MM2M3. Let us define
ζ= 2π/(M∆). The mesh of the ξ-grid equals ζ1=ζ/M2. Hence the length of the
ξ-grid equals M3·(2π/∆). Explicitly, the ξ-grid is given by
~
ξ= (ξk)M1
k=1, ξk=ξ1+ (k−1)ζ1=ξ1+ (k−1) ·ζ
M2
.
3.7.2. Implementing FFT in the new setup. We remain in the setup of §3.7.1. Let
f(x) be a function whose domain contains the points of the grid ~x. We would like to
calculate the values of Ffastf(defined by (3.7)) at all the points of the grid ~
ξ.
To this end, we represent ~
ξas a disjoint union of M2·M3grids, each of which
has Mpoints and mesh ζ. It is convenient to arrange these grids in a rectangular
pattern, as follows (each row has M2items, and there are M3rows in total):
ξM2·(k−1)+1M
k=1,ξM2·(k−1)+2M
k=1, . . . , ξM2·kM
k=1,
ξM2·(k−1+M)+1M
k=1,ξM2·(k−1+M)+2M
k=1, . . . , ξM2·(k+M)M
k=1,
. . . ,
ξM2·(k−1+(M3−1)M)+1M
k=1, . . . , ξM2·(k+(M3−1)M)M
k=1.
More succinctly, if ~
ξ(j, `) is the sub-grid of ~
ξlabeled by 1 ≤j≤M3and 1 ≤`≤M2,
then
~
ξ(j, `) = ξM2·(k−1+(j−1)M)+`M
k=1.
Since M∆ζ= 2πby construction, and the grid ~
ξ(j, `) has mesh ζ, the values of Ffastf
on ~
ξ(j, `) can be computed using the algorithm of §B.2. Performing M2·M3such
computations, working with one pair (j, `) at a time, we obtain the desired values of
Ffastfon the grid ~
ξ.
3.7.3. Implementing iFFT in the new setup. Let g(ξ) be a function whose domain
contains the grid ~
ξ. We would like to calculate the values of the function F−1
fastg
(defined by (3.8), with M1in place of M) on the grid ~x. To this end, we let gj,` be
the restriction of gto the grid ~
ξ(j, `) defined in §3.7.2, for each 1 ≤j≤M3and

22 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
each 1 ≤`≤M2. The values of F−1
fastgj,` on the grid ~x can be calculated using the
algorithm of §B.2, and it follows immediately from the definitions that
F−1
fastg=1
M2
M3
X
j=1
M2
X
`=1
F−1
fast(gj,`),
which allows us to calculate the values of F−1
fastgon the grid ~x.
4. Practical implementation of the algorithms
In this section we use Carr’s randomization method, recalled in §2, as well as the
technical tools described in §3, to produce efficient algorithms for the calculation
of the prices, deltas, gammas and thetas of down-and-out barrier put options and
down-and-in first-touch digital options.
4.1. General setup. We begin by describing the initial data and the steps of the
algorithm that are common to both types of options. The basic assumptions on the
market are the same as in §1. In particular, we recall that we are considering options
on a non-dividend paying stock whose price process is of the form St=eXt, where
X={Xt}t≥0is a 1-dimensional L´evy process of exponential type (λ−, λ+), with
λ−<−1<0< λ+. (The reader will not lose very much by assuming that Xis a
process of one of the types listed in the examples in §1.4.)
We now commence a step-by-step description of the algorithm.
I. One must describe the market by giving the riskless rate r > 0 and a formula
for the characteristic exponent, ψ(ξ), of the process X. The EMM condition,
r+ψ(−i) = 0, must hold (see §1.2 for the details).
II. One must specify the maturity date, T, and the barrier, H, of the option. In
the case of a barrier option, one must also specify its strike price, K > H .
III. We will use Carr’s randomization in the situation where the maturity period,
[0, T ], of the option is divided into subintervals of equal lengths10. Thus we only
need to specify the number, N, of time steps. Set ∆t=T /N and q=r+ ∆−1
t.
These steps conclude the input of the initial data for the algorithms. From now on
we will write V(x) = V0(x) for the Carr randomization approximation to the value
function of the option, calculated using the procedure described in §2.2 (for a barrier
option) or §2.3 (for a first-touch digital option). The next portion of the algorithm
deals with the control parameters and the auxiliary calculations.
10As we will see momentarily, this assumption increases the accuracy of the calculations, because
it allows us to rewrite the algorithm in such a way that only one application of E+
qand one application
of E−
qare required. We borrowed this idea from the FWHF method of [20].

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 23
IV. Choose a uniformly spaced grid ~x = (xj)M
j=1, where xj=x1+ (j−1)∆ and
∆>0. This is the grid of points where the values of V(x) will be calculated.
Our algorithm is organized in such a way that the optimal choice of x1is
h= ln H. (Note that there is no need to calculate V(x) for x<h, because
V≡0 on (∞, h] for a down-and-out barrier option, and V≡1 on (∞, h] for
a down-and-in first-touch digital option.)
V. Set ζ= 2π/(M∆) and choose positive integers M2,M3so that the dual grid
~
ξ= (ξk)M1
k=1, ξk=−M1ζ1/2+(k−1)ζ1,
where M1=M·M2·M3and ζ1=ζ/M2, is sufficiently long and sufficiently
fine. Since one of the subsequent steps uses FFT for arrays of length 2M1, we
recommend making the choices so that M,M2and M3are powers of 2.
VI. Calculate the values of φ+
q(ξ) on the grid ~
ξusing the algorithm of §3.5. Then
find the values of φ−
q(ξ) on the ~
ξusing the identity φ+
q(ξ)φ−
q(ξ) = q(q+ψ(ξ))−1.
VII. Calculate the convolution coefficients c0
`, c1
`, c∆
`, c1
−,`, c−
`used for the enhanced
realization of the operators Eq(see §3.6.1) and E−
q(see §3.6.3).
To complete the calculation of prices using Carr’s randomization procedure, we
must now consider the two types of options separately.
4.2. Pricing down-and-out barrier put options. For a down-and-out barrier put
option, the remaining steps in the calculation of V0(x) are as follows:
VIII. For 0 ≤s≤N, let Vs(x) be defined as in §2.2. For 0 ≤s≤N−1, put
Ws(x) = [h,+∞)(x)·(E+
qVs+1)(x),
where h= ln H(and His the barrier for the option). Calculate the values of
WN−1(x) on the grid ~x using the next identity (which follows from (A.8)):
WN−1(x) = [h,+∞)(x)· F−1
ξ→xφ+
q(ξ)·b
VN(ξ),
where the Fourier transform of VN(x) = [h,+∞)(x)·(K−ex)+equals11
b
VN(ξ) = K·e−iξ ln H−e−iξ ln K
iξ +e(1−iξ) ln H−e(1−iξ) ln K
1−iξ .
We remark that this step of the algorithm uses the refined implementation of
iFFT, described in §3.7.3 above.
11The apparent singularities of b
VN(ξ) at the points ξ= 0 and ξ=−iare, in fact, removable.

24 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
IX. In a cycle with respect to s=N−2, N −3, . . . , 0, find the values of Ws(x) on
the grid ~x using the realization of Eqas a convolution operator (see (3.2) and
Proposition 3.4) and the next formula (which results from (2.5), the definition
of Ws(x), and the operator form, (A.15), of Wiener-Hopf factorization):
Ws(x) = (1 + r∆t)−1·[h,+∞)(x)·(EqWs+1)(x).
X. Finally, compute the values of V(x) on the grid ~x using the realization of E−
q
as a convolution operator (see (3.2) and §3.6.3) and the formula
V(x) = (1 + r∆t)−1·(E−
qW0)(x).
This is the desired approximation to the value function of the option.
4.3. Pricing down-and-in first-touch digitals. For a down-and-in first-touch dig-
ital option, the steps listed in §4.2 must be replaced with the following ones.
VIII0. For 0 ≤s≤N−1, let Vs(x) be as in §2.3. It will be more convenient for us
to work with the function12 Us(x) = 1 −Vs(x). For 0 ≤s≤N−2, put
Ws(x) = [h,+∞)(x)·(E+
qUs+1)(x),
where h= ln H(and His the barrier for the option). Calculate
WN−2(x) = [h,+∞)(x)· Eq[h,+∞)(x)
(this identity follows from (2.7) and (A.15)).
IX0. In a cycle with respect to s=N−3, N −4, . . . , 0, find the values of Ws(x)
on the grid ~x using the realization of Eqas a convolution operator (see (3.2)
and Proposition 3.4) and the next formula (which can be easily deduced from
(2.8), the definition of Ws(x), and (A.15)):
Ws(x) = (1 + r∆t)−1·[h,+∞)(x)· EqWs+1(x) + r∆t·[h,+∞)(x).
X0. Finally, compute the values of V(x) on the grid ~x using the realization of E−
q
as a convolution operator (see (3.2) and §3.6.3) and the formulas
U0(x) = (1 + r∆t)−1·[h,+∞)(x)· E−
qW0(x) + r∆t·[h,+∞)(x),
V(x) = 1 −U0(x)
This is the desired approximation to the value function of the option.
12It simplifies some of the formulas and has the advantage that it vanishes below h.

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 25
4.4. Deltas and gammas. As we mentioned before, our algorithm uses formulas
(2.9) and numerical differentiation in order to calculate the deltas and the gammas of
the option. In other words, let ~
V= (V(xj))M
j=1 denote the array of approximations to
the value function of the option obtained at step X or X0of the algorithm described in
§4.1 and §4.2 (respectively, §4.1 and §4.3). The corresponding approximations, V∆(xj)
and VΓ(xj), to the values of ∂V
∂s0s0=exjand ∂2V
∂s2
0s0=exj, are computed as follows:
V∆(xj) = e−xj·V(xj+1)−V(xj)
∆,1≤j≤M−1;
VΓ(xj) = e−xj·V∆(xj+1)−V∆(xj)
∆,1≤j≤M−2.
4.5. Thetas. In order to calculate the thetas of the options under consideration,
we use formula (2.14), together with the enhanced FFT realization of Fx→ξV(x, T )
described in §3.4 and the refined FFT and iFFT techniques recalled in §3.7.
Let us make a few technical remarks. If we are in the situation of §4.2, we directly
use the array of values of V(x) on the grid ~x computed at step X in order to calculate
the values of FenhVon the dual grid ~
ξ, and then we use the algorithm of §3.7.3 to
calculate the values of ∂V /∂T . If, instead, we are in the situation of §4.2, then we
replace V(x, T ) with the function U(x, T ) = 1 −V(x, T ), use the identity
∂V
∂T =−(r−L)(1 −U(x, T )) = −r+ (r−L)U(x, T )
(which results from (2.11)), and proceed to calculate (r−L)U(x, T ) as above.
Our examples in §5 show that sometimes integration by parts in formula (2.14)
leads to an integral that converges faster and results in a noticeable improvement of
the precision of the calculation. We briefly mention the idea of this integration by
parts and give the final answer, leaving all the intermediate details to the interested
reader. One begins by representing the integrand as
eiξx(r+ψ(ξ))b
V(ξ)dξ =eihξ(r+ψ(ξ))b
V(ξ)dei(x−h)ξ
i(x−h),
where we write b
V(ξ) = Fx→ξV(x, T ). Next, one integrates by parts and finds that
∂V
∂T (x, T ) = 1
2π(x−h)Z∞
−∞hiψ0(ξ)b
V(ξ)+(r+ψ(ξ)) · Fy→ξ(y−h)V(y, T )ieiξx dξ
for all x > h. The right hand side of the last formula can then be calculated in the
same manner as before (using enhanced and refined FFT and iFFT).

26 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
5. Numerical examples for barrier options
All calculations, the results of which are presented in this work, were performed
in MATLAB c
7.3.0 (R2006b), on a PC with characteristics Intel R
CoreTM 2 Duo
T7200 (2.00GHz, 4MB L2 Cache, 667MHz FSB), under the Genuine Windows R
XP
Professional operating system.
5.1. Barrier options under a NIG process. In this subsection we compare the
performance of our algorithm with the performance of the method developed in [18]
for calculating the prices and sensitivities of down-and-out barrier put options. As
in op. cit., we assume that under a chosen EMM, the log-spot price, Xt= ln St,
of the underlying follows a NIG process (see §1.4(6)) with parameters α= 8.858,
β=−5.808, δ= 0.174. We assume that the riskless rate is r= 0.03, which allows us
to find the remaining parameter, µ≈0.161, from the EMM condition r+ψ(−i) = 0
(where ψ(ξ) is the characteristic exponent of {Xt}, given by (1.7)).
For this market, we calculate the prices, deltas, gammas and thetas of a down-
and-out barrier put option on the stock St=eXt, with strike price K= 3500, barrier
H= 2100 and maturity date T= 1 year.
The results of our calculations are presented in graphical form in Figure 2, and are
recorded numerically in Tables 1 and 2 at the end of this article. (We use the acronym
FWHF to label the results obtained using our method because it is a realization of the
fast Wiener-Hopf factorization method introduced in [20].) The auxiliary parameters
of our algorithm (see §4.1) are specified in the captions to the figure and the tables.
The total amount of CPU time spent calculating the prices, deltas and gammas
of the option at all the points of the x-grid was 6.9 seconds. The two different
calculations of the thetas (see §4.5) took 1.5 seconds in total. This represents an
immense gain in computational speed by comparison with the 55 seconds needed for
the calculation of a single option price in a C++ program, reported in [18].
One can observe that the agreement between our results and the results obtained
using the method of op. cit. is quite good in the region where the spot price of the
underlying is at least 20% away from the barrier. Closer to the barrier, there is a
more noticeable discrepancy, which is as one may expect. Indeed, as we explained in
the Introduction, the value function of the option in the NIG model has a singularity
at the barrier, namely, the delta of the option approaches +∞as the current spot
price of the underlying approaches the barrier (see Figure 1). On the other hand,
the method of op. cit. approximates the NIG model with a hyper-exponential jump-
diffusion that has a tangible diffusion component. For models of the latter type,
the delta of the option has a finite limit as the current spot price of the underlying
approaches the barrier. This explains why, closer to the barrier, the prices of the
option calculated using our method are noticeably higher than the prices calculated
using the method of op. cit.

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 27
2000 2500 3000 3500 4000 4500
0
100
200
300
400
500
600
Spot price
Option prices
FWHF
JP
MC
2000 2500 3000 3500 4000 4500
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Spot price
Option deltas
FWHF
JP
MC
2000 2500 3000 3500 4000 4500
−10
−8
−6
−4
−2
0
2x 10−3
Spot price
Option gammas
FWHF
JP
MC
2000 2500 3000 3500 4000 4500
−400
−300
−200
−100
0
100
Spot price
Option thetas
FWHF
JP
MC
Figure 2. Prices and sensitivities of a down-and-out barrier put option in the
NIG model. The solid lines represent the results obtained using our method (for
the calculation of thetas, we used the second approach explained in §4.5, involving
integration by parts). Crosses and circles represent the results obtained using the
method of [18] and the Monte-Carlo method, respectively (these results are taken
from the tables in op. cit.). NIG parameters: α= 8.858, β=−5.808, δ= 0.174,
µ≈0.161. Option parameters: K= 3500, H= 2100, r= 0.03, T= 1. Algorithm
parameters: ∆ = 0.001, M= 4096, M2= 2, M3= 16, ζ1≈0.77, m= 8 (for the
calculation of the Wiener-Hopf factors: see §3.5), N= 800 (number of time steps).
5.2. Barrier options under a KoBoL process. Next, let us assume that under
a chosen EMM, the log-spot price, Xt= ln St, of the underlying follows a KoBoL
process (see §1.4(4)) with parameters ν= 0.5, c= 1, λ+= 9, λ−=−8. (These
parameters are taken from the examples that appear in [20, 4].) As in §5.1, we
assume that the riskless rate is r= 0.03, which allows us to find the remaining
parameter, µ≈ −0.0423, from the EMM condition.
For this market, we calculate the prices, deltas, gammas and thetas of a down-and-
out barrier put option on the stock St=eXt, with strike price K= 3500, barrier H=
2100 and maturity date T= 0.1 years. The results of our calculations are recorded
in Table 5 at the end of this article. The auxiliary parameters of our algorithm (see
§4.1) are specified in the captions to the table. (We checked independently that the
two methods of calculating thetas explained in §4.5 produce almost identical results.)
The calculation took a total of 15 seconds.

28 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
6. Numerical examples for first-touch digitals
6.1. First-touch digitals under a V.G. process. In this subsection we compare
the performance of our algorithm with the performance of the method developed
in [18] for calculating the prices and sensitivities of down-and-in first-touch digital
options. As in op. cit., we assume that under a chosen EMM, the log-spot price,
Xt= ln St, of the underlying follows a V.G. process (see §1.4(5)) with parameters
c= 0.925, λ+= 4.667, λ−=−11.876. We assume that the riskless rate is r= 0.03,
which allows us to find the remaining parameter, µ≈0.128, from the EMM condition.
For this market, we calculate the prices, deltas, gammas and thetas of a down-
and-in first-touch digital option on the stock St=eXt, with barrier H= 2100 and
maturity date T= 1 year. The results of our calculations are presented in graphical
form in Figure 3, and are recorded numerically in Tables 3 and 4 at the end of this
article. The auxiliary parameters of our algorithm (see §4.1) are specified in the
captions to the figure and the tables.
The total amount of CPU time spent calculating the prices, deltas and gammas of
the option at all the points of the x-grid was 96 seconds. The two different calculations
of the thetas (see §4.5) took 50 seconds in total. Thus our method still yields a
significant gain in computational speed by comparison with the 40 seconds needed for
the calculation of a single option price in a C++ program, reported in [18] (especially
taking into account the fact that our calculations were done in MATLAB, which is
known to be dozens of times slower than C++). Our algorithm works slower for V.G.
processes than for NIG processes because the characteristic exponent of the former
behaves less regularly at infinity, which forces us to choose a much longer ξ-grid in
order to guarantee the desired accuracy of the calculations.
One can see that the agreement between our results and the results of [18] is
quite good at all spot prices of the underlying; in particular, it is noticeably better
than the agreement that was observed in §5.1 for down-and-out barrier put options
under a NIG process. This, too, can be explained. Just like a NIG process, a V.G.
process has zero diffusion component and infinite activity jump component, whereas
a hyper-exponential jump-diffusion has finite activity jump component. However,
unlike NIG processes, V.G. processes are on the border between processes with finite
activity jump component and processes with infinite activity jump component. Thus
one may expect that using hyper-exponential jump-diffusions to approximate V.G.
processes leads to more accurate results than in the case of NIG processes.
6.2. First-touch digitals under a KoBoL process. Next we consider the same
market as the one described in §5.2. We calculate the prices, deltas, gammas and
thetas of a down-and-in first-touch digital option on the stock St=eXt, with barrier
H= 2100 and maturity date T= 0.1 years. The results of are also recorded in Table
5 at the end of this article. The calculation took a total of 14 seconds.

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 29
2000 2500 3000 3500 4000 4500
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Spot price
Option prices
FWHF
JP
MC
2000 2500 3000 3500 4000 4500
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0x 10−3
Spot price
Option deltas
FWHF
JP
MC
2000 2500 3000 3500 4000 4500
0
1
2
3
4
5x 10−6
Spot price
Option gammas
FWHF
JP
MC
2000 2500 3000 3500 4000 4500
0
0.05
0.1
0.15
0.2
Spot price
Option thetas
FWHF
JP
MC
Figure 3. Prices and sensitivities of a down-and-in first-touch digital option in
the V.G. model. The solid lines represent the results obtained using our method (for
the calculation of thetas, we used the second approach explained in §4.5, involving
integration by parts). Crosses and circles represent the results obtained using the
method of [18] and the Monte-Carlo method, respectively (these results are taken
from the tables in op. cit.). V.G. parameters: c= 0.925, λ+= 4.667, λ−=−11.876,
µ≈0.128. Option parameters: H= 2100, r= 0.03, T= 1. Algorithm parameters:
∆ = 0.001, M= 4096, M2= 2, M3= 256, ζ1≈0.77, m= 8 (for the calculation of
the Wiener-Hopf factors: see §3.5), N= 800 (number of time steps).
Appendix A. Wiener-Hopf factorization
The Wiener-Hopf factorization (WHF) method was applied to option pricing in a
series of works by S.I. Boyarchenko and Levendorski˘
i, including [6, 7, 8, 27, 9]. In
this appendix we summarize the main definitions and facts used in this method.
A.1. Setup. Throughout the appendix, we work with a fixed 1-dimensional L´evy
process X={Xt}t≥0, a number q > 0, and an exponentially distributed random
variable Tqwith mean q−1that is independent of the process X. We let ψ(ξ) denote
the characteristic exponent of X(cf. §1.2). The supremum and infimum processes,
Xand X, of X, are defined by (2.2).
A.2. EPV operators. The EPV operators Eqand E±
qare defined by (3.1) and (2.3),
respectively. Since the probability distribution of Tqis qe−qt [0,+∞)(t)dt, and since

30 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
Tqis independent of X, we obtain
(A.1) (Eqf)(x) = ExZ∞
0
qe−qt f(Xt)dt,
(A.2) (E+
qf)(x) = ExZ∞
0
qe−qt f(Xt)dt
and
(A.3) (E−
qf)(x) = ExZ∞
0
qe−qt f(Xt)dt,
which explains the terminology “EPV operators”: the operator Eq(respectively, E+
q
or E−
q) calculates the (normalized) expected present value of a perpetual stream of
profits f(Xt) (respectively, f(Xt) or f(Xt)) under the assumption that the riskless
rate at which one can borrow or lend money is equal to q.
A.3. Realization of the EPV operators via Fourier transforms. Formula
(A.1) and the identity ExeiξXt=eixξe−tψ(ξ), which follows from the definition of
the characteristic exponent, ψ(ξ), implies that
(A.4) Eqeixξ=q·(q+ψ(ξ))−1·eixξ.
Decomposing a more general function f(x) as a Fourier integral yields
(A.5) (Eqf)(x) = 1
2πZ∞
−∞
eiξx qb
f(ξ)
q+ψ(ξ)dξ,
where b
f(ξ) = Fx→ξfis the Fourier transform of f. Identity (A.5) can be justified
under rather weak regularity assumptions; we refer the reader to [7, §2.3.3] for the
details (with the notation of loc. cit., we have Eq=qUq, where Uqis referred to as
the resolvent operator, or the q-potential operator, of X).
To obtain similar expressions for the action of E±
q, let us define functions φ±
q(ξ) by
(A.6) φ+
q(ξ) = EeiξXTq, φ−
q(ξ) = EeiξXTq.
It follows immediately from (2.3) that
(A.7) E±
qeiξx=φ±
q(ξ)·eiξx.
As above, we deduce that
(A.8) (E±
qf)(x) = 1
2πZ∞
−∞
eiξxφ±
q(ξ)b
f(ξ)dξ.

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 31
A.4. Convolution realization of the EPV operators. As in [4, §4.2], formulas
(A.5) and (A.8) can be used to obtain alternative expressions for the action of the
operators Eqand E±
q. Starting with (A.5) and applying Fubini’s theorem, we get
(Eqf)(x) = 1
2πZ∞
−∞
eiξx qb
f(ξ)
q+ψ(ξ)dξ
=1
2πZ∞
−∞
eiξx ·q
q+ψ(ξ)·Z∞
−∞
e−iξy f(y)dy dξ
=1
2πZ∞
−∞
f(y)Z∞
−∞
qe−iξ(y−x)
q+ψ(ξ)dξ dy
=1
2πZ∞
−∞
f(y+x)Z∞
−∞
qe−iξy
q+ψ(ξ)dξ dy,
where the last identity used the change of variables y7→ y+x. Finally, we arrive at
(A.9) (Eqf)(x) = Z∞
−∞
f(x+y)gq(y)dy,
where the convolution kernel gq(y) is given by
(A.10) gq(y) = 1
2πZ∞
−∞
qe−iyξ dξ
q+ψ(ξ).
Completely analogous calculations show that
(A.11) (E±
qf)(x) = Z∞
−∞
f(x+y)g±
q(y)dy,
with the convolution kernels g±
q(y) being given by the formulas
(A.12) g±
q(y) = 1
2πZ∞
−∞
φ±
q(ξ)e−iyξ dξ.
A.5. Three forms of WHF. The form of the Wiener-Hopf factorization (WHF)
formula that is commonly used in probability theory is as follows:
(A.13) EeiξXTq=EeiξX Tq·EeiξX Tq∀ξ∈R,
where Tqis as in §A.1. This formula immediately follows from the obvious identity
XTq= (XTq−XTq) + XTqand two important facts [33, p. 81]:
•the random variables XTqand XTq−XTqare independent; and
•the random variables XTqand XTq−XTqare identical in law.

32 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
Next, identities (A.13), (A.4) and the definitions (A.6) imply the analytical form
of the Wiener-Hopf factorization formula:
(A.14) q
q+ψ(ξ)=φ+
q(ξ)φ−
q(ξ).
Since the trajectories of the supremum process Xare nondecreasing, φ+
q(ξ) admits
analytic continuation into the upper half-plane, in view of (A.6). Similarly, φ−
q(ξ)
admits analytic continuation into the lower half-plane. Thus, equation (A.14) is a
special case of the Wiener-Hopf factorization of a function defined on the real axis,
which historically was the first form of the WHF formula.
Finally, (A.4), (A.7) and (A.14) imply the operator form of the WHF formula:
(A.15) Eq=E+
qE−
q=E−
qE+
q.
A.6. Integral formulas for the WH factors. We now suppose that there exist
λ−<0< λ+such that the L´evy process Xis of exponential type (λ−, λ+) (cf. §1.4).
Under a certain regularity assumption on the characteristic exponent ψ(ξ) of X(see,
e.g., [7, Theorem 3.2]), S.I. Boyarchenko and Levendorski˘
i obtained integral formulas
for the Wiener-Hopf factors φ±
q(ξ). This assumption holds in all model examples of
L´evy processes of exponential type, including those listed in §1.4, so we prefer not to
state it for the sake of space. The formulas [7, Eqns. (3.58), (3.60)] are as follows:
(A.16) φ±
q(ξ) = exp ±1
2πi ZIm η=ω∓
ξ·ln(1 + q−1ψ(η))
η(ξ−η)dη.
Here, ω±are real numbers such that λ−< ω−<0< ω+< λ+, and such that there
exists δ > 0 with Re(q+ψ(η)) ≥δwhenever Im η∈[ω−, ω+].
Under some additional assumptions on ψ(ξ), one can improve the convergence of
the integral in formula (A.16), i.e., reduce its evaluation to the calculation of another
one, where the integrand decays even faster at infinity. The details of this approach
are explained in [4, §4.3]. We note, however, that the required assumption does not
hold for Variance Gamma processes. For this reason, and for the sake of simplicity,
we will only use (A.16) in the present paper. The numerical examples in §§5–6 show
that formula (A.16) can be used to guarantee the desired accuracy of computations
at the expense of a moderate increase in computational cost by comparison with the
improved method of loc. cit.
A.7. Proof of Lemma 2.1. Let us recall the necessary notation: r, T, H > 0 are
fixed, h= ln H, and G(x) is the terminal payoff function of the option. Further, T0
denotes an exponentially distributed random variable with mean T, independent of
the process X, and we set q=r+ 1/T .

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 33
Since the probability distribution of T0is given by e−t/T
T[0,+∞)(t)dt, and since T0
is independent of X, we have, in view of (2.1),
V(x, T 0) = ExZ∞
0
1
T·e−rte−t/T G(Xt){τh>t}dt
=1
T(r+T−1)Z∞
0
(r+T−1)·e−(r+T−1)tG(Xt){τh>t}dt
= (1 + r T )−1·ExG(XTq){τh>Tq},
where Tqis as in §A.1. Next, we observe that the identity τh> Tqis equivalent to
XTq> h, by definition. Thus we can write
ExG(XTq){τh>Tq}=ExG(XTq+XTq−XTq){XTq>h}.
Using the two facts recalled in §A.5, we see that
ExG(XTq+XTq−XTq){XTq>h}=E−
q[h,+∞)(x)·(E+
qG)(x),
which proves Lemma 2.1.
A.8. Proof of Lemma 2.3. We keep all the notation of §A.7. In view of (2.6) and
the definition of T0, we have
V(x, T 0) = ExZ∞
0
1
T·e−rτhe−t/T {τh≤t}dt
def
=Ex1
T·e−rτh·Z∞
τh
e−t/T dt
=Exe−qτh,
where q=r+ 1/T . Now e−qτhis a random variable with values in [0,1]. Introducing
an auxiliary variable λ∈[0,1], we can write
Exe−qτh=Z1
0
Ex{e−qτh≥λ}dλ.
Next, we make the change of variables t=−(ln λ)/q in the last integral (so that
λ=e−qt and dλ =−qe−qtdt), which transforms it into
Z∞
0
qe−qt Ex{τh≤t}dt.
Using Fubini’s theorem and the fact that τh≤tif and only if Xt≤h, we can rewrite
the last integral as
ExZ∞
0
qe−qt {Xt≤h}dt=Ex{XTq≤h}def
=E−
q(−∞,h](x),
which proves Lemma 2.3.

34 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
A.9. On methods that use Laplace transform with respect to time. We are
now in a position to explain the ideas of the methods of pricing barrier and first-touch
digital options that use Laplace transforms with respect to the maturity date, such as
the one used in [18]. For simplicity, we restrict attention to a down-and-in first-touch
digital option with barrier H, and let V(x, T ) denote its value function, where xis
the current log-spot price of the underlying and Tis the maturity date. Thus V(x, T )
is given by formula (2.6), where h= ln H. We will hold xfixed in what follows.
The Laplace transform, LTV, of V(x, T ) with respect to T, is defined as
(LTV)(q) = Z∞
0
e−qT V(x, T )dT ∀q > 0.
We calculate it using Fubini’s theorem, as in §A.8:
(LTV)(q) = ExZ∞
0
e−qT e−rτh{τh≤T}dT 
=Exe−rτhZ∞
τh
e−qT dT 
=1
q·Exe−(r+q)τh=1
q· E−
r+q(−∞,h](x),
where at the last step we used an identity obtained in the course of §A.8.
Let us now suppose that the L´evy process Xis hyper-exponential (see [1, 26, 18]
and also §1.4(3)). Then, in particular, the characteristic exponent, ψ(ξ), of Xis
a rational function, which allows us to calculate the Wiener-Hopf factors φ±
r+q(ξ)
explicitly, provided we can find the zeros and poles of the function q+r+ψ(ξ) (see
[26], or [18], or [9, Ch. 11] for the details). In turn, the explicit formula for φ−
q+r(ξ)
allows us to calculate explicitly the kernel of E−
q+rgiven by (A.12), which can then
be used to explicitly calculate E−
r+q(−∞,h](x).
This results in an explicit formula for the Laplace transform (LTV)(q) that involves
the roots of the characteristic equation q+r+ψ(ξ) = 0. It is not hard to check that
this formula is the same as the formula given in [18, §4.1]; in fact, even the derivation
we presented is more or less equivalent to the argument used in op. cit.
Note, however, that in order to use this method for the calculation of V(x, T ),
a numerical Laplace inversion must be performed (as was done in op. cit.). This
procedure requires the evaluation of (LTV)(q) at many different points q. In turn,
this means that the equation q+r+ψ(ξ) = 0 must be solved for many different values
of q. Unfortunately, even for Kou’s model [23], the characteristic equation amounts
to a polynomial equation of degree 4, whereas for more general hyper-exponential
jump-diffusion models, the degree of the equation becomes larger (with the notation
of §1.4(3), the degree is equal to 2+n++n−), so that we have no choice but to find the
roots of the characteristic equation numerically (and approximately). The necessity

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 35
of solving many different polynomial equations of large degree is what accounts for
the high computational cost of the method we described.
Appendix B. Applications of FFT
The goal of this appendix is to recall the standard definitions of fast Fourier and
inverse Fourier transforms used in modern programming languages, and explain how
these procedures can be used for efficient numerical calculation of the sums appearing
in formulas such as (3.7), (3.8) and (3.6). (This appendix corresponds more or less
to [4, §6], but the format in which it is written is somewhat different.)
B.1. Discrete Fourier transforms. If ~
f= (fj)M
j=1 is an array of complex numbers,
the discrete Fourier transform of ~
fis another array of complex numbers, fft(~
f), of
the same length, whose entries are given by the formula
(B.1) fft(~
f)k=
M
X
j=1
fj·e−2πi(j−1)(k−1)/M ,1≤k≤M.
Similarly, the inverse discrete Fourier transform of an array ~g = (gk)M
k=1 is given by
(B.2) ifft(~g)j=1
M
M
X
k=1
gk·e2πi(j−1)(k−1)/M ,1≤j≤M.
The reason we use the notation fft and ifft for the discrete Fourier transforms
is that most of the standard numerical algorithms for computing fast Fourier and
inverse Fourier transforms are designed for the calculation of (B.1) and (B.2).
B.2. FFT and iFFT. We assume that two uniformly spaced grids, ~x = (xj)M
j=1 and
~
ξ= (ξk)M
k=1, are given, so that
xj=x1+ (j−1)∆,1≤j≤Mand ξk=ξ1+ (k−1)ζ, 1≤k≤M,
where ∆, ζ > 0 are fixed. We also assume that M∆ζ= 2π.
If f(x) is a function whose domain contains the points xj, and (Ffastf)(ξ) is defined
by (3.7), the array of values (Ffastf)(ξk)M
k=1 can be computed as follows.
•Calculate the array ~
f= (fj)M
j=1 whose entries are given by
fj=f(xj)·exp(−i·∆·ξ1·(j−1)),1≤j≤M.
•The desired values of Ffastfare given by
(Ffastf)(ξk) = ∆ ·exp(−i·x1·ξk)·fft(~
f)k,1≤k≤M.
If g(ξ) is a function whose domain contains the points ξk, and (F−1
fastg)(x) is defined
by (3.8), the array of values (F−1
fastg)(xj)M
j=1 can be computed as follows.

36 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
•Calculate the array ~g = (gk)M
k=1 whose entries are given by
gk=g(ξk)·exp(i·ζ·x1·(k−1)),1≤k≤M.
•The desired values of F−1
fastgare given by
(F−1
fastg)(xj) = (1/∆) ·exp(i·ξ1·xj)·ifft(~g)j,1≤j≤M.
B.3. Discrete convolution. The enhanced realization of EPV operators described
in §§3.3, 3.6, and the algorithm for the approximate calculation of the Wiener-Hopf
factors presented in §3.5, involve the calculation of sums of the form
(B.3) hk=
M
X
j=1
fjgk−j,
where ~
f= (fj)M
j=1 and ~g = (g`)M−1
`=1−Mare arrays of complex numbers of lengths M
and 2M−1, respectively. In this subsection we explain how the calculation of the
sums (B.3) can be reduced to three applications of FFT to arrays of length 2M.
•Let e
fbe the array of length 2Mwith entries
e
fj=(fj,1≤j≤M;
0, M + 1 ≤j≤2M.
•Let egbe the array of length 2Mwith entries
g0, g1, . . . , gM−1,0, g1−M, g2−M, . . . , g−1
•Calculate the array e
h= (e
h`)2M
`=1 with entries
e
h`=fft(e
f)`·fft(eg)`.
•The sums (B.3) are given by
hk=ifft(e
h)k,1≤k≤M.
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PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 39
Table 1. Prices of a down-and-out barrier put option in the NIG
model: comparison with the results of Jeannin and Pistorius [18]
Spot Option price Relative difference between FWHF and
price FWHF JP (MC Down, MC Up) JP MC Down MC Up
64% 507.0212 486.8291 (488.3106, 493.8222) 0.0415 0.0383 0.0267
66% 554.2226 532.6638 (534.0392, 539.1428) 0.0584 0.0378 0.0280
68% 573.7149 551.8006 (552.8031, 557.6677) 0.0397 0.0378 0.0288
70% 574.2871 552.6467 (540.5887, 558.0167) 0.0392 0.0380 0.0292
72% 561.3327 540.3645 (518.1208, 545.3937) 0.0388 0.0384 0.0292
74% 538.5875 518.5689 (518.1208, 523.0244) 0.0386 0.0395 0.0298
76% 508.8566 490.0120 (489.2809, 494.3067) 0.0384 0.0400 0.0294
78% 474.3919 456.9306 (455.8267, 460.9730) 0.0382 0.0407 0.0291
80% 437.1020 421.2329 (419.7600, 425.0064) 0.0377 0.0413 0.0285
82% 398.6511 384.5789 (382.8580, 388.1767) 0.0366 0.0413 0.0270
84% 360.4770 348.3821 (346.4850, 351.8264) 0.0347 0.0404 0.0246
86% 323.7562 313.7675 (311.6338, 316.9683) 0.0318 0.0389 0.0214
88% 289.3576 281.5287 (279.2026, 284.4766) 0.0278 0.0364 0.0172
90% 257.8240 252.1222 (249.6634, 254.8398) 0.0226 0.0327 0.0117
92% 229.3982 225.7084 (223.1217, 228.1809) 0.0163 0.0281 0.0053
94% 204.0858 202.2236 (199.4732, 204.4055) 0.0092 0.0231 -0.0016
96% 181.7321 181.4602 (178.6514, 183.4198) 0.0015 0.0172 -0.0092
98% 162.0915 163.1318 (160.2075, 164.8221) -0.0064 0.0118 -0.0166
100% 144.8760 146.9159 (143.9324, 148.3939) -0.0139 0.0066 -0.0237
102% 129.8093 132.5359 (129.6358, 133.9306) -0.0206 0.0013 -0.0308
104% 116.6071 119.8498 (117.0598, 121.2020) -0.0271 -0.0039 -0.0379
106% 105.0261 108.6566 (105.9225, 109.8985) -0.0334 -0.0085 -0.0443
108% 94.8474 98.7645 (96.1249, 99.9680) -0.0397 -0.0133 -0.0512
110% 85.8807 90.0030 (87.4760, 91.1173) -0.0458 -0.0182 -0.0581
112% 77.9615 82.2233 (79.7728, 83.3392) -0.0518 -0.0227 -0.0645
114% 70.9488 75.2968 (72.9390, 76.3846) -0.0577 -0.0273 -0.0712
116% 64.7221 69.1130 (66.8413, 70.1647) -0.0635 -0.0317 -0.0776
118% 59.1783 63.5768 (61.3948, 64.5947) -0.0692 -0.0361 -0.0839
120% 54.2293 58.6068 (56.4842, 59.5707) -0.0747 -0.0399 -0.0897
122% 49.7997 54.1331 (52.0623, 55.0605) -0.0801 -0.0435 -0.0955
124% 45.8249 48.3773 (46.9334, 49.8212) -0.0528 -0.0236 -0.0802
126% 42.2494 44.7756 (43.3803, 46.1709) -0.0564 -0.0261 -0.0849
The first column contains the spot price as a percentage of 3500, and the second one contains the
option prices calculated using the algorithm of §4. The third (respectively, fourth) column
contains the option prices calculated using the method of [18] (respectively, the Monte-Carlo
method); they are taken from the tables in op. cit. The Monte-Carlo results for the prices are
reported in the form of a 95% confidence interval. The relative difference between quantity Aand
quantity Bis defined as (A−B)/B (this explains how columns 5–7 should be interpreted).
NIG parameters: α= 8.858, β=−5.808, δ= 0.174, µ≈0.161.
Option parameters: K= 3500, H= 2100, r= 0.03, T= 1.
Algorithm parameters: ∆ = 0.001, M= 4096, M2= 2, M3= 16, ζ1≈0.77, m= 8 (for the
calculation of the Wiener-Hopf factors: see §3.5), N= 800 (number of time steps).

40 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
Table 2. The greeks of a down-and-out barrier put option in the NIG
model: comparison with the results of Jeannin and Pistorius [18]
Spot Delta Gamma Theta
price FWHF JP MC FWHF JP MC FWHF1FWHF2JP MC
64% 0.929 0.907 0.953 -0.00877 -0.00856 -0.00891 -334.0 -342.5 -359.8 -350.8
66% 0.443 0.437 0.447 -0.00546 -0.00532 -0.00556 -350.1 -359.2 -373.2 -350.1
68% 0.123 0.127 0.123 -0.00377 -0.00368 -0.00371 -346.4 -355.0 -368.6 -350.0
70% -0.102 -0.092 -0.099 -0.00271 -0.00265 -0.00262 -332.4 -340.4 -354.0 -328.7
72% -0.265 -0.251 -0.264 -0.00196 -0.00193 -0.00210 -312.6 -320.0 -333.5 -334.1
74% -0.383 -0.366 -0.378 -0.00140 -0.00137 -0.00113 -289.1 -296.0 -308.9 -307.8
76% -0.465 -0.445 -0.452 -0.00095 -0.00092 -0.00098 -262.7 -269.1 -281.1 -282.6
78% -0.518 -0.496 -0.505 -0.00056 -0.00053 -0.00054 -233.6 -239.6 -250.2 -249.0
80% -0.545 -0.521 -0.528 -0.00022 -0.00019 -0.00013 -202.1 -207.6 -216.7 -216.3
82% -0.551 -0.524 -0.528 0.00007 0.00010 0.00012 -168.6 -173.8 -181.2 -179.4
84% -0.537 -0.508 -0.514 0.00031 0.00033 0.00028 -134.4 -139.1 -145.0 -139.0
86% -0.509 -0.479 -0.487 0.00049 0.00049 0.00050 -100.8 -105.2 -109.9 -109.0
88% -0.471 -0.441 -0.449 0.00059 0.00058 0.00060 -69.7 -73.8 -77.5 -77.6
90% -0.428 -0.399 -0.407 0.00064 0.00061 0.00061 -42.3 -46.1 -49.2 -48.4
92% -0.383 -0.356 -0.365 0.00064 0.00060 0.00057 -19.3 -22.8 -15.6 -15.2
94% -0.339 -0.315 -0.324 0.00060 0.00056 0.00061 -9.3 -4.2 -6.7 -6.7
96% -0.299 -0.279 -0.286 0.00055 0.00050 0.00047 13.2 10.2 18.0 17.5
98% -0.262 -0.246 -0.254 0.00049 0.00043 0.00047 23.7 20.8 24.9 23.8
100% -0.229 -0.218 -0.221 0.00043 0.00036 0.00046 31.1 28.4 26.9 25.2
102% -0.201 -0.193 -0.193 0.00039 0.00035 0.00037 36.2 33.6 32.4 32.1
104% -0.176 -0.170 -0.170 0.00033 0.00030 0.00027 39.4 37.0 36.2 37.8
106% -0.154 -0.150 -0.151 0.00028 0.00027 0.00030 41.2 38.9 38.4 40.9
108% -0.136 -0.133 -0.132 0.00025 0.00023 0.00024 41.9 39.8 39.7 40.0
110% -0.120 -0.118 -0.117 0.00021 0.00020 0.00018 42.0 39.9 40.1 33.0
112% -0.106 -0.105 -0.104 0.00018 0.00017 0.00019 41.4 39.4 39.9 35.4
114% -0.094 -0.093 -0.092 0.00016 0.00015 0.00015 40.5 38.6 39.3 38.2
116% -0.084 -0.084 -0.083 0.00014 0.00013 0.00013 39.3 37.5 38.4 34.6
118% -0.075 -0.075 -0.074 0.00012 0.00012 0.00010 38.0 36.2 37.4 29.3
120% -0.067 -0.067 -0.067 0.00011 0.00010 0.00010 36.5 34.8 36.1 26.0
122% -0.060 -0.061 -0.061 0.00009 0.00009 0.00008 35.0 33.4 34.8 24.4
124% -0.054 -0.055 -0.055 0.00008 0.00008 0.00009 33.5 31.9 33.5 22.2
126% -0.048 -0.050 -0.049 0.00007 0.00007 0.00012 32.0 30.4 32.1 22.1
The first column contains the spot price as a percentage of 3500. Columns labeled “FWHF”
contain the results obtained using the algorithm of §4, where the subscript “1” (resp., “2”) means
that the first (resp., second) method of calculating the theta of the option explained in §4.5 was
used. Columns labeled “JP” (resp., “MC”) contain the results obtained using the method of [18]
(resp., the Monte-Carlo method); both are taken from the tables in op. cit.
NIG parameters: α= 8.858, β=−5.808, δ= 0.174, µ≈0.161.
Option parameters: K= 3500, H= 2100, r= 0.03, T= 1.
Algorithm parameters: ∆ = 0.001, M= 4096, M2= 2, M3= 16, ζ1≈0.77, m= 8 (for the
calculation of the Wiener-Hopf factors: see §3.5), N= 800 (number of time steps).

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 41
Table 3. Prices of a down-and-in first-touch digital option in the V.G.
model: comparison with the results of Jeannin and Pistorius [18]
Spot Option price Relative difference between FWHF and
price FWHF JP (MC Down, MC Up) JP MC Down MC Up
64% 0.3997 0.3989 (0.3952, 0.4012) 0.0021 0.0115 -0.0036
66% 0.3357 0.3350 (0.3309, 0.3370) 0.0022 0.0146 -0.0037
68% 0.2854 0.2847 (0.2804, 0.2864) 0.0026 0.0179 -0.0034
70% 0.2448 0.2441 (0.2399, 0.2458) 0.0029 0.0204 -0.0041
72% 0.2114 0.2106 (0.2069, 0.2126) 0.0039 0.0219 -0.0055
74% 0.1837 0.1827 (0.1793, 0.1848) 0.0055 0.0245 -0.0060
76% 0.1604 0.1593 (0.1562, 0.1615) 0.0071 0.0271 -0.0066
78% 0.1408 0.1396 (0.1368, 0.1419) 0.0083 0.0290 -0.0080
80% 0.1240 0.1227 (0.1203, 0.1252) 0.0108 0.0310 -0.0093
82% 0.1097 0.1084 (0.1063, 0.1109) 0.0121 0.0320 -0.0108
84% 0.0974 0.0960 (0.0942, 0.0986) 0.0144 0.0338 -0.0123
86% 0.0867 0.0854 (0.0837, 0.0879) 0.0157 0.0363 -0.0098
88% 0.0775 0.0761 (0.0744, 0.0784) 0.0184 0.0416 -0.0115
90% 0.0694 0.0681 (0.0666, 0.0704) 0.0197 0.0427 -0.0136
92% 0.0624 0.0611 (0.0598, 0.0635) 0.0213 0.0454 -0.0173
94% 0.0562 0.0550 (0.0538, 0.0573) 0.0223 0.0451 -0.0188
96% 0.0508 0.0496 (0.0484, 0.0517) 0.0239 0.0493 -0.0177
98% 0.0460 0.0449 (0.0437, 0.0469) 0.0241 0.0523 -0.0195
100% 0.0417 0.0407 (0.0395, 0.0425) 0.0253 0.0565 -0.0181
102% 0.0380 0.0370 (0.0358, 0.0387) 0.0258 0.0602 -0.0192
104% 0.0350 0.0337 (0.0325, 0.0353) 0.0265 0.0644 -0.0200
106% 0.0316 0.0307 (0.0296, 0.0323) 0.0292 0.0674 -0.0218
108% 0.0289 0.0281 (0.0270, 0.0296) 0.0289 0.0708 -0.0232
110% 0.0265 0.0257 (0.0246, 0.0271) 0.0314 0.0775 -0.0219
112% 0.0243 0.0236 (0.0225, 0.0249) 0.0316 0.0821 -0.0222
114% 0.0224 0.0217 (0.0207, 0.0229) 0.0323 0.0822 -0.0218
116% 0.0206 0.0200 (0.0190, 0.0212) 0.0324 0.0867 -0.0261
118% 0.0191 0.0184 (0.0174, 0.0195) 0.0360 0.0955 -0.0225
120% 0.0176 0.0170 (0.0160, 0.0180) 0.0368 0.1016 -0.0208
122% 0.0163 0.0158 (0.0148, 0.0167) 0.0332 0.1030 -0.0225
124% 0.0151 0.0146 (0.0136, 0.0155) 0.0371 0.1133 -0.0231
126% 0.0141 0.0135 (0.0126, 0.0144) 0.0419 0.1163 -0.0233
The first column contains the spot price as a percentage of 3500, and the second one contains the
option prices calculated using the algorithm of §4. The third (respectively, fourth) column
contains the option prices calculated using the method of [18] (respectively, the Monte-Carlo
method); they are taken from the tables in op. cit. The Monte-Carlo results for the prices are
reported in the form of a 95% confidence interval. The relative difference between quantity Aand
quantity Bis defined as (A−B)/B (this explains how columns 5–7 should be interpreted).
V.G. parameters: c= 0.925, λ+= 4.667, λ−=−11.876, µ≈0.128.
Option parameters: H= 2100, r= 0.03, T= 1.
Algorithm parameters: ∆ = 0.001, M= 4096, M2= 2, M3= 256, ζ1≈0.77, m= 8 (for the
calculation of the Wiener-Hopf factors: see §3.5), N= 800 (number of time steps).

42 MITYA BOYARCHENKO AND SERGEI LEVENDORSKI˘
I
Table 4. The greeks of a down-and-in first-touch digital option in the
V.G. model: comparison with the results of Jeannin and Pistorius [18]
Spot Delta×104Gamma×107Theta
price FWHF JP MC FWHF JP MC FWHF1FWHF2JP MC
64% -10.34 -10.43 -10.72 42.06 43.21 44.35 0.1901 0.1901 0.1901 0.1644
66% -8.01 -8.03 -8.20 27.04 27.36 27.80 0.1856 0.1856 0.1855 0.1721
68% -6.41 -6.42 -6.51 19.35 19.35 20.49 0.1767 0.1767 0.1765 0.1756
70% -5.22 -5.25 -5.26 14.59 14.55 15.19 0.1659 0.1659 0.1656 0.1557
72% -4.32 -4.35 -4.34 11.34 11.35 11.01 0.1544 0.1544 0.1540 0.1494
74% -3.61 -3.64 -3.64 9.01 9.06 9.14 0.1428 0.1428 0.1424 0.1274
76% -3.04 -3.07 -3.05 7.26 7.34 7.69 0.1317 0.1317 0.1313 0.1123
78% -2.58 -2.60 -2.58 5.92 6.02 5.77 0.1211 0.1211 0.1207 0.1128
80% -2.20 -2.22 -2.20 4.88 4.97 5.13 0.1113 0.1113 0.1109 0.1117
82% -1.89 -1.90 -1.88 4.05 4.13 3.97 0.1021 0.1021 0.1018 0.1022
84% -1.63 -1.64 -1.63 3.38 3.46 3.17 0.0938 0.0938 0.0934 0.0983
86% -1.41 -1.41 -1.43 2.85 2.90 2.50 0.0861 0.0861 0.0857 0.0873
88% -1.23 -1.23 -1.24 2.41 2.45 3.05 0.0791 0.0791 0.0787 0.0739
90% -1.07 -1.07 -1.05 2.05 2.08 2.20 0.0726 0.0726 0.0722 0.0680
92% -0.94 -0.93 -0.93 1.75 1.77 1.43 0.0670 0.0668 0.0664 0.0624
94% -0.83 -0.82 -0.83 1.50 1.52 1.34 0.0615 0.0615 0.0611 0.0594
96% -0.73 -0.72 -0.73 1.29 1.30 1.36 0.0566 0.0566 0.0563 0.0560
98% -0.64 -0.64 -0.65 1.11 1.12 1.14 0.0522 0.0522 0.0519 0.0470
100% -0.57 -0.56 -0.57 0.97 0.97 1.05 0.0482 0.0482 0.0479 0.0515
102% -0.51 -0.50 -0.51 0.84 0.84 0.73 0.0445 0.0445 0.0442 0.0457
104% -0.45 -0.45 -0.46 0.73 0.73 0.75 0.0412 0.0412 0.0409 0.0387
106% -0.40 -0.40 -0.40 0.64 0.64 0.75 0.0381 0.0381 0.0378 0.0375
108% -0.36 -0.36 -0.36 0.56 0.56 0.47 0.0353 0.0353 0.0351 0.0391
110% -0.32 -0.32 -0.32 0.50 0.49 0.57 0.0327 0.0328 0.0325 0.0294
112% -0.29 -0.29 -0.29 0.44 0.43 0.38 0.0304 0.0304 0.0302 0.0313
114% -0.26 -0.26 -0.26 0.39 0.38 0.44 0.0283 0.0283 0.0281 0.0277
116% -0.24 -0.23 -0.24 0.34 0.34 0.25 0.0263 0.0263 0.0262 0.0234
118% -0.21 -0.21 -0.22 0.30 0.30 0.33 0.0245 0.0245 0.0244 0.0242
120% -0.19 -0.19 -0.20 0.27 0.27 0.28 0.0228 0.0228 0.0227 0.0208
122% -0.18 -0.17 -0.17 0.24 0.24 0.31 0.0213 0.0213 0.0212 0.0178
124% -0.16 -0.16 -0.16 0.22 0.21 0.08 0.0199 0.0199 0.0198 0.0193
126% -0.15 -0.14 -0.15 0.19 0.19 0.22 0.0186 0.0186 0.0186 0.1067
The first column contains the spot price as a percentage of 3500. To save space, we rescaled the
deltas and gammas by multiplying them by 104and 107, respectively. Columns labeled “FWHF”
contain the results obtained using the algorithm of §4, where the subscript “1” (resp., “2”) means
that the first (resp., second) method of calculating the theta of the option explained in §4.5 was
used. Columns labeled “JP” (resp., “MC”) contain the results obtained using the method of [18]
(resp., the Monte-Carlo method); both are taken from the tables in op. cit.
V.G. parameters: c= 0.925, λ+= 4.667, λ−=−11.876, µ≈0.128.
Option parameters: H= 2100, r= 0.03, T= 1.
Algorithm parameters: ∆ = 0.001, M= 4096, M2= 2, M3= 256, ζ1≈0.77, m= 8 (for the
calculation of the Wiener-Hopf factors: see §3.5), N= 800 (number of time steps).

PRICES AND SENSITIVITIES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS 43
Table 5. The prices and greeks of a down-and-out barrier put option
and a down-and-in first-touch digital in the KoBoL model
Spot Down-and-out barrier put Down-and-in first-touch digital
price Price Delta Gamma×102Theta Price Delta×103Gamma×106Theta
61% 610.1590 8.0264 -16.2458 -3850.70 0.5148 -6.4238 12.0623 2.6180
62% 815.8835 4.1794 -7.3147 -3874.95 0.3451 -3.6184 5.2440 2.6082
63% 925.9785 2.2773 -3.9535 -3321.84 0.2442 -2.2652 2.7901 2.2121
64% 985.5060 1.2006 -2.3517 -2738.61 0.1791 -1.5090 1.6430 1.8035
65% 1015.2731 0.5400 -1.4916 -2236.87 0.1349 -1.0490 1.0345 1.4558
66% 1026.2855 0.1115 -0.9910 -1828.27 0.1036 -0.7527 0.6835 1.1748
67% 1024.8908 -0.1780 -0.6819 -1501.04 0.0809 -0.5534 0.4683 0.9512
68% 1014.9842 -0.3799 -0.4823 -1239.85 0.0641 -0.4149 0.3303 0.7738
69% 999.0708 -0.5242 -0.3487 -1030.86 0.0514 -0.3162 0.2383 0.6327
70% 978.8220 -0.6295 -0.2566 -862.81 0.0416 -0.2443 0.1754 0.5201
71% 955.3867 -0.7075 -0.1916 -726.79 0.0340 -0.1910 0.1311 0.4297
72% 929.5746 -0.7660 -0.1447 -615.89 0.0280 -0.1508 0.0994 0.3568
73% 901.9688 -0.8104 -0.1103 -524.74 0.0233 -0.1202 0.0763 0.2976
74% 872.9970 -0.8444 -0.0845 -449.13 0.0195 -0.0966 0.0592 0.2493
75% 842.9785 -0.8705 -0.0650 -385.79 0.0165 -0.0782 0.0464 0.2097
76% 812.1555 -0.8905 -0.0499 -332.12 0.0140 -0.0637 0.0367 0.1771
77% 780.7148 -0.9059 -0.0381 -286.07 0.0120 -0.0522 0.0292 0.1501
78% 748.8027 -0.9176 -0.0286 -245.97 0.0103 -0.0430 0.0234 0.1276
79% 716.5361 -0.9262 -0.0209 -210.48 0.0090 -0.0356 0.0189 0.1089
80% 684.0101 -0.9324 -0.0145 -178.49 0.0078 -0.0296 0.0153 0.0932
81% 651.3042 -0.9365 -0.0090 -149.05 0.0069 -0.0248 0.0125 0.0800
82% 618.4863 -0.9388 -0.0041 -121.39 0.0061 -0.0208 0.0103 0.0688
83% 585.6164 -0.9395 0.0004 -94.80 0.0054 -0.0175 0.0085 0.0594
84% 552.7497 -0.9386 0.0046 -68.68 0.0048 -0.0148 0.0070 0.0514
85% 519.9382 -0.9363 0.0088 -42.47 0.0043 -0.0126 0.0058 0.0446
86% 487.2336 -0.9325 0.0131 -15.65 0.0039 -0.0107 0.0049 0.0387
87% 454.6887 -0.9271 0.0177 12.26 0.0036 -0.0091 0.0041 0.0337
88% 422.3596 -0.9201 0.0227 41.74 0.0033 -0.0078 0.0034 0.0294
89% 390.3077 -0.9112 0.0284 73.25 0.0030 -0.0067 0.0029 0.0257
90% 358.6022 -0.9002 0.0349 107.25 0.0028 -0.0058 0.0024 0.0225
91% 327.3229 -0.8867 0.0425 144.16 0.0026 -0.0050 0.0021 0.0197
92% 296.5636 -0.8703 0.0517 184.39 0.0025 -0.0043 0.0018 0.0173
93% 266.4366 -0.8503 0.0629 228.28 0.0023 -0.0038 0.0015 0.0152
94% 237.0793 -0.8260 0.0768 276.03 0.0022 -0.0033 0.0013 0.0134
95% 208.6615 -0.7963 0.0943 327.59 0.0021 -0.0028 0.0011 0.0118
96% 181.3976 -0.7596 0.1167 382.43 0.0020 -0.0025 0.0010 0.0104
97% 155.5620 -0.7141 0.1458 439.13 0.0019 -0.0022 0.0008 0.0092
98% 131.5104 -0.6569 0.1836 494.55 0.0019 -0.0019 0.0007 0.0082
99% 109.7049 -0.5848 0.2311 542.23 0.0018 -0.0017 0.0006 0.0072
100% 90.7149 -0.4956 0.2756 569.88 0.0017 -0.0015 0.0005 0.0064
101% 75.0251 -0.3995 0.2597 562.87 0.0017 -0.0013 0.0005 0.0056
102% 62.4906 -0.3180 0.2044 527.61 0.0016 -0.0011 0.0004 0.0050
103% 52.4852 -0.2550 0.1564 481.02 0.0016 -0.0010 0.0004 0.0044
104% 44.4210 -0.2067 0.1204 432.52 0.0016 -0.0009 0.0003 0.0039
105% 37.8510 -0.1693 0.0939 386.20 0.0015 -0.0008 0.0003 0.0035
106% 32.4452 -0.1400 0.0742 343.67 0.0015 -0.0007 0.0002 0.0031
107% 27.9581 -0.1167 0.0593 305.40 0.0015 -0.0006 0.0002 0.0027
108% 24.2049 -0.0980 0.0479 271.33 0.0015 -0.0006 0.0002 0.0024
109% 21.0442 -0.0828 0.0391 241.16 0.0015 -0.0005 0.0002 0.0021
110% 18.3664 -0.0703 0.0321 214.54 0.0014 -0.0004 0.0001 0.0018
The first column contains the spot price as a percentage of 3500.
KoBoL parameters: ν= 0.5, c= 1, λ+= 9, λ−=−8, µ≈ −0.0423.
Option parameters: K= 3500 (for the barrier put), H= 2100, r= 0.03, T= 0.1.
Algorithm parameters: ∆=0.00025, M= 16384, M2= 2, M3= 8, ζ1≈0.77, m= 8 (for the calculation of the
Wiener-Hopf factors: see §3.5), N= 80 (number of time steps).