An Efficient Transform Method For Asian Option Pricing

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AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION
PRICING
J. LARS KIRKBY
Abstract. This paper introduces a novel method to price arithmetic Asian options in
Levy-driven models, with discrete and continuous averaging, by expanding on the approach
of sequential characteristic function recovery. By utilizing frame duality and a FFT-based
implementation of density projection, we obtain rapidly converging value approximations
to high precision, consistently resulting in a 10- to 100-fold time reduction compared to
state-of-the-art procedures. Theoretical convergence rates are confirmed by an in-depth
analysis of error propagation. Formulas for Greeks are provided, in addition to generalized
averaging and in-progress option pricing.
1. Introduction
Since their introduction in 1987, Asian options (known also as average rate or average
price options) have provided a popular means of risk management in a variety of markets.
For example, Eydeland and Wolyniec (2003) document their importance in mitigating the
delivery risks present in gas markets. Since Asian options have payoffs that are contingent
on the average price of an underlying asset (index, interest rate, exchange rate, commodity,
etc.) over a given time horizon, their prices are less sensitive to price manipulations, and they
become easier to hedge towards the option’s expiry. By taking an average of the underlying,
these options are typically much cheaper than standard European contracts. Moreover, their
relative stability has led to the hybridization of exotic options that contain an Asian type
specification towards the end of the contract, known as an “Asian tail”.
As is generally the case with path-dependent contracts, robust pricing of Asian options is
very challenging and computationally demanding. Even in the Black-Scholes-Merton (BSM)
framework, no analytical formulas exist for the pricing of arithmetic Asian options. The
computational approaches can be categorized as analytical approximations and bounds [1, 2,
29, 32], partial differential equation (PDE) methods [3, 4, 15, 38], lattices [13], Monte Carlo
[25,33], and transform methods [6, 10, 11, 14, 39], to which our approach belongs. Alternative
methods include Taylor expansion [24], perturbation [40], direct iterated integration [21],
and maturity randomization [20]. In terms of both speed and accuracy, the transform based
approaches are generally superior for models with Levy (log) returns, including BSM.
By working in the Fourier domain, we develop a fast and highly accurate method for
pricing generalized Asian options in exponential Levy models, which we call APROJ1. This
includes discretely monitored contracts as well as the continuously monitored options that
pervade foreign exchange markets. In-progress option prices and Greeks are also determined
efficiently. Compared to state-of-the-art-methods, the APROJ method provides a 10- to
100-fold improvement in terms of cpu time to reach the same (or better) accuracy. This is
confirmed for the methods of [6,10,11, 30,39], most notably the improved convolution method
of Cerny and Kyriakou [11], the ASCOS method of Zhang and Oosterlee [39], and the inverse
Date: 2016.
2010 Mathematics Subject Classification. 62P05, 60E10, 91G20, 91B25, 91G60, 65C20, 65T50, 65D07,
42C15, 65T40, 65T60.
Key words and phrases. arithmetic Asian options, fast Fourier transform, Levy processes, basis, character-
istic function, Carverhill-Clewlow factorization, PROJ, COS, FFT, frame projection, option pricing, B-spline,
exotic options.
The author wishes to thank Shijie Deng, Richard Birge, and Mike Staunton for fruitful discussions.
1APROJ is short for Asian PROJection, due to its use of a biorthogonal projection method.
1

2 J. LARS KIRKBY
Fourier transform method of Levendorskii and Xie [30], which are (to our knowledge) the
fastest available pricing methods for discretely monitored arithmetic Asian options under
Levy dynamics.
The paper is organized as follows. Section 2 reviews exponential Levy models and the
method of density projection by frame duality. The problem of arithmetic Asian option
pricing is formulated in Section 3, along with a derivation of the APROJ method. Section
4 develops extensions to in-progress option pricing and Greeks, generalized averaging, and
continuous averaging. An in-depth analysis of error propagation and terminal valuation error
is given in Section 5, after which Section 6 demonstrates the accuracy and efficiency of the
method with a series of numerical experiments. Comparisons are made to existing methods
with parameter sets from the literature. Finally, Section 7 concludes the paper.
2. Density Projection Method
The projection method described in this section applies whenever the characteristic func-
tion of the underlying random variable is known, which is the case for the family of Levy
processes. Since the variance gamma (VG) model was introduced in 1990 to price deriva-
tives [31], the versatility and tractability of Levy processes as generalizations of the BSM
framework have generated a surge of research and modeling success. While application of the
VG model itself has waned, subsequent developments such as the KoBoL [7,8] model (with
CGMY [9] as a special case) as well as the NIG [5] model have proven to be excellent alter-
natives which calibrate well to market data [9,22], and the exponential (semi-heavy) decay of
their tails engenders a significant computational advantage over the VG model.
2.1. Exponential Levy Models. Suppose L(t), t ≥0,is a Levy process, which is a stochas-
tically continuous process with stationary and independent increments. We denote its Levy
symbol by ψL(ξ), where by the Levy-Khintchine theorem the characteristic function (ChF)
satisfies
φL(t)(ξ) := E[eiL(t)ξ] = etψL(ξ), t ≥0.
Figure 6 in the appendix provides some of the more popular Levy symbols used in financial
modeling, along with any parameter restrictions2.
To model the underlying randomness on which Asian options are contracted, we consider
exponential Levy processes of the form
S(t) = S(0)eY(t)=S(0)e(r−q+ω)t+L(t), ω =−ψL(−i),
where r, q ≥0 are the interest rate and dividend yield. Here ωis a “convexity correction”
that is used to ensure that discounted asset processes (with reinvested dividends) behave as
martingales. That is, E[S(t+ ∆t)|S(t)] ≡S(t)E[eR∆t] = S(t)e(r−q)∆t, ∆t, t ≥0, where
R∆t:= log(S(t+ ∆t)/S(t)) d
= (r−q+ω+L(1))∆t, t, ∆t>0.
The ChF of R∆tis given by
φR∆t(ξ) = eiξ(r−q+w)∆teψL(ξ)∆t,∆t>0.
Note that the underlying Levy processes satisfies an exponential moment condition E[e−αL(t)]<
∞,∀t≥0, where IL= (λ−, λ+) denotes the set of all such α. Here −∞ ≤ λ−≤0≤λ+≤ ∞
with possible inclusion of the endpoints. As a function of z=ξ+iw,ψL(z) is analytic in the
strip D(λ−,λ+):= {z∈C:=(z)∈(λ−, λ+)}. With the exception of the pure jump VG (ie
when σ= 0), the Levy processes of interest in finance satisfy the following bound for some
c, κ > 0 and ν∈(0,2]
(1) |φR∆t(ξ)|=|eψL(ξ)∆t| ≤ κe−∆tc|ξ|ν.
2If no restriction is given, the permissible parameter values are taken to be the real line.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 3
2.2. Density Recovery and Option Pricing by Frame Projection. In [26], a method
of European option pricing, called PROJ, is derived from the theory of frames and Riesz
bases. The insight is to project the risk-neutral log return density, given in terms of its ChF,
onto a tractable basis of compactly supported functions. The basis is formed by scaling and
shifting a fixed generator or scaling function. The resulting method produces highly accurate
localized approximations at low resolutions, where the number of basis elements grows with
the resolution. The reader is referred to [26] for more details on the PROJ method, in
particular the derivation of dual bases. We refer the reader to [12, 23] for an introduction to
frame theory (also see [28] for applications to static hedging).
The B-spline bases of order pare of particular interest, and can be derived as follows.
Starting with the Haar scaling function defined by ϕ[0](y) :=
1
[−1
2,1
2](y), the p-th order B-
spline scaling functions are derived successively by the convolution
(2) ϕ[p](x) = ϕ[0] ? ϕ[p−1](x) = Z∞
−∞
ϕ[p−1](y−x)
1
[−1
2,1
2](y)dy.
With p= 1, the linear B-spline basis is generated by
ϕ[1](x) := (1 − |x|)+= (1 − |x|)
1
[−1,1](x),
while for p= 2 we obtain the quadratic scaling function
ϕ[2](y) = 




y2/2+3y/2+9/8, y ∈[−3/2,−1/2]
3/4−y2, y ∈[−1/2,1/2]
y2/2−3y/2+9/8, y ∈[1/2,3/2] .
To ease notation, we will write ϕ=ϕ[p]when the context is clear.
Given a resolution a, and a grid xn=x1+ (n−1)/a, the approximation space for a fixed
generator ϕis given by the span of ϕa,n(x) = a1/2ϕ(a(x−xn)),which is centered over xn. To
derive finite dimensional approximations in terms of {ϕa,n}N
n=1 for Nfixed, we will truncate
the corresponding projections onto the infinite dimensional space Ma:= span{ϕa,n}n∈Z,
using the fact that ϕsatisfies the frame bounds
(3) Akfk2≤X
n∈Z|hf, ϕa,n i|2≤Bkfk2,∀f∈L2(R),
for some 0 < A ≤B(independent of a).
2.2.1. Density Projection by Duality. Given a random variable X, with unknown density3
fX, we utilize the frame representation theorem [12, 23] which states that the orthogonal
projection PMafXof fXonto Mais given by
PMafX=X
n∈ZhfX,eϕa,niϕa,n ,
where {eϕa,n}n∈Zis the dual basis, which is guaranteed to exist in some form. As shown in [26],
if the ChF φX(ξ) := E[eiXξ ] is known, the projection coefficients satisfy for 1 ≤n≤N
(4) hfX,eϕa,ni=E[eϕa,n (X)] = a−1/2
π<Z∞
0
exp(−ixnξ)·φX(ξ)b
eϕξ
adξ,
where
b
eϕ(ξ) = Feϕ(ξ) = ZR
eiξx eϕ(x)dx.
When b
eϕ(ξ) is known, as for the linear and quadratic generators [26]
(5) b
eϕ[1](ξ) = 12 sin2(ξ/2)
ξ2(2 + cos(ξ)),b
eϕ[2](ξ) = 480 sin3(ξ/2)
ξ3(26 cos(ξ) + cos(2ξ) + 33),
3Levy models, with the exception of the compound Poisson process (ie no diffusion component and finite
jump activity), possess a continuous density [34].

4 J. LARS KIRKBY
the coefficients can thus be calculated efficiently using the fast Fourier transform (FFT), as
described next.
When φX(ξ) satisfies a growth estimate of the form of equation (1), the truncation error
from numerically integrating (4) will decay exponentially, and polynomially otherwise. Even
so, multiplication of the chf by b
eϕ(ξ) in equation (4) has a damping effect which reduces
aliasing caused by an otherwise insufficient choice of a(the discrete Fourier transform implies
a truncation interval of 2πa in Fourier space). This is one factor which contributes to accurate
approximations at low resolutions.
2.2.2. Coefficient Approximation. To recover the orthogonal projection of the density of a
random variable X, the first step is to set a resolution, for example a= 2Pfor P∈N. By
further specifying ¯
P∈N, which determines the support width of the projected density, and
x1, which determines its location in log return space, a conceptual grid xn=x1+ (n−1)/a,
n= 1, . . . , N , is designated where
N= 2P+¯
P=a2¯
P:= a¯a,
where the choice of parameters is discussed in Section 3.6. For example, if E[X] := µX, then
to center the grid over µX, set x1=µX−N
2−1∆ (where ∆ := 1/a), so that µX=xN
2.
The density is then recovered on
[x1, x1+ ¯a−∆] ≈[µX−¯a/2, µX+ ¯a/2].
To discretize the integral in equation (4), by the Nyquist frequency requirement ∆∆ξ=
2π/N the grid in frequency space is set to ξj= (j−1)∆ξ,j= 1, . . . , N, where ∆ξ= 2πa/N =
2π/¯a. It is shown in [26] that the truncated true projection e
fX(x) is well represented by the
numerical approximation ˘
fX(x), defined respectively by4
e
fX(x) :=
N
X
n=1hfX,eϕa,niϕa,n (x),˘
fX(x) :=
N
X
n=1 a1/2Ca,N ˘
βX
a,nϕa,n (x),
where the coefficients hfX,eϕa,ni ≈ a1/2Ca,N ˘
βX
a,n are calculated by the discrete Fourier trans-
form, in the absence of ChF error5:
(6) a1/2Ca,N ·˘
βX
a,n =a−1/2
π<(N
X
j=1
exp(−ixnξj)·φX(ξj)b
eϕξj
avj∆ξ),
where νj:= 1−(δj,1+δj,N )/2 and Ca,N is a constant which depends on the selected generator
ϕ. The full set of {˘
βX
a,n}N
n=1 are computed with complexity O(Nlog2(N)) by the FFT.
As long as the numerical error is controlled, the overall convergence of the APROJ algo-
rithm will be at least of the order of projection convergence. Define H(Dd) to be the set of
analytic functions in the strip Dd={z∈C:=(z)∈(−d, d)}which satisfy
Zd
−d|h(x+iy)|dy →0,as |x|→∞.
For h∈ H(Dd), we define the norm
khkHd:= lim
→0+ZR|h(x+i(d−))|dx +ZR|h(x−i(d−))|dx.
We have the following result for pth order B-spline generators.
4The term a1/2will be absorbed by an intermediate calculation.
5Error in the characteristic functions will be introduced.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 5
Proposition 2.1. Suppose that φX(ξ)∈ H(Dd)for some d > 0, and let ¯µ= ¯µXbe an
approximation to E[X]. Fix a= 2Pand N=a·¯a, where ¯a= 2 ¯
Pfor ¯
P > 1 + log2|¯µ|. Fix
x1= ¯µ+1−N
21
a. Then for some 0< γ ≤d
sup
1≤n≤Na1/2Ca,N ·˘
βX
a,n − hfX,eϕa,ni≤a−1/2
πC[p]
γ(φX)e−(¯a−2|¯µ|)γ/2
1−e−¯aγ +τa(X),
where C[p]
γ(φX)is a constant. If for some c, κ > 0and ν∈(0,2], the tail of φXsatisfies
(7) |φX(ξ)| ≤ κexp(−tc|ξ|ν), ξ ∈R,
where t > 0is some fixed time, then
(8) τa(X) = O(aexp(−tc ·(2πa)ν).
In this case, the largest trapezoidal error converges exponentially in ¯a, while the truncation
error is exponential in a. Moreover, when a > 2d,γ=d.
Proof. See appendix. 
Note that for the linear basis we have the bound C[1]
γ(φX)≤24kφXkHand τa(X)≤6κ
π·
aexp(−tc·(2πa)ν), although the specific constants will not be required for our implementation.
2.2.3. Quadratic Basis Implementation. To implement the APROJ algorithm, we fix the qua-
dratic basis, although the method applies more generally to pth order B-splines and other
generators as well. In particular,
(9) fX(x)≈a1/2Ca,N
N
X
n=1
˘
βX
a,nϕ[2]
a,n(x), Ca,N := 960a3
N.
The coefficients a1/2Ca,N ˘
βX
a,n are found using the discretization in equation (6). From the
dual generator transform b
eϕ[2](ξ) in equation (5), we define
(10) H1= 1/(960a3), Hj=φX(ξj)ζjexp(−ix1ξj),2≤j≤N,
where
(11) ζj:= (sin(ξj/2a)/ξj)3
26 cos(ξj/a) + cos(2ξj/a) + 33,2≤j≤N.
The coefficients ˘
βX={˘
βX
n}N
n=1 are recovered by the discrete Fourier transform (DFT)
(12) ˘
βX:= <[D{Hj}N
j=1],Dn{Hj}:=
N
X
j=1
e−i2π
N(j−1)(n−1)Hj, n = 1, . . . , N ,
For φXanalytic in a strip containing D(−d,d)with d > 0, trapezoidal approximations to the
DFT converge exponentially with respect to a, ¯a, by Proposition 2.1.
2.3. Arithmetic Asian Options. Our main goal is to price discretely monitored arith-
metic Asian options, which are contracts on the average over an observed set of prices
of an underlying, with observations taken at a discrete set of M+ 1 monitoring dates,
{0 = t0, t1, . . . , tM=T}, with S0=S(t0) observed upon entering the contract. We as-
sume a uniform spacing between observations6,tm=m∆t=mT
M,m= 0, . . . , M . If the
density of AM:= 1
M+1 PM
m=0 Smis known, say fAM, then the initial value of an option
paying g(AM) at time Tmust initially satisfy V ◦ g(S0) = e−rT RRg(u)fAM(u;S0)du.
Fixed strike vanilla Asian options (calls and puts) are priced according to the terminal
payoffs with strike W > 0
(13) g(AM) := 

1
M+1 PM
m=0 Sm−W+,for a call,
W−1
M+1 PM
m=0 Sm+,for a put.
6This assumption is easily relaxed at a modest increase in cpu time.

6 J. LARS KIRKBY
By considering a change of numeraire, floating strike arithmetic options can be priced using
an analogous formula, but only at inception [16]. On the other hand, frame projection can be
used to efficiently obtain bounds on the prices of floating strike arithmetic options in terms
of their geometrically averaged counterparts.
3. Mean Adjusted APROJ Method
This section details the APROJ method, which combines elements of several different
methods to produce a highly efficient pricing algorithm. The first step is to reduce the problem
dimension by employing a technique known as the Carverhill-Clewlow-Hodges factorization
[10], which has been utilized as well by [11,20, 21, 39]. The factorization results in a recursive
scheme to recover a single state variable, YM, defined by a sequence of intermediate variables
{Ym}M
m=1. As in [39], we focus on the ChF of this process, which we extend to generalized
averaging and in-progress contracts. Analyticity of the chf of Ymat each stage is proved. To
reduce the computational cost and improve accuracy, we explicitly account for the shifting
mean of Ym, by employing an alternative to the lower bound grid shift algorithm proposed
in [6]. In particular, we derive upper and lower bounds on the mean of Ym, and devise an
efficient grid shift scheme.
To derive the ChF, we extend the PROJ method of [26]. By utilizing the orthogonally
projected density, PROJ obtains highly accurate approximations even at low resolutions. This
phenomenon is explained in [36], where for modest resolutions the least squares projection
behaves like an interpolation with twice the order of accuracy. Consequently, the use of
projected densities results in a substantial reduction in overall cost. Transitioning between
time states mrequires the calculation of a series of complex valued integrals, for which we
derive accurate closed form approximations, taking advantage of the compactly supported
basis elements of the PROJ method. In contrast, the globally supported basis elements of a
cosine series expansion, for example, require a much more expensive procedure to evaluate the
analogous integrals. The resulting algorithm achieves high accuracy at a low computational
cost compared with existing methods. Parameters are determined by an iterative procedure
which uses the transform method of [18, 19] to estimate truncation error, as well as a proxy
for the integration error incurred at each step. Greeks are obtained at a negligible added cost.
3.1. Change of Variables. The idea behind the Carverhill-Clewlow-Hodges factorization is
to express the average in terms of a random variable YM, defined below, so that
(14) AM=1
M+ 1
M
X
m=0
Sm=S0
M+ 1 (1 + exp(YM)) .
Given an approximation of the density fYM, the value of a payoff g(YM;S0) satisfies
V ◦ g(S0) = e−rT ZR
g(y;S0)fYM(y)dy,
where for vanilla options
(15) g(y) := 








S0(1 + exp(y))
M+ 1 −W+
,for a call,
W−S0(1 + exp(y))
M+ 1 +
,for a put.
In this way, pricing of a path-dependent Asian option is reduced to the valuation of a European
option on the variable YM. As will be demonstrated, such a variable can also be found for
generalized Asian options with fixed strikes, and for geometric Asian options with fixed and
floating strikes (see [26] for the PROJ implementation for geometric Asian options).

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 7
The insight of [10] is that the arithmetic average can be expressed as
AM=S0
M+ 1 1 + S1
S01 + S2
S1···SM−1
SM−21 + SM
SM−1
=S0
M+ 1 1 + eR11 + eR2···eRM−11 + eRM
=S0
M+ 1 (1 + exp(R1+ log (1 + exp(R2+ log (···RM−1+ log (1 + exp(RM)))))
where the log return increments are defined by7
Rm:= log(Sm/Sm−1), m = 1, . . . , M,
where we have suppressed the dependence of Rmon the time step ∆t=T /M. By introducing
the sequence {Ym}M
m=1, defined recursively by
(16) Y1:= RM, Ym:= RM+1−m+ log(1 + exp(Ym−1)), m = 2, . . . , M,
we have
(17) Ym= log 1
S(M−m)
m
X
j=1
S(M−m)+j!,
from which it follows that exp(YM) = 1
S0PM
m=1 Sm, and so equation (14) holds. As in [39],
we recover the ChF of φYMby computing the ChFs of the sequence {Ym}M
m=1.
3.2. The Basic Recursion. With Zm:= log(1 + exp(Ym)),the characteristic function of
YMis found recursively from Y1=RMby the equation
(18) Ym=RM+1−m+Zm−1, m = 2, . . . , M.
Assuming exponential Levy dynamics, the log return increments Rmare independent, from
which independence of RM+1−mand log(1 + exp(Ym−1)) follow. Moreover, stationarity (and
uniform monitoring) implies that RM+1−m=Rin law for all m, where Rhas known ChF for
many Levy processes. Hence, starting with φY1(ξ) = φR(ξ),the ChF of Ymis derived from
that of Ym−1using equation (18):
φY1(ξ) = φR(ξ), φYm(ξ) = φR(ξ)φZm−1(ξ), m = 2, . . . , M.
Specifically,
(19) φZm−1(ξ) := Eheiξ log(1+exp(Ym−1))i=ZR
(ey+ 1)iξfYm−1(y)dy,
where fYm−1is approximated using φYm−1.
The next result will ensure that the DFT errors, which are incurred at each density pro-
jection step, converge exponentially with respect to a, ¯a.
Proposition 3.1. Suppose that φR(z)is analytic in the strip Dd:= {z∈C:=(z)∈(−d, d)},
for some d > 0, and satisfies equation (1) for some κ, c > 0and ν∈(0,2]. If {Ym}M
m=1 are
defined by equation (16), then the ChFs satisfy
(i) φYmis analytic in Dd,1≤m≤M, and
(ii) |φYm(ξ)| ≤ κe−∆tc|ξ|ν,ξ∈R,1≤m≤M.
Hence, the domain of analyticity and the decay of φYmare independent of m.
Proof. See appendix. 
It should also be noted that fYm(y)∼e−d|y|as |y| → ∞, ie the densities have exponentially
decaying tails8, determined by the tail behavior of fR. This follows since analyticity of φYm
in Ddimplies that E[eηYm]<∞for η∈(−d, d). In particular, we are dealing with densities
of rapid decrease.
7We reserve the notation Rto denote the return distribution over a time increment of size ∆t, while Rm
denotes the return random variable itself. To make the dependence on ∆texplicit, we will at times use R∆t
to denote a generic return increment.
8The rate of decay could be faster than d, but this gives a conservative estimate.

8 J. LARS KIRKBY
0 10 20 30 40 50
m
-1
0
1
2
3
4
5M=52, BSM
0 50 100 150 200 250
m
-1
0
1
2
3
4
5
6
7M=252, CGMY
Figure 1. Plot of ˜µm, the approximated mean of Ym, as a function of mwith r=.05,
q= 0 in the BSM σ= 0.3 model (Left) and the CGMY = (0.27,17.5,54.8,0.8) model
(Right). The bounds ¯µm±¯a/2 are given by dashed lines, where ¯a= 2.
3.3. APOJ Algorithm Overview. Before developing the APROJ algorithm in detail, we
present the main blocks with references to their derivation in the text:
(1) To account for the shifting mean of Ym, a grid shift algorithm is derived in Section
3.4
(2) The initial ChF φZ1is obtained in terms of the closed form ChF φRin Section 3.5.1,
where we introduce the integral matrix Ψ
(3) The ChFs φZm−1are obtained recursively in Section 3.5.2
(4) Given φZm−1, we obtain φYmin Section 3.5.3
(5) An automated method of parameter selection is detailed in Section 3.6, which is
summarized by initialization Subroutine 1
(6) An approximation of the integral matrix Ψ is given in Section 3.7, which is summarized
by Subroutine 2
(7) The final valuation step (which applies to general payoffs) is presented in Section 3.8,
after recovering φYM
(8) Formulas for vanilla option Greeks are provided in Section 3.10
After developing the main algorithm blocks, in Section 3.9 we summarize the routine in
Algorithm 3, which calls initialization Subroutine 1 to determine parameters, and Subroutine
2 to populate the integral matrix Ψ.
3.4. Mean-adjusted Grid. We employ a grid shift to ensure that we capture to within a
set tolerance the mass of fYm−1, while the grid specific to each Ymwill belong to a single
enlarged grid, for m= 1, . . . , M −1. The final grid corresponding to YMwill vary slightly
according to the payoff to be priced. Since the distribution of Y1=R∆tis roughly centered
about its mean, a natural starting grid in log return space is fixed by centering about
E[R∆t]=(r−q+ω+E[L(1)])∆t=c1∆t,
where c1=E[log(St+1/St)] is the first cumulant of log return over a unit interval, and is
provided in Table 6 for common processes. For example, the Black-Scholes-Merton (BSM)
model satisfies E[R∆t] = (r−q−σ2/2)∆t, where σis the rate of volatility.
The approach of Benhamou [6] is to approximate the mean E[Ym] = E[R∆t] + E[log(1 +
eYm−1)] by
(20) µB
1:= E[R∆t], µB
m=µB
1+ log 1 + eµB
m−1, m = 2, . . . , M.
By convexity of log(1 + ey), Jensen’s inequality implies log(1 + exp(E[Ym−1])) ≤E[log(1 +
exp(Ym−1))], so the mean shift underestimates the true mean. We employ an alternative
grid-shift scheme, described next.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 9
3.4.1. Grid Shift and Bounds. As an alternative to the grid adjustment of [6], and to bound
the growth of the grid shift, we derive an upper bound on E[Ym] by applying Jensen’s in-
equality(for concave functions) to equation (17):
(21) E[Ym]≤log 

m
X
j=1
ES(M−m)+j
S(M−m)
= (r−q)∆t+ log exp ((r−q)∆tm)−1
exp ((r−q)∆t)−1,
since e(r−q)j∆t=EhS(M−m)+j
S(M−m)FM−mi=EhS(M−m)+j
S(M−m)i, where the first equality follows from
the martingale property and the second from the fact that Levy increments are independent
of the current filtration, FM−m. Similarly,
Ym≥log(m) + 1
m
m
X
j=1
log S(M−m)+j
S(M−m),
from which we derive E[Ym]≥log(m) + E[R∆t]m+1
2. In particular, we obtain a set of upper
and lower bounds on the growth of E[Ym].
Proposition 3.2. With µB
mdefined by equation (20), and θ:= (r−q)∆twe have
E[R∆t]m+ 1
2+ log(m)≤E[R∆t] + log 1 + eµB
m−1
(22)
≤E[Ym]
≤θ+ log exp (θm)−1
exp (θ)−1≤log(m) + θ(m
1
r≥q+
1
r<q ).(23)
With µB
0:= 0, these bounds hold for all 1≤m≤M.
Proof. Both inequalities in equation equality in equation (23) follow from equation (21). To
prove equation (22), define θm= log(m) + ρm+1
2, where ρ:= E[R∆t]. We show that µB
m≥θm
by proving exp(µB
m−θm)≥1 inductively, where the case of m= 1 holds trivially. For m≥2,
exp(µB
m−θm) = eρ1 + eµB
m−1e−ρ(m+1)/21
m
≥eρ1 + eρm/2+log(m−1)e−ρ(m+1)/21
m
=eρ/2e−ρm/21
m+m−1
m
(24)
where the inequality follows by the inductive hypothesis. For m= 2, equation (24) becomes
exp(µB
2−θ2)≥eρ/2e−ρ1
2+1
2= cosh(ρ)≥1.
It is thus sufficient to show that the lower bound in (24) is nondecreasing in m. In particular,
d
dmeρ/2e−ρm/21
m+m−1
m=eρ(1−m)/2
2m22eρm/2−1−ρm.
Since eρ(1−m)/2/2m2>0, the result follows from the fact that the second term 2 eρm/2−1−
ρm := 2(eλ/2−1) −λhas a global minimum at λ= 0. That is, for any ρ6= 0 fixed (the case
of ρ= 0 follows immediately), the derivative is a nondecreasing function of m, and equation
(22) is proved. 
An immediate consequence of Proposition 3.2 is that we obtain a priori a corridor in which
E[Ym] lies for all 1 ≤m≤M, in terms of the mean return and (r−q):
(25) E[R∆t]m+ 1
2≤E[Ym]−log(m)≤ |r−q|Tm
M,∀m≤M.
Hence, E[Ym] = log(m) + O(m|r−q|∆t) and the growth in E[Ym] is no faster than log(m),
independently of M(the second term is always bounded by |r−q|T). We also note that the
upper bounds in equation (23) can be applied when E[R∆t] is unknown.

10 J. LARS KIRKBY
3.4.2. Grid Shift Algorithm. The APROJ grid shift is implemented by combining the inner-
most upper and lower bounds of Proposition 3.2. In particular, with µB
1=E[R∆t] = c1∆t
(see Table 6), and for m= 2, . . . , M
µB
m:= µB
1+ log 1 + eµB
m−1, µU
m:= (r−q)∆t+ log exp ((r−q)∆tm)−1
exp ((r−q)∆t)−1,
we define our grid as the lower-upper bound average
(26) ˜µ1:= c1∆t,˜µm:= (µB
m+µU
m)/2, m = 2, . . . , M,
with maximum grid shift error |E[Ym]−˜µm| ≤ (µU
m−µB
m)/2.
In order to reduce the computations required below (namely in computing a matrix Ψ),
we perturb each ˜µmslightly to obtain ¯µm, which belongs to an extension of the initial grid
defined by ˜µ1:
(27) ¯µm:= ˜µ1+Nm∆, Nm:= ba( ˜µm−˜µ1)c, m = 2, . . . , M ,
and ¯µ1≡˜µ1,N1:= 0. Hence, we define the mean-adjusted grids
(28) xm
n=xm
1+ (n−1)∆, xm
1:= ¯µm+ (1 −N/2) ∆, m = 1, . . . , M −1,
each corresponding to a subset of the linear basis {ϕa,n}N+NM−1
n=1 , with ϕa,1centered over x1
1.
In particular, the density of Ymis recovered over [¯µm−¯a/2,¯µm+ ¯a/2], m = 1, . . . , M, which
is illustrated in Figure 1. The choice of xM
1will be detailed in Section 3.8. To implement the
algorithm, only {xm
1}M
m=1 and {Nm}M
m=1 are needed (there is no need to actually generate
the grids at each stage).
3.5. Characteristic Function Recovery. We now derive the ChF recovery by successive
PROJ expansions on the mean-adjusted grid. The algorithm is summarized in Section 3.9,
along with a discussion of its complexity. In the algorithm description, we will denote by ¯
βX
the DFT approximation in the presence of ChF error, to distinguish it in the error analysis
from ˘
βX(which is absent ChF error).
3.5.1. Initialization. To initialize the characteristic function recursion we need
φZ1(ξ) := Eheiξ log(1+exp(R))i=ZR
(ey+ 1)iξfR(y)dy.
Since φR(ξ) is known, φZ1(ξ) is approximated by a (quadratic) PROJ expansion of fR(y),
with coefficients ˘
βn=˘
βa,n to yield9
φZ1(ξ)≈ZR
(ey+ 1)iξ a1/2Ca,N
N
X
n=1
˘
βnϕa,n(y)!dy
=Ca,N
N
X
n=1
˘
βn·a1/2ZIn
(ey+ 1)iξϕa,n (y)dy
≈Ca,N
N
X
n=1
¯
β1
n·¯
Ψ(ξ, n) := ¯
φZ1(ξ),(29)
where for the quadratic basis In:= [x1
n−3∆/2, x1
n+ 3∆/2] and Ca,N = 960a3/N.
With the initial grid implied by the choice of x1
1=E[R] + (1 −N/2)∆, so that φZ1is
approximated by a projected expansion of fRabout E[R], the column vector ¯
β1is determined
by
(30) ¯
β1:= <[D{H1
j}N
j=1], H 1
j:= φR(ξj)·ζjexp(−ix1
1ξj), j = 2, . . . , N,
where H1
1= 1/(960a3) and ζjis defined in equation (11). Further,
(31) Ψ(ξ, n) := a1/2ZIn
(ey+ 1)iξϕa,n (y)dy, n = 1, . . . , N +NM−1,
9We use the notation ¯
β1here to be consistent with ¯
βm,m≥2, although it should be noted that ¯
β1=˘
β1
in this case since φRis known.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 11
0 50 100 150 200 250 300 350 400
ξ∈[0,2πa)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
¯
Ψ(ξ, n)
xn = 0
xn = .5
xn = 1
xn = 1.5
xn = 3
θ(ξ)
Figure 2. Convergence in xnof Ψ(ξ, n) to a1/2F[ϕa,n](ξ), a plot of the modulus.
and ¯
Ψ(ξ, n) denotes a Newton-Cotes approximation to Ψ(ξ, n) (discussed in Section 3.7).
From here we obtain ¯
φY2(ξ) = φR(ξ)¯
φZ1(ξ), which concludes the initialization.
Remark 1.As demonstrated in Figure 2, for increasing xnthe columns in Ψ(ξ, n) become
progressively closer to the values of a1/2F[ϕa,n] on [0,2πa). This is illustrated with the linear
basis in terms of the scaled modulus
θ(ξ) := sin(ξ/2a)
ξ/(2a)2
=a1/2a−1/2eixnξsin(ξ/2a)
ξ/(2a)2=a1/2|F[ϕa,n]|,
and reflects the fact that
ZIn
(ey+ 1)iξϕa,n (y)dy −ZIn
eiξy ϕa,n(y)dy→0,as xn→+∞.
For a pth order B-spline basis, we have the following characterization for large xn.
Lemma 3.1. With a > 0fixed, the elements of ¯
Ψbehave as
Ψ(ξ, n)∼eixnξsin(ξ/2a)
ξ/(2a)(p+1)
+O(a·exp(−xn−1)),
when xnis large, with respect to the B-spline basis of order p.
Proof. See appendix. 
Especially when Mis large (in which case a significant portion of ¯
Ψ will be well approxi-
mated by Lemma 3.1), the algorithm can be improved to use this result.
3.5.2. Recovery of φZm−1.From the definition of Zm−1, the characteristic function is approx-
imated in terms of the PROJ expansion of fYm−1, recovered over [¯µm−1−¯a
2,¯µm−1+¯a
2], and
corresponding to the subset of basis elements ϕa,n,n=Nm−1+ 1, . . . , Nm−1+N:
φZm−1(ξ) = ZR
(ey+ 1)iξfYm−1(y)dy
≈ZR
(ey+ 1)iξ a1/2Ca,N
N
X
n=1
¯
βm−1
nϕa,Nm−1+n(y)!dy
≈Ca,N
N
X
n=1
¯
βm−1
n·¯
Ψ(ξ, Nm−1+n) := ¯
φZm−1(ξ).(32)

12 J. LARS KIRKBY
−400 −300 −200 −100 0 100 200 300 400
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
m = 1
m = 10
m = 20
m = 50
Figure 3. Modulus of φYmwith ∆t= 1/50 for (C, G, M, Y ) =
(.0244, .0765,7.5515,1.2945), and r=.0367. The x-axis: ξ∈[−2πa, 2πa], ∆ξ= 2π/¯a,
where a= 26, ¯a= 23.
As before, the grid is fixed by xm−1
1, and the column vector ¯
βm−1:= <[D{Hm−1
j}N
j=1] is
determined via
(33) Hm−1
1= 1/(960a3), Hm−1
j:= ¯
φYm−1(ξj)·ζjexp(−ixm−1
1ξj), j = 2, . . . , N.
In fact, we only need the values of ¯
φYm(ξ) for the discrete set of points ξj= (j−1)∆ξ,
j= 1, . . . , N . Accordingly, if we define the N×(NM−1+N) matrix ¯
Ψ by
¯
Ψ(j, n) := ¯
Ψ(ξj, n), j, n = 1, . . . , NM−1+N,
the computation at each stage can be represented as
(34) ¯
ΦZm−1=Ca,N ¯
Ψm−1¯
βm−1
where ¯
ΦZm−1= ( ¯
φZm−1(ξ1),..., ¯
φZm−1(ξN))>, and for m= 2, . . . , M ,
¯
Ψm−1(j, n) = ¯
Ψ(j, Nm−1+n), j, n = 1, . . . , N.
Here, ¯
Ψm−1is defined for notational compactness and to indicate that only a subset of ¯
Ψ
takes part in the matrix-vector product.
3.5.3. Recovery of φYm.To determine ¯
ΦYm, equation (32) yields
¯
φYm(ξ) = ¯
φZm−1(ξ)φR(ξ) = Ca,N
N
X
n=1
¯
Ψ(ξ, Nm−1+n)·¯
βm−1
n·φR(ξ).(35)
In matrix form the algorithm reads
(36) ΦC
R:= Ca,N ΦR,¯
ΦYm=¯
Ψm−1¯
βm−1◦ΦC
R, m = 2, . . . , M,
where ◦denotes the Hadamard product and ΦR= (φR(ξ1), . . . , φR(ξN))>.
An example of the modulus of recovered ChFs for the CGMY model with M= 50 is given
in Figure 3, where the line corresponding to m= 1 is just |φR(ξ)|. Notice how the ChFs
collapse about the origin as mapproaches M. The reflects the fact that, as mincreases, the
density of fYmbecomes less peaked (ie smoother), which translates into a more rapid decay
of φYm.
3.6. Parameter Selection. The two parameters required to apply the APROJ method are
¯aand N(or equivalently ∆). For several experiments in the numerical section, we fix a value
of ¯a= 2 ¯
P(often excessively large to isolate the resolution error) and increase the parameter

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 13
a= 2P, which allows us to illustrate the convergence behavior as a function of resolution.10
For example, Figure 7 in appendix illustrates the convergence in afor several levels of ¯afixed.
This section provides an automated approach to parameter selection, requiring no user
input, which should facilitate implementation in practical pricing scenarios. We first fix an
initial value for Nand truncation multiplier L1. For ∆t≥1/80 we find that N= 26and
L1= 12 provide good starting values. Similarly we initialize N= 27and L1= 16 for
∆t<1/80. We then initialize ¯agiven according to the cumulants of R∆t, as proposed in [17]
(without the lower bound):
¯a←2·max n1, L1qc2∆t+pc4∆to,
and set ∆ ←¯a/N (see Table 6 for cn). Finally, we estimate the truncation error, with a
tolerance 1, and a proxy for the valuation error, with a tolerance 2, increasing the values
for Nand ¯aaccording to a set of rules.
First we estimate the truncation error. As shown in [18] (see also [19]), the probability
mass of a random variable over and interval [l, u] is given by
P[l < R < u] = Z∞
−∞
e−iξ(u+l)/2sin(ξ(u−l)/2)
πξ φR(ξ)dξ.
Fixing N > 0 and ∆ξ>0, we have the approximation
P[l < R < u]≈F∆ξ,N (l, u) := ∆ξ
π"γ1+X
1≤|n|≤N−1
e−iγ2(n∆ξ)sin(γ1(n∆ξ))
n∆ξ
φR(n∆ξ)#
where γ1= (u−l)/2 and γ2= (u+l)/2. From Section 3.4.2, we know the grid shift error
is bounded by |E[Ym]−˜µm| ≤ (µU
m−µB
m)/2 := τm, and in practice we find that τMis
the largest such error. Hence, given a grid estimate (l, u), we estimate the mass of fRon
(˜
l, ˜u) = (l+τM, u −τM). If |1−F∆ξ,N (˜
l, ˜u)|> 1, we double the grid size N, set ¯a←√2¯a,
and reestimate.
As a second verification, by the martingale property of e−(r−q)tSt, we can utilize the
following estimate to obtain a proxy for integration error incurred at each step:
EN:= Ca,N ·ϑ[2]
∗·
N
X
n=1
¯
β1
nexp(x1
1+ (n−1)∆) = Z˘
fR(x)exdx(37)
where ϑ[2]
∗is defined in Table 1. ENapproximates E[exp(R∆t)] = exp((r−q)∆t) using the
projected density. Hence, once the truncation criterion is satisfied, we will further double
the grid size as long as |EN−exp((r−q)∆t)| · M > 2.The multiplier Mis to account for
the number of such approximations made during the algorithm. The resulting initialization
routine is summarized in Subroutine 1. After the main algorithm, a final check will be made
(see Remark 3).
Note that the parameter 1= 5e-04, along with 2= 5e-04 are set in Subroutine 1 to satisfy
an overall valuation error tolerance of TOL:= 5e-04 or better, uniformly across models, and
tends to be conservative. This is illustrated in Table 9 of the numerical section.
3.7. Approximation of Ψ.We now discuss the numerical integration of the matrix Ψ. From
equation (31), for j= 1, . . . , N , From equation (31), for j= 1, . . . , N,
Ψ(j, n) := a1/2ZIn
(ey+ 1)iξjϕa,n(y)dy, n = 1, . . . , N +NM−1,
10One could use the value of ¯a= 2 ¯
Pprescribed in Corollary 5.1 which ensures ¯a−2|¯µM|>0, and hence
the exponential convergence in ¯a; it is usually around ¯
P= 3 for M≤50, or ¯
P= 4 when M= 250. Since
this controls the largest coefficient error, it tends to be conservative although robust for heavy tailed returns
(for BSM, ¯
P= 2 is more than sufficient for M≤250 and σ≤.5, and practical accuracy of greater than e-04
is achieved with ¯
P= 0 ∼1). In practice, a conservative rule of thumb is to choose ¯
P= 4 for heavy tailed
distributions, and ¯
P= 1 for diffusion models.

14 J. LARS KIRKBY
Subroutine 1 Initialization by automated parameter selection
For ∆t≥1/80, Set: L1= 12, N = 26; For ∆t<1/80, Set: L1= 16, N = 27
Set error tolerances 1= 5e-04; 2= 5e-04
Calculate cumulants c1, c2, c4of R1(see Table 6)
˜µ1←c1∆t;θ←(r−q)∆t;µB
1←c1∆t
Initialize ¯a←2·max 1, L1pc2∆t+√c4∆t
Set(∆, a, ∆ξ): ∆ ←¯a/N;a←1/∆; ∆ξ←2π/2¯a
for m= 2 . . . M do
µB
m←µB
1+ log 1 + eµB
m−1; ˜µm←1
2µB
m+θ+ log exp(θm)−1
exp(θ)−1
end for
Max grid shift error: τM:= 1
2θ+ log exp(θM )−1
exp(θ)−1−µB
M
x1
1←˜µ1+ (1 −N/2)∆
l←x1
1+τM;u←(x1
1+ ¯a)−τM;γ1←u−l
2;γ2←u+l
2
while |1−F∆ξ,N (l, u)|> 1do
N←2N; ¯a←√2¯a; Set(∆, a, ∆ξ)
x1
1←˜µ1+ (1 −N/2)∆; Update: l, u, γ1, γ2
end while
{ξj}N
j=1 = (j−1)∆ξ,Φ← {φR(ξj)}N
j=1; Calculate {ζj}N
j=2 from (11)
Calculate {Hj}N
j=1 from (30); {¯
βn}N
n=1 ← <{FFT{Hj}N
j=1}
Calculate ENfrom (37)
while |EN−exp(θ)| · M > 2do
N←2N; ¯a←√2¯a; Set(∆, a, ∆ξ)
Recalculate: {ξj}N
j=1,Φ,{ζj}N
j=2,{Hj}N
j=1,{¯
βn}N
n=1 and EN
end while
Nm← ba(˜µm−˜µ1)c;xm
1←(˜µ1+Nm∆) + (1 −N/2)∆, m = 1, . . . , M
which we approximate by ¯
Ψ using Newton-Cotes quadrature. By fixing the grids with
{xm
1}M−1
m=1 defined by equation (28), each can be considered as a subset of x1
1+ (n−1)∆,
n= 1, . . . , N +NM−1, so quadrature points (and function evaluations) can be reused in sub-
sequent approximations. Moreover, the induced grid overlap reduces the computation11 of ¯
Ψ
from N×((M−1)N) elements to N×(N+NM−1)≤N×(log(M−1)N) (see Section 3.9).
To obtain the matrix ¯
Ψ we evaluate the integrals by applying a seven point Newton-Cotes
rule to each subinterval Il
n,l= 1,2,3, where
In:= [xn−3∆/2, xn−∆/2] ∪[xn−∆/2, xn+ ∆/2] ∪[xn+ ∆/2, xn+ 3∆/2] := I1
n∪I2
n∪I3
n.
Combined with the known values of ϕ[2](y) at each quadrature point, this results in the
(composite) seven-point rule on In
Q(ν) = 1
840n3ν1+ν17 + 25(ν5+ν13) + 46(ν7+ν11 )
+27
18ν2+ν16 + 4(ν4+ν14) + 13(ν8+ν10 )
+ 34 [ν3+ν15 + 6ν9] + 41 [ν6+ν12]o,
where νis defined in Subroutine 2, and represents generic values of the integrand for some
(j, n) fixed.12.
11For example, when N= 211 and M= 250, the size of ¯
Ψ is reduced from 1.04×108to 7.08 ×105elements.
12Note that Q(ν) requires only 17 points to evaluate to populate Ψ(j, n), since xn−∆/2 and xn+ ∆/2
are each shared by two subintervals, and on the boundaries ϕ[2](y)=0

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 15
Subroutine 2 Calculation of ¯
Ψ
Nη= 17 + 6(N+NM−1−1)
ηk←x1
1+ (k−9)∆/6, k = 1, . . . , Nη
θk←exp (i∆ξlog (1 + exp(ηk))) , k = 1, . . . , Nη
η←θ
¯
Ψ(1, n)←1, n= 1, . . . , N +NM−1
for j= 2 . . . N do
for n= 1, . . . , N +NM−1do
νk←ηk+6(n−1), k = 1,...,17
¯
Ψ(j, n)←Q(ν)
end for
η←η◦θ
end for
1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
BSM
NIG
CGMY
Figure 4. Plotted densities fYM,M= 12, recovered by PROJ for the models:
BSM(.17801), NIG(6.1882, -3.8941, .1622), CGMY(.6509, 5.853, 18.27, .8) in section 6.
To calculate all integrals in Ψ(j, ·) for jfixed thus requires a full grid {ηk}Nη
k=1 of size
Nη= 17 + 6(N+NM−1−1),where13
{ηk}Nη
k=1 =x1
1−8∆/6, . . . , x1
1+ (N+NM−1−1)∆ + 8∆/6, ηk−ηk−1= ∆/6.
Using the fact that
(ey+ 1)iξj= exp (iξjlog (1 + ey)) = exp (i(ξj−1+ ∆ξ) log (1 + ey))
= exp (i∆ξlog (1 + ey)) ·exp (iξj−1log (1 + ey)) ,
we obtain Subroutine 2 for ¯
Ψ, where η◦θdenotes the Hadamard product14. Since the quadra-
ture rule is fixed (e.g. seven-point in our case, although alternative quadratures can be used as
well), no user-supplied inputs are required. This simplifies the implementation as compared
to a procedure such as ASCOS [39], which requires a specification of nq(quadrature points
for the Clenshaw-Curtis integration rule), which can vary substantially from one application
to the next.
13This grid is used to initialize the algorithm, after which the value of ηis updated.
14To evaluate the complexity, ηrequires on the order of O(Nη) operations to initialize (as θ), followed by
N−1 Hadamard products for a total cost of O((N−1)Nη) operations. Each quadrature application across
a row ¯
Ψ(j, ·) of ¯
Ψ, of which there are N−1, requires O(N+NM−1) operations. Hence, ¯
Ψ is populated at a
cost of O((N−1)(N+NM−1)) operations.

16 J. LARS KIRKBY
3.8. The Valuation Step. Given the approximation ¯
ΦYM, the final step is analogous to the
valuation problem for a European option. Rather than specify xM
1as before, the valuation
accuracy can be further improved by perturbing the terminal grid so that the vanilla option
“kink”, defined by
(38) y∗:= log (M+ 1)W/S0−1,
is a member. In this case, equation (15) can be expressed as
(39) g(y) := 






S0(1 + exp(y))
M+ 1 −W
1
[y≥y∗],for a call,
W−S0(1 + exp(y))
M+ 1 
1
[y≤y∗],for a put.
Initially defining ˜xM
1= ˜µ1+NM∆ + (1 −N/2)∆ and n∗=b(y∗−˜xM
1)a+ 1c, we set
(40) xM
1:= y∗−(n∗−1)∆, xM
n=xM
1+ (n−1)∆, n = 1, .., N.
from which y∗=xM
n∗. If we then define the terminal basis {ϕM
a,n(y)}N
n=1 where ϕM
a,n(y) is
centered over xM
n, the density is approximated by
fYM(y)≈1
2π
N
X
n=1h¯
φYM,b
eϕM
a,ni · ϕM
a,n(y)≈a1/2Ca,N
N
X
n=1
¯
βM
nϕM
a,n(y)
where ϕM
a,N/2(y) is roughly centered over the mean of YM, and ¯
βM:= <[D{HM
j}N
j=1] is
determined using
(41) HM
1= 1/(960a3), HM
j:= ¯
φYM(ξj)ζjexp(−ixM
1ξj), j = 2, . . . , N.
The final step is to approximate the initial value by integrating the terminal payoff against
the PROJ expansion of fYM(see Figure 4):
V ◦ g(S0) = e−rT ZR
g(y;S0)fYM(y)dy ≈e−rtMCa,N
n∗+1
X
n=1
¯
βM
ngn,(42)
where the terminal payoff coefficients are defined for n= 1, . . . , N by
gn:= a1/2ZxM
n+3∆/2
xM
n−3∆/2
ϕM
a,n(y)g(y)dy =Z3/2
−3/2
ϕ(y)gxM
n+y
ady.(43)
Remark 2.For a general payoff g(y), equation (43) can be numerically integrated, by taking
into account the piecewise definition of ϕand any payoff discontinuities. In general, even when
analytical formulas for gnare known, closed form quadrature rules (such as those in Table
1 for put options) provide more numerically stable coefficients as the resolution is refined
(see [27] for more discussion).
As for European options, put-call parity can be used to price Asian call options (see
equation (45)). This approach is preferred numerically since the put has a bounded payoff.
For vanilla options defined in equation(39), define C:= S0
M+1 and D:= W−C, and
En:= exp(xM
n) = exp(xM
1+ (n−1)∆), n = 1, . . . , n∗+ 1.
The payoff coefficients of a put option are given by gput
n= 0 for n=n∗+ 2, . . . , N , and
(44) gput
n:= 












D·¯
ϑ[2]
∗−C·ϑ[2]
∗·Enn= 1, . . . , n∗−2
D·¯
ϑ[2]
−1−C·ϑ[2]
−1·Enn=n∗−1
D·¯
ϑ[2]
0−C·ϑ[2]
0·Enn=n∗
D·¯
ϑ[2]
1−C·ϑ[2]
1·Enn=n∗+ 1

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 17
Quadratic ¯
ϑ[2]
jϑ[2]
j
j=∗11
51
2+1
9(cosh(5∆/4) + 7 cosh(∆/2) + 22cosh(∆/4)) + cosh(3∆/4) + 1
6cosh(∆)
j=−147
48
1
10 1 + 1
9e−5∆/4+ 7e−∆/2+ 44 cosh(∆/4)+e−3∆/4+1
6e−∆
+7
12 e∆/2+49
72 e5∆/8+25
72 e7∆/8+3
16 e3∆/4+7
144 e∆
j= 0 1
2
1
10 7
24 +1
9e−5∆/4+1
6e−∆+e−3∆/4+7
12 e−∆/2+13
12 e−3∆/8+11
24 e−∆/4+47
36 e−∆/8
j= 1 1
48
1
80 e−9∆/8+1
6e−5∆/4+1
9e−11∆/8+7
18 e−∆
Table 1. Stable closed form coefficient approximations using Boole’s rule for use with
terminal payoffs.
where ϑ[2]
jand ¯
ϑ[2]
j, derived in [27], are provided in Table 1 for reference. The value is
then approximated by substituting gput
nfor gnin equation (42). Once the put value Vput is
determined, the call value Vcall satisfies (see Section 4.3)
(45) Vcall =Vput +S0e−rT
M+ 1 e(r−q)∆t(M+1) −1
e(r−q)∆t−1−e−rT W.
Remark 3.While the two checks in Section 3.6 are designed to prevent an insufficient choice
of ∆ and ¯aat initialization, we can use the following quantity
E[eYM] = M+ 1
S0
E[AM]−1 = e(r−q)∆t(M+1) −1
e(r−q)∆t−1−1
to estimate the final valuation error. In particular, the error in estimating E[eYM],
(46) EM:= E[eYM]−Ca,N ·ϑ[2]
∗·
N
X
n=1
¯
βM
nexp(xM
n)
serves as a proxy for the error in V ◦ g(S0).Given an value error tolerance TOL = 5e-04,
we set a mean error tolerance for EMof 3:= TOL/10 = 5e-03. If EM< 3, the algorithm
terminates. Otherwise, if this threshold is exceeded, we reenter the main loop in Algorithm
3. We will then have the new value estimate, VN, and the previous estimate VN/2. Hence,
the new stopping criteria becomes |VN− VN/2|<TOL.
3.9. The Algorithm and its Complexity. We now summarize the proceeding steps which
define the quadratic APROJ algorithm, while alternative bases can be accommodated simi-
larly. The algorithm calls initialization Subroutine 1, although one can instead select Nand
∆ directly. After Subroutine 2 is called to compute ¯
Ψ, the main loop begins. Note that we
have designed the routine for memory efficiency by reusing the arrays Hand ¯
β.
3.9.1. Complexity. We begin with cost of initializing the matrix ¯
Ψ. From Section 3.7, for a
given quadrature rule the complexity associated with calculating ¯
Ψ is O((N−1)(N+NM−1)).
From equation (25), we can bound the growth of NM−1, and hence the dimensions of ¯
Ψ. With
˜µmdefined in equation (26), it follows that
˜µM−1−˜µ1≤log(M−1) + T
M((M−2)(r−q)−(w+E[L(1)]))
≤2 log(M−1),
for sufficiently large M, by Proposition 3.2. For ¯a≥2,
(47) NM−1=ba(˜µM−1−˜µ1)c≤b2Nlog(M−1)/¯ac ≤ Nlog(M−1).
Thus, N+NM−1≤(N+1) log(M−1) = O(Nlog(M)), so the complexity of ¯
Ψ is O(N2log(M)).
Given that the computational cost of determining xm
1and Hm,m= 1, . . . , M , is on the order
O(MN), and the final value cost is O(N), the remaining contribution to the algorithm’s

18 J. LARS KIRKBY
Algorithm 3 Main Algorithm
Value error tolerance TOL:=5e-04
Call Subroutine 1 to obtain:
Input 1: Final parameters N, ∆,¯a, ∆ξ
Input 2: Grids {ξj}N
j=1,{Nm}M
m=1,{xm
1}M
m=1
Input 3: Coefficient input Φ,{ζj}N
j=2,{¯
βn}N
n=1
Call Subroutine 2 to compute ¯
Ψ
Φ←Ca,N Φ; C1←1/(960a3)
Hj←Φj·PN
n=1 ¯
Ψj,n ¯
βn, j = 1, . . . , N
¯
β←H
for m= 3, . . . , M :do
H1←C1;Hj←ζj·¯
βj·exp(−iξj·xm−1
1), j = 2, . . . , N
¯
β← <[FFT(H)]
Hj←Φj·PN
n=1 ¯
Ψj,Nm−1+n¯
βn, j = 1, . . . , N
end for
Redefine xM
1by equation (40)
H1←C1;Hj←ζj·¯
βj·exp(−iξj·xM
1), j = 2, . . . , N
¯
β← <[FFT(H)]
Find put value Vput using equation (42) with gput
ndefined in (44)
For a call, use put-call parity equation (45)
Compute final error proxy EMin eq. (46), and proceed as directed in Remark 3
complexity resides in the cost of ¯
βm,m= 1, . . . , M , which is on the order O(MN log2(N)
when the fast Fourier transform is utilized, the matrix vector multiplications ¯
Ψm−1¯
βm−1,
m= 2, . . . , M , at a cost of O((M−1)N2), and the Hadamard products ¯
Ψm−1¯
βm−1◦ΦC
R,
m= 2, . . . , M , at a cost of O((M−1)N). Hence, the total cost is O(MN log2(N) +
N2log(M) + MN2) = O(MN2).
3.10. Greeks. We now demonstrate how price sensitivities are calculated at almost no addi-
tional cost from the valuation algorithm. Consider first the put option payoff g(y;S0) defined
in equation (39), where y∗=y∗(S0) = log (M+ 1) W
S0−1. First we observe that YMis
independent of S0. Indeed,
exp(YM) = 1
S0
M
X
m=1
Sm=1
S0
M
X
m=1
S0exp m
X
k=1
Rk!=
M
X
m=1
exp m
X
k=1
Rk!.
From equation (42), Leibniz rule is used to determine the put option Delta, noting that
g(y∗(S0), S0) = 0:
∆:= ∂V ◦ g
∂S0
=e−rT Zy∗(S0)
−∞
∂g(y;S0)
∂S0
fYM(y)dy =−e−rT
M+ 1 Zy∗(S0)
−∞
(1 + ey)fYM(y)dy.
Using quantities that were already computed during the valuation stage, we find that
(48) ∆put ≈Ca,N −e−rT
M+ 1
n∗+1
X
n=1
¯
βM
ngn(∆put).
The coefficients gn(∆put) are defined similarly to equation (44), but instead of D·¯
ϑ[2]
j−C·
ϑ[2]
j·Enfor j∈ {∗,−1,0,1}, we use gn(∆put) = ¯
ϑ[2]
j+ϑ[2]
j·En. To determine the call Delta,
equation (45) leads to the put-call parity formula
∆call =∆put +e−rT
M+ 1 e(r−q)∆t(M+1) −1
e(r−q)∆t−1!

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 19
Likewise, the put (and call) option Gamma is given by
Γ:= ∂2V ◦ g
∂S2
0
=−e−rT
M+ 1
∂
∂S0Zy∗(S0)
−∞
(1 + ey)fYM(y)dy
=−e−rT
M+ 1 1 + ey∗fYM(y∗)∂y∗(S0)
∂S0
=W
S02(M+ 1) ·e−rT fYM(y∗)
W(M+ 1) −S0
.(49)
For the quadratic basis we use the approximation15
fYM(y∗)≈a·Ca,N ·ϕ[2](0) ¯
βM
n∗+ϕ[2](1)¯
βM
n∗−1+¯
βM
n∗+1,
where ϕ[2](0) = 3/4, ϕ[2] (1) = 1/8 and n∗is given in the previous subsection. Thus ∆and Γ
are computed as byproducts of the pricing algorithm.
4. Extensions
In this section we illustrate in-progress option pricing, generalized arithmetic averaging
and continuously monitored option pricing.
4.1. In-Progress Options: Pricing and Greeks. Only a slight modification is required to
price Asian options at arbitrary times after averaging has begun. With the arithmetic average
AMdefined in equation (14), then for τ∈[Ms∆t, (Ms+ 1)∆t), for some Ms< M −1, we
find a variable YM−Mssuch that
AM=1
M+ 1 "Ms
X
m=0
Sm+S(τ)·exp(YM−Ms)#.
That is, Msindexes the most recent monitoring date, and UMs:= PMs
m=0 Smas well as S(τ)
are known at the time of pricing. Noting that for h:= (Ms+ 1)∆t−τ,SMs+1 =S(τ)eR(h)
where R(h)d
= log(St+h/St), it follows from stationarity and independence of increments that
YM−Mscan be found recursively by
φY1=φR,¯
φYm=φR·¯
φZm−1, m = 2, . . . , M −Ms−1,
¯
φYM−Ms=φR(h)·¯
φZM−Ms−1.
When τ=Ms∆t,φR≡φR(h). As before, the final grid defined by xM−Ms
1is shifted so that
the kink point
(50) y∗:= log ((M+ 1)W−UMs)−log(S(τ))
is a member. For in-progress vanilla options, the payoff is expressed as
(51) g(y) := 

1
M+1 (UMs+S(τ)ey)−W
1
[y≥y∗],for a call,
W−1
M+1 (UMs+S(τ)ey)
1
[y≤y∗],for a put,
and payoff coefficients are derived analogously. Perhaps of even more interest than the price
for an in-progress option are the Greeks. For the fixed strike Asian put,
∆:= ∂Vτ◦g
∂S(τ)=−e−r(T−τ)
M+ 1 Zy∗
−∞
eyfYM−Ms(y)dy,
where Vτ◦g(UMs, S(τ)) = e−r(T−τ)E[g(AM)|UMs, S (τ)]. Similarly, the put (and call) option
Gamma is given by
Γ:= ∂2Vτ◦g
∂S2(τ)=e−r(T−τ)
(S(τ))2fYM−Ms(y∗)W−UMs
M+ 1,
where fYM−Ms(y∗) is calculated as before.
15For the linear basis, fYM(y∗)≈a·Ca,N ·¯
βM
n∗

20 J. LARS KIRKBY
4.2. Generalized Arithmetic Asian Pricing. By a slight modification of the original
algorithm, the ARPOJ method is capable of pricing payoffs on generalized averages of the
underlying16
(52) Aλ
M:= 1
M+ 1
M
X
m=0
λmSm,
where λm>0, m= 0, . . . , M . We have the following extension, which is proved in a similar
fashion to the Carverhill-Clewlow result, noting that
Aλ
M=S0
M+ 1 λ0+eR1λ1+eR2···eRM−1λM−1+λMeRM.
In alternative representation is provided in Corollary 4.1, which prevents the matrix ¯
Ψ from
becoming stage dependent, and results in an efficient algorithm.
Corollary 4.1. Fix a set of positive weights λ={λm}M
m=0, and define Xm:= λm
λm−1exp(Rm),
m= 1, . . . , M , where Rm= log(Sm/Sm−1). Set Y1= log(XM) = log(λM/λM−1) + RM, and
define recursively
Ym= log λM+1−m
λM+1−(m−1) +RM+1−m+Zm−1, m = 2, . . . , M,
where Zm:= log(1 + exp(Ym)). Then
(53) Aλ
M≡λ0S0
M+ 1 (1 + exp(YM)) .
Proof. The proof relies on an equivalent factorization of Aλ
M,
Aλ
M=λ0S0
M+ 1 1 + λ1S1
λ0S01 + λ2S2
λ1S1···λM−1SM−1
λM−2SM−21 + λMSM
λM−1SM−1,
which can be verified by multiplying each of the terms. The remainder of the proof is similar
to standard construction, and follows algebraically. 
This form of the recursion requires that λm>0 for each m, in which case the structure of
the APROJ algorithm is unaffected. Namely, the matrix ¯
Ψ is the same for each m, and the
only real change is the grid shift, where we add ˜
λm:= log(λM+1−m/λM+1−(m−1)) to each
˜µm. The perturbed grid shifts ¯µmare defined still by equation (27).
4.3. Put-Call Parity. Just as for vanilla European options, put-call parity can be used to
price Asian call options in terms of puts and conversely (this will be used for all numerical
experiments). In the generalized setting, we have
e−rT E(Aλ
M)+−(−Aλ
M)+=e−rT
M+ 1E"M
X
m=0
λmSm#=S0e−rT
M+ 1
M
X
m=0
λme(r−q)∆tm,
where qis the continuous dividend yield, and ∆t=T/M in the uniform case. Considering
the fixed and floating strikes17 together, with α=±1,
(α(AM−K1ST−K2))+= (αAλ
M)+,
where
λ0= 1 −K2
S0
(M+ 1), λM= 1 −(M+ 1)K1, λm= 1, m = 1, . . . , M −1.
In this setting, with CM(S0, T ) and PM(S0, T ) denoting the call and put prices,
CM(S0, T )−PM(S0, T ) = S0e−rT
M+ 1 e(r−q)T(M+1)
M−1
e(r−q)T
M−1!−S0K1e−qT −e−rT K2,
16We include the term 1/(M+ 1) so that the standard average is obtained when all λm= 1.
17For example, a floating strike call has payoff (AM−K1ST)+.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 21
W= 90 W= 100
dASCOS APROJ ASCOS APROJ
1 – 12.67415 – 5.11827
2 – 12.67441 – 5.11855
3 – 12.67443 – 5.11859
4 12.6748 12.67443 5.1191 5.11859
5 12.6744 12.67443 5.1186 5.11859
6 12.6743 12.67443 5.1185 5.11859
Table 2. Continuously monitored Asian option values by Richardson Extrapolation. NIG
model with parameters from [21]. Values obtained by quadratic APROJ with P= 7,¯
P= 4,
and seven point rule. ASCOS values given in [39].
from which the fixed and floating strike parities are derived. Moreover, the forward contract,
g({Sm}) = AM−K2, is priced immediately by setting K1= 0.
It should be noted that put-call parity is a useful tool for maintaining robustness when
pricing call options. Since the density of YMis recovered approximately over [¯µM−¯a
2,¯µM+¯a
2],
this implies a lower bound on the truncation error for pricing call options:
trunc ≥e−rT S0(1 + exp(¯µM+ ¯a/2))
M+ 1 −WZ∞
¯µM+¯a/2
fYM(y)dy.
For a heavy-tailed density, the implied truncation error can be unacceptable, in which case
put-call parity can be used to price call options in terms of the bounded put prices.
4.4. Continuous Monitoring. As a final extension, we consider the case of continuously
monitored contracts, with terminal payoffs
g(S) = 

1
TRT
0S(t)dt −W+for a call,
W−1
TRT
0S(t)dt+for a put.
Let VN(M) denote the discretely monitored value approximation with Mmonitoring dates,
and with Nfixed. By fixing a positive integer d, the continuously monitored option value can
be approximated by a four-point Richardson extrapolation:
V∞
N(d) := 1
21 64VN(2d+3)−56VN(2d+2 ) + 14VN(2d+1)− VN(2d),
as demonstrated in [39]. We compare the extrapolation procedure18, when applied with
APROJ, to the values obtained by [39] in Table 2. For both strikes, agreement in prices is to
at least three decimals.
5. Error Analysis
In this section, we provide a stability analysis of the error propagation of ChFs for 1 ≤
m≤M, after which we conclude with the terminal valuation error for pricing options on the
arithmetic average.
Recall that the characteristic functions for Levy processes of interest satisfy
(54) |φR∆t(ξ)| ≤ κexp(−∆tc|ξ|ν), ξ ∈R.
For the BSM, KOU (double exponential), and MJD (Merton’s Jump Diffusion) models
from Table 6, the ChF of log return satisfies |φR∆t(ξ)| ≤ exp −∆tσ2
2|ξ|2, so equation
(1) holds with ν= 2 and c=σ2
2. For the CGMY model, ν=Yand ccan be taken as
c= 2C|Γ(−Y) cos(πY /2)| · , for any ∈(0,1). With the Normal Inverse Gaussian (NIG)
model, ν= 1 and c=δ. For the pure jump VG, |φR∆t(ξ)| ≤ κ|ξ|−2∆t/ν, so that φR∆tfails
18For greatest efficiency, a common ¯
Ψ can be used for all four settings of Min the extrapolation procedure,
by perturbing the means slightly so they align.

22 J. LARS KIRKBY
to be integrable for ∆t≤ν/2. However, by adding a Brownian motion component, −σ2
2ξ2,
equation (1) is satisfied with ν= 2. We have the following Corollary of Proposition 2.1.
Corollary 5.1. Suppose that φR∆t(ξ)∈ H(Dd)for some d > 0. Fix a= 2Pand N=a·¯a,
where ¯a= 2 ¯
Pfor ¯
P > 1+log2|¯µM|. Assume for some c, κ > 0and ν∈(0,2],φR∆t(ξ)satisfies
equation(54). Then for some 0< γ ≤d, and a constant CM=O(maxm=1,...,M kφYmkHd),
sup
1≤n≤Na1/2Ca,N ·˘
βm
a,n − hfYm,eϕa,ni≤a−1/2
πCM
e−(¯a−2|¯µM|)γ /2
1−e−¯aγ +τa(R∆t)
(55)
independently of 1≤m≤Mwhere τa(R∆t) = O(aexp(−∆tc ·(2πa)ν)) is as in equation (8).
For large enough a > 0, and d < ∞,γwill approach d.
5.1. Error Propagation. We can now state the core result concerning the propagation of
ChF error for a given number of monitoring dates M.
Proposition 5.1. Suppose that φR∆t(ξ)∈ H(Dd)for some d > 0, and consider a pth order
B-spline basis generated by ϕ. Fix a= 2Pand N=a·¯a, where ¯a= 2 ¯
Pfor ¯
P > 1 + log2|¯µM|.
Assume for some c, κ > 0and ν∈(0,2], the tail of φR∆t(ξ)satisfies equation (54). The
terminal ChF error satisfies (¯
φYM(ξ1)) = 0 and
(56) |(¯
φYM(ξj))|=O∆(p+1) ·e−˜c∆t(j−1)
¯aν
¯a1/2kξ(p+1)φR∆t(ξ)k2,2≤j≤N,
where ˜c:= (2π)νc. The dependence on Mis governed by the behavior of φR∆t.
Proof. Fix any ξ≥0, and let G:= ∪m=1,..,M Gmthe full truncated integration range implied
by ¯
P, where Gm= [¯µm−¯a
2,¯µm+¯a
2]. To manage notation, we will suppress the dependence of
certain objects on m. For example, we assume by the indexing on ¯
βm
nthat the corresponding
elements ϕa,n have been shifted appropriately.
We start by fixing m≥3, for which
(¯
φZm−1(ξ)) := φZm−1(ξ)−¯
φZm−1(ξ)
=ZR
(ey+ 1)iξfYm−1(y)dy −Ca,N
N
X
n=1
¯
βm−1
n¯
Ψ(ξ, n)
=ZR/Gm−1
(ey+ 1)iξfYm−1(y)dy
+ ZGm−1
(ey+ 1)iξfYm−1(y)dy −Ca,N
N
X
n=1
βm−1
nΨ(ξ, n)!
+Ca,N
N
X
n=1
βm−1
n(Ψ(ξ, n)−¯
Ψ(ξ, n)) + Ca,N
N
X
n=1
¯
Ψ(ξ, n)(βm−1
n−¯
βm−1
n)
:= τ(Gm−1) + Jm−1
1(ξ) + Jm−1
2(ξ)+Jm−1(ξ),
where the error term Jm−1(ξ) will be further split into two components. Here we have defined
βm−1
nso that a1/2Ca,N βm−1
n=hfYm−1,eϕa,ni, from which
e
fYm−1(y) := a1/2Ca,N
N
X
n=1
βm−1
nϕa,n(y)
is the true projection truncated to the set {ϕa,n }N
n=1.
Since |(ey+ 1)iξ|=|exp(iξ log(1 + ey))|= 1,the truncation error satisfies
τ(Gm−1) = ZR/Gm−1
(ey+ 1)iξfYm−1(y)dy ≤ZR/Gm−1
fYm−1(y)dy ≤τM(G),
for m= 1, . . . , M , where τM(G) bounds the largest such truncation error (typically, τM(G)≈
τ(G1), since fRhas the heaviest tails). The next result characterizes the convergence of
Jm−1
1, which is governed by the projection error.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 23
Lemma 5.2. For ξ∈R,1≤m≤M, and C1(R∆t) := C1(ϕ)· kξ2φR∆t(ξ)k2/(2π),Jm−1
1
satisfies
|Jm−1
1(ξ)| ≤ √¯a·C1(R∆t)∆(p+1),
with the constant C1(ϕ)from (58), independent of φR∆t.
Proof. In particular, by Cauchy-Schwartz
Jm−1
1(ξ) = ZGm−1
(ey+ 1)iξfYm−1(y)dy −Ca,N
N
X
n=1
βm−1
nΨ(ξ, n)
=ZGm−1
(ey+ 1)iξ fYm−1(y)−a1/2Ca,N
N
X
n=1
βm−1
nϕa,n(y)!dy
≤ k(ey+ 1)iξkGm−1
2· kfYm−1−e
fYm−1kGm−1
2
≤ k(ey+ 1)iξkGm−1
2· kfYm−1−PMafYm−1kR
2.
To characterize the convergence rate of density projections onto B-spline bases, we note that
ϕis Riesz generator which satisfies
(57) bϕ(0) = 1,and for m∈ {0,1},bϕ(m)(2πk) = 0, k ∈Z/{0},
where bϕ(m)denotes the mth derivative of ϕ. In particular, the pth order B-spline generator ϕ
is a (p+ 1)th order Riesz generator. It then follows that for any fX∈L2(R), the projection
error satisfies
(58) inf
fa∈MakfX−fak2≤ kfX−PMafXk2≤C1(ϕ)a−(p+1)kf(p+1)
Xk2,
where C1(ϕ) is a constant independent of fX(see [37]). It follows that
(59) kf(p+1)
Ymk2=1
2πkF[f(p+1)
Ym]k2=1
2πk(−iξ)(p+1) φYm(ξ)k2≤1
2πkξ(p+1)φR∆t(ξ)k2<∞,
since for ξ∈R,|φYm(ξ)|≤|φR∆t(ξ)|, and the (p+ 1)th moment is finite by exponential decay
of φR∆t(ξ). Thus if we define C1(R∆t) as in the statement of the Lemma,
kfYm−1−PMafYm−1kR
2≤C1(R∆t)∆(p+1),∀m≥2.
Hence, for m≥2 and ξ∈R
|Jm−1
1(ξ)|≤k(ey+ 1)iξ kGm−1
2C1(R∆t)∆(p+1) ≤√¯a·C1(R∆t)∆(p+1),
since |(ey+ 1)i2ξ|= 1 and |Gm−1| ≤ ¯a.
Remark 4.We should note that, while the bound in (59) is chosen to be independent of m,
the behavior of this term is truly a decreasing function of m, although is difficult to quantify.
This can be seen by examining the behavior of φYmfrom the approximations given in Figure
3 for a CGMY model.
The next source of error materializes from the approximation of Ψ by ¯
Ψ.
Lemma 5.3. For ξ∈R1≤m≤M, and C2(R∆t) := C2(ϕ)kφR∆tk2/2π,
(60) |Jm−1
2(ξ)| ≤ √¯a·(¯
Ψ)C2(R∆t),
where the constant C2(ϕ)is the lower frame bound defined in equation (3) for the piecewise
linear basis, and
(¯
Ψ) := sup{|Ψ(ξj, n)−¯
Ψ(ξj, n)|: 1 ≤j≤N, 1≤n≤N+NM−1}.

24 J. LARS KIRKBY
Proof. By the discrete version of Cauchy-Schwartz,
Jm−1
2(ξ) = Ca,N
N
X
n=1
βm−1
n(Ψ(ξ, n)−¯
Ψ(ξ, n))
≤a−1/2 N
X
n=1 Ψ(ξ, n)−¯
Ψ(ξ, n)2!1/2 N
X
n=1 a1/2Ca,N βm−1
n2!1/2
≤√¯a·(¯
Ψ) ·X
n∈Z|hfYm−1,eϕa,ni|21/2≤√¯a·(¯
Ψ) ·C2(ϕ)kfYm−1k2.
The term C2(ϕ)kfYm−1k2follows from Bessel’s inequality, which is the upper frame bound
corresponding to the piecewise linear basis. Noting that
kfYm−1k2=kφYm−1k2/2π≤ kφR∆tk2/2π,
we have
|Jm−1
2(ξ)| ≤ √¯a·(¯
Ψ) ·C2(ϕ)kφR∆tk2/2π.

Remark 5.While the definition of (¯
Ψ) is made so that we obtain an overall convergence rate
in ∆ when ¯ahas been fixed and a sufficiently accurate quadrature rule has been selected, the
the error in ¯
Ψ tends to be much smaller for ξj∈[0,2πa] near zero than for ξjnear 2πa. If
we define
j(¯
Ψm−1) := sup
1≤n≤N|Ψ(ξj, Nm−1+n)−¯
Ψ(ξj, Nm−1+n)|
then Jm−1
2(ξj)≤j(¯
Ψm−1)√¯a·C2(R∆t). This is more than offset, however, when multiplied
by φR∆t(ξj) to obtain the error in ¯
φYm, since φR∆t(ξj) is close to one for ξjnear zero, and
decays exponentially for larger ξj. In practice, the contribution of (¯
Ψ) is dominated by the
projection error when using a seven-point Newton-Cote’s rule. Although Boole’s rule is often
sufficient (and cheaper) for M≤52, we opt for the more conservative seven-point rule in
general.
For the final term, which reflects the discrete Fourier transform error inherent in ¯
βm, we
have
Jm−1(ξ) := Ca,N
N
X
n=1
¯
Ψ(ξ, n)(βm−1
n−¯
βm−1
n) = a−1/2
N
X
n=1
¯
Ψ(ξ, n)·(¯
βm−1
n),
where (¯
βm−1
n) := a1/2Ca,N (βm−1
n−¯
βm−1
n).
Lemma 5.4. The error source Jm−1(ξ)can be bounded by
(61) |Jm−1(ξ)| ≤ ¯a
πM(a, ¯a) + C(J4)·(¯
φZm−2)a−1/2|¯
φZ1(ξ)|
where C(J4)is a constant, and
(62) M(a, ¯a) := CM
e−(¯a−2|¯µM|)γ /2
1−e−¯aγ +τa(R∆t).
Proof. Splitting (¯
βm−1
n) in terms of the discrete Fourier transform and ChF errors, where
a1/2Ca,N ˘
βm−1
nis the discrete Fourier transform approximation using the true φYm−1(see
equation (6)), it follows that
(¯
βm−1
n) = hfYm−1,eϕa,ni − a1/2Ca,N ˘
βm−1
n+a1/2Ca,N ˘
βm−1
n−¯
βm−1
n
:= 1(¯
βm−1
n) + 2(¯
βm−1
n).
Hence,
(¯
φZm−1(ξ)) = τ(Gm−1) + Jm−1
1(ξ) + Jm−1
2(ξ) + Jm−1
3(ξ)+Jm−1
4(ξ),

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 25
where we have defined
Jm−1
3(ξ) := a−1/2
N
X
n=1
¯
Ψ(ξ, n)·1(¯
βm−1
n), Jm−1
4(ξ) := a−1/2
N
X
n=1
¯
Ψ(ξ, n)·2(¯
βm−1
n).
Moreover, for the Newton-Cotes quadrature rules, |¯
Ψ(ξj, n)| ≤ 1 for any 1 ≤j, n ≤N, and
by Corollary 5.1
|1(¯
βm−1
n)| ≤ a−1/2
πM(a, ¯a),
where M(a, ¯a) is defined in equation (62).
Hence,
|Jm−1
3(ξj)| ≤ a−1/2
πM(a, ¯a)·a−1/2
N
X
n=1 |¯
Ψ(ξj, n)| ≤ ¯a
πM(a, ¯a)
Note that Jm−1
4(ξ) alone depends on (¯
φZm−2(ξj)), since
2(¯
βm−1
n) = a−1/2
π<
∆ξ
N
X
j=1
0ha,n(ξj)φYm−1(ξj)−¯
φYm−1(ξj)

=a−1/2
π<
∆ξ
N
X
j=1
0ha,n(ξj)φR∆t(ξj)(¯
φZm−2(ξj))
,(63)
where ha,n(ξ) and ha(ξ) are defined in equation (68) for the linear basis (and in general is
determined by b
eϕ(ξ)), and P0indicates that the first and last terms in the sum are halved.
If we define (¯
φZm−2) := max1≤j≤N|(¯
φZm−2(ξj))|, it follows that
< ∆ξ
N
X
j=1
0ha,n(ξj)φR∆t(ξj)(¯
φZm−2(ξj))!≤(¯
φZm−2)∆ξ
N
X
j=1
0ha(ξj)<(φR∆t(ξj)),
which is bounded above for all Nand a, since <(φR∆t) admits an upper frame bound. To
derive an upper bound on Jm−1
4(ξ), we recall the dependence of ¯
Ψ and ha,n on m−1 (through
the shift xm−1
1, denoted by hm−1
a,n ), from which equation (63) yields
Jm−1
4(ξ) = a−1/2
N
X
n=1
¯
Ψm−1(ξ, n)·2(¯
βm−1
n)
=a−1/2
N
X
n=1
¯
Ψm−1(ξ, n)a−1/2
π< ∆ξ
N
X
j=1
0hm−1
a,n (ξj)φR∆t(ξj)(¯
φZm−2(ξj))!
=O (¯
φZm−2)
a1/2
N
X
n=1
¯
Ψm−1(ξ, n)a−1/2
π<
∆ξ
N
X
j=1
0hm−1
a,n (ξj)φR∆t(ξj))
!
=O (¯
φZm−2)
a1/2
N
X
n=1
¯
Ψm−1(ξ, n)·a1/2Ca,N ¯
β1
Nm−1+n!.
As a final simplification, we note that
Jm−1
4(ξ) = O (¯
φZm−2)a−1/2
N
X
n=1
¯
Ψ(ξ, n)a1/2Ca,N ¯
β1
n!≤C(J4)
a1/2·(¯
φZm−2)|¯
φZ1(ξ)|,
for some C(J4). To see that C(J4) can be chosen independently of Nm−1, from the decay of
φR∆t(ξ), it follows that fR∆t∈C∞(R) has exponential decay at infinity, along with all of its
derivatives [7] (see also [34]). 

26 J. LARS KIRKBY
Summarizing the obtained bounds, it follows that
|(¯
φZm−1(ξ))|=τ(Gm−1) + Jm−1
1(ξ) + Jm−1
2(ξ) + Jm−1
3(ξ)+Jm−1
4(ξ)
≤CM(a, ¯a) + B(a, ξ)(¯
φZm−2).
where
(64) CM(a, ¯a) := τM(G) + √¯a·C1(R∆t)∆(p+1) +√¯a·C2(R∆t)(¯
Ψ) + ¯a
πM(a, ¯a),
and B(a, ξ) := C(J4)|¯
φZ1(ξ)|a−1/2. Iterating from M−1 we obtain
|(¯
φZM−1(ξ))| ≤ CM(a, ¯a)
M−3
X
j=0
B(a, ξ)j+B(a, ξ)M−2(¯
φZ1)
=CM(a, ¯a)1−B(a, ξ)M−2
1−B(a, ξ)+B(a, ξ)M−2(¯
φZ1).
Moreover, the error in ¯
φZ1satisfies
(¯
φZ1(ξ)) := φZ1(ξ)−¯
φZ1(ξ)
=ZR
(ey+ 1)iξfR∆t(y)dy −Ca,N
N
X
n=1
¯
β1
n¯
Ψ(ξ, n)
=ZR/G1
(ey+ 1)iξfR∆t(y)dy
+ ZG1
(ey+ 1)iξfR∆t(y)dy −Ca,N
N
X
n=1
β1
nΨ(ξ, n)!
+Ca,N
N
X
n=1
β1
n(Ψ(ξ, n)−¯
Ψ(ξ, n)) + Ca,N
N
X
n=1
¯
Ψ(ξ, n)(β1
n−¯
β1
n)
:= τ(G1) + J1
1(ξ) + J1
2(ξ)+J1
3(ξ),
where we note that ¯
β1
n=˘
β1
n, since φR∆t(ξ) is known exactly. Hence (¯
φZ1)≤CM(a, ¯a), from
which we derive
|(¯
φZM−1(ξ))| ≤ CM(a, ¯a)
1−B(a, ξ)1−B(a, ξ)M−2+ (1 −B(a, ξ))B(a, ξ)M−2
≤CM(a, ¯a)1−B(a, ξ)M−1
1−B(a, ξ)≤2CM(a, ¯a),(65)
for asufficiently large.
The behavior of CM(a, ¯a) can be characterized by noting that with ¯achosen sufficiently
large, the truncation error τM(G) is dominated by the other sources. Further, as (¯
Ψ) can
be made negligible by a sufficient choice of quadrature, and M(a, ¯a) converges exponentially
in ¯a, a, the error behaves like O(∆(p+1)), which is the projection convergence with respect to
the B-spline basis of order p. In particular, from equation (64) we have
CM(a, ¯a) = O(√¯a·C2(R∆t)∆(p+1)).
Recalling that ¯
φYM=¯
φZM−1φR∆t,
|(¯
φYM(ξj))| ≤ 2CM(a, ¯a)· |φR∆t(ξj)|=O(√¯a·∆(p+1)|φR∆t(ξj)|),2≤j≤N,
where we note that (¯
φYM(ξ1)) = 0, since ¯
φYM(ξ1) = 1 is enforced by the algorithm. Equation
(56) then follows from the assumed decay of φR∆t.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 27
5.2. Valuation Error. The terminal valuation error for a contract on the arithmetic average
is now characterized. We show that for bounded payoffs19 (as for a put, with put-call parity
to price a call), the error converges on the order of projection error, O(∆(p+1)). If we define
E(VN) := erT (V ◦ g(S0)− VN◦g(S0)), we obtain
E(VN) = ZR
g(y)fYM(y)dy −Ca,N
N
X
n=1
¯
βM
ngn
=ZR/GM
g(y)fYM(y)dy + ZGM
g(y)fYM(y)dy −Ca,N
N
X
n=1
βM
ngn!
+Ca,N
N
X
n=1 βM
n−˘
βM
n+˘
βM
n−¯
βM
ngn:= ˜τM(G) + E1+E2.
Assuming gis bounded, we have
˜τM(G) = ZR/GM
g(y)fYM(y)dy ≤ kgk∞·P[YM∈Gc
M] = kgk∞·τM(G),
which is controlled by the choice of ¯asufficiently large.20
Since the coefficients gnare exact21, the second source of error satisfies
E1=ZGM
g(y)fYM(y)dy −Ca,N
N
X
n=1
βM
ngn
=ZGM
g(y)fYM(y)dy −
N
X
n=1hfYM,eϕa,niZGM
g(y)ϕa,n(y)dy
=ZGM
g(y) fYM(y)−
N
X
n=1hfYM,eϕa,niϕa,n (y)!dy
≤ kgkGM
2· kfYM−PMafYMkR
2=O(∆(p+1)).(66)
The third source of error, which accounts for the trapezoidal approximation to the projec-
tion coefficients as well as the terminal ChF error, satisfies
E2=Ca,N
N
X
n=1 βM
n−˘
βM
n+˘
βM
n−¯
βM
ngn
=
N
X
n=1
a1/2Ca,N βM
n−˘
βM
n+˘
βM
n−¯
βM
n·ZGM
g(y)ϕa,n(y)dy.
We will need the following result.
Lemma 5.5. For any NM∈Z, it holds that
N
X
n=1
Ca,N ˘
β1
NM+n=O(1),
where ˘
β1
NM+nare the DFT coefficients of fR∆tcorresponding to xM
n, which are absent of ChF
error since φR∆tis known (these are not calculated explicitly). For ¯asufficiently large, ˜a > 0
can be chosen so that the sum is strictly less than one ∀a≥˜a.
19This assumption is not essential, although it simplifies the analysis.
20For unbounded g, as long as the price is finite, the integral RRg(y)fYM(y)dy < ∞, hence
RR/GMg(y)fYM(y)dy →0 as GM↑R. That is, τM(G)→0 as the truncation error decreases.
21For numerical reasons we have elected instead to use a more stable approximation than the exact
coefficients.

28 J. LARS KIRKBY
Proof. We have the following bound
N
X
n=1
Ca,N ˘
β1
NM+n≤ZGM X
n∈Z
a1/2Ca,N |˘
β1
n| · ϕa,n(y)!dy
=ZGM|˘
fR(y)|dy ≤ZGM|fR(y)|dy +ZGM|˘
fR(y)−fR(y)|dy,
and the result follows from Corollary 5.1 after applying Cauchy-Swartz inequality to the
second integral, and a similar argument as the proof of Lemma 5.2. 
We can now provide the convergence rate for the third error source.
Lemma 5.6. The term E2is characterized by
(67) E2=OCM(a, ¯a),
where CM(a, ¯a)is defined in equation (64).
Proof. We consider each error 1(¯
βM
n) := a1/2Ca,N (βM
n−˘
βM
n) and 2(¯
βM
n) := a1/2Ca,N (˘
βM
n−
¯
βM
n) in turn. Noting that, by Corollary 5.1
sup
1≤n≤N|1(¯
βM
n)|=a1/2Ca,N sup
1≤n≤NβM
n−˘
βM
n≤a−1/2
πM(a, ¯a),
where M(a, ¯a) is defined in equation (62), we have
N
X
n=1
a1/2Ca,N βM
n−˘
βM
nZGM
g(y)ϕa,n(y)dy
≤ N
X
n=1 a−1/2
πM(a, ¯a)2!1/2 N
X
n=1 ZGM
g(y)ϕa,n(y)dy2!1/2
=¯a1/2
πM(a, ¯a)· N
X
n=1 |hg
1
GM, ϕa,ni|2!1/2
≤¯a1/2
πM(a, ¯a)·C3(ϕ)· kgkGM
2,
where C3(ϕ) is the upper frame bound of the dual basis,{eϕa,n}n∈Z, and is the inverse of the
lower from bound of the “primal” basis22. Since kgkGM
2≤ kgk∞¯a1/2(for bounded payoffs),
the final inequality is on the order O(¯aM(a, ¯a)).
Considering 2(¯
βM
n), we have (noting the dependence of hM
a,n(ξj) on the grid {xM
n})
2(¯
βM
n) = a−1/2
π<(∆ξ
N
X
j=1
0hM
a,n(ξj)φR∆t(ξj)(¯
φZM−1(ξj)))
=O(¯
φZM−1)
a1/2π<n∆ξ
N
X
j=1
0hM
a,n(ξj)φR∆t(ξj)o=O(¯
φZM−1)√aCa,N ˘
βNM+n,
where (¯
φZM−1) := sup1≤j≤N|(¯
φZM−1(ξj))|. Hence,
N
X
n=1
gn
2(¯
βM
n)
a1/2=O (¯
φZM−1)
N
X
n=1 ZGM
g(y)ϕa,n(y)dya1/2Ca,N ˘
βNM+n!
=kgkGM
∞(¯
φZM−1)O N
X
n=1
a1/2Ca,N ˘
βNM+n!=OkgkGM
∞(¯
φZM−1)
by Lemma 5.5, and RGMg(y)ϕa,n (y)dy≤ kgkGM
∞RGMϕa,n(y)dy =a−1/2kgkGM
∞.
22Duality is used here to obtain a tighter bound, by a factor of ¯a1/2, than if the standard techniques were
applied, which in turn allows us to dominate this source of error by the one derived from 2(¯
βM
n) next.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 29
234567
log10 |err|
-10
-8
-6
-4
-2
0M=12, BSM
Linear
Quadratic
234567
log10 |err|
-10
-8
-6
-4
-2
0M=250, BSM
234567
log10 |err|
-10
-8
-6
-4
-2
0M=12, KOU
234567
log10 |err|
-10
-8
-6
-4
-2
0M=250, KOU
P= log2(a)
234567
log10 |err|
-10
-8
-6
-4
-2
0M=12, MJD
P= log2(a)
234567
log10 |err|
-10
-8
-6
-4
-2
0M=250, MJD
Figure 5. Convergence of linear vs. quadratic APROJ. Parameters as in [20]. Errors are
max over strikes {90,100,110}. For MJD and BSM, ¯
P:= log2(¯a) = 3; for KOU ¯
P= 4.
Reference values by linear APROJ with P= 10 given in Table 8.
For bounded payoffs, kgkGM
∞≤ kgk∞<∞, and (¯
φZM−1) = O(CM(a, ¯a)) by equation
(65). Hence, by the definition of CM(a, ¯a),
E2=OCM(a, ¯a) + ¯aM(a, ¯a)=OCM(a, ¯a).

Remark 6.Combining equations (66) and (67), and assuming that ¯ais chosen to make τM(G)
(and hence ˜τM(G)) negligible, we conclude
V ◦ g(S0)− VN◦g(S0) = Oe−rT ·CM(a, ¯a)=O(∆(p+1) ).
This of course requires that the error contributed by ¯
Ψ has been controlled by the choice of
quadrature, a choice which may vary by basis. Figure 5 illustrates the difference in conver-
gence rates for the APROJ method with linear and quadratic B-splines.
6. Numerical Experiments
A major improvement over the breakthrough pricing methods of Clewlow (1990), Ben-
hamou (2002), and later Fusai and Meucci (2008), referred to as FM, was the improved
convolution method of Cerny and Kyriakou (2009), referred to as CK. The method of CK
represented a major improvement in speed23, but also demonstrated that references prices
reported by the other three are less precise than the four to five decimal places claimed, often
23The results for CK were obtained in MATLAB 7.2 on Dell Latitude 620 Intel(R) Dual Core T7200,
2GHz, 2Gb RAM.

30 J. LARS KIRKBY
Model vol Calibrated Strike
(Param.) Parameters 90 100 110
0.1 (0.1) 11.581134 3.338617 0.273759
BSM 0.3 (0.3) 13.669816 7.698599 3.896399
(σ) 0.5 (0.5) 17.192393 12.091536 8.314413
0.1 (0.1222, 0.0879, -0.1364) 11.640247 3.323853 0.158354
NIG 0.3 (0.1222, 0.2637, -0.4091) 13.700850 7.342655 3.278604
(ν, σ, θ) 0.5 (0.1222, 0.4395, -0.6819) 16.763062 11.235866 7.168361
0.1 (0.2703, 17.56, 54.82, 0.8) 11.639881 3.324584 0.157877
CGMY 0.3 (0.6509, 5.853, 18.27, 0.8) 13.701604 7.347424 3.283082
(C, G, M, Y ) 0.5 (0.9795, 3.512, 10.96, 0.8) 16.768352 11.244236 7.176236
Table 3. Calibrated parameters from [11]; values reported here to an additional decimal,
obtained by quadratic APROJ with P= 9,¯
P= 3. Other parameters: M= 50, r=.04,
q= 0, T= 1.
correct to only two or three decimals. The ASCOS method of [39] is capable of obtaining
precise estimates of prices, but it does not seem to compete with CK in terms of cpu time24.
The primary drawbacks of ASCOS are its global basis functions, which require several hun-
dred quadrature points per element of a matrix analogous to ¯
Ψ, and the fixed truncation
support (no mean-adjustment)25. We also compare to the recent method of Levendorskii and
Xie [30], denoted LX, which takes two forms: LX(f) for the flat iFT method, and LX(p) for
the parabolic iFT method26
Through numerical experiments27 we demonstrate that APROJ is not only highly accurate
(on the level of CK and LX), but is also faster than the state-of-the-art methods to obtain
the same or superior accuracy, typically on the order of a 10- to 100-fold improvement. This
is true for both linear and quadratic implementations. Given that the initial peak of fR
is quickly softened to obtain fYm, we find that quadratic APROJ is remarkably accurate
for Asian option pricing, and is presented next. Numerical results for linear APROJ (not
presented), are also impressive.
In the first few sets of experiments, to isolate the rate of convergence of APROJ, we
conservatively fix ¯
P= 3 for pure diffusion models, and ¯
P= 3 ∼4 for heavy-tailed models,
such as CGMY and NIG. For most cases, a smaller value of ¯
Pwould have sufficed (especially
with BSM experiments), and reduced the computation time.28 Sensitivity of APROJ with
respect to the choice of ¯
Pis illustrated in Figure 7. The final set of experiments investigates
the automated approach to parameter selection which is often much more efficient, improving
cpu times even further.
Our first set of experiments compares the convergence and cpu time of APROJ against
the method of CK for M= 50 and strikes {90,100,110}. The specifications considered
are the log-normal, ie Black-Scholes-Merton (BSM), the Normal Inverse Gaussian (NIG),
and the Carr-Geman-Madan-Yor (CGMY) model. Three test cases are considered for each
model, with parameters calibrated by [11] to a fixed volatility (vol) in the set {0.1,0.3,0.5}.
Recovered values, as well as calibrated parameters are provided for each strike in Table 3.
For the NIG model, we use the alternative ChF form with parameters (ν, σ, θ) to maintain
24The results for ASCOS were obtained in MATLAB 7.7 with Intel(R) Core(TM)2 Duo CPU E6550,
2.33GHz and 4MB cache size.
25The author’s indicate that a grid adjustment is possible, but to do so would require re-computation of
the matrix analogous to ¯
Ψ at each step (or every several steps), and would incur a substantial cost.
26The results of LX were obtained in MATLAB 7.11.0, with an Intel(R) Celeron(R) Processor T1600,
1.66GHz, 667MHz FSB and 1MB cache.
27The results for APROJ were obtained in MATLAB 8.0 with Intel(R) Core(TM) i5-3470T CPU, 2.90GHz
with 3MB cache size.
28Moreover, Boole’s rule, which is faster than the seven point rule, obtains nearly identical results in many
of the cases. However, it is safer in practice to use the more accurate method, so this is how we present the
results.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 31
Quadratic APROJ, P= log2(a) APROJ CK
vol 1 2 3 4 5 6 7 cpu(sec) cpu(sec)
.1 4.8e+00 1.1e+00 1.6e-01 7.3e-03 3.1e-05 2.4e-07 3.1e-09 .008/.202 1.0
BSM .3 3.8e+00 9.2e-02 8.8e-04 5.6e-06 6.2e-08 8.8e-10 1.3e-11 .003/.009 .3
.5 1.7e+00 4.8e-03 3.0e-05 2.9e-07 3.9e-09 5.8e-11 9.5e-13 .001/.008 .3
.1 4.3e+00 1.1e+00 1.5e-01 1.7e-02 2.2e-04 1.3e-06 1.1e-08 .026/.203 3.7
NIG .3 2.9e+00 1.8e-01 6.4e-03 4.4e-05 1.4e-07 2.6e-09 3.9e-11 .003/.028 1.8
.5 2.6e+00 2.1e-02 2.2e-04 1.6e-06 5.9e-09 6.8e-09 4.8e-09 .003/.009 1.8
.1 3.3e+00 1.1e+00 1.5e-01 1.6e-02 2.2e-04 1.3e-06 9.9e-09 .027/.201 8.5
CGMY .3 2.9e+00 1.7e-01 6.2e-03 4.2e-05 1.1e-07 2.4e-09 3.6e-11 .004/.027 4.1
.5 3.0e+00 2.1e-02 2.1e-04 1.5e-06 3.9e-08 1.4e-08 7.7e-09 .004/.009 2.1
Table 4. Parameters from CN [11]. For ARPOJ with ¯
P= 3 and the seven point rule,
each cpu pair ·/·reports the time to achieve an error of TOL1/TOL2, where TOL1is
on the order of e-03∼e-04 and TOL2is on the order of e-06∼e-07. The error is taken as
the maximum abs. error over the strike set {90,100,110}. CK prices are on the order of
e-05∼e-06. Ref prices are provided in Table 3.
Quadratic APROJ, P= log2(a) NIG
Mstrike 1 2 3 4 5 6 7 reference
90 5.3e-01 2.0e-01 6.4e-03 1.7e-04 3.4e-08 2.2e-09 3.8e-11 12.62243
12 100 1.9e+00 4.0e-01 1.9e-03 8.4e-04 8.4e-06 7.2e-08 1.1e-09 5.06060
110 1.5e+00 3.0e-01 9.2e-03 9.1e-04 1.6e-05 6.3e-08 1.3e-09 1.01355
90 6.6e-01 4.2e-01 4.3e-02 2.5e-03 1.9e-07 1.1e-09 6.6e-11 12.66126
50 100 2.3e+00 5.0e-01 1.4e-02 8.3e-03 2.7e-05 1.0e-07 1.9e-09 5.10370
110 2.1e+00 6.1e-01 9.8e-02 2.3e-03 2.0e-04 7.6e-07 4.4e-09 1.03770
90 3.3e-01 4.4e-01 5.6e-02 6.4e-03 6.6e-05 6.0e-07 2.3e-08 12.67176
250 100 1.9e+00 3.9e-01 3.6e-02 1.8e-02 3.6e-05 2.9e-06 1.7e-07 5.11556
110 1.8e+00 6.2e-01 1.3e-01 1.1e-02 2.5e-04 3.7e-06 3.0e-07 1.04448
Table 5. NIG parameters from FM [21], (α, β, δ ) = (6.1882,−3.8941,0.1622),and r=
0.0367, q = 0, T = 1, S0= 100. Convergence for quadratic APROJ with ¯
P= 4 and seven-
point rule. Reference values obtained by quadratic APROJ with P= 9,¯
P= 4, and seven
point rule.
consistency with [11], which has Levy symbol
ψL(ξ) = 1
ν1−p1−2θνiξ +νσ2ξ2.
In Table 4, we see rapid convergence of the quadratic APROJ method, which is imple-
mented with the seven point rule and ¯
P= 3. By P= 5, accuracy on the order e-07∼e-09 is
achieved for vol ∈ {0.3,0.5}and for all models. With P= 7, accuracy on the order e-08 is
achieved for all models and levels of vol. In the far right column of Table 4 we provide the
cpu times reported by [11] to achieve within four to five correct decimals, which are at least
a factor of 10 more than the time required for APROJ to reach e-06∼e-07 accuracy (with
only one exception), and are often more than 100 times that of APROJ. This is consistent
across all models and specifications as well as strikes tested. Similar results hold for the linear
implementation of APROJ.
For the set of experiments in Table 4 involving the CMGY (KoBoL) model, we can also
compare our results to those of LX [30], using the parabolic method LX(p). When vol = 0.1,
they report a max error of 6.7e-05 over strikes in {90,100,110}at a cost of 1.581 seconds
(compared to an APROJ accuracy of 1.3e-06 in 0.201 seconds). When vol =0.3, they achieve
3.9e-06 in 1.037 seconds (compared to 1.2e-07 in 0.027 seconds), and when vol =0.5 they
achieve 4.6e-06 in 0.684 seconds (compared to 1.5e-06 in 0.009 seconds). In each of these
cases, APROJ obtains greater accuracy and at the same time provides a 7.8, 38, and 76-fold
time reduction respectively.
The second set of experiments compares the convergence and cpu time of APROJ against
the ASCOS method for M∈ {12,50,250}and strikes {90,100,110}. For this case, we specify
the NIG model with parameter set in [21],
(α, β, δ ) = (6.1882,−3.8941,0.1622), r = 0.0367, q = 0, T = 1, S0= 100,

32 J. LARS KIRKBY
ASCOS Quadratic APROJ
N= 128 N= 256 N= 384 ¯
P= 4
M nq= 200 nq= 400 nq= 600 Seven-Point
12 |err.|2.0e-03 1.74e-04 5.16e-06 9.1e-04 1.6e-05 6.3e-08
(sec) (2.41) (15.13) (46.09) (.017) (.085) (.314)
50 |err.|2.26e-04 6.94e-05 2.17e-06 2.0e-04 7.6e-07 4.4e-09
(sec) (2.43) (15.16) (46.22) (.190) (.731) (2.94)
250 |err.|7.8e-03 9.33e-05 8.49e-06 2.5e-04 3.7e-06 2.8e-07
(sec) (2.42) (15.23) (46.68) (.717) (2.94) (11.42)
Table 6. NIG parameters from FM [21], (α, β, δ ) = (6.1882,−3.8941,0.1622),and r=
0.0367, q = 0, T = 1, S0= 100. APROJ with ¯
P= 4, seven point rule. Corresponding
values of Pfor each accuracy are given in Table 5. Absolute errors for strike W= 110.
Quadratic APROJ, P= log2(a) LX(f) LX(p)
strike Ref. 2 3 4 5
90 14.795530855 6.349e-02 1.161e-04 1.136e-05 9.312e-09 2.1e-07 2.1e-07
100 8.281218252 2.973e-02 2.641e-04 3.467e-05 7.533e-08 7.8e-07 7.8e-07
110 3.718094231 1.523e-01 1.040e-03 1.951e-04 5.002e-06 1.7e-06 1.8e-06
cpu (sec) 0.003 0.007 0.016 0.082 27.77 0.792
Table 7. CGMY (KoBoL) Parameters from Levendorskii and Xie [30]: S0= 100, M= 12,
T= 1, r= 0.04, q= 0, CGM Y = (1.1136,3,10,0.2); in terms of KoBoL parameterization,
(c, λ−, λ+, ν) = (1.1136,−10,3,0.2). Convergence for quadratic APROJ with ¯
P= 4 and
seven-point rule. Reference values obtained by quadratic APROJ with P= 11,¯
P= 5, and
seven point rule, and verified to seven decimals with prices of [30]. The LX(f) and LX(p)
methods are respectively the flat and parabolic Fourier transform methods of [30].
and ChF given in Table 6 of the appendix. Table 5 reports the convergence of quadratic
APROJ, along with the reference prices. Reference prices as well as reported cpu times are
provided for ¯
P= 4 and the seven point rule29. In Table 6 the performance of APROJ is
compared to ASCOS, with similar findings as in the first set of experiments. For example,
when M= 12, ASCOS requires 15.13 seconds to achieve 1.74e-04 accuracy, while APROJ
reaches 1.6e-05 accuracy in 0.085 seconds, an almost 200-fold improvement. To reach 6.3e-08
accuracy takes APROJ 0.314 seconds compared to 46.09 seconds for ASCOS to reach 5.15e-
06. For other cases of comparable accuracy, the improvement is by at least a factor of 10 or
more.
We next consider a KoboL (CGMY) model from Levendorskii and Xie [30], with pa-
rameters CGM Y = (1.1136,3,10,0.2), or in terms of the KoBoL [7, 8] parameterization
(c, λ−, λ+, ν) = (1.1136,−10,3,0.2). As demonstrated in Table 7, the APROJ method con-
verges rapidly to high accuracy. Two methods from [30] are provided for comparison, the
LX(f) method and LX(p), neither of which seems to dominate the other in terms of speed or
accuracy from the experiments provided in [30]. In this case, LX(f ) is slower to converge (in
terms of cpu), but for strikes {90,100,100}, both methods of [30] reach an accuracy of about
(2.1e-07,7.8e-07,1.7e-06) respectively. The APROJ method with P= 5 achieves accuracy of
(9.3e-09,7.5e-08,5.0e-06), with a cpu time reduction factor of 9.65 for the LX(p) method and
a 338-fold reduction for LX(f).
Now we consider the BSM model, Merton’s Jump Diffusion (MJD), and Kou’s double
exponential (KOU) model, which characteristic functions given in Figure 6. Parameters are
as in [21] (later used in [20]), which are provided in Table 8 along with reference values. The
parameter setting for BSM is also considered in [11]. Convergence is compared for the linear
and quadratic implementation of APROJ in Figure 5.
The first observation is that the prices obtained for BSM agree with those of CK [11] to
7 decimals (the other two models are not reported in [11]), while the method of FM [21] is
accurate to only about 2-3 decimals in most cases with cpu times in excess of 5 seconds (this
is pointed out as well in [11]). Greater accuracy is obtained by APROJ in just milliseconds.
29The necessarily larger value of ¯
Pis detected by recovering the value of ¯
β1
1prior to the algorithm’s
initialization.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 33
Reference Values
Model Parameters Strike M= 12 M= 250
σ= 0.17801 90 11.9049157 11.9405632
BSM 100 4.8819616 4.9521569
110 1.3630380 1.4133670
σ= 0.120381 90 12.713070 12.753177
KOU λ= 0.330966, p= 0.2071 100 5.017859 5.070220
η1= 9.65997, η2= 3.13868 110 1.041531 1.076568
σ= 0.126349 90 12.710669 12.749182
MJD λ= 0.174814 100 5.011290 5.063823
µJ=−0.390078, σJ= 0.338796 110 1.051633 1.087406
Table 8. Model parameters from [20,21], and r= 0.0367, q = 0, T = 1, S0= 100. Refer-
ence values by Linear APROJ, P= 10. For MJD and BSM, ¯
P= 3; for KOU ¯
P= 4.
Model W M Ref |Err|cpu(sec) NL1NL2log2(N)
CGMY 90 12 11.999099 2.33e-06 0.006 1 1 6
(0.2703, 17.56, 54.82, 0.8) 100 250 3.643684 7.01e-07 0.083 2 1 8
CGMY 90 12 15.061188 5.63e-04 0.005 1 1 6
(1.1136, 3, 10, 0.2) 100 250 8.644264 7.80e-06 0.471 2 2 9
MJD 90 12 13.134793 6.10e-04 0.005 1 1 6
(0.13, 0.17, -0.39, 0.34) 100 250 5.480458 1.02e-04 0.103 1 2 8
MJD 90 12 12.704098 1.02e-06 0.005 1 1 6
(0.1, 3, -0.05, 0.086) 100 250 5.620436 6.79e-06 0.030 1 1 7
BSM 90 12 11.949574 4.22e-07 0.006 1 1 6
σ= 0.1 100 250 3.639486 2.02e-06 0.029 1 1 7
BSM 90 12 13.854399 1.96e-06 0.006 1 1 6
σ= 0.3 100 250 7.939288 4.92e-06 0.028 1 1 7
NIG 90 12 12.290729 5.89e-05 0.005 1 1 6
(8, -1, 0.2) 100 250 4.610758 1.70e-07 0.423 3 1 9
Kou 90 12 13.564345 9.71e-04 0.005 1 1 6
(0.15, 0.4, 0.2, 9, 3) 100 250 6.297930 2.33e-05 0.473 2 2 9
Table 9. Call price errors for quadratic APROJ with automated parameter selection.
Cpu times represent full cost including parameter determination. Columns NL1and NL2
are the number of loops required in initialization (Subroutine 1) and the main algorithm
(Algorithm 3) before tolerance is met, where 1= 5e-04, 2= 5e-04, and 3= 5e-03 in
Algorithm 3. Nis the final grid size. In all cases, S0= 100, r= 0.05, q= 0, T= 1. MJD
params: (σ, λ, µJ, σJ). Kou params: (σ, λ, p, η1, η2). NIG params: (α, β , δ)
When M= 250, K= 100, the price to seven decimals is given by 4.9521569, as computed
by CK and APROJ. FM obtains 4.95233, while the maturity randomization methods of
Fusai, Marazzina and Marena (FMM) [20] report prices of 4.95212 and 4.95242, using density
recursion and price recursion respectively, with cpu times of 38.32 seconds and 95.80 seconds30.
We find similar results for the models of KOU and MJD, where the prices of FMM agree with
those computed by ARPOJ (given in Table 8) to 3 or 4 decimals with FMM cpu times in the
dozens of seconds, compared to milliseconds for APROJ.
The previous experiments illustrate the convergence of APROJ as a function of the reso-
lution when the grid width is fixed a priori. The final set of experiments analyzes the ability
of the APROJ algorithm to accurately select parameters without user input, as described
in Section 3.6 (and implemented in Subroutine 1) to achieve a practical accuracy of about
TOL = 5e-04 or better.31 Table 9 considers several models and settings for Mand W, with
reference prices obtained by APROJ with N= 213. Based on the prescription given in Sec-
tion 3.6, for M= 12 the algorithm is initialized with N= 26and and grid width multiplier
L1= 12, while for M= 250 we set N= 27and L1= 16. The column labeled log2(N) reports
the final value after satisfying all error tolerances. The column labeled NL1is the number
30The results for FMM were obtained in MATLAB 7.4 on a personal computer with Intel(R) Core 2 Quad
Q6600, 2.4GHz, 4Gb RAM.
31We have selected to the parameters 1, 2, 3to attain an accuracy of TOL = 5e-04 or better. However,
these parameters, as well as the initial value of Nand L1can be increased if the desired accuracy is beyond
what is required in practice.

34 J. LARS KIRKBY
of iterations required in initialization Subroutine 1 before the error tolerances 1and 2were
satisfied (so NL1= 1 implies that the initial estimate of Nand ¯awere sufficient). Column
NL2is the number of loops in the main Algorithm 3 before the terminal valuation criteria
was satisfied.
Ideally, since the cost of Subroutine 1 is negligible, we would prefer it to identify insufficient
settings of Nand ¯aprior to entering Algorithm 3. Either way we see that these three
consistency checks are more than sufficient to achieve high accuracy. Column cpu(sec) reports
the time in seconds for the full procedure, which is generally fractions of a second, including
the cost of demeriting initial values for Nand ¯a. We conclude that APROJ is capable of
obtaining accurate prices at a very small computational cost when the algorithm, rather
than the user, determines the required values of Nand ¯aneeded to achieve the designated
tolerance.
7. Conclusions
In this article, we introduced a novel method, APROJ, for pricing arithmetic Asian options
driven by exponential Levy processes. This method is based on a recursive characteristic
function recovery by density projection, using frame duality on a shifted grid. Continuously
monitored Asian options are also priced in this framework. After an extensive investigation of
its theoretical behavior, numerical experiments demonstrate the rapid convergence of APROJ,
for both the linear and quadratic implementations. A variety of models and parameter settings
from the literature are considered.
Compared to recently developed breakthrough methods, APROJ achieves higher accuracy
at a fraction of the cost, consistently reducing cpu times by a factor of 10-100, and often much
greater. Moreover, the algorithm is able to accurately select the required parameter settings
needed to achieve a supplied tolerance. The computational cost of pricing and calculating
sensitivities of an important path-dependent derivate is now within milliseconds of the cost
associated with vanilla European options. The extension to discretely monitored barrier
options facilitates a similar cost reduction, and will appear in a subsequent work.
Appendix A. Proofs
Proof of Proposition 3.1. We proceed by induction where m= 1 follows from Y1=RM. Fix
m≥2 and assume (i) and (ii) hold for m−1. First we show finiteness of φZm−1(z) for any
fixed z=x+iη ∈ Dd. Consider the case of η∈(−d, 0) (finiteness for η∈[0, d) follows
immediately). Since φYm−1(x+iη) = RRei(x+iη)yfYm−1(y)dy < ∞, ie the integral exists and is
finite, ei(x+iη)yfYm−1(y) must be absolutely integrable in y, from which RRe−ηyfYm−1(y)dy <
∞. Note that
ZR|ei(x+iη) log(1+ey)fYm−1(y)|dy ≤e−ηlog(2) Z0
−∞
fYm−1(y)dy
+Z∞
0
e−ηlog(1+ey)fYm−1(y)dy.
To bound the second integral, note that ∃˜η∈(−d, η ), and τ > 0 s.th ∀y > τ ,−˜ηy >
−ηlog(1 + ey). Hence,
Z∞
τ
e−ηlog(1+ey)fYm−1(y)dy ≤Z∞
τ
e−˜ηyfYm−1(y)dy ≤ZR
e−ηy fYm−1(y)dy < ∞,
so φZm−1(z) exists and is finite ∀z∈ Dd. To prove continuity, fix any {zn} ∈ Ddwith
zn→z∈ Dd. Let G⊂ Ddbe a bounded open set containing the tail of {zn}, so ¯
G⊂ Dd.
With ¯η:= max{|η|:z=x+iη ∈¯
G}, note that for any z∈¯
Git holds
|eiz log(1+ey)fYm−1(y)| ≤ e¯ηlog(1+ey)fYm−1(y) = |ei¯zlog(1+ey)fYm−1(y)|,
where ¯z=x−i¯ηfor arbitrary x∈R, since log(1 + ey)≥0 for all y∈R. Hence
sup
z∈¯
G|φZm−1(z)|≤|φZm−1(¯z)|<∞,

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 35
Model ψL(ξ), ω =−ψL(−i),Param. Restrictions Cumulants
ψL(ξ) = −σ2
2ξ2c1=γ
BSM ω=−σ2
2c2=σ2
σ > 0, IL=Rc4= 0
ψL(ξ) = −σ2
2ξ2+λexp(iξµJ−σ2
J
2ξ2)−1c1=γ+λµJ
MJD ω=−σ2
2−λexp(µJ+σ2
J
2)−1c2=σ2+λ(µ2
J+σ2
J)
λ, σJ, σ > 0, IL=Rc4=λ(µ4
J+ 6σ2
Jµ2
J+ 3σ4
Jλ)
ψL(ξ) = CΓ(−Y)(M−iξ)Y−MY+ (G+iξ)Y−GYc1=γ+CΓ(1 −Y)(MY−1−GY−1)
CGMY ω=−CΓ(−Y)((M−1)Y−MY+ (G+ 1)Y−GY)c2=CΓ(2 −Y)(MY−2+GY−2)
C, G > 0, M > 1, Y∈(0,1) ∪(1,2), IL= [−M, G]c4=CΓ(4 −Y)(MY−4+GY−4)
ψL(ξ) = −δpα2−(β+iξ)2−pα2−β2c1=γ+δβ /pα2−β2
NIG ω=δpα2−(β+ 1)2−pα2−β2c2=δα2(α2−β2)−3/2
α, δ > 0, β∈(−α, α −1), IL= [β±α]c4= 3δα2(α2+ 4β2)(α2−β2)−7/2
ψL(ξ) = −σ2
2ξ2+λpη1
η1−iξ +(1−p)η2
η2+iξ −1c1=γ+λp
η1−λ(1−p)
η2
KOU ω=−σ2
2−λpη1
η1−1+(1−p)η2
η2+1 −1c2=σ2+ 2λp
η2
1+ 2λ(1−p)
η2
2
λ, σ > 0, p ∈[0,1], η1>1, η2>0, IL= (−η1, η2)c4= 24λ(p
η4
1+1−p
η4
2)
ψL(ξ) = −σ2
2ξ2−1
νlog 1−iνθξ +νσ2
V
2ξ2c1=γ+θ
VG ω=−σ2
2+1
νlog 1−νθ −νσ2
V
2c2=σ2+σ2
V+νθ2
ν, σV>0, σ≥0, IL=hθ
σ2
V±qθ2
σ4
V
+2
νσ2
Vic4= 3(σ4
Vν+ 2θ4ν3+ 4σ2
Vθ2ν2)
Figure 6. Symbols ψL(ξ), cumulants cnof log(St+1/St), parameter restrictions and strip
of analyticity ILfor tractable Levy processes. γ:= r−q−ψL(−i) = r−q+ω. Note that
E[log(St+1/St)] = c1=r−q+w+E[L(1)],and E[R∆t] = c1∆t.
where finiteness of ¯z∈¯
G⊂ Ddwas proved above, so by dominated convergence
lim
zn→zφZm−1(zn) = ZR
lim
zn→zexp (iznlog(1 + ey)) fYm−1(y)dy =φZm−1(z).
Analyticity is now proved as follows. Fix any positively oriented triangle Γ ∈ Dd. By Fubini’s
theorem ZΓ
φZm−1(z)dz =ZR
fYm−1(y)ZΓ
exp (iz log(1 + ey)) dzdy = 0,
where the final equality holds by Cauchy’s theorem. Hence, by Morera’s theorem, we conclude
that φZm−1(z) is analytic on Dd, and so too is φYm(z) = φR(z)φZm−1(z). The growth estimate
(ii) follows immediately from |φZm−1(ξ)| ≤ 1 for ξ∈R.
Proof of Lemma 3.1. Let [−λ, λ] be the support of ϕ. For a > 0, ξ∈[0,2πa),
a1/2F[ϕa,n](ξ)−¯
Ψ(ξ, n)≤a1/2Zϕa,n (y)eiξy −eiξ log(1+exp(y))dy
=Zλ
−λ
ϕ(y)eiξ(xn+y
a)1−eiξ(log(1+exp(xn+y
a))−(xn+y
a))
≤ |ξ|Zλ
−λ
ϕ(y)log 1 + exp xn+y
a−xn+y
ady
≤2πa (log(1 + exp(xn−1)) −xn−1)Zλ
−λ
ϕ(y)dy
= 2πa (log(1 + exp(xn−1)) −xn−1),

36 J. LARS KIRKBY
234567
log10 |err|
-10
-8
-6
-4
-2
0
2M=12, BSM
¯
P= 0
¯
P= 1
¯
P= 2
¯
P= 3
234567
log10 |err|
-10
-8
-6
-4
-2
0
2M=12, NIG
¯
P= 1
¯
P= 2
¯
P= 3
¯
P= 4
234567
log10 |err|
-10
-8
-6
-4
-2
0
2M=50, BSM
234567
log10 |err|
-10
-8
-6
-4
-2
0
2M=50, NIG
P= log2(a)
234567
log10 |err|
-10
-8
-6
-4
-2
0
2M=250, BSM
P= log2(a)
234567
log10 |err|
-10
-8
-6
-4
-2
0
2M=250, NIG
Figure 7. Convergence in ¯
Pof quadratic APROJ prices for BSM and NIG models (one
legend for each model). Parameters and reference values as in Table 8, strike W= 100.
where the next to last line follows since log(1+exp(x))−xis strictly decreasing. An asymptotic
expansion yields
log(1 + exp(xn−1)) −xn−1∼e−xn−1−e−2xn−1/2 + O(e−3xn−1),
and the result follows from F[ϕa,n](ξ). 
Proof of Proposition 2.1. We provide a proof here for linear case, with a bound on the term
C[1]
γ(φX). The more general case of a pth order basis is discussed in [26]. First define
(68) ha,n(ξ) := 12 sin2(ξ/2a)
(ξ/a)2(2 + cos(ξ/a)) exp(−ixnξ) := ha(ξ) exp(−ixnξ),
and ξj= (j−1)∆ξwhere ∆ξ= 2πa/N. We have that (˘
βX
a,n) := a1/2Ca,N ·¯
βX
a,n −hfX,eϕa,k i
satisfies
(˘
βX
a,n) = a−1/2
π<
∆ξ
N
X
j=1
νjφX(ξj)ha,n(ξj)−Z∞
0
φX(ξ)ha,n(ξ)dξ

=a−1/2
π<
∆ξ
∞
X
j=1
˜νjφX(ξj)ha,n (ξj)−Z∞
0
φX(ξ)ha,n(ξ)dξ
+∆ξ
∞
X
j=N
¯νjφX(ξj)ha,n (ξj)
:= a−1/2
π(trap(a, ¯a) + τa(X)) ,
where νj:= 1 −(δj,1+δj,N )/2, ˜νj= 1 −δj,1/2, and ¯νj= 1 −δj,N /2. To apply Theorem 3.2.1
in [35], we must show that the presence of ha(ξ) does not affect the integrand’s analyticity

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING 37
or the finiteness of the Hardy norm, both of which will follow if we can bound ha(ξ) in a
strip contained within Dd(note that Proposition 3.1 of [26] demonstrates the existence of a
bound). Consider b
eϕ(ξ) = 12 sin2(ξ/2)
ξ2(2+cos(ξ)) =ha(aξ), and let z=x+iy. Note first that
|2 + cos(x+iy)|=1
24 + e−y(cos(x) + isin(x)) + ey(cos(x)−isin(x))
=sinh2(y) sin2(x) + (cosh(y) cos(x) + 2)21/2.
For |y| ≤ 1/2, cosh(y)≤3/2, from which (cosh(y) cos(x) + 2)2≥1/4, and |2 + cos(x+iy)| ≥
1/2, uniformly in x. Similarly, for |y| ≤ 1,

sin x+iy
2
x+iy 
2
=sinh2y
2cos2x
2+ cosh2y
2sin2x
2
y2+x2≤1,
uniformly in x. Hence, ∀|y| ≤ 1/2, |b
eϕ(x+iy)| ≤ 24, so for |y| ≤ a/2, |b
eϕ((x+iy)/a)| ≤ 24,
∀x∈R. Thus, φX·ha,n ∈ H(Dγ) where γ=γ(a) = d∧a/2, and Cγ(φX) := kφX·ha,nkHγ≤
24kφXkHγ. For asufficiently large, the integrand is bounded within Dd(for any finite d > 0).
Moreover, since ¯
P > 1 + log2|¯µ|, it holds that ¯a/2>|¯µ|and so |xn| ≤ |¯µ|+ ¯a/2<¯a,
∀1≤n≤N. Thus by Theorem 3.2.1 in [35], trap(a, ¯a) converges exponentially in ¯a, according
to the bound given.
The truncation error depends on the tail behavior of φX. Since |ha,n(ξ)| ≤ 12a2
ξ2, and
|φX(ξ)|satisfies equation (7), the truncation error is bounded by
∆ξ
∞
X
j=N
¯νjφX(ξj)ha,n (ξj)≤12κa2Z∞
2πa
e−tc|ξ|ν
ξ2dξ ≤12κa2e−tc|2πa|νZ∞
2πa
1
ξ2dξ,
and the result follows after simplifying. 
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School of Industrial and Systems Engineering, Georgia Institute of Technology
E-mail address:jkirkby3@gatech.edu