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A General Framework for Pricing Asian Options Under
Markov Processes
Ning Cai, Yingda Song, Steven Kou
To cite this article:
Ning Cai, Yingda Song, Steven Kou (2015) A General Framework for Pricing Asian Options Under Markov Processes. Operations
Research 63(3):540-554. http://dx.doi.org/10.1287/opre.2015.1385
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OPERATIONS RESEARCH
Vol. 63, No. 3, May–June 2015, pp. 540–554
ISSN 0030-364X (print) — ISSN 1526-5463 (online)
http://dx.doi.org/10.1287/opre.2015.1385
© 2015 INFORMS
Downloaded from informs.org by [137.132.123.69] on 02 July 2015, at 02:16 . For personal use only, all rights reserved.
A General Framework for Pricing Asian
Options Under Markov Processes
Ning Cai
Department of Industrial Engineering and Logistics Management, The Hong Kong University of Science and Technology,
Kowloon, Hong Kong, [email protected]
Yingda Song
Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China, [email protected]
Steven Kou
Risk Management Institute and Department of Mathematics, National University of Singapore, Singapore 119077, [email protected]
A general framework is proposed for pricing both continuously and discretely monitored Asian options under onedimensional Markov processes. For each type (continuously monitored or discretely monitored), we derive the double
transform of the Asian option price in terms of the unique bounded solution to a related functional equation. In the special
case of continuous-time Markov chain (CTMC), the functional equation reduces to a linear system that can be solved
analytically via matrix inversion. Thus the Asian option prices under a one-dimensional Markov process can be obtained
by first constructing a CTMC to approximate the targeted Markov process model, and then computing the Asian option
prices under the approximate CTMC by numerically inverting the double transforms. Numerical experiments indicate that
our pricing method is accurate and fast under popular Markov process models, including the CIR model, the CEV model,
Merton’s jump diffusion model, the double-exponential jump diffusion model, the variance gamma model, and the CGMY
model.
Subject classifications: finance: asset pricing; probability: stochastic model applications.
Area of review: Financial Engineering.
History: Received December 2013; revision received October 2014; accepted March 2015. Published online in Articles in
Advance May 13, 2015.
1. Introduction
our paper provides a unified framework for pricing both
continuously and discretely monitored Asian options under
general one-dimensional Markov process models. Recently,
Novikov and Kordzakhia (2014) propose a unified approach
to deriving upper and lower bounds for Asian-type options,
including discretely and continuously monitored Asian
options and options on volume-weighted average price,
under general semimartingale models. Fusai and Kyriakou
(2014) develop a unified method to obtain very accurate
bounds for both discretely and continuously monitored
Asian option prices under a wide range of models, including exponential Lévy models, stochastic volatility models,
and the constant elasticity of variance (CEV) model. In
comparison, our paper is focused on deriving the prices
rather than the bounds, and our method is completely different from theirs.
The contribution of this paper is threefold.
1. Under general one-dimensional Markov processes, we
derive the double transforms of the Asian option prices,
either discretely or continuously monitored, in terms of the
unique bounded solutions to related functional equations;
see §§2 and 3.
2. In the special case of continuous-time Markov chain
(CTMC), we show that the functional equations reduce to
Asian options, whose payoffs are contingent on the arithmetic average of the underlying asset prices over a prespecified period, are among the most popular path-dependent
options that are actively traded in the financial markets.
The average of the underlying asset prices can be computed
either discretely, for which the average is taken over the
asset prices at discrete monitoring time points, or continuously, for which the average is calculated via the integration
of asset prices over the monitoring time period. The valuation of Asian options is challenging since the arithmetic
average usually does not have a simple distribution. There
is a vast literature on the pricing of Asian options under the
Black-Scholes model (BSM); we refer to Fu et al. (1999)
and Linetsky (2004) for detailed literature reviews.
Results on the valuation of Asian options beyond the
Black-Scholes framework are less extensive. Table 1 summarizes some papers on the pricing of either discretely or
continuously monitored Asian options under certain special one-dimensional Markov processes, which are closely
related to the study of our paper.
It can be seen that the existing literature is usually
focused on one type of Asian option (discrete or continuous) and on particular Markov processes. By contrast,
540
Cai, Song, and Kou: A General Framework for Pricing Asian Options
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Operations Research 63(3), pp. 540–554, © 2015 INFORMS
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Table 1.
Some literature on Asian option pricing under one-dimensional Markov process models beyond the BSM.
Discretely monitored
Continuously monitored
One-dimensional diffusions
Fusai et al. (2008) (CIR)
Cai et al. (2014) (general)
Fusai et al. (2008) (CIR)
Exponential Lévy processes
Fusai and Meucci (2008) (general)
Fusai et al. (2011) (general)
Cai and Kou (2012) (HEM)
This paper (general)
This paper (general)
One-dimensional Markov processes
Notes. Fusai et al. (2008) provide a single transform formula for the discretely monitored Asian option price under the Cox-Ingersoll-Ross (CIR) model.
Cai et al. (2014) derive closed-form analytical expansions for the discretely monitored Asian option prices under general one-dimensional diffusion models.
Fusai and Meucci (2008) and Fusai et al. (2011) develop recursive algorithms to evaluate discretely monitored Asian options under general exponential
Lévy models, by using convolution and maturity randomization, respectively. Cai and Kou (2012) obtain a closed-form double Laplace transform for the
continuously monitored Asian option price under the hyperexponential jump diffusion model (HEM).
linear systems that can be solved analytically via matrix
inverses, thanks to the Lévy-Desplanques theorem (see,
e.g., Horn and Johnson 1985, Corollary 5.6.17); see §4.
3. By first constructing a CTMC to approximate the targeted Markov process via the elegant technique in Mijatović and Pistorius (2013) and then numerically inverting
the double transforms related to the constructed CTMC,
we can compute the Asian option prices under general
one-dimensional Markov process models. Numerical results
demonstrate that our method is accurate and fast under popular Markov process models, including the CIR model, the
CEV model, Merton’s jump diffusion model (MJD), the
double-exponential jump diffusion model (DEJD), the variance gamma model (VG), and the Carr-German-Madan-Yor
(CGMY) model; see §5.
Pricing Asian options under a CTMC is quite different
from pricing other path-dependent options. For example,
to price barrier options, one needs to study the first passage times under a CTMC, which have analytical solutions
via a matrix obtained by deleting some rows and columns
in the transition rate matrix of the CTMC (see Mijatović
and Pistorius 2013). Unfortunately, it is not easy to find
such simple matrix operations for Asian options. We overcome this difficulty by working on the double transform of
the continuously monitored (respectively, discretely monitored) Asian option price with respect to the strike price
and the maturity (respectively, the number of monitoring
time points), in the spirit of Fu et al. (1999). Then we show
that the double transforms are the unique bounded solutions
to related functional equations, which can be solved analytically in the case of CTMCs, using the strictly diagonally
dominant matrix and the Lévy-Desplanques theorem.
The remainder of this paper is organized as follows.
Some preliminary results regarding the double transforms
of Asian option prices are given in §2. In §3 we derive
the double transforms for Asian option prices under general
one-dimensional Markov processes in terms of the unique
bounded solutions to related functional equations. Then we
apply the general results to the CTMC in §4 and solve the
related functional equations explicitly. We summarize our
pricing method in two steps and provide numerical results
in §5. Error analysis of our pricing method is conducted
in §6. For error analysis related to Asian option pricing, we
refer to, e.g., Zhang and Oosterlee (2013) about a Fourier
cosine expansion method for Asian option pricing under
exponential Lévy models.
2. Preliminary Results
For simplicity, we consider only Asian call options, while
the put options can be dealt with similarly. Throughout
this paper, we shall work under the risk-neutral probability space 4ì1 F1 8Ft 9t¾0 1 5. This implies that for t ¾ 0,
Ɛx 6St 7 = xe4r−d5t , where r is the risk-free rate, d is the dividend rate, and Ɛx 6 · 7 denotes the expectation taken under
the risk-neutral measure given the initial price S0 = x. The
price of a continuously monitored Asian call option at time
0 is given by
+ 1
Vc 4T 1 K3 x5 = e−rT Ɛx
AT − K
1
T
Z T
with AT 2=
St dt1
0
where T is the maturity and K the strike price. Similarly,
the price of a discretely monitored Asian call option at time
0 is given by
+ 1
−rT x
B −K
1
Vd 4n1 K3 x5 = e Ɛ
n+1 n
with Bn 2=
n
X
Siã 1
i=0
where ã is the length of the monitoring time interval and
the n+1 monitoring dates are assumed to be equally spaced
with nã = T . For convenience, define
vc 4t1 k3 x5 = Ɛx 64At − k5+ 7 and
vd 4n1 k3 x5 = Ɛx 64Bn − k5+ 70
(1)
Note that Vc 4T 1 K3 x5 = 4e−rT /T 5vc 4T 1 TK3 x5 and
Vd 4n1 K3 x5 = 4e−rT /4n + 155vd 4n1 4n + 15K3 x5. Then the
pricing problems of Asian options reduce to the computation of vc 4t1 k3 x5 and vd 4n1 k3 x5.
The following proposition provides a model-free result
for the double transforms of vc 4t1 k3 x5 with respect to t
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542
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
and k and vd 4n1 k3 x5 with respect to n and k. Fu et al.
(1999) use the same transform in the pricing of continuously monitored Asian options under the BSM. Here we
extend it to general Markov processes for both continuously
and discretely monitored Asian options.
(ii) The proof is similar to that for (i) and is thus
omitted. ƒ
Proposition 1 (Double Transforms of Asian Option
Prices). (i) (Discretely Monitored Asian Options)
Let Ld 4z1 ˆ3 x5 be the Z-Laplace transform
k3 x5
P of nvRd 4n1
+ˆ −ˆk
with respect to n and k, i.e., Ld 4z1 ˆ3 x5 = +ˆ
e
·
n=0 z
0
vd 4n1 k3 x5 dk. Then for any complex z and ˆ such that —z— <
min8e−4r−d5ã 1 19 and Re4ˆ5 > 0, we have
1
1
x
Ld 4z1ˆ3x5 = 2 l4z1ˆ3x5− 2
+
1
ˆ
ˆ 41−z5 ˆ41−z541−zerã 5
where
+ˆ
X
l4z1 ˆ3 x5 2=
zn Ex 6e−ˆBn 70
(2)
According to Proposition 1, to derive the double transforms
of discretely and continuously monitored Asian option
prices, it suffices to compute the functions l4z1 ˆ3 x5 and
m4Œ1 ˆ3 x5. In this section, we will show that for onedimensional time-homogeneous Markov processes, the two
functions l4z1 ˆ3 x5 and m4Œ1 ˆ3 x5 are the unique bounded
solutions to certain functional equations.
Suppose 8St 1 t ¾ 09 is a nonnegative one-dimensional
time-homogeneous Markov process. For continuously monitored Asian options, we require 8St 9 to have the infinitesimal generator G (the existence of infinitesimal generator
can be ensured by the Feller property; see, e.g., Ethier and
Kurtz 2005, Chap. 4). Infinitesimal generators of some popular models are listed in Table 2.
n=0
(ii) (Continuously Monitored Asian Options) Let
Lc 4Œ1 ˆ3 x5 be the double-Laplace transform
of Rvc 4t1 k3 x5
R +ˆ
+ˆ −Œt
with respect to t and k, i.e., Lc 4Œ1 ˆ3 x5 = 0
e
·
0
−ˆk
e vc 4t1 k3 x5 dk dt. Then for any complex Œ and ˆ such
that Re4Œ5 > max8r − d1 09 and Re4ˆ5 > 0, we have
1
1
x
1
Lc 4Œ1 ˆ3 x5 = 2 m4Œ1 ˆ3 x5 − 2 +
ˆ
ˆ Œ ˆŒ4Œ − r5
where
Z +ˆ
m4Œ1 ˆ3 x5 2=
e−Œt Ex 6e−ˆAt 7 dt0
(3)
0
Proof. (i) First, the function l4z1 ˆ3 x5 is well defined
for any —z— < 1 and Re4ˆ5 > 0 because —l4z1 ˆ3 x5— ¶
P
ˆ
n
n=0 —z— < ˆ. By Fubini’s theorem,
Z B
ˆ
n
X
Ld 4z1 ˆ3 x5 =
zn Ɛx
e−ˆk 4Bn − k5 dk
0
n=0
=
ˆ
X
n=0
=
zn Ɛx
Bn 1 − e−ˆBn
−
ˆ
ˆ2
ˆ
1 X
zn 4Ɛx 6e−ˆBn 7 + ˆƐx 6Bn 7 − 15
ˆ 2 n=0
ˆ
1
1
1X
+
zn Ɛx 6Bn 70
= 2 l4z1 ˆ3 x5 − 2
ˆ
ˆ 41 − z5 ˆ n=0
Note that under the risk-neutral measure,
n
X
1 − e4n+154r−d5ã
Ɛx 6Bn 7 = x e4r−d5ti = x
0
1 − e4r−d5ã
i=0
Then it follows that for any complex z and ˆ such that
—z— < e−4r−d5ã and Re4ˆ5 > 0,
1
1
Ld 4z1 ˆ3 x5 = 2 l4z1 ˆ3 x5 − 2
ˆ
ˆ 41 − z5
+
=
ˆ
X
x
zn 61 − e4n+154r−d5ã 7
ˆ41 − e4r−d5ã 5 n=0
1
1
l4z1 ˆ3 x5 − 2
2
ˆ
ˆ 41 − z5
x
+
0
ˆ41 − z541 − ze4r−d5ã 5
3. Functional Equations for
Markov Processes
Theorem 1 (Functional Equations: Uniqueness and
Stochastic Representations). (i) (Discretely Monitored Asian Options) Given ã > 0, for any z and ˆ such
that —z— < 1 and Re4ˆ5 > 0, if there exists a function f 4x5
that solves the functional equation
eˆx f 4x5 − zƐx 6f 4Sã 57 = 11
(4)
and is bounded, i.e., supx∈601 ˆ5 —f 4x5— ¶ C < ˆ for some
constant C > 0, then we must have f 4x5 = l4z1 ˆ3 x5, where
l4z1 ˆ3 x5 is defined in (2).
(ii) (Continuously Monitored Asian Options) Furthermore, assume that 8St 1 t ¾ 09 has right-continuousand-left-limit paths and the infinitesimal generator G, and
Ɛx 6St1+… 7 < ˆ for some … > 0. For any Œ and ˆ such that
Re4Œ5 > 0 and Re4ˆ5 > 0, if there exists a function f 4x5
that solves the functional equation
4ˆx + Œ − G5f 4x5 = 11
(5)
and is bounded, i.e., supx∈601 ˆ5 —f 4x5— ¶ C < ˆ for some
constant C > 0, then we must have f 4x5 = m4Œ1 ˆ3 x5,
where m4Œ1 ˆ3 x5 is defined in (3).
Proof. See Appendix A.
ƒ
4. Analytical Solutions to the Functional
Equations Under Continuous-Time
Markov Chains
It turns out that the two functional Equations (4) and (5) in
Theorem 1 can be solved analytically for a special family
of Markov processes, i.e., the CTMCs. Consider a nonnegative CTMC 8St 1 t ¾ 09 with finite state space 8x1 1 0 0 0 1 xN 9,
whose transition probability matrix P4t5 = 4pij 4t55N ×N and
transition rate matrix G = 4qij 5N ×N are defined as
pij 4t5 = 4St+u = xj — Su = xi 5 and
qij = pij0 4051
1 ¶ i1 j ¶ N 1 t1 u ¾ 00
Cai, Song, and Kou: A General Framework for Pricing Asian Options
543
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
Table 2.
The infinitesimal generators of some popular models.
Gf 4x5
Model
BSM
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CIR
CEV
DEJD
MJD
VG
1
4r − d5xf 0 4x5 + ‘ 2 x2 f 00 4x5
2
1
4r − d5xf 0 4x5 + ‘ 2 xf 00 4x5
2
1
4r − d5xf 0 4x5 + ‘ 2 x241+‚5 f 00 4x5
2
Z
1 2 2 00
0
4r − d5xf 4x5 + ‘ x f 4x5 + 6f 4xey 5 − f 4x5 − xf 0 4x54ey − 157de 4dy5
2
Z
1
4r − d5xf 0 4x5 + ‘ 2 x2 f 00 4x5 + 6f 4xey 5 − f 4x5 − xf 0 4x54ey − 157mj 4dy5
2
Z
4r − d5xf 0 4x5 + 6f 4xey 5 − f 4x5 − xf 0 4x54ey − 157vg 4dy5
CGMY
4r − d5xf 0 4x5 +
Z
6f 4xey 5 − f 4x5 − xf 0 4x54ey − 157cgmy 4dy5
Notes. For the DEJD model, de 4dy5 = ‹4p‡e−‡y I8y¾09 +41−p5ˆeˆy I8y<09 5dy, with 0 ¶ p ¶ 1 and ‡ > 11ˆ1‹ > 0; for the
√
2 −44y−52 /2„2 5 dy, with „1‹ > 0; for the VG model,  4dy5 = 41/4v—y—55exp44ˆ/„2 5y −
MJD
vg
p model, mj 4dy5 = 4‹/ 2„ 5e
2
2
4 2/v +ˆ /„ /„5—y—5dy, with „1 v > 0; for the CGMY model, cgmy 4dy5 = 4C/—y—1+Y 5 exp444G − M5/25y −
44G + M5/25—y—5dy, with C > 0, G1 M ¾ 0 and Y < 2.
Define x = 4x1 1 0 0 0 1 xN 5T , D = 4dij 5N ×N as a diagonal
matrix with djj = xj for j = 11 0 0 0 1 N , I = IN as the identity
matrix, and 1 as an N × 1 constant vector with all entries
equal to 1. Theorem 1 immediately implies the following
result.
Corollary 1 (Uniqueness and Stochastic Representation Under CTMCs). (i) (Discretely Monitored
Asian Options) Given ã > 0, for any z and ˆ such that
—z— < 1 and Re4ˆ5 > 0, if there exists an N × 1 vector f that
solves the linear system
4eˆD − zP4ã55 · f = 11
(6)
then we have f = l4z1ˆ3x5 2= 4l4z1ˆ3x1 510001l4z1ˆ3xN 55T ,
where l4z1ˆ3x5 is defined in (2) for the CTMC 8St 9.
(ii) (Continuously Monitored Asian Options) For any Œ
and ˆ such that Re4Œ5 > 0 and Re4ˆ5 > 0, if there exists
an N × 1 vector f that solves the linear system
4ˆD + ŒI − G5 · f = 11
then f = m4Œ1ˆ3x5 2= 4m4Œ1ˆ3x1 510001m4Œ1ˆ3xN 55 , where
m4Œ1ˆ3x5 is defined in (3) for the CTMC 8St 9.
A square matrix A = 4aij 5N ×N with complex entries is
said to be strictly diagonally dominant, if
N
X
—ajk —
Proof. The two matrices eˆD − zP4ã5 and ˆD + ŒI − G
are both strictly diagonally dominant because for all j =
11 0 0 0 1 N , we have
—eˆdjj −zpjj 4ã5— ¾ —eˆdjj —−—z—pjj 4ã5 ¾ 1−—z—pjj 4ã5
> —z—41−pjj 4ã55 = —z—
N
X
pjk 4ã5 and
k=1
k6=j
—ˆdjj +Œ−qjj — ¾ Re4ˆdjj +Œ−qjj 5
(7)
T
—ajj — >
Theorem 2 (Analytical Solutions for l4z1 ˆ3 x5 and
m4Œ1 ˆ3 x5 Under CTMCs). (i) (Discretely Monitored
Asian Options) For any z and ˆ such that —z— < 1
and Re4ˆ5 > 0, we have l4z1 ˆ3 x5 2= 4l4z1 ˆ3 x1 51 0 0 0 1
l4z1 ˆ3 xN 55T = 4eˆD − zP4ã55−1 · 1.
(ii) (Continuously Monitored Asian Options) For
any Œ and ˆ such that Re4Œ5 > 0 and Re4ˆ5 > 0,
we have m4Œ1 ˆ3 x5 2= 4m4Œ1 ˆ3 x1 51 0 0 0 1 m4Œ1 ˆ3 xN 55T =
4ˆD + ŒI − G5−1 · 1.
for all j = 11 0 0 0 1 N 0
k=1
k6=j
The Lévy-Desplanques theorem (see, e.g., Horn and Johnson 1985, Corollary 5.6.17) states that if A is strictly diagonally dominant, then it is invertible. Applying the LévyDesplanques theorem, we can solve (6) and (7) by inverting
the matrices.
= Re4ˆ5djj +Re4Œ5−qjj > −qjj =
N
X
qjk 0
k=1
k6=j
Therefore, both of them are invertible thanks to the LévyDesplanques theorem. Hence the solutions to (6) and (7)
are f = 4eˆD − zP4ã55−1 · 1 and f = 4ˆD + ŒI − G5−1 1,
respectively. By Corollary 1, we complete the proof immediately. ƒ
5. Pricing Asian Options by
CTMC-Approximation
According to Proposition 1 and Theorem 2, if we can construct a CTMC properly to approximate a general onedimensional time-homogeneous Markov process, then the
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Operations Research 63(3), pp. 540–554, © 2015 INFORMS
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double transform inversion-based analytical solutions to
Asian option prices under this constructed CTMC can serve
as approximations for Asian option prices under the original Markov process model. Mijatović and Pistorius (2013)
develop an elegant technique to construct a sequence of
CTMCs that weakly converge to a one-dimensional Markov
process with generator G given by
Gf 4x5 = GD f 4x5 + GJ f 4x51
where
GD f 4x5 = 4r − d5xf 0 4x5 + 21 ‘4x52 x2 f 00 4x5 and
Z ˆ
GJ f 4x5 =
6f 4x41 + y55 − f 4x5 − f 0 4x5xy74x1 dy50
−1
Here
R ˆ 2for any x ¾ 0, 4x1 dy5 is a Lévy measure such that
y 4x1 dy5 < ˆ.
−1
More precisely, based on the technique of Mijatović
and Pistorius (2013), we can use a CTMC with state
space 8x1 1 0 0 0 1 xN 9 and transition rate matrix G = åD + åJ
to approximate the targeted Markov process. Here åJ =
4åJij 5N ×N is constructed explicitly by
åJij =


01
i = 1 or N 1





Z i 4j5




4x1 dy51 i =
6 11 N and i 6= j1
i 4j−15



N

X


−
åJ 1


 j=1 ij

i 6= 11 N and i = j1
j6=i
where i 405 = −1, i 4N 5 = +ˆ, and i 4j5 is any number
between xj /xi − 1 and xj+1 /xi − 1 for j = 11 0 0 0 1 N − 1.
Moreover, åD = 4åDij 5N ×N is a tridiagonal matrix determined implicitly by
N
X
åDij = 01
j=1
N
X
åDij 4xj − xi 5 = 4r − d5xi −
j=1
N
X
N
X
åJij 4xj − xi 51
j=1
åDij 4xj − xi 52 = xi2 ‘4xi 52 +
j=1
−
N
X
Z
ˆ
y 2 4x1 dy5
−1
åJij 4xj − xi 52 1
j=1
when i 6= 11 N , and åDij = 0 when i = 1 or N . Then G
and P4ã5 = Exp8ãG9 will be used to calculate l4z1 ˆ3 x5
and m4Œ1 ˆ3 x5 in Theorem 2, where Exp8 · 9 denotes the
matrix exponential. Notice that when the original Markov
process is a diffusion, G ≡ GD is simply a tridiagonal matrix. In our numerical implementation, we choose
x1 = 10−3 S0 and xN = 4S0 , and generate other states
8x2 1 0 0 0 1 xN −1 9 based on the procedure in Mijatović and Pistorius (2013). More specifically, we set xk = S0 + sinh441 −
4k − 15/4N /2 − 155 arsinh4x1 − S0 55 for 2 ¶ k ¶ N /2,
and xk = S0 + sinh444k − N /25/4N /255 arsinh4xN − S0 55
for N /2 < k ¶ N − 1. Intuitively, these N states are
nonuniformly distributed over 6x1 1 xN 7 and are placed more
densely around S0 .
Since the Asian option payoffs are continuous with
respect to the sample paths (see, e.g., Prigent 2003, §1.2.2),
we can show that the Asian option prices under the constructed CTMC converge to the prices under the targeted
Markov process models, thanks to the continuous mapping
theorem and the dominated convergence theorem.
In summary, our pricing algorithm consists of two steps:
Step 1. Construct a CTMC to approximate the targeted
Markov process model via the technique in Mijatović and
Pistorius (2013).
Step 2. Invert the analytical double transforms of Asian
option prices under the approximate CTMC numerically to
obtain the approximations for Asian option prices under the
original Markov process model. For the numerical inversion
of the double transforms, we use the algorithms in Choudhury et al. (1994); the details are given in Appendix B.
In the following subsections, numerical results will be
provided to illustrate the performance of our method under
different models, including the CIR model, the CEV model,
the DEJD model, the MJD model, the VG model, and the
CGMY model. These models are selected as representatives for different types of Markov process models, namely,
diffusion models (CIR and CEV), jump diffusion models
(DEJD and MJD), and pure jump models (VG and CGMY).
Our method appears to be accurate and fast under these
models. All the computations are conducted using MATLAB 7 on a desktop with an Intel Core i7 3.40 GHZ
processor.
5.1. The CIR Model
Table 3 provides numerical prices (denoted by “CTMC”)
for both discretely and continuously monitored Asian
option prices under the CIR model obtained via our
method. The parameter settings are the same as in Fusai
et al. (2008) for the commodity market, and the results
obtained from their analytical solution are used as benchmarks (denoted by “Fusai et al.”). As shown in Table 3,
our method is highly accurate for both discretely monitored Asian options with different monitoring frequencies
(n = 12, 25, 50, 100 and 250) and continuously monitored Asian options (n = +ˆ). The average (respectively,
the maximum) absolute error is approximately 0000020
(respectively, 0.00095) and the average (respectively, the
maximum) relative error is approximately 0.11% (respectively, 0.44%). Besides, we report that the CPU times to
generate one numerical price via our method are about 0.2,
0.4, 0.7, 1.5, and 3.7 seconds for n = 12, 25, 50, 100, and
250, respectively, and about 0.12 seconds for continuously
monitored Asian options (i.e., n = +ˆ).
Cai, Song, and Kou: A General Framework for Pricing Asian Options
545
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
Table 3.
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K
Pricing Asian options under the CIR model.
Fusai et al.
CTMC
0.90
0.95
1.00
1.05
1.10
0021279
0018659
0016282
0014140
0012223
0021257
0018638
0016264
0014126
0012213
0.90
0.95
1.00
1.05
1.10
0021501
0018883
0016505
0014359
0012434
0.90
0.95
1.00
1.05
1.10
0021560
0018943
0016565
0014418
0012492
Abs. err.
Rel. err. (%)
Fusai et al.
CTMC
0010
0011
0011
0010
0009
0021428
0018810
0016432
0014287
0012365
0021406
0018789
0016414
0014273
0012355
0044
0011
0011
0010
0008
0021538
0018920
0016542
0014395
0012470
0010
0011
0011
0010
0009
0021575
0018958
0016580
0014433
0012506
n = 12
−0000022
−0000021
−0000018
−0000014
−0000010
n = 50
0021406
−0000095
0018862
−0000021
0016487
−0000018
0014344
−0000015
0012424
−0000010
n = 250
0021537
−0000023
0018922
−0000021
0016547
−0000018
0014403
−0000015
0012481
−0000011
Abs. err.
n = 25
−0000022
−0000021
−0000018
−0000014
−0000010
n = 100
0021515
−0000023
0018899
−0000021
0016524
−0000018
0014381
−0000014
0012460
−0000010
n = +ˆ
0021552
−0000023
0018937
−0000021
0016562
−0000018
0014418
−0000015
0012496
−0000010
Rel. err. (%)
0010
0011
0011
0010
0008
0011
0011
0011
0010
0008
0010
0011
0011
0010
0008
Notes. Pricing Asian options under the CIR model via our CTMC approximation with N = 50. The parameter settings are the same as in Fusai et al.
(2008), i.e., r = 0004, d = 0, ‘ = 007, S0 = 1, and T = 1. The columns “Fusai et al.” are taken from Fusai et al. (2008) and are obtained from their
analytical solution. The CPU times to generate one numerical price via our method are about 0.2, 0.4, 0.7, 1.5, and 3.7 seconds for n = 12, 25, 50, 100,
and 250, respectively, and about 0.12 seconds for n = +ˆ, i.e., for continuously monitored Asian options.
5.2. The CEV Model
Table 4 gives numerical prices (denoted by “CTMC”) for
both discretely and continuously monitored Asian options
under the CEV model obtained via our method (we refer
to Davydov and Linetsky 2001 for analytical pricing of
Table 4.
other path-dependent options such as continuously monitored lookback and barrier options. Also see Sesana et al.
2014 for a recursive algorithm for the pricing of discretely
monitored Asian options under the CEV model). Here
we consider three cases with ‚ = 00251 −0025 and −005,
Pricing Asian options under the CEV model.
(I) Discretely monitored Asian options
under the CEV model (n = 250)
K
Cai et al.
80
90
100
110
120
21060167
13015550
6084034
3007180
1022841
80
90
100
110
120
21067122
13026903
6084853
2092962
1004072
80
90
100
110
120
21071428
13032877
6085365
2086119
0095542
CTMC
Abs. err.
‚ = 0025
21060974
0000807
13015548
−0000002
6082619
−0001415
3005691
−0001489
1022497
−0000344
‚ = −0025
21067979
0000857
13026768
−0000135
6083407
−0001446
2091597
−0001365
1004152
0000080
‚ = −005
21072237
0000809
13032675
−0000202
6083904
−0001461
2084823
−0001296
0095803
0000261
(II) Continuously monitored Asian options
under the CEV model
Rel. err. (%)
MC value
Std. err.
0004
0000
0021
0048
0028
21059408
13015109
6083859
3007333
1023175
0000468
0000425
0000340
0000239
0000154
0004
0001
0021
0047
0008
21066618
13026741
6085150
2093166
1004453
0000464
0000417
0000327
0000221
0000131
0004
0002
0021
0045
0027
21071118
13032850
6085984
2086666
0095995
0000465
0000416
0000324
0000215
0000122
CTMC
‚ = 0025
21061076
13015931
6083128
3006138
1022762
‚ = −0025
21068104
13027147
6083920
2092049
1004429
‚ = −005
21072370
13033052
6084420
2085276
0096084
Abs. err.
Rel. err. (%)
0001667
0000822
−0000731
−0001195
−0000413
0008
0006
0011
0039
0034
0001486
0000407
−0001230
−0001116
−0000025
0007
0003
0018
0038
0002
0001252
0000202
−0001564
−0001390
0000089
0006
0002
0023
0048
0009
Notes. Part (I) compares our numerical prices via CTMC approximation (N = 50) with asymptotic expansion results in Cai et al. (2014) for discretely
monitored Asian options under the CEV model. It takes around 3.6 seconds to generate one price via our method. Part (II) compares our numerical prices
via CTMC approximation (N = 50) with Monte Carlo simulation estimates (denoted by “MC value.” The sample size is 110001000 and the number of
time steps is 101000) for continuously monitored Asian options under the CEV model. It takes around 0.12 seconds to generate one price via our method.
The unvarying parameters are r = 0005, d = 0, S0 = 100, T = 1, and ‘S0‚ = 0025.
Cai, Song, and Kou: A General Framework for Pricing Asian Options
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546
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
respectively. Part (I) in Table 4 provides our numerical
results for daily monitored (n = 250) Asian option prices
as well as the benchmark prices (denoted by “Cai et al.”)
that are taken from Cai et al. (2014) and obtained from
asymptotic expansion. It can be seen that our method
is very accurate with average (respectively, maximum)
absolute error 0.00798 (respectively, 0.01489) and average
(respectively, maximum) relative error 0.19% (respectively,
0.48%). Besides, it takes around 3.6 seconds to generate one price via our method. For continuously monitored
Asian options, we construct benchmarks using Monte Carlo
simulation with 1,000,000 sample paths and 10,000 time
steps. From part (II) of Table 4 we can see that our method
is also very accurate with average (respectively, maximum)
absolute error 0.00906 (respectively, 0.01667) and average
(respectively, maximum) relative error 0.17% (respectively,
0.48%). Besides, it takes around 0.12 seconds to generate
one price via our method.
Table 5.
5.3. The Double-Exponential Jump
Diffusion Model
Part (I) in Table 5 shows the comparison of discretely
monitored Asian option prices obtained via our method
(denoted by “CTMC”) and those obtained by the recursive algorithm in Fusai and Meucci (2008) under the DEJD
model (see Kou 2002). We can see that our method is quite
accurate for different monitoring frequencies (n = 12, 50,
and 250) because the average (respectively, the maximum)
absolute error is around 0.00325 (respectively, 0.00458)
and the average (respectively, maximum) relative error is
around 0.09% (respectively, 0.15%). Besides, it takes about
0.2, 0.7, and 3.6 seconds to generate one price via our
method for n = 12, 50, and 250, respectively. For continuously monitored Asian options, we compare our numerical
results with those obtained via transform-based analytical
solution in Cai and Kou (2012); see part (II) in Table 5. It
can be seen that the average (respectively, the maximum)
Pricing Asian options under the DEJD model.
(I) Discretely monitored Asian options under the DEJD model
n
K
Fusai and Meucci
CTMC
Abs. err.
Rel. err. (%)
12
90
100
110
90
100
110
90
100
110
12071236
5001712
1004142
12074369
5005809
1006878
12075241
5006949
1007646
12070857
5001254
1003988
12074016
5005358
1006725
12074875
5006491
1007489
−0000379
−0000458
−0000154
−0000353
−0000451
−0000153
−0000366
−0000458
−0000157
0003
0009
0015
0003
0009
0014
0003
0009
0015
50
250
(II) Continuously monitored Asian options under the DEJD model
K
Cai and Kou
90
95
100
105
110
13047952
9016588
5038761
2072681
1028264
90
95
100
105
110
14017380
10053795
7048805
5009001
3032061
90
95
100
105
110
16081490
13087995
11033257
9016131
7034063
CTMC
Abs. err.
‘ = 0005
13046823
−0001129
9018472
0001884
5037399
−0001362
2071628
−0001053
1030224
0001960
‘ = 002
14017589
0000209
10053824
0000029
7048621
−0000184
5008708
−0000293
3031802
−0000259
‘ = 004
16081130
−0000360
13087460
−0000535
11032581
−0000676
9015366
−0000765
7033266
−0000797
Rel. err. (%)
Cai and Kou
0008
0021
0025
0039
1053
13055964
9041962
5091537
3035071
1074896
0001
0000
0002
0006
0008
15033688
12010723
9035336
7008059
5026109
0002
0004
0006
0008
0011
18046259
15075006
13036027
11027716
9047826
CTMC
Abs. err.
‘ = 001
13056418
0000454
9042931
0000969
5091365
−0000172
3034830
−0000241
1075431
0000535
‘ = 003
15033545
−0000143
12010414
−0000309
9034883
−0000453
7007520
−0000539
5025561
−0000548
‘ = 005
18045288
−0000971
15073859
−0001147
13034737
−0001290
11026330
−0001386
9046389
−0001437
Rel. err. (%)
0003
0010
0003
0007
0031
0001
0003
0005
0008
0010
0005
0007
0010
0012
0015
Notes. Part (I) compares our numerical prices via CTMC approximation (N = 50) with those obtained via the recursive algorithm in Fusai and Meucci
(2008) for discretely monitored Asian options under the DEJD model. The unvarying parameters are the same as in Fusai and Meucci (2008), i.e.,
r = 000367, d = 0, S0 = 100, T = 1, ‘ = 00120381, p = 002071, ‡ = 9065997, ˆ = 3013868, and ‹ = 00330966. It takes about 0.2, 0.7, and 3.6 seconds to
generate one price via our method for n = 12, 50, and 250, respectively. Part (II) compares our numerical prices via CTMC approximation (N = 100) with
the double-Laplace inversion results in Cai and Kou (2012) for continuously monitored Asian options under the DEJD model. The unvarying parameters
are r = 0009, d = 0, S0 = 100, T = 1, p = 006, ‡ = ˆ = 25, and ‹ = 5. It takes about 1.1 seconds to obtain one price via our method.
Cai, Song, and Kou: A General Framework for Pricing Asian Options
547
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
absolute error is around 0.00736 (respectively, 0.01960)
and the average (respectively, maximum) relative error is
around 0.14% (respectively, 1.53%). Besides, it takes about
1.1 seconds to obtain one price via our method.
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5.4. Merton’s Jump Diffusion Model
In Table 6, we compare in part (I) the numerical results
of discretely monitored Asian option prices obtained via
our method with those obtained via the recursive method
in Fusai and Meucci (2008), and we compare in part (II)
our numerical results for continuously monitored Asian
option prices with Monte Carlo simulation estimates under
the MJD model (see Merton 1976). It can be seen that
our method is quite accurate in both cases. Indeed, in
the discrete monitoring case, the average (respectively,
maximum) absolute error is around 0.00417 (respectively,
0.00592) and the average (respectively, maximum) relative
error is around 0.12% (respectively, 0.21%). In the continuous monitoring case, the average (respectively, maximum)
absolute error is around 0.00235 (respectively, 0.00822)
and the average (respectively, maximum) relative error is
around 0.27% (respectively, 0.76%). In addition, the CPU
Table 6.
Pricing Asian options under the MJD model.
(I) Discretely monitored Asian options
under the MJD model
n
K
Fusai and
Meucci
CTMC
Abs. err.
Rel.
err. (%)
12
90
100
110
90
100
110
90
100
110
12071066
5001127
1005162
12074093
5005246
1007959
12074917
5006381
1008740
12070620
5000539
1004941
12073659
5004654
1007736
12074485
5005790
1008515
−0000446
−0000588
−0000221
−0000434
−0000592
−0000223
−0000432
−0000591
−0000225
0004
0012
0021
0003
0012
0021
0003
0012
0021
50
250
Table 7.
Case no.
1
2
3
4
90
100
110
MC
value
Std. err.
CTMC
Abs. err.
Rel.
err. (%)
12074587
5005974
1008413
0000371
0000399
0000280
12074705
5005740
1009235
0000117
−0000235
0000822
0001
0005
0076
Notes. Part (I) compares our numerical prices via CTMC approximation
(N = 50) with those obtained via the recursive algorithm in Fusai and
Meucci (2008) for discretely monitored Asian options under the MJD
model. It takes about 0.2, 0.7, and 3.6 seconds to generate one price
via our method for n = 12, 50, and 250, respectively. Part (II) compares
our numerical prices via CTMC approximation (N = 50) with Monte
Carlo simulation estimates (denoted by “MC value.” The sample size is
110001000 and the number of time steps is 101000) for continuously monitored Asian options under the MJD model. It takes about 0.3 seconds to
obtain one price via our method. For both parts, the unvarying parameters
are the same as in Fusai and Meucci (2008) and obtained from calibration,
i.e., r = 000367, d = 0, S0 = 100, T = 1, ‘ = 00126349,  = −00390078,
„ = 00338796, and ‹ = 00174814.
r
d
„

ˆ
000533
000536
000549
000541
00011
00012
00011
00012
0017875
0018500
0019071
0020722
0013317
0022460
0049083
0050215
−0030649
−0028837
−0028113
−0022898
Note. These parameter settings are obtained from calibration for S&P 500
on June 30, 1999 by Hirsa and Madan (2004).
times for producing one numerical result via our method
are 0.2 seconds for n = 12, 0.7 seconds for n = 50, 3.6 seconds for n = 250, and 0.3 seconds for continuously monitored Asian options.
5.5. The Variance Gamma Model
In this subsection, we consider the VG model, a pure jump
model, under the four cases of parameter settings calibrated for S&P 500 on June 30, 1999 by Hirsa and Madan
(2004) except T = 1 (see Table 7). Table 8 provides the
numerical results for both discretely and continuously monitored Asian options obtained via our method as well as the
benchmark prices computed by Monte Carlo simulation. We
can see that the average (respectively, maximum) absolute
error is around 0.0099 (respectively, 0.0187) and the average (respectively, maximum) relative error is around 0.17%
(respectively, 0.30%). Besides, the CPU times for producing one numerical result via our method are 0.2 seconds for
Table 8.
n
12
50
(II) Continuously monitored Asian options
under the MJD model
K
Test cases of parameter settings for Asian
options under the VG model.
250
ˆ
Pricing Asian options under the VG model.
Case
no.
MC
values
Std.
err.
CTMC
Abs.
err.
Rel.
err. (%)
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
505193
507773
603873
601591
505740
508455
604541
602146
505975
508606
604752
602246
505932
508585
604687
602289
000034
000036
000039
000040
000034
000037
000040
000040
000034
000037
000040
000041
000034
000037
000040
000041
505068
507692
603802
601404
505655
508301
604436
602011
505815
508467
604610
602177
505869
508533
604673
602295
−000125
−000082
−000071
−000187
−000085
−000154
−000105
−000135
−000160
−000139
−000143
−000069
−000063
−000052
−000014
000007
0023
0014
0011
0030
0015
0026
0016
0022
0029
0024
0022
0011
0011
0009
0002
0001
Notes. Pricing Asian options under the VG model via CTMC approximation (N = 50). The columns “MC value” and “Std. err.” denote the
Monte Carlo simulation estimates and associated standard errors obtained
by simulating 110001000 sample paths and using 10,000 time steps for
continuous monitoring cases. The four cases of parameter settings are
taken from Table 1 of Hirsa and Madan (2004) except T = 1 (see Table 7
in this paper) and obtained from calibration of S&P 500 on June 30, 1999.
The CPU times for producing one numerical result via our method are 0.2
seconds for n = 12, 0.8 seconds for n = 50, 3.8 seconds for n = 250, and
0.3 seconds for continuously monitored Asian options (i.e., n = +ˆ).
Cai, Song, and Kou: A General Framework for Pricing Asian Options
548
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
Table 9.
for n = 12, 0.8 seconds for n = 50, 3.6 seconds for n = 250,
and 0.3 seconds for continuously monitored Asian options.
Pricing Asian options under the CGMY
model.
Downloaded from informs.org by [137.132.123.69] on 02 July 2015, at 02:16 . For personal use only, all rights reserved.
(I) Discretely monitored Asian options
under the CGMY model
5.7. Computing the Greeks
n = 12, 0.8 seconds for n = 50, 3.8 seconds for n = 250,
and 0.3 seconds for continuously monitored Asian options.
Our method can also be applied to compute the sensitivities
of the option prices approximately. Below we take the delta
as an example. To approximate the delta ã4S0 5 of either
continuously or discretely monitored Asian option price for
an initial stock price S0 , we first use our CTMC approximation method to obtain the Asian option prices V 4S0 +…5 and
V 4S0 − …5 with the initial stock price equal to S0 + … and
S0 − …, respectively (here … is a small number). Then ã4S0 5
can be approximated by ã4S0 5 ≈ 41/42…554V 4S0 +…5−
V 4S0 −…55.
Table 10 gives the numerical results for the delta
(denoted by “CTMC”) under the CEV model obtained by
our method. The parameter settings are the same as in
Table 4, and the benchmarks are constructed using the finite
difference approximation of Monte Carlo simulation estimates. It can be seen that our results are very accurate.
For the daily monitored (n = 250) Asian options, part (I)
in Table 10 shows that our numerical results of delta have
average (respectively, maximum) absolute error 0.00092
(respectively, 0.00170) and average (respectively, maximum) relative error 0.33% (respectively, 1.15%). Besides,
it takes around 7.1 seconds to generate one numerical
result of delta via our method. For continuously monitored Asian options, as shown in part (II) of Table 10, the
average (respectively, maximum) absolute error is 0.00086
(respectively, 0.00149), and the average (respectively, maximum) relative error is 0.27% (respectively, 0.96%). It takes
around 0.2 seconds to generate one numerical result of delta
via our method. For a detailed study for computing greeks
of Asian options under exponential Lévy models, we refer
to the paper by Ballotta et al. (2014).
5.6. The CGMY Model
5.8. Pricing of Floating Strike Asian Options
In this subsection, we consider another pure jump model,
the CGMY model (see Carr et al. 2002). In Table 9,
we compare in part (I) the numerical results of discretely
monitored Asian option prices obtained via our method
with those obtained via the recursive method in Fusai and
Meucci (2008), and we compare in part (II) our numerical
results for continuously monitored Asian option prices with
those generated by the Monte Carlo simulation method of
Madan and Yor (2008). It can be seen that our method
is quite accurate in both cases. In the discrete monitoring case, the average (respectively, maximum) absolute
error is around 0.00573 (respectively, 0.00941) and the
average (respectively, maximum) relative error is around
0.28% (respectively, 0.64%). In the continuous monitoring
case, the average (respectively, maximum) absolute error
is around 0.00388 (respectively, 0.00846) and the average (respectively, maximum) relative error is around 0.13%
(respectively, 0.21%). In addition, the CPU times for producing one numerical result via our method are 0.2 seconds
In the previous sections, we focus our discussions on the
Asian options with fixed strike prices. There also exist
floating strike Asian options. For instance, the price of a
continuously monitored floating strike Asian put option at
time 0 is given by
+ 1
AT − ŽST
1
Ṽc 4T 1 Ž3 x5 = e−rT Ɛx
T
Z T
with AT 2=
St dt1
n
K
Fusai and
Meucci
12
90
100
110
90
100
110
90
100
110
12070625
5003492
1002115
12073854
5007570
1004674
12074737
5008694
1005389
50
250
CTMC
Abs. err.
Rel.
err. (%)
12070406
5002551
1001464
12073745
5006651
1004012
12074653
5007783
1004725
−0000219
−0000941
−0000651
−0000109
−0000919
−0000662
−0000084
−0000911
−0000664
0002
0019
0064
0001
0018
0063
0001
0018
0063
(II) Continuously monitored Asian options
under the CGMY model
K
90
100
110
MC
values
Std. err.
CTMC
Abs. err.
Rel.
err. (%)
12074788
5008865
1005810
0000396
0000405
0000280
12074689
5008019
1006028
−0000099
−0000846
0000218
0001
0017
0021
Notes. Part (I) compares our numerical prices via CTMC approximation
(N = 50) with those obtained via the recursive algorithm in Fusai and
Meucci (2008) for discretely monitored Asian options under the CGMY
model. It takes about 0.2, 0.8, and 3.6 seconds to generate one price
via our method for n = 12, 50, and 250, respectively. Part (II) compares
our numerical prices via CTMC approximation (N = 50) with Monte
Carlo simulation estimates (denoted by “MC value.” The sample size is
110001000 and the number of time steps is 101000) for continuously monitored Asian options under the CGMY model. It takes about 0.3 seconds to
obtain one price via our method. For both parts, the unvarying parameters
are the same as in Fusai and Meucci (2008) and obtained from calibration, i.e., r = 000367, d = 0, S0 = 100, T = 1, C = 000244, G = 000765,
M = 705515, and Y = 102945.
0
and the price of a discretely monitored floating strike Asian
put option at time 0 is given by
+ 1
−rT x
Ṽd 4n1 Ž3 x5 = e Ɛ
B − ŽST
1
n+1 n
with Bn 2=
n
X
i=0
where Ž is a positive constant.
Siã 1
Cai, Song, and Kou: A General Framework for Pricing Asian Options
549
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
Table 10.
Computing deltas for discretely and continuously monitored Asian options under the CEV model.
(I) Discretely monitored Asian options
under the CEV model (n = 250)
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K
MC value
80
90
100
110
120
0094395
0081128
0056919
0032310
0015470
80
90
100
110
120
0093839
0081468
0058549
0033351
0015113
80
90
100
110
120
0093583
0081682
0059358
0033836
0014866
CTMC
Abs. err.
(II) Continuously monitored Asian options
under the CEV model
Rel. err. (%)
MC value
CTMC
0004
0017
0000
0036
1002
0094395
0081126
0056844
0032315
0015487
0094415
0081250
0056923
0032212
0015338
0005
0016
0000
0035
1008
0093815
0081458
0058509
0033375
0015080
0093873
0081582
0058552
0033252
0014977
0006
0016
0000
0035
1015
0093586
0081732
0059266
0033802
0014829
0093624
0081791
0059364
0033735
0014725
‚ = 0025
0094429
0000033
0081269
0000141
0056920
0000002
0032193
−0000118
0015312
−0000158
‚ = −0025
0093884
0000045
0081601
0000133
0058547
−0000002
0033234
−0000116
0014950
−0000163
‚ = −005
0093634
0000052
0081811
0000129
0059359
0000001
0033718
−0000118
0014696
−0000170
Abs. err.
‚ = 0025
0000020
0000123
0000080
−0000102
−0000149
‚ = −0025
0000058
0000124
0000043
−0000122
−0000102
‚ = −005
0000038
0000059
0000097
−0000067
−0000104
Rel. err. (%)
0002
0015
0014
0032
0096
0006
0015
0007
0037
0068
0004
0007
0016
0020
0070
Notes. Part (I) and (II) compare the numerical results of delta via our CTMC approximation method (N = 50) with Monte Carlo simulation estimates
(denoted by “MC value”) under the CEV model. For the simulation method, the sample size is 110001000, and the number of time steps for approximating
continuously monitored Asian options is 101000; the deltas are calculated by the central difference scheme with step size 0.50. It takes around 7.1 seconds
and 0.2 seconds to generate one numerical result via our method for the discretely and continuously monitored Asian options, respectively. The unvarying
parameters are r = 0005, d = 0, S0 = 100, T = 1, and ‘S0‚ = 0025.
When the underlying asset follows an exponential Lévy
model, Eberlein and Papapantoleon (2005) prove that under
some regularity conditions, a floating strike Asian option
is equivalent to a fixed strike Asian option under another
closely related exponential Lévy model. More precisely,
suppose 8St 1 t ¾ 09 follows an exponential Lévy model with
the infinitesimal generator
1
4r − d5xf 0 4x5 + ‘ 2 x2 f 00 4x5
2
Z
+ 6f 4xey 5 − f 4x5 − xf 0 4x54ey − 1574dy51
and assume there exists a constant M > 1 such that
Ɛx 6Stb 7 < ˆ1
Asian call (respectively, put) option with fixed strike ŽS0
under 8St∗ 1 t ¾ 09.
As a consequence, this correspondence enables us to
price the floating strike Asian options under an exponential Lévy model 8St 1 t ¾ 09 by pricing the corresponding
fixed strike Asian options under the closely related exponential Lévy model 8St∗ 1 t ¾ 09 via our CTMC approximation method. Table 11 provides some numerical results
for pricing Asian options with floating strike under Merton’s jump diffusion model. We can see that the average
(respectively, maximum) absolute error is around 0.0062
(respectively, 0.0114), and the average (respectively, maximum) relative error is around 0.19% (respectively, 0.35%).
Besides, the CPU times for producing one numerical result
for any —b— < M0
Furthermore, consider another process
infinitesimal generator
Table 11.
8St∗ 1 t
¾ 09 with the
1
4d − r5xf 0 4x5 + ‘ 2 x2 f 00 4x5
2
Z
y
+ 6f 4xe 5 − f 4x5 − xf 0 4x54ey − 157 ∗ 4dy51
n
12
50
250
ˆ
Pricing floating strike Asian options under
the MJD model.
MC value
Std. err.
CTMC
Abs. err.
Rel. err. (%)
303002
303463
303676
303658
000041
000041
000042
000042
303116
303565
303688
303678
000114
000103
000012
000020
0035
0031
0004
0006
where  ∗ 4dy5 = e−y 4−dy5. In other words, 8St∗ 1 t ¾ 09
follows another exponential Lévy model with risk free
rate d, dividend rate r, and Lévy measure  ∗ . Then the
price of an (either discretely or continuously monitored)
Asian put (respectively, call) option with floating strike and
multiplier Ž under 8St 1 t ¾ 09 is equal to the price of an
Notes. Pricing Asian put options with floating strikes (Ž = 1) under Merton’s jump diffusion model via our CTMC approximation with N = 50.
The parameters are the same as in Table 6. The columns “MC value”
are generated by Monte Carlo simulation, for which the sample size is
1,000,000 and the number of time steps for the continuously monitored
Asian option is 10,000. The CPU times to generate one numerical price
via our method are about 0.2, 0.7, and 3.8 seconds for n = 12, 50, and
250 respectively, and about 0.3 seconds for n = +ˆ, i.e., for continuously
monitored Asian options.
Cai, Song, and Kou: A General Framework for Pricing Asian Options
550
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
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via our method are 0.2, 0.7, and 3.8 seconds for n = 12,
50 and 250, respectively, and 003 seconds for continuously
monitored Asian options. Under general Markov processes,
there may exist no such simple relationship between floating strike and fixed strike Asian option prices, and the problem of using CTMC approximation to price floating strike
Asian options remains an interesting research topic.
Since —vc 4t1 k3 x5— = —Ɛx 64At − k5+ 7— ¶ Ɛx 6At 7 =
4x/4r − d554e4r−d5t − 15, for any A1 > max824r − d5t1 09
and A2 > 0, we can also derive a computable error bound
for Error c :
—Error c —
¶
6. Error Analysis for Our Pricing Method
This section is devoted to the analysis on the errors associated with our pricing method, including the CTMC approximation error in Step 1 and the transform inversion error in
Step 2.
6.1. The Transform Inversion Error in Step 2
According to Choudhury et al. (1994), the numerical inversion of double transforms in Step 2 may incur two types of
errors: the discretization error and the truncation error. For
simplicity, we assume r 6= d in the following analysis. For
the discretely monitored Asian options, the discretization
error for the double inversion (B2) is given by
Error d =
ˆ X
ˆ
X
e−Aj 2ln vd 441 + 2l5n1 41 + 2j5k3 x50
j=0 l=0
not j=l=0
Note that —vd 4n1 k3 x5— = —Ɛx 64Bn − k5+ 7— ¶ Ɛx 6Bn 7 =
x44e4r−d5ã4n+15 − 15/4e4r−d5ã − 155. Then for any  ∈
401 min8e−4r−d5ã 1 195 and A > 0, we can obtain the following computable error bound for Error d :
—Error d —
ˆ X
ˆ
X
x
¶
=
e4r−d5ã −1
e−Aj 2ln 6e4r−d5ã641+2l5n+17 −17
j=0 l=0
not j=l=0
x
e4r−d5ã −1
e−A +4e4r−d5ã 52n −e−A 4e4r−d5ã 52n
41−e−A 561−4e4r−d5ã 52n 7
e−A +2n −e−A 2n
−
1
(8)
41−e−A 541−2n 5
· e4r−d5ã4n+15
which implies that —Error d — = O4e−A 5 + O42n 5 as A →
+ˆ and  → 0. Therefore the discretization error can be
controlled by properly specifying inversion parameters A
and . For the numerical examples in this paper, we choose
A = 20 and  such that e−A = 2n . It follows from (8) that
this setting is able to bound the discretization errors of the
option prices by 10−6 .
Similarly, for the continuously monitored Asian options,
the discretization error for the double inversion (B1) is
given by
Error c =
ˆ X
ˆ
X
j=0 l=0
not j=l=0
e−4A1 j+A2 l5 vc 441 + 2j5t1 41 + 2l5k3 x50
ˆ X
ˆ
x X
e−4A1 j+A2 l5 6e4r−d541+2j5t − 17
r − d j=0 l=0
not j=l=0
x
e−A1 +24r−d5t + e−A2 − e−A1 −A2 +24r−d5t
e4r−d5t
r −d
41 − e−A1 +24r−d5t 541 − e−A2 5
e−A1 + e−A2 − e−A1 −A2
0
(9)
−
41 − e−A1 541 − e−A2 5
=
It follows that —Error c — = O4e−A1 5 + O4e−A2 5 as A1 → +ˆ
and A2 → +ˆ. Namely, the discretization error has an
exponential decay as A1 and A2 go to +ˆ. In our implementation, we set A1 = A2 = 20 based on (9) such that the
discretization error is no greater than 10−6 .
The truncation errors are related to the parameters m1
and m2 used in the Euler transformation E4m1 1 m2 5 in (B3),
which is applied to accelerate the convergence of the partial
sums to the corresponding infinite series in (B1) and (B2).
As suggested by Abate and Whitt (1992), the truncation
errors can be estimated by —E4m1 1 m2 5 − E4m1 − 11 m2 5—
and —E4m1 1 m2 + 15 − E4m1 1 m2 5—.
Figure 1 plots the estimated truncation errors with respect to different choices of m1 (with the fixed m2 = 15)
under the CEV model, Merton’s jump diffusion model, and
the VG model and for different monitoring frequencies,
n = 12, 50, 250, and +ˆ (continuously monitored). As
we can see, the estimated truncation errors decay very fast
as m1 increases. For instance, when m1 ¾ 30, all the errors
are less than 10−6 except the continuously monitored Asian
option under the VG model. In the latter case, the error is
less than 10−5 that is still sufficiently small in practice. This
might be due to the power decay of the moment generating
function of the VG process. Nonetheless, this effect seems
not significant in the discretely monitored cases under the
VG model for commonly used discretely monitoring frequencies such as n = 12, 50, and 250.
6.2. The CTMC Approximation Error in Step 1
In Step 1, we approximate the targeted Markov process model with a CTMC. With the inversion errors in
Step 2 controlled at a low level, the approximation error in
Step 1 decreases as the number of states for the approximate CTMC increases. Figure 2 plots the absolute pricing errors for both discretely and continuously monitored
Asian options obtained via our method with respect to an
increasing number of states in the approximate CTMC,
under the CIR model, Merton’s jump diffusion model
and the VG model, respectively. The errors are estimated
by Error4N 5 = —Price4N 5 − Price4N ∗ 5—, where Price4N 5
Cai, Song, and Kou: A General Framework for Pricing Asian Options
551
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
(Color online) The decay of the truncation errors of the double inversions.
%STIMATEDTRUNCATIONERROR
#%6MODEL
¾n
n
n
n
n∞
-ERTONSMODEL
¾n n
n
n
n∞
6'MODELDISCRETELYMONITORED
!SIANOPTIONS
n
n
n
%STIMATEDTRUNCATIONERROR
%STIMATEDTRUNCATIONERROR
¾n
m
m
¾n 6'MODELCONTINUOUSLYMONITORED
!SIANOPTIONS
n∞
m
m
Notes. The estimated truncation errors of the double inversions with respect to different choice of m1 and fixed m2 = 15. All the options computed in these
figures are at the money. Other parameters are set as follows: the parameters for the CEV model are the same as in Table 4 (‚ = −0025); the parameters
for Merton’s jump diffusion model are the same as in Table 6; the parameters for the VG model are the same as in Table 8 (case no. 2).
represents the price computed by CTMC approximation
with N states and N ∗ is a fixed large natural number. Here
we set N ∗ = 11000. It can be seen from Figure 2 that for
the discretely monitored Asian options, the logarithms of
(Color online) The decay of the CTMC approximation errors for pricing Asian options.
Error decay for ATM Asian options
under CEV model
Error decay for ATM Asian options
under VG model
Error decay for ATM Asian options
under Merton’s model
10–1
10–1
Absolute error
n = 12
n = 50
n = 250
n=∞
10–2
10–3
10–4 1
10
absolute pricing errors are approximately linear in the logarithm of N and the slopes of the lines are about −2, which
suggests that the absolute pricing errors appear to be proportional to 1/N 2 . For the continuously monitored Asian
102
N
103
10–1
10–2
Absolute error
Figure 2.
Absolute error
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%STIMATEDTRUNCATIONERROR
Figure 1.
10–3
10–4
10–5 1
10
102
N
103
10–2
10–3
10–4 1
10
102
103
N
Notes. The absolute errors of discretely and continuously monitored Asian option prices obtained via our CTMC approximation with the number of states
N = 25, 50, 100, 200, and 400. Left: The CEV model with the same parameters as in Table 4 (‚ = −0025). Middle: Merton’s jump diffusion model with
the same parameters as in Table 6. Right: The VG model with the same parameters as in Table 8 (case no. 2).
Cai, Song, and Kou: A General Framework for Pricing Asian Options
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552
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
options, the slopes of the decay lines in Figure 2 vary from
−106 to −208, which suggests that the decay of the absolute pricing errors of our method appears to be at the speed
of 1/N 106 to 1/N 208 . This is similar to the observation by
Mijatović and Pistorius (2013) in the CTMC pricing of continuously monitored double barrier options, where they find
that the absolute pricing error is approximately proportional
to 1/N 102 to 1/N 2 under different models.
(ii) Suppose that f is any bounded solution to the functional
Equation (5). Define
Z t
Mt 2= f 4St 5e−Œt−ˆAt + e−Œu−ˆAu du for t ¾ 00
Acknowledgments
d x
Ɛ 6f 4St+’ 5e−Œ4t+’5−ˆAt+’ — Ft 7
d’
d
= Ɛy 6f 4S’ 5e−Œ’−ˆA’ 7—y=St ·e−Œt−ˆAt
d’
The authors thank the area editor Benjamin Van Roy, the associate
editor, and three anonymous referees for their helpful comments.
The first and third authors are grateful for the financial support
from the General Research Fund of the Hong Kong Research
Grants Council with project No. 16200614. The research of the
second and third authors are supported in part by National University of Singapore Research Grants [R-146-000-180-133 and
C-146-000-038-001].
Appendix A. Proof of Theorem 1
Mn 2= f 4Stn 5eˆStn · zn e−ˆBn +
zk e−ˆBk
= −Ɛy 6e−Œ’−ˆA’ 7—y=St ·e−Œt−ˆAt = −Ɛx 6e−Œ4t+’5−ˆAt+’ — Ft 71
where the second equality holds because of (A2). Then it follows
that
Ɛx 6f 4St+u 5e−Œ4t+u5−ˆAt+u — Ft 7 − Ɛx 6f 4St 5e−Œt−ˆAt 7
Z u
=−
Ɛx 6e−Œ4t+’5−ˆAt+’ — Ft 7 d’0
Therefore,
Ɛx 6Mt+u — Ft 7 − Mt = −
for n = 01 11 0 0 0 0
n−1
X
—zk e−ˆBk —
k=0
n−1
X
—zk e−ˆBk —
1
< +ˆ1
1 − —z—
(A1)
where Pã f 4x5 2= Ɛx 6f 4Sã 57. On the other hand, for n ¾ 1, since
eˆStn e−ˆBn = e−ˆBn−1 ,
Ɛx 6Mn — F4n−15 7 − Mn−1
= zn e−ˆBn−1 Pã f 4Stn−1 5 + zn−1 e−ˆBn−1
− f 4Stn−1 5eˆStn−1 · zn−1 e−ˆBn−1
= zn−1 e−ˆBn−1 6zPã f 4Stn−1 5 + 1 − f 4Stn−1 5eˆStn−1 7 = 00
Therefore, 8Mn 1 n ¾ 09 is a martingale with respect to 8F4n5 1
n ¾ 09, and hence Ɛx 6Mn 7 = M0 = f 4x5. Letting n → +ˆ
P
k −ˆBk
and noticing that limn→+ˆ Mn = +ˆ
almost surely, we
k=0 z e
obtain
h
i
f 4x5 = lim E x 6Mn 7 = Ɛx lim Mn
n→+ˆ
=Ɛ
+ˆ
X
x
Also, noticing that —Mt — ¶ C + Re4Œ5−1 < +ˆ, we conclude that
8Mt 1 t ¾ 09 is a martingale with respect to 8Ft 1 t ¾ 09. Then by
the dominated convergence theorem, we obtain
h
i
f 4x5 = M0 = lim Ɛx 6Mt 7 = Ɛx lim Mt
t→ˆ
t→ˆ
Z ˆ
= Ɛx
e−Œu−ˆAu du = m4Œ1 ˆ3 x50
0
k=0
¶1+C +
u
0
t
Then we claim that 8Mn 2 n ¾ 09 is a martingale with respect to
8F4n5 1 n ¾ 09, where F4n5 2= Fnã . Indeed, on the one hand,
= —1 + zPã f 4Stn 5——zn e−ˆBn — +
Z
Ɛx 6e−Œ4t+’5−ˆAt+’ — Ft 7 d’
Z t+u
+ Ɛx
e−Œs−ˆAs ds — Ft = 00
k=0
—Mn — ¶ —f 4Stn 5eˆStn ——zn e−ˆBn — +
(A2)
0
Proof. (i) Suppose that f is any bounded solution to the functional Equation (4). Define
n−1
X
0
We claim that
d x
Ɛ 6f 4St 5e−Œt−ˆAt 7 = −Ɛx 6e−Œt−ˆAt 70
dt
If (A2) holds, we have
n→+ˆ
zk e−ˆBk = l4z1 ˆ3 x51
k=0
where the second equality holds because of (A1) and the dominated convergence theorem. This implies that l4z1 ˆ3 x5 is the
unique bounded solution to (4).
To prove (A2), first note that for any fixed t ¾ 0,
Ɛx 6f 4St+u 5e−Œ4t+u5−ˆAt+u 7 − Ɛx 6f 4St 5e−Œt−ˆAt 7
u
1
x
−Œ4t+u5−ˆAt+u
− f 4St+u 5e−Œt−ˆAt 7
= · 4Ɛ 6f 4St+u 5e
u
+ Ɛx 6f 4St+u 5e−Œt−ˆAt − f 4St 5e−Œt−ˆAt 75
f 4St+u 54e−Œ4t+u5−ˆAt+u − e−Œt−ˆAt 5
= Ɛx
u
4f 4St+u 5 − f 4St 55e−Œt−ˆAt
+ Ɛx
0
u
(A3)
We intend to let u ↓ 0 on both sides of (A3). For the first term
on the right-hand side (RHS) of (A3), when u ∈ 401 u0 5 with any
fixed u0 > 0,
f 4St+u 54e−Œ4t+u5−ˆAt+u − e−Œt−ˆAt 5 u
Ru
f 4St+u 5 0 4−Œ − ˆSt+s 5e−Œ4t+s5−ˆAt+s ds = u
CZ u
¶
—Œ + ˆSt+s — · —e−Œ4t+s5−ˆAt+s — ds
u 0
¶ C · —Œ— + —ˆ— max Ss ¶ C · —Œ— + —ˆ— max Ss 0
t¶s¶t+u
0¶s¶t+u0
Cai, Song, and Kou: A General Framework for Pricing Asian Options
553
Operations Research 63(3), pp. 540–554, © 2015 INFORMS
Appendix B. Double Transform Inversion
Formulas
By Doob’s inequality,
Ɛx
h
i
max Ss ¶ 41 − e−1 5−1 41 + Ɛx 6—St+u0 log St+u0 —75
0¶s¶t+u0
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1+…
¶ C1 + C2 · Ɛx 6St+u
7 < +ˆ1
0
for some C1 1 C2 > 0, where the second inequality holds because
—x log x— ¶ C3 + —x—1+… for some C3 > 0. Therefore, applying the
dominated convergence theorem yields
lim Ɛx
u↓0
f 4St+u 54e−Œ4t+u5−ˆAt+u − e−Œt−ˆAt 5
u
(A4)
For the second term on the RHS of (A3), by Proposition 1.1.5 in
Ethier and Kurtz (2005) we obtain
4f 4St+u 5 − f 4St 55e−Œt−ˆAt
u
e−Œt−ˆAt
= Ɛx 4Ɛx 6f 4St+u 5 — Ft 7 − f 4St 55
u
Z u
−Œt−ˆAt
e
= Ɛx
Ɛx 6Gf 4St+s 5 — Ft 7 ds
u
0
Z u
−Œt−ˆAt
e
= Ɛx
Gf 4St+s 5 ds
0
u
0
4f 4St+u 5 − f 4St 55e−Œt−ˆAt
u
Ru
Gf 4St+s 5 ds −Œt−ˆAt
x
0
= Ɛ lim
e
u↓0
u
= Ɛ 6Gf 4St 5 · e
70
ˆ X
ˆ
X
e−4A1 j+A2 k5 f 441 + 2j5t1 1 41 + 2k5t2 50
j=0 k=0
not j=k=0
ˆ
n−1
X
eA/2 X
4−15j+1
4−15k
n
4n t j=−ˆ
k=−n
A i ij
−
−
1 e4ik5/n − eg 1
·g̃
2t
t
t
(B2)
eg =
ˆ X
ˆ
X
e−Aj 2kn g441 + 2j5t1 41 + 2k5n50
j=0 k=0
not j=k=0
−Œt−ˆAt
ef =
where
Then it follows from the dominated convergence theorem that
x
(B1)
where
g4t1 n5 =
Z u
−Œt−ˆAt ¶ C —Œ— + —ˆ— max Ss + 10
Gf 4St+s 5 ds e
0
t¶s¶t+u
u
u↓0
ˆ
ˆ
X
e4A1 +A2 5/2 X
4−15j+1
4−15k+1
4t1 t2 j=−ˆ
k=−ˆ
A1 i ij A2 i ik
· f˜
− −
1
− −
−ef 1
2t1 t1
t1 2t2 t2
t2
Similarly, we use the formula below to evaluate g4t1 n5 from
its Z-Laplace transform g̃4t1 n5
Since —Gf 4x5— = —4Œ + ˆx5f 4x5 − 1— ¶ 4—Œ— + —ˆx—5C + 1, we have
lim Ɛx
f 4t1 1t2 5 =
= Ɛx 6f 4St 54−Œ − ˆSt 5e−Œt−ˆAt 70
Ɛx
To numerically invert the double transforms involved in our pricing algorithms for both discretely and continuously monitored
Asian options, we shall use the transform inversion algorithms
proposed by Choudhury et al. (1994). Specifically, to evaluate
f 4t1 1 t2 5 from its double-Laplace transform f˜4t1 1 t2 5, we have
The involved parameters, i.e., A1 > 0 and A2 > 0 in (B1) and
A > 0 and  ∈ 401 15 in (B2), are chosen to control the errors of
the corresponding inversion formulas.
P
j
Besides, when evaluating series of the form +ˆ
j=0 4−15 aj in
the formulas (B1) and (B2), we can accelerate the convergence via
P
j
Euler transformation. Namely, we approximate +ˆ
j=0 4−15 aj by
E4m1 1 m2 5 2=
(A5)
m1
X
m1 !
2−m1 Sm2 +j 1
j!4m
−
j5!
1
j=0
(B3)
Now letting u ↓ 0 on both sides of (A3) and using (A4) and (A5)
yields
Pj
where Sj 2= k=0 4−15k ak . We set two parameters m1 = 10 and
m2 = 15 in our implementation. See Abate and Whitt (1992) for
more details about Euler transformation.
d x
Ɛ 6f 4St 5e−Œt−ˆAt 7
dt
References
Ɛx 6f 4St+u 5e−Œ4t+u5−ˆAt+u 7 − Ɛx 6f 4St 5e−Œt−ˆAt 7
≡ lim
u↓0
u
= Ɛx 6f 4St 54−Œ − ˆSt 5e−Œt−ˆAt 7 + Ɛx 6Gf 4St 5 · e−Œt−ˆAt 7
= −Ɛx 6e−Œt−ˆAt 71
where the last equality follows from the functional Equation (5). ƒ
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Ning Cai is an associate professor in the Department of Industrial Engineering and Logistics Management at the Hong Kong
University of Science and Technology. His research interests
include financial engineering, stochastic modeling, and applied
probability.
Yingda Song is an associate professor in the Department of
Statistics and Finance at the University of Science and Technology
of China. His research interests focus on financial engineering,
simulation, stochastic modeling, and applied probability.
Steven Kou is the director of the Risk Management Institute
and a Provost’s Chair Professor of Mathematics at the National
University of Singapore. His research interests are in quantitative
finance, applied probability, and statistics. He is the winner of the
2002 Erlang Prize awarded by the Applied Probability Society of
INFORMS. He is currently the Vice Chair of Applied Probability
for the Financial Service Section of INFORMS.