Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models

Abstract

Utilizing frame duality and a FFT-based implementation of density projection we develop a novel and efficient transform method to price Asian options for very general asset dynamics, including regime switching Lévy processes and other jump diffusions as well as stochastic volatility models with jumps. The method combines Continuous-Time Markov Chain (CTMC) approximation, with Fourier pricing techniques. In particular, our method encompasses Heston, Hull-White, Stein-Stein, 3/2 model as well as recently proposed Jacobi, α-Hypergeometric, and 4/2 models, for virtually any type of jump amplitude distribution in the return process. This framework thus provides a 'unified ' approach to pricing Asian options in stochastic jump diffusion models and is readily extended to alternative exotic contracts. We also derive a characteristic function recursion by generalizing the Carverhill-Clewlow factorization which enables the application of transform methods in general. Numerical results are provided to illustrate the effectiveness of the method. Various extensions of this method have since been developed, including the pricing of barrier, American, and realized variance derivatives.

Full text

Efficient Asian option pricing under regime switching jump diffusions
and stochastic volatility models
J. Lars Kirkby ∗Duy Nguyen†
April 25, 2020
Abstract
Utilizing frame duality and a FFT-based implementation of density projection we develop a novel
and efficient transform method to price Asian options for very general asset dynamics, including
regime switching L´evy processes and other jump diffusions as well as stochastic volatility models
with jumps. The method combines Continuous-Time Markov Chain (CTMC) approximation, with
Fourier pricing techniques. In particular, our method encompasses Heston, Hull-White, Stein-Stein,
3/2 model as well as recently proposed Jacobi, α-Hypergeometric, and 4/2 models, for virtually any
type of jump amplitude distribution in the return process. This framework thus provides a ‘unified’
approach to pricing Asian options in stochastic jump diffusion models and is readily extended to
alternative exotic contracts. We also derive a characteristic function recursion by generalizing the
Carverhill-Clewlow factorization which enables the application of transform methods in general.
Numerical results are provided to illustrate the effectiveness of the method. Various extensions of
this method have since been developed, including the pricing of barrier, American, and realized
variance derivatives.
Keywords: Asian options, jump diffusion, stochastic volatility, regime switching, Markov chain,
CTMC, Fourier, exotic option
AMS subject classifications: 91G80, 93E11, 93E20
JEL classifications: C00, C02, G12, G13
∗School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30318, Email:
jkirkby3@gatech.edu
†Department of Mathematics, Marist College, Poughkeepsie, NY 12601, USA, Email:nducduy@gmail.com

1 Introduction
Asian options are financial derivatives whose payoffs depend on the average of the underlying asset
price over a pre-specified time period. Since their introduction in 1987, they have provided a popular
means of risk management in a variety markets. As a consequence of averaging, Asian options are
less sensitive to price manipulation, and become easier to hedge toward the option’s expiry. How-
ever, pricing Asian options poses a great challenge mainly because the distribution of the average
asset price is usually not available analytically, even under the simple Geometric Brownian Motion
(GBM) model. Hence, numerical methods must be used to price and hedge these financial derivatives.
The computational approaches can be categorized as Monte-Carlo methods Kemna and Vorst (1990);
Broadie and Glasserman (1996); Broadie et al. (1999); Dinge¸c et al. (2014), analytical approxima-
tions and bounds Fusai and Kyriakou (2016), asymptotic approximations Pirjol and Lingjiong (2016),
numerical partial differential equation (PDE) techniques in Ingersoll (1987); Zvan et al. (1998), di-
mension reduction techniques proposed in Rogers and Shi (1995); Vecer (2001), perturbation methods
developed in Zhang (2001, 2003), spectral expansion of Linetsky (2004), the Laplace inversion method
of German and Yor (1993), Boyarchenko and Levendorskii (2015) and Cui and Nguyen (2017), and
Fourier transform methods Cerny and Kyriakou (2011); Zhang and Oosterlee (2013); Kirkby (2016);
Levendorskii (2018); Leitao et al. (2018); Corsaro et al. (2019).
Despite its popularity, it is well known that the standard Black-Scholes (BS) model suffers from
several deficiencies, such as inconsistencies with the market-observed implied volatility smile (or skew).
Many extensions have been introduced in the literature to provide more realistic descriptions for asset
price dynamics and to account for the empirical behavior of implied volatility smile (smirk). Among
the most popular extensions are jump diffusions, stochastic volatility, and regime switching models.
By allowing the model features such as volatility and jump activity to behave differently across a
set of prescribed market states, regime switching models can capture more accurately the change in
macro-market conditions, while, at the same time, preserving to a certain degree the tractability of
the model (see (Hamilton, 1994)). In addition, as illustrated in Yao et al. (2006) for example, regime
switching models are able to generate the implied volatility smiles commonly observed in practice.
Regime-switching studies include modeling electricity prices (Weron et al., 2004), short rates (Siu,
2010), portfolio selection (Zhang et al., 2012), and option pricing (Elliott et al., 2007; Yuen and Yang,
2009; Liu, 2010; Ramponi, 2012; Zhang et al., 2012; Fard and Siu, 2013; Florescu et al., 2013; Costabile
et al., 2014; Jiang et al., 2016; Elliott et al., 2005; Zhang and Chan, 2016). While the literature on
pricing European/American options under regime switching models is quite rich, notably Buffington
and Elliot (2002); Boyarchenko and Levendorskii (2007, 2008, 2009, 2013), only a handful of papers
discuss the pricing of Asian options under regime switching models, such as Boyle and Draviam (2007);
Yuen and Yang (2012), and very recently, Zhou and Ma (2016).
Another important extension is to augment the GBM with jumps, as pioneered in Merton (1976).
1

Compared with the simple BS model, the literature on pricing Asian options under jump models
is relatively under-developed. Research in this area includes (d’Halluin et al., 2005), where a semi-
Lagrangian method is proposed to solve the partial integro-differential equation (PIDE) directly, and
Vecer and Xu (2004), where a representation for Asian options under semi-martingale models is
developed via analysis of the PIDE. Building on Vecer and Xu (2004), the technique proposed in
Bayraktar and Xing (2011) involves constructing a sequence of functions that converges uniformly
to the option price. Fusai and Kyriakou (2016) 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 L´evy and CEV models. Levendorskii (2018) introduced a backward
induction procedure in the dual space for pricing Asian option in L´evy models. Recently, Kirkby
(2016) introduced a novel Fourier-based method to price generalized arithmetic Asian options in
L´evy-driven models, with discrete and continuous averaging. The method obtains rapidly converging
value approximations to high precision, resulting in a substantial time reduction compared with state-
of-the-art transform methods, such as Zhang and Oosterlee (2013), which is in theory exponentially
convergent, as well as Levendorskii and Xie (2012).
Along a different line of research, a double Laplace transform method for pricing Asian options
under the double exponential jump diffusion model of Kou (2002) is developed in Cai and Kou (2012)
and Cai et al. (2015). Cui et al. (2018b) are able to reduce the double Laplace transform of Cai et al.
(2015) to a single Laplace transform, which significantly reduces the computational effort (see also Cui
et al. (2018a)). In Funahashi and Kijima (2017), the authors propose a unified approximation method
for the pricing of such options when the underlying process is a diffusion using an approximation of
the density function. In contrast to these works, which focus on one dimensional Markov processes,
the approach developed in this paper applies to two dimensional problems, namely regime switching
Markov processes and stochastic volatility.
Despite the intense research efforts in the area of Asian option pricing, we emphasize the following
important shortcomings in the existing literature which formed the motivation for this work. Firstly,
although there has been considerable interest in applying regime switching jump diffusion models to
various financial problems, the existing option pricing literature under these models focuses primarily
on European/American options. Very few papers (for example Dang et al. (2016)) in the open liter-
ature consider the problem of pricing Asian options under such models. Similarly, the literature on
Asian option pricing in stochastic volatility models is sparse, even in the absence of jumps. This mo-
tivates us to extend the pricing literature to a very general class of models, which includes stochastic
volatility models with arbitrary correlation in the two driving Brownian motions, as well as regime
switching diffusions, allowing for jumps in either case.
In this paper, by working in the Fourier domain, we develop a fast and highly accurate method for
pricing Asian options under a general class of models, which we call APROJ (for Asian PROJection).
2

This includes discretely monitored contracts as well as the continuously monitored options common in
foreign exchange markets. Extending further, we provide a unified approach to pricing Asian options
under general stochastic volatility jump diffusion models. In short, the contribution of our paper is
threefold:
1. We extend the realm of transform-based pricing methods to incorporate Asian options under
very general regime switching jump diffusion models, which permit different jump distributions
in each regime. Capitalizing on the rapid convergence of frame projection with the cubic basis,
we devise an efficient and accurate pricing algorithm.
2. By adopting a Markov chain approximation of the variance diffusion combined with a novel
decorrelation method for stochastic volatility models, we are able to reduce the problem of
pricing Asian options in stochastic volatility models to that of a regime switching jump diffusion
model. The approach thus provides a unified framework for pricing Asian options in many well-
known stochastic volatility jump diffusion models which include those of Heston, Hull-White,
Stein-Stein, Jacobi, α-Hypergeometric, 3/2 and 4/2 models with virtually any jump component
in the asset price. This new methodology has the potential to simplify pricing for many exotic
contracts under stochastic volatility.
3. We provide numerical results for Asian options under regime switching jump diffusion models and
under some stochastic volatility models which have never been reported in the literature. Hence,
these results can be used as references for future research. In particular, our extensive numerical
experiments for Heston’s model demonstrate the stability and robustness of our method.
The paper is organized as follows. The Markov chain and regime switching framework is developed in
Section 2, where in Section 2.1 we derive the main recursive algorithm in the Fourier domain, which can
be implemented with standard Fourier techniques. In Section 3, we introduce the stochastic volatility
framework and our “decorrelation” approach which enables a regime switching approximation for
general stochastic volatility models. Section 4 details a particular implementation based on the Fourier
method of frame duality. Section 5 provides numerical examples to illustrate the effectiveness of the
method. Appendix B provides an error analysis of the proposed method, and the paper is concluded
in Section 6.
2 Asian Option Pricing Under Regime Switching
We first describe the proposed methodology for a Regime Switching (RS) jump diffusion, and extend
it later to the case of stochastic volatility. This framework is general, and includes models with
jumps in the underlying. Moreover, it has the potential to unify the pricing of a large number of
exotic contracts under very general dynamics. First recall that a finite state CTMC on (Ω,F,Q)
3

is a stochastic process {α(t), t ≥0}which transitions between states M:= {1,2, . . . , m0}, where M
indexes the set of variance states of vα(t)in our example. The transition dynamics of α(t) are described
by a generator Q= [qjk]m0×m0, whose elements qjk satisfy (i) qjj ≤0, and qjk ≥0, if j6=k, and (ii)
Pkqjk = 0, ∀j∈ M. In terms of qjk, the Markov chain makes transitions according to
Q[α(t+ ∆t) = j|α(t) = k, α(t0),0≤t0≤t] = δjk +qjk ∆t+o(∆t),(1)
where δjk = 1 if j=k, and zero otherwise.
Given a CTMC {α(t), t ≥0}on M:= {1,2, . . . , m0}with generator Q, an RS jump diffusion
model for S(t) under Qis described by
dSt
St−
= (r−q−λα(t)κα(t))dt +σα(t)dW ∗(t) + ZR
[ey−1]πα(dy, dt),(2)
where W∗
tis a standard Brownian motion independent of α(t), παis the random jump measure with
corresponding L´evy measure να, and κα(t)=RR[ey−1]να(t)(y)dy is finite for each regime.1The log
return process Xt:= log(St/S0), t ∈[0, T ] satisfies (see for example Ramponi (2012) for more details)
Xt=Zt
0
µα(s)ds +Zt
0
σα(s)dW ∗(s) + Zt
0ZR
yπα(dy, ds),(3)
where µα(t)=r−q−λα(t)κα(t)−1
2σ2
α(t). Each state j= 1, . . . , m0is characterized by a particular model
for dynamics in that state, or equivalently by its characteristic function. Given a time increment of
size δ > 0, and ξ∈R, define for each state j= 1, ..., m0
e
φj
Xδ(ξ) := E[eiξXδ|α(0 ≤s≤δ) = j] := exp(ψj(ξ)δ),(4)
where e
φj
Xδ(ξ) is the characteristic function of a process which spends the entirety of δin state j, and
ψj(ξ) is its L´evy symbol. In general, a full model of the process is specified by describing each ψj(ξ)
along with generator Q. For example, with ξ∈R, an RS jump diffusion is characterized by the set of
ψj(ξ)=iξµj−1
2ξ2σ2
j−λj(φj(ξ)−1), j = 1, . . . , m0
where φj(ξ) is the characteristic function of jump magnitude in state j. Table 1 includes several
common jump distributions. The next result, which follows from the work of Buffington and Elliot
(2002); Chourdakis (2002); Ramponi (2012), connects the characteristic functions e
φj
X∆t(ξ) with that
of X∆t, given the possibility of visiting multiple states within ∆t.
Corollary 1 The characteristic function of X∆tsatisfies
E[exp(iX∆tξ)|α(0) = j] = 10E(ξ)·ej,
where we define
E(ξ) := exp ∆tQ0+diag(ψ1(ξ), . . . , ψm0(ξ),(5)
and 1∈Rm0×1a vector of ones, and ej∈Rm0×1a unit vector with a one in the jth position.
1In between regime switching times, the asset price process follows a regular jump diffusion with constant drift rate
and instantaneous volatility rate. Note that this formulation allows for a different jump distribution in each regime.
4

Jump Model ν(y)φ(ξ)
Normal 1
√2πb e−(x−a)2/2b2eiξa−1
2ξ2b2
DE pη1e−η1yI{y≥0}+ (1 −p)η2eη2yI{y<0}pη1
η1−iξ+ (1 −p)η2
η2+iξ
Mixed Normal p1
√2πb1e−(y−a1)2
2b2
1+ (1 −p)1
√2πb2e−(y−a2)2
2b2
2peiξa1−1
2ξ2b2
1+ (1 −p)eiξa2−1
2ξ2b2
2
Table 1: L´evy measure ν(y) and characteristic function φ(ξ) of common jump distributions. With
jump intensity λ, the characteristic exponent of the jump contribution is λ(φ(ξ)−1). Note that
κ:= φ(−i)−1, and λκ is the drift compensator.
2.1 Pricing by Characteristic Function
In this section we derive a general characteristic function recursion which facilitates transform-based
Asian option pricing under regime switching (RS) L´evy processes, including jump diffusions. We
then show in Section 3 how to extend the method to stochastic volatility. We consider discrete and
continuously monitored call options of the form
1
M+ 1
M
X
m=0
Stm−K!+
and 1
TZT
0
Stdt −K+
where {tm}M
m=0 is a set of monitoring dates on [0, T ], with tm:= m∆t, and ∆t:= T /M. Put options
are defined similarly. Define the log returns Xmbetween monitoring dates by
Xm:= log Sm
Sm−1, Sm:= Stm, m = 1, ..., M, (6)
and recall the factorization of Carverhill and Clewlow (1990), which decomposes the average AM
AM=1
M+ 1
M
X
m=0
Sm(7)
=S0
M+ 1 (1 + exp(X1+ ln (1 + exp(X2+ ln (···XM−1+ ln (1 + exp(XM))))) .(8)
In particular, if we define the sequence
Y1:= XM, Ym:= XM−(m−1) + log(1 + exp(Ym−1))
:= XM−(m−1) +Zm−1, m = 2, ..., M. (9)
it holds that exp(YM) = 1
S0PM
m=1 Sm=M+1
S0AM−1. Then next result presents the recursion we
use to price Asian options under RS jump diffusions and stochastic volatility (in which case the price
obtained is that of the RS approximation).
Proposition 2 Consider a RS jump diffusion with log return process (Xt)t≥0described in (3). Sup-
pose that G(AM)is a European style payoff on the arithmetic average AMdefined in (7), and let
g(y;S0) := G(S0(1 + exp(y))/(M+ 1)). The value at time zero satisfies
V ◦ g(S0) = e−rT ZR
g(y;S0)fα(0)
YM(y)dy,
5

where the density fα(0)
YMof YM, conditional on α(0) = j, is given in terms of its characteristic function
φj
YM(ξ) := E[eiYMξ|α(0) = j], derived by the following recursion:
m=1: φj
Y1(ξ) = X
k=1,...,m0Ek,j (ξ), j = 1, . . . , m0
m= 2, . . . , M :φj
Zm−1(ξ) = ZR
(ey+ 1)iξfj
Ym−1(y)dy, j = 1, . . . , m0
φj
Ym(ξ) = X
k=1,...,m0
φk
Zm−1(ξ)Ek,j (ξ), j = 1, . . . , m0
where the matrix E(ξ)=[Ek,j(ξ)] is defined in (5).
Proof: Consider the case of a pure RS jump diffusion with log return process (Xt)t≥0described
in (3), corresponding to the price process (St)t≥0, which is monitored at the points tm=m∆t, with
∆t:= T/M. Let Fmdenote the filtration generated by {Sm0}0≤m0≤m. To prove the conditional ChF
recursion which leads to φj
YM(ξ) := E[eiYMξ|α(0) = j], we start with the definition of Y1=XMin (9).
To initialize the recursion, note first that for j= 1, ..., m0,
φj
Y1(ξ) := EM−1[eiξY1|αM−1=j]
=EM−1[eiξXM|αM−1=j] = E[eiξX∆t|α(0) = j],
which follows from stationarity and independence of returns. Combining with Corollary 1, we have
φj
Y1(ξ) = E[eiξX∆t|α(0) = j]
=10E(ξ)·ej
=X
k=1,...,m0Ek,j (ξ),
where ej∈Rm0×1is a vector with a one in position j, and zeros otherwise. Define the transition
probabilities
P∆t
jk := P[α(t+ ∆t) = k|α(t) = j], j, k = 1, . . . , m0.
Now for m= 2, . . . , M, recall that
Ym:= XM−(m−1) + log(1 + exp(Ym−1)) := XM−(m−1) +Zm−1,(10)
which yields
φj
Ym(ξ) := EM−m[eiξYm|αM−m=j]
=EM−m[eiξXM−m+1 ·eiξln(1+eYm−1)|αM−m=j]
=X
k=1,..,m0
EM−m[eiξXM−(m−1) ·eiξZm−1|αM−m=j, αM−(m−1) =k]·P∆t
jk
=X
k=1,..,m0
EM−m[eiξXM−(m−1) |αM−m=j, αM−(m−1) =k]·E[eiξZm−1|αM−(m−1) =k]·P∆t
jk
(11)
6

where the second equality holds from (10) and the third applies the tower property of conditional
expectation. The final equality results from conditional independence of XM−(m−1) and Zm−1, given
{αM−m=j, αM−(m−1) =k}, and from the fact that E[eiξZm−1|αM−m=j, αM−(m−1) =k] =
E[eiξZm−1|αM−(m−1) =k], due to the increasing filtration.
Continuing from (11), we have
φj
Ym(ξ) = X
k=1,..,m0
E[eiξX1|α0=j, α1=k]·E[eiξZm−1|αM−(m−1) =k]·P∆t
jk
=X
k=1,..,m0
E[eiξX∆t|α(0) = j, α(∆t) = k]·φk
Zm−1(ξ)·P∆t
jk
=X
k=1,..,m0
E[eiξX∆t|Gjk]·φk
Zm−1(ξ)·P∆t
jk ,(12)
which follows from the stationary and independent increments of X∆t, conditional on α(0) and α(∆t),
and from the definitions Gjk := {α(0) = j, α(∆t) = k}and φk
Zm−1(ξ) := E[eiξZm−1|αM−(m−1) =k].
Hence we have separated the two variables XM−(m−1) and Zm−1, which are not independent since
information about the state α(t) is coupled in the returns. Let {Tl}={T∆t
l}denote the sojourn times
which represent the time spent in state lduring an arbitrary period of length ∆t
Tl=T∆t
l:= Z∆t
0{α(t)=l}dt, l = 1, ..., m0,(13)
and note that
X∆t
d
=
m0
X
l=1
XTl.
If we condition on the sojourn times on [0,∆t], the conditional ChF satisfies
EheiξX∆t{Tl}m0
l=1i=EheiξPm0
l=1 XTl{Tl}m0
l=1i
=EhePm0
l=1 Tlψl(ξ){Tl}m0
l=1i,(14)
due to the L´evy increments, where ψl(ξ) is the L´evy symbol defined in (35).
Using the fact that for two sigma-algebras G1⊂ G2,E[X|G1] = E[E[X|G2]|G1], we obtain
E[eiξX∆t|Gjk] = EheiξX∆tα(0) = j, α(∆t) = ki
=EhEheiξX∆t{Tl}m0
l=1, α(0) = j, α(∆t) = kiα(0) = j, α(∆t) = ki
=EhEheiξPm0
l=1 XTl{Tl}m0
l=1iα(0) = j, α(∆t) = ki
=EhePm0−1
l=1 Tlψl(ξ)+(∆t−Pm0−1
l=1 Tl)ψm0(ξ)α(0) = j, α(∆t) = ki
=e∆tψm0(ξ)EhePm0−1
l=1 Tl(ψl(ξ)−ψm0(ξ))α(0) = j, α(∆t) = ki
=e∆tψm0(ξ)
P∆t
jk
EhePm0−1
l=1 Tle
ψl(ξ){α(∆t)=k}α(0) = ji,
which follows from (14) and algebra, where e
ψl(ξ) := ψl(ξ)−ψm0(ξ). Hence, by Proposition 3.1 of
Ramponi (2012) (which extends Buffington and Elliot (2002) and Chourdakis (2002) to include jump
7

diffusions), we have
E[eiξX∆t|Gjk] = e∆tψm0(ξ)
P∆t
jk he∆t(Q0+diag(˜
ψ1(ξ),..., ˜
ψm0−1(ξ),0)) jik
=1
P∆t
jk he∆t(Q0+diag(ψ1(ξ),...,ψm0(ξ))) jik:= 1
P∆t
jk Ejk(ξ).(15)
Substituting (15) into the expression for φj
Ym(ξ) in (12), we arrive at
φj
Ym(ξ) = X
k=1,...,m0
E[eiξX∆t|Gjk]·φk
Zm−1(ξ)P∆t
jk
=X
k=1,...,m0
φk
Zm−1(ξ)Ek,j (ξ).
Finally, note that
φj
Zm−1(ξ) = E[eiξZm−1|αM−(m−1) =j] = ZR
(ey+ 1)iξfj
Ym−1(y)dy,
since exp(iξln(1 + exp(y))) = (1 + exp(y))iξ.
Remark 1 To apply Proposition 2 to price a vanilla Asian option with strike K > 0, note that
g(y;S0) = g(y) := 






S0(1 + exp(y))
M+ 1 −K[y≥y∗],for a call,
K−S0(1 + exp(y))
M+ 1 [y≤y∗],for a put,
(16)
where
y∗:= ln ((M+ 1)K/S0−1) .(17)
The next result characterizes the decay of φj
Ym(ξ) for large ξ∈R, in terms of e
φj
X∆t(ξ) defined in
equation (4). The exponential L´evy (and jump diffusion) models of interest satisfy
|e
φj
X∆t(ξ)|=|exp(ψj(ξ)∆t)| ≤ κjexp (−∆tcj|ξ|νj), j = 1, . . . , m0,(18)
for some κj, cj>0 and νj∈(0,2] which defines the decay rate. When νj= 1 (e.g. a Normal-Inverse-
Gaussian process), the decay is exponential. Pure diffusions and jump diffusions (e.g. Merton’s model
or Kou’s double exponential model) satisfy the bound with νj= 2, with greater than exponential
decay.
Lemma 1 Assume that for sufficiently large ξ, there exists 1≤¯
k≤m0such that |e
φj
X∆t(ξ)| ≤
|e
φ¯
k
X∆t(ξ)|for 1≤j≤m0, with the parameters as in (18). Then φj
Ym(ξ)satisfies
|φj
Ym(ξ)| ≤ κ¯
kexp (−∆tc¯
k|ξ|ν¯
k), j = 1, . . . , m0, m = 1, . . . , M,
which holds for large |ξ|.
8

Proof. Consider the case of m≥2, where m= 1 follows similarly. For this proof we use the notation
k·k to denote modulus to distinguish from conditioning. For fixed j,


φj
Ym(ξ)

≤X
k=1,...,m0

E[eiξX∆t|Gjk]

·

φk
Zm−1(ξ)

·P∆t
jk
≤max
k=1,...,m0

E[eiξX∆t|Gjk]

,
where Gjk := {α(0) = j, α(∆t) = k}. For j, k fixed, and Tl≡T∆t
ldefined in (13),


E[eiξX∆t|Gjk]

=

EhEheiξX∆t|{Tl}m0
l=1i|Gjki


≤Eh

EhePm0
k=1 Tlψl(ξ)|{Tl}m0
l=1i

|Gjk i
≤E[kexp (ψ¯
k(ξ)∆t)k|Gjk ] = kexp (ψ¯
k(ξ)∆t)k,
for sufficiently large ξ, since for any realization of {Tl}m0
l=1 ={tl}m0
l=1,




exp m0
X
l=1
tlψl(ξ)



≤



exp ψ¯
k(ξ)
m0
X
l=1
tl



=kexp (ψ¯
k(ξ)∆t)k.
Since this bound is independent of j, k, the result follows. 
3 Stochastic Volatility
The methodology introduced in Section 2 can now be applied to price Asian options under stochastic
volatility. Our general approach is to transform a stochastic volatility model into a process that
is well approximated by a much simpler regime-switching model, which is accomplished as follows.
First, we derive a transformation which “decorrelates” the stochastic volatility and underlying asset
processes. Next we approximate the stochastic volatility diffusion using a locally consistent Markov
chain. We then model the asset processes as a regime switching jump diffusion, where the regimes
represent a discretized state-space for the variance process. Once the regime switching approximation
has been formed, we transition to the Fourier domain and apply a recently developed frame projection
methodology to obtain prices.
3.1 Decorrelating a stochastic volatility model
Consider the following stochastic volatility model with a jump component in the stock price









dSt
St−
= Γdt +κ(vt)dW 1
t+d

N(t)
X
i=1
(eJi−1)
,
dvt= ˆµ(vt)dt + ˆσ(vt)dW 2
t,
(19)
where E[dW 1
tdW 2
t] = ρdt with ρ∈(−1,1), and all expectations are with respect to the risk-neutral
measure on (Ω,F,Q). Here Ntis a Poisson process with rate λ, and Ji∼Jis the jump amplitude with
κ:= E[eJ−1], and they are independent of W1
tand W2
t. The drift Γ := r−q−λκ compensates the
9

jump process under Q, and κ(vt) is defined so that (19) has a unique strong solution. Our first goal
is to obtain an equivalent representation for the process such that the correlated Brownian motions
driving dStand dvtare separated.
Let Sc
tdenote the continuous part of the stock price. An application of Ito’s Lemma for jump
processes (see Cont and Tankov (2003)) yields
dlog(St) = 1
St
dSc
t−1
2S2
t
(dSc
t)2+d
X
0<s≤t
[log(Ss)−log(Ss−)]

=Γ−1
2κ2(vt)dt +κ(vt)dW 1
t+d
X
0<s≤t
[log(Ss)−log(Ss−)]
.(20)
The decorrelation approach is as follows. With can arbitrary constant define
ˆ
f(x) := Zx
c
κ(u)
ˆσ(u)du, h(x) := L(ˆ
f(x)) = ˆµ(x)ˆ
f0(x) + 1
2ˆσ2(x)ˆ
f00(x),(21)
and let f(vt, v0) := ρ(ˆ
f(vt)−ˆ
f(v0)), which satisfies the dynamics
df(vt, v0) = ρh(vt)dt +ρκ(vt)dW 2
t.(22)
Further define W∗
t:= W1
t−ρW 2
t
√1−ρ2. One can easily verify that W∗
tis a standard Brownian motion and
E[dW ∗
tdW 2
t] = 0, from which the two Brownian motions W∗
tand W2
tare independent. Next, substitute
(22) into (20), and obtain
dlog(St) = Γ−1
2κ2(vt)dt +κ(vt)(ρdW 2
t+p1−ρ2dW ∗
t) + d
X
0<s≤t
[log(Ss)−log(Ss−)]

=Γ−1
2κ2(vt)dt +df(vt, v0)−ρh(vt)dt +p1−ρ2κ(vt)dW ∗
t+d

N(t)
X
i=1
Ji
.(23)
With e
Xt= log(St/S0)−f(vt, v0), we can reformulate the model as



de
Xt=Γ−1
2κ2(vt)−ρh(vt)dt +p1−ρ2κ(vt)dW ∗
t+dPN(t)
i=1 Ji,
dvt= ˆµ(vt)dt + ˆσ(vt)dW 2
t.
(24)
In particular, (24) provides us with a decorrelated representation of the dynamics in (19), in terms of
the new auxiliary process e
Xt.
3.1.1 Example: 4/2 model with jumps
Many prominent examples fall within the framework of equation (19), including those of Heston,
Hull-White, Stein-Stein, Jacobi, α-Hypergeometric, 3/2 and 4/2 models for virtually any type of
jump amplitude distribution in the return process. We illustrate the transform required to obtain a
decorrelated representation for the 4/2 model. The reader is invited to refer to Appendix C for a list
of additional models considered in this paper.
10

The 4/2 stochastic volatility model (without jumps) was recently proposed by Grasselli (2017),
with the important property that the instantaneous volatility can be uniformly bounded away from
zero (unlike Heston’s model, for example). We extend the 4/2 model of Grasselli by adding a jump
component in the underlying process, which results in the dynamics
4/2: 


dSt
St−= (r−q−λκ)dt + [a√vt+b
√vt]dW 1
t+dPN(t)
i=1 (eJi−1),
dvt=η(θ−vt)dt +σv√vtdW 2
t.
(25)
For this model, it is assumed that Feller’s condition ηθ > 1
2σ2
vis satisfied, and for a, b > 0, the
volatility component [a√vt+b
√vt] is uniformly bounded away from zero2, see Grasselli (2017). The
change of variable that will help us to remove the correlation between the two stochastic processes
W1
t,W2
tin (25) is given by
e
Xt= log St
S0−ρ
σva(vt−v0) + b(log(vt)−log(v0)).(26)
Therefore, if we let µ(t) = h(aρη
σv−a2
2)vt+ (ρbσv−b2
2−bρηθ
σv)1
vt+ρη
σv(b−aθ) + r−q−λκ −abi, then



de
Xt=µ(t)dt + [a√vt+b
√vt]p(1 −ρ2)dW ∗
t+dPN(t)
i=1 Ji,
dvt=η(θ−vt)dt +σv√vtdW 2
t.
(27)
As discussed in Appendix C.2, the 4/2 model nests the 3/2 model and Heston’s model as a special case.
Details on the decorrelated formulation are provided in Appendix C for the other models considered.
3.2 A Markov chain approximation for the variance
We can now recast the original stochastic volatility model from (19) in the framework of regime
switching. Given a decorrelated representation of the stochastic volatility dynamics, the next step is
to approximate vtby a m0−state continuous time Markov chain (CTMC) vα(t). Suppose the variance
process under the pricing probability measure follows a diffusion



dvt= ˆµ(vt)dt + ˆσ(vt)dW 2
t,
v(0) = v0.
(28)
In the following, we will detail a construction of a continuous time Markov chain vα(t)which converges
weakly to vt. To this end, we note that the infinite generator Lof vtis given by
LΠ(v) = ˆµ(v)∂Π(v)
∂v +1
2ˆσ2(v)∂2Π(v)
∂v2,(29)
for Π in the domain of L. Let v={v1, v2, . . . , vm0}be the truncated domain of vt(the choice of grid is
detailed in Section A.1 using the moments of vt), and k={k1, k2, . . . , km0−1}the set of grid spacings
2which can be seen by applying Cauchy’s inequality to [a√vt+b
√vt]≥2qa√vtb
√vt= 2√ab > 0 for a, b > 0
11

with ki=vi+1 −vi. A non-uniform finite discretization of Lis given by
ˆµ(vi)−ki
ki−1(ki−1+ki)Π(vi−1) + ki−ki−1
kiki−1
Π(vi) + ki−1
ki(ki−1+ki)Π(vi+1)
+ˆσ2(vi)
22
ki−1(ki−1+ki)Π(vi−1)−2
ki−1ki
Π(vi) + 2
ki(ki−1+ki)Π(vi+1)
=qi,i−1Π(vi−1) + qi,iΠ(vi) + qi,i+1 Π(vi+1) =: Lm0
α(t)Π(v),
where qi,j’s are chosen as in Lo and Skindilias (2014),
qij =



















ˆµ−(vi)
ki−1
+ˆσ2(vi)−(ki−1ˆµ−(vi) + kiˆµ+(vi))
ki−1(ki−1+ki),if j=i−1,
ˆµ+(vi)
ki
+ˆσ2(vi)−(ki−1ˆµ−(vi) + kiˆµ+(vi))
ki(ki−1+ki),if j=i+ 1,
−qi,i−1−qi,i+1,if j=i,
0,if j6=i−1, i, i + 1.
(30)
Moreover, if kis chosen such that
0<max
1≤i≤m0−1{ki} ≤ min
1≤i≤m0ˆσ2(vi)
|ˆµ(vi)|
then we have
ˆσ2(vi)≥max
1≤i≤m0−1{ki}·|ˆµ(vi)| ≥ max
1≤i≤m0−1{ki} · (ˆµ+(vi) + ˆµ−(vi))
≥ki−1ˆµ−(vi) + kiˆµ+(vi).(31)
Hence we have qij ≥0,∀1≤i6=j≤m, and Pm0
j=1 qij = 0, i = 1, . . . , m0.
Now let vα(t)be a continuous time Markov chain taking values in {v1, v2, . . . , vm0}with the gen-
erator Q= [qij]m0×m0as in (30), then we have
Lm0
α(t)Π(v)−→ LΠ(v), as max{|ki|:i= 1, . . . , m0−1} → 0.(32)
By Theorem 5.1 of Mijatovi´c and Pistorius (2013), vα(t)will converge weakly to its continuous coun-
terpart vt. We then replace the diffusive variance process vtwith the CTMC approximation vα(t), to
obtain a regime switching approximation. More details on the convergence of CTMC approximations
and grid design can be found in Li and Zhang (2017b) and Li and Zhang (2017a).
3.3 Characteristic Function of Auxiliary Process
Recall the decorrelated stochastic volatility representation in (24), where f(·) was chosen so that
log(St/S0) = e
Xt+f(vt, v0),
where the auxiliary system satisfies



de
Xt=Γ−1
2κ2(vt)−ρh(vt)dt +p1−ρ2κ(vt)dW ∗
t+dPN(t)
i=1 Ji,
dvt= ˆµ(vt)dt + ˆσ(vt)dW 2
t,
(33)
12

with W∗
tand W2
tindependent. After approximating the variance process by a CTMC with states
{v1, . . . , vm0}indexed by the chain α(t) on j= 1, . . . , m0described in Section 3.2, the term e
Xtcan be
cast in the form of an RS jump diffusion process
de
Xt=ζα(t)dt +σα(t)dW ∗
t+d

N(t)
X
i=1
Ji
,(34)
where 


ζα(t)= (r−q−λκ)−1
2κ2(vα(t))−ρh(vα(t)),
σα(t)=p1−ρ2κ(vα(t)),
and W∗
tis a standard Brownian motion independent of W2
tthat drives the variance process. In
particular, the the decorrelated representation is vital as α(t) must be independent of W∗
t.
Hence, as α(t) tracks movements in the variance process, the dynamics of e
Xtin state jare described
by the characteristic exponent
ψj(ξ)=iζjξ−1
2ξ2σ2
j+λ(φ(ξ)−1), j = 1, . . . , m0(35)
where φ(ξ) is the characteristic function of jump magnitude, which is state-independent. With X∆t∼
ln(St+∆t/St) and Gj,k := {α(0) = j, α(∆t) = k}, it follows that
E[eiξX∆t|Gj,k ] = E[eiξe
X∆t|Gj,k]·exp(iξf (vk, vj))
=Ek,j (ξ)·exp(iξf(vk, vj)).
To handle applications for pure RS models as well as stochastic volatility, we introduce the notation
e
Ek,j (ξ) := 


Ek,j (ξ),regime-switching model
Ek,j (ξ)·exp(iξf(vk, vj)),stochastic volatility model,
(36)
where Eis defined in equation (5). The recursion in Proposition 2 is then applied to e
E. We summarize
the methodology presented thus far in Algorithm 1, which forms the initialization step of the APROJ
algorithm. Note that the input grid {ξn}N
n=1 will be discussed in section 4.1. In the RS case, the
generator matrix Qis input as part of the model, while it is constructed in the SV case. After this
initialization step, which returns the matrix e
Edefined in (36), as well as the initial chain index j0, the
main pricing algorithm will proceed identically in either case, and is detailed in the following sections.
4 The APROJ Algorithm
The previous sections derived a unified approach to pricing Asian options under general dynamics in
the Fourier domain. We next derive an implementation of this approach using the Fourier method of
Kirkby (2015, 2017a, 2018).
13

Algorithm 1 Initialization
Regime Switching Case:
Input: {ξn}N
n=1, a grid in frequency domain
Input: L´evy symbols, {ψj}m0
j=1, described in Section 2
Input: generator matrix Q= [qj k]m0×m0of regime switching
Input: initial state, j0∈ {1, . . . , m0}
Compute e
E(ξn), n= 1, . . . , N by (36)
return e
E, j0
Stochastic Volatility Case:
Input: {ξn}N
n=1, a grid in frequency domain
Input: L´evy symbols, {ψj}m0
j=1, described in Section 2
Input: SV model parameters for auxiliary process in (24)
Construct variance grid {vj}m0
j=1 with initial state j0as prescribed in Appendix A.1
Construct generator matrix Q= [qjk ]m0×m0using (30)
Compute e
E(ξn), n= 1, . . . , N by (36)
return e
E, j0
4.1 Biorthogonal Projection
The method of frame projection introduced in Kirkby (2015) represents the probability density of a
random variable as the orthogonal projection onto a suitably chosen Riesz basis (a special class of
frame).3Consider a random variable Xwith (unknown) density fX. Given a compactly supported
generator ϕ(x), a resolution a > 0, and a reference point x1, we form the infinite dimensional space
Ma:= span{ϕa,n}n∈Zwhere ϕa,n(x) := a1/2ϕ(a(x−x1)). The orthogonal projection PMafXof fX
onto Mais given in terms of a dual basis {eϕa,n}n∈Zby
PMafX(x) = X
n∈ZhfX,eϕa,niϕa,n(x),
where the dual basis is biorthogonal in the sense that hϕa,k,eϕa,ni={k=n}(orthogonal bases are
self-dual). While the form of eϕa,n is generally unknown, PMafXcan still be determined using the
Fourier transform b
eϕof eϕ:
hfX,eϕa,ni=a−1/2
π<Z∞
0
exp(−ixnξ)·φX(ξ)b
eϕξ
adξ,(37)
assuming the characteristic function φX(ξ) is known. In this work, we utilize the cubic B-spline
generator
ϕ[3](y) =















(y+ 2)3/6, y ∈[−2,−1]
2/3−y3/2−y2, y ∈[−1,0]
2/3 + y3/2−y2, y ∈[0,1]
(2 −y)3/6, y ∈[1,2].
(38)
First we restrict Mato the finite set {xn}N
n=1, where xn=x1+ (n−1)/a, where the choice of a, N
and x1will be discussed below. If we define an N-point frequency grid
∆ξ= 2πa/N, ξn= (n−1)∆ξ, n = 1, ..., N, (39)
3For more details on basis theory and its applications in finance, see Kirkby and Deng (2019).
14

and constant Υa,N := 32a4/N, the coefficients a1/2Υa,N ·¯
βa,k ≈βa,k =hfX,eϕa,niare found using the
exponentially convergent discretization
¯
βa,kN
k=1 := <{D{Hj}} ,Dn{Hj}=
N
X
j=1
e−i2π
N(j−1)(n−1)Hj, n = 1, ..., N , (40)
where Dis the discrete Fourier transform (DFT). The DFT input vector {Hj}N
j=1 is defined by
H1:= 1/32a4, Hn:= φX(ξn)·ζn·exp(−iξn·x1), n ≥2.
where4
ζn:= 2520(sin(ξn/(2a))/ξn)4
1208 + 1191 cos(ξn/a) + 120cos(2ξn/a) + cos(3ξn/a), n ≥2.(41)
4.2 Characteristic Function Recovery
Consider an RS model, which may be the result of the CTMC approximation of stochastic volatility
described above. Let {ψj(ξ)}m0
j=1 be the set of characteristic exponents for each of the m0states,
defined in Section 2.1. Let e
Ek,j (ξ) be defined as in (36). According to Proposition 2, at time step
m= 1 the characteristic function recursion is initialized along the frequency grid (39) with
φj
Y1(ξn) = X
k=1,...,m0e
Ek,j (ξn), j = 1, . . . , m0, n = 1, . . . , N,
which is available in closed-form.
For each m= 2, . . . , M, and j= 1, . . . , m0, we first require
φj
Zm−1(ξn) = ZR
(ey+ 1)iξnfj
Ym−1(y)dy, n = 1, . . . , N,
where fj
Ym−1(y) is known only by its characteristic function φj
Ym−1(ξ), approximated by ¯
φj
Ym−1(ξ) in
the previous time step. We find fj
Ym−1(y) by projecting onto the cubic basis as in Section 4.1, using a
grid budget of size N(an N-point basis projection is defined for each jin each stage m). The budget is
determined by two parameters, P, ¯
P∈N+, where Pdetermines the resolution through ∆ := 2P, and
¯
Pdetermines the truncated density support of width ¯a= 2 ¯
P. With P, ¯
Pchosen, we define N:= 2P+¯
P.
For enhanced accuracy and efficiency, we employ a stage-dependent grid shift defined by
¯µm:= µ1+Nm∆, Nm:= a·log exp(mθ)−1
exp(θ)−1, m = 2, ..., M, (42)
where N1= 0, θ:= (r−q)∆tand µ1=θ. Once Nmis known, all that is needed to define the grids is
the set of left grid points
xm
1= ¯µm+ (1 −N/2)∆, m = 1, ..., M −1,(43)
where xM
1is defined in Section 4.3. By design, the grids at each stage overlap, and are shifted to the
right to account for the mean growth of fj
Ym(y).
4Details on the derivation of b
eϕ(ξ) for the cubic basis can be found in Kirkby (2017b).
15

Projecting fj
Ym−1(y) onto the cubic basis, with a grid centered over [¯µm−1−¯a/2,¯µm−1+ ¯a/2] we
obtain
fj
Ym−1(y)≈¯
fj
Ym−1(y) := a1/2Υa,N
N
X
k=1
¯
βj
Ym−1,kϕa,k (y)
where n¯
βj
Ym−1,koN
k=1 := <DnHj
Ym−1,noN
n=1is defined in terms of the DFT input
Hj
Ym−1,1:= 1/32a4, Hj
Ym−1,n =¯
φj
Ym−1(ξn)·ζn·exp(−ixm−1
1ξn), n ≥2,
and the vector {ζn}N
n=2 is defined in equation (41). Note that an expansion for Ym−1is with respect
to a basis {ϕa,k}={ϕm−1
a,k }, where ϕm−1
a,k is centered over xm−1
1+ (k−1)∆. The superscript m−1 is
suppressed to ease notation. Hence, we have
φj
Zm−1(ξn)≈ZR
(ey+ 1)iξn¯
fj
Ym−1(y)dy =ZR
(ey+ 1)iξna1/2Υa,N
N
X
k=1
¯
βj
Ym−1,kϕa,k (y)dy
= Υa,N
N
X
k=1
¯
βj
Ym−1,k ·a1/2ZR
(ey+ 1)iξnϕa,k(y)dy
≈Υa,N
N
X
k=1
¯
βj
Ym−1,k ¯
Ψ(ξn, Nm−1+k) := ¯
φj
Zm−1(ξn).
Define the set x∗
k=x1
1+ (k−1)∆ for k= 1, . . . , N +NM−1, so that for m= 1, . . . , M , each subgrid
{xm
k}N
k=1 is contained within {x∗
k}N+NM−1
k=1 . With Ik:= [x∗
k−2∆, x∗
k+ 2∆], the matrix ¯
Ψ(ξn, k) is an
approximation to
Ψ(ξn, k) := a1/2ZIk
(ey+ 1)iξnϕa,k(y)dy, k = 1, ..., N +NM−1,(44)
discussed in Section A.2. In particular, Ψ(ξn, k) is independent of j= 1, ..., m0, and is computed
at initialization. Only the subset {Ψ(ξn, Nm−1+k)}N
k=1 is required at staged m. Finally, given an
approximation ¯
φj
Zm−1(ξn), we define for j= 1, . . . , m0:
¯
φj
Ym(ξn) = X
k=1,...,m0
¯
φk
Zm−1(ξn)e
Ek,j (ξn), n = 1, . . . , N.
Remark 2 In Algorithm 2, there is no need to store the vectors Hj
Ym−1,¯
βj
Ym−1,¯
φj
Zm−1and ¯
φj
Ym−1
at each time step. Instead, we define Hj,¯
βj,¯
φj
Zand ¯
φj
Yfor j= 1, . . . , m0which are overwritten in
each stage to conserve storage. Moreover, in the last stage M, we only need to recover ¯
φj
YM(ξn)for
the initial state j0rather than j= 1, . . . , m0.
4.3 Final Stage Valuation
In the final stage M, the recursion terminates with ¯
φj
YM(ξn). From equation (16) with y∗defined in
(17), we define the final grid to include y∗. Hence
xM
1:= y∗−(k∗−1)∆, k∗:= ba(y∗−(¯µM+ (1 −N/2)∆)) + 1c.(45)
16

Assuming an initial state α0=j0, the valuation formula for an Asian put becomes
V ◦ g(S0, α0) = e−rT ZR
g(y;S0)fj0
YM(y)dy ≈e−rT Υa,N X
1≤k≤k∗+1
¯
βj0
Ym−1,k ·gput
k,(46)
where with C:= S0/(M+ 1), D := K−C, and
Ek:= Cexp(xM
k) = Cexp(y∗+ (k−k∗)∆), k = 1, ..., k∗+ 1,
the coefficients satisfy
gput
k:=

















¯
ϑ[3]
∗D−ϑ[3]
∗Ekk= 1, ..., k∗−2
¯
ϑ[3]
−1D−ϑ[3]
−1Ekk=k∗−1
¯
ϑ[3]
0D−ϑ[3]
0Ekk=k∗
¯
ϑ[3]
1D−ϑ[3]
1Ekk=k∗+ 1
(47)
The constants, ϑ[3]
k,¯
ϑ[3]
kare provided in Table 15 of Appendix A, which are stable five-point Newton-
Cotes approximations to the true integrals.5
For numerical robustness, call options (which have unbounded payoffs) with strike Kare priced
through put options by the following parity
CM(S0, K, T )−PM(S0, K, T ) = S0e−rT
M+ 1 e(r−d)T(M+1)
M−1
e(r−d)T
M−1!−e−rT K, (48)
where CM(S0, K, T ) and PM(S0, K, T ) denote the call and put prices, and d≥0 a continuous dividend
yield. The methodology is summarized in Algorithm 2, which applies to both RS and SV models.
The only difference in the pricing algorithm for these models is in the initialization step, provided in
Algorithm 1.
5 Numerical Examples
In the sections below, we present numerical results obtained for different stochastic volatility and
regime switching jump diffusion models. While our focus is primarily on fixed-strike Asian options, it
would be very interesting to extend the results of Henderson and Wojakowski (2002), and of Eberlein
and Papapantoleon (2005) to price floating-strike Asian options under stochastic volatility framework
(see also Henderson et al. (2007)), which we leave as an interesting question for future studies. In the
remaining sections, we choose q= 0.
5.1 A 2-state regime switching jump diffusion
As a first example, we consider two-state regime switching jump diffusion models. For these exper-
iments, we use the model parameters from Florescu et al. (2013): K= 100, r= 0.05, σ1= 0.15
σ2= 0.25, the process α(t) has the generator Q=
−0.5 0.5
0.5−0.5
.
5For stochastic volatility models, the initial variance state v0is not necessarily a member of the variance grid. One
approach is to apply linear interpolation with equation (46) for j=j0and j0+ 1, where vj0≤v0< vj0+1 . However, the
grid can be easily adjusted so that v0=vj0is a member.
17

Algorithm 2 Main Algorithm
Input: M, K, P, ¯
P
Set N= 2P+¯
P;γ:= 1/32a4; Υa,N := 32a4/N
Compute {ξn}N
n=1, the grid in frequency domain from (39)
Obtain j0,e
Efrom initialization Algorithm 1 (using input {ξn}N
n=1)
Compute grid shifts: {xm
1}M−1
m=1 ,{Nm}M
m=1 (eq. (42),(43))
Compute ζn,n= 1, . . . , N (eq. (41))
Compute ¯
Ψ(ξn, k), n= 1, . . . , N, k = 1, . . . , N +NM−1(Section A.2)
Initialiaze ChF: φj
Y(ξn) = Pm0
k=1 e
Ek,j (ξn) = 0e
E(ξn)jj= 1, . . . , m0, n = 1, . . . , N
for m= 2, . . . , M :do
for j= 1, . . . , m0do
Hj
1=γ;Hj
n=¯
φj
Y(ξn)·ζn·exp(−ixm−1
1ξn), n = 2, . . . , N
¯
βj=<[FFT{Hj
n}N
n=1]
¯
φj
Z(ξn) = Υa,N PN
l=1 ¯
βj
l·¯
Ψ(ξn, Nm−1+l), n = 1, . . . , N
end for
for j= 1, . . . , m0do
¯
φj
Y(ξn) = Pm0
k=1 ¯
φk
Z(ξn)·e
Ek,j (ξn)n= 1, . . . , N
end for
end for
Valuation:
Define y∗(eq (17)) and final grid shift xM
1, k∗(eq. (45))
Hj0
1=γ;Hj0
n=¯
φj0
Y(ξn)·ζn·exp(−ixM
1ξn), n = 2, . . . , N
¯
βj0=<[FFT{Hj0
n}N
n=1]
Value Put: V ◦ g(S0)≈e−rT Υa,N Pk∗+1
n=1 ¯
βj0
n·gput
n
For call, use put-call-parity (eq. (48))
18

V1V2
S0APROJ MC 95CI APROJ MC 95CI
92 10.9208 10.9170 [10.8635, 10.9705] 7.8275 7.8275 [7.7895, 7.8656]
96 13.0186 13.0197 [12.9606, 13.0788] 9.7930 9.8012 [9.7581, 9.8444]
100 15.3130 15.3195 [15.2544, 15.3846] 12.0203 12.0248 [11.9763, 12.0733]
104 17.7867 17.7790 [17.7078, 17.8502] 14.4887 14.4895 [14.4354, 14.5436]
108 20.4220 20.4308 [20.3532, 20.5083] 17.1727 17.1711 [17.1110, 17.2311]
Table 2: Asian call prices with normal jumps: K= 100, r = 0.05, T = 1, M = 500. Diffusion parameters:
σ1= 0.15, σ2= 0.25. Normal jumps: a1=a2=−0.10, b1=b2= 0.30, λ1= 5, λ2= 2. Vjdenotes the option
value in regime j.
In Table 2, we consider Asian call options with normal jump Normal(−0.1,0.32). Since there is
no result in the literature to compare with, we compare our results with those obtained by Monte
Carlo simulations. For Monte Carlo simulation, we use Euler scheme with 106sample paths and 1000
timesteps. We report the mean (the “MC” column) and the 95% confidence interval (the “95CI”
column), obtained by taking the average of 10 simulations. The two regimes have the same jump
distribution (i.e. ν1(y) = ν2(y)), but with arrival rates λ1= 5 and λ2= 2, and diffusion volatilities
σ1= 0.15, σ2= 0.25. In each case, the discrepancy with MC is small (usually less than a penny).
As illustrated in Table 4 below, the APROJ algorithm is very efficient, requiring only fractions of a
second when the number of regimes is about ten or less.
V1V2
S0APROJ MC 95CI APROJ MC 95CI
92 17.8773 17.8995 [17.7360, 18.0629] 14.9958 14.9841 [14.8850, 15.0831]
96 19.8945 19.9069 [19.7330, 20.0809] 17.1689 17.1727 [17.0686, 17.2768]
100 22.0555 22.0599 [21.8832, 22.2366] 19.5192 19.5537 [19.4428, 19.6645]
104 24.3602 24.3538 [24.1641, 24.5434] 22.0360 22.0450 [21.9268, 22.1633]
108 26.8040 26.8382 [26.6377, 27.0387] 24.7047 24.6930 [24.5698, 24.8161]
Table 3: Asian call prices with different jump distributions: K= 100, r = 0.05, T = 1, M = 500. Diffusion
parameters: σ1= 0.15, σ2= 0.25. Normal jumps in regime 1: a1=−0.10, b1= 0.30, λ1= 5, DE jumps in
regime 2: η1,2= 3.0465, η2,2= 3.0775, p2= 0.3445, λ2= 2.
In Table 3 we consider Asian call options with a different jump distribution in each regime. In
regime 1, jumps are modeled by a mixture of normal random distributions, with a1,2= 0.3753,
b1,2= 0.18, a2,2=−0.5503, and b2,2= 0.6944. In regime 2, jumps are modeled by a double exponential
(DE) distribution with η1,1= 3.0465, η2,1= 3.0775. The jump arrival rates for the two regimes are
λ1= 5 and λ2= 2, and the probability of having an upward jump in each regime is p1=p2= 0.3445.
Note the clear difference among option values obtained for different jump distributions. When the
19

jump is modeled using a mixture of two normal distributions (in Table 3) with the parameters chosen,
the resulting option price is considerably higher than those obtained in the normal case in Table 2.
As before, MC confirms the reasonableness of our estimates.
P
234567
log10 |err|
-12
-10
-8
-6
-4
-2
0
M=52
None
DE
Normal
Mixed
NIG
P
234567
log10 |err|
-12
-10
-8
-6
-4
-2
0
M=252
None
DE
Normal
Mixed
NIG
Figure 1: Two Regime Convergence. Common parameters: S0=K= 100, r= 0.05, T= 1, k0= 1.
Figure 1 compares the convergence rate of APROJ as a function of the resolution parameter
Pfor five models. The first model (labeled None) is a jumpless two state diffusion with (σ1, σ2) =
(0.10,0.25), which are also the volatility rates for the jump diffusions models DE (double exponential),
Normal, and mixed (mixture of normals). These models share common jump rates (λ1, λ2) = (5,2)
and jump probability (p1, p2) = (0.35,0.35) for DE and mixture of normals. The DE parameters are
η11 =η12 = 15 and η21 =η22 = 5. For the normal jumps, a11 =a12 =−0.1 and b11 =b12 = 0.2. For
the mixture of normals, a11 = 0.1, a21 =−0.1, b11 = 0.15, b21 = 0.25, and the same holds in regime
two. Finally, we consider the Normal Inverse Gaussian model (NIG), described by (α1= 15, β1=
−5, δ1= 0.5) and (α2= 10, β2=−2, δ2= 0.25). The risk neutral L´evy symbol for this model in state
jis
ψj(ξ) = −δjqα2
j−(βj+ iξ)2−qα2
j−β2
j+ iξr+δjqα2
j−(βj+ 1)2−qα2
j−β2
j
Reference values for M= 52 are respectively [4.029586,11.067220,11.399949,12.191974,5.508016],
and for M= 252 values are [4.035733,11.100209,11.435851,12.228826,5.522320]. For the DE model,
¯
P= 4 is used to obtain e−13 accuracy, while ¯
P= 3 is sufficient for e−09. For all other models
we set ¯
P= 3. On average, doubling the resolution (ie incrementing Pby one) reduces the error by
around 100-fold. In practice, a value of ¯
P= 2 ∼3 with P= 4 ∼5 is recommended for basis point
accuracy.
20

M= 12 M= 52 M= 252
m01 2 3 1 2 3 1 2 3
7 0.006 0.025 0.030 0.009 0.036 0.052 0.021 0.106 0.176
log2(N) 8 0.026 0.067 0.078 0.035 0.101 0.136 0.072 0.274 0.427
9 0.087 0.188 0.223 0.142 0.355 0.480 0.394 1.162 1.756
Table 4: Cpu times in seconds for regime switching model averaged over 100 trials each. Total asset space grid
budget N= 2P+¯
P.
Table 4 illustrates the low cost of the APROJ method required to obtain very high precision, where
all experiments are conducted in Matlab 8.5 on a personal computer with Intel(R) Core(TM) i7-6700
CPU @3.40GHz. For each of monthly (M= 12), weekly (M= 52), and yearly (M= 252) monitoring,
we consider the regime switching algorithm for m0= 1,2,3 states, where for m0= 1 the standard
L´evy APROJ algorithm is applied for comparison. Times are reported for N= 2P+¯
P. For example,
with ¯
P= 3 fixed, the bottom row of Table 4 represents the times required for each m0and Mto
obtain an accuracy of about e−07 ∼e−11. Setting ¯
P= 2, the top row corresponds to e−03 ∼e−06.
KAPROJ PP MC 95CI
90 10.5499 10.5534 10.5439 [10.5329,10.5550]
95 6.0221 6.0272 6.0168 [6.0069,6.0267]
100 2.6027 2.6175 2.6026 [2.5953,2.6098]
105 0.7886 0.7998 0.7902 [0.7862,0.7943]
110 0.1626 0.1594 0.1622 [0.1604,0.1639]
Table 5: Options under Heston’s model without jumps: S0= 100, r = 0.05, M = 200, T = 0.25. Heston
parameters: η= 3, θ= 0.04, ρ=−0.10, σv= 0.10, v0= 0.04.
5.2 Asian options in Heston’s model
In this section, we provide numerical examples for Asian call options in Heston’s model. For com-
parison, we include the general lower bound method of Fusai and Kyriakou (2016), as well as the
simulation method of Pag`es and Printems (2005) which is implemented by the authors at Corlay et al.
(2005)6(which applies only to Heston’s model without jumps), and is presented as “PP” in the tables
below. Additional benchmarks are obtained by standard Monte Carlo simulation, for which we employ
the simulation scheme of Andersen (2008) using 106sample paths. We report the mean (the “MC”
column) and the 95% confidence interval (the “95CI” column), obtained by taking the average of 10
simulations. To reduce bias, paths are generated on a finer grid than determined by M.
To obtain benchmark values for the APROJ method in this section, we fix P= 5 and ¯
P= 3, and
variance grid parameters γ= 4.5 and m0= 200 (which we find to be conservative). Moreover, M= 200
6Accessed on September 20th 2017.
21

as for MC. From Table 5, all of results obtained by APROJ are within the 95% confidence interval of
those obtain by Monte Carlo simulations. In contrast, for the method of Pag`es and Printems (2005),
out of five values only one lies within the 95% confidence interval. Additional numerical examples
are provided in Table 6 with similar findings. For ATM and ITM contracts, their method provides
reasonable estimates (within a few cents), but at a high computational cost. On the other hand, we
find that APROJ is very stable and robust, even for OTM contracts.
T= 0.50 T= 1.0
KAPROJ PP MC 95CI APROJ PP MC 95CI
92 9.4738 9.4882 9.4736 [9.4591,9.4880] 11.0066 11.0291 11.0056 [10.9860, 11.0251]
96 6.3342 6.3553 6.3329 [6.3201,6.3457] 8.1413 8.1687 8.1292 [8.1115, 8.1469]
100 3.8498 3.8657 3.8513 [3.8407, 3.8618] 5.7581 5.7764 5.7510 [5.7355, 5.7666]
104 2.1095 2.1112 2.1055 [2.0975,2.1135] 3.8902 3.9049 3.8784 [3.8662, 3.8906]
108 1.0393 1.0547 1.0331 [1.0276, 1.0386] 2.5116 2.5156 2.5021 [2.4933, 2.5109]
Table 6: Options under Heston’s model (without jumps): S0= 100, r = 0.05, M = 200. Heston parameters:
η= 3, θ = 0.04, ρ =−0.10, σv= 0.10, v0= 0.04.
Next, we consider Asian call option in Heston’s model with jumps, which are modeled by a normal
distribution and by an double exponential distribution. In this case, there are no numerical results in
the literature to verify against, so we use Monte Carlo simulation for comparison, implemented with
the Euler scheme with 106sample paths and M= 200 (with a finer time grid to reduce bias). The
results are reported in Table 7 and Table 8 for a different set of volatility parameters, but the same
jump distributions. Also note the increase in time to maturity.
Normal Jumps Double Exponential Jumps
KAPROJ MC 95CI APROJ MC 95CI
90 13.7048 13.7025 [13.6713,13.7338] 12.4783 12.4812 [12.4606,12.5019]
95 10.1260 10.1156 [10.0863,10.1448] 8.5718 8.5540 [8.5355,8.5725]
100 7.0985 7.0728 [7.0459,7.0997] 5.2842 5.2845 [5.2695, 5.2995]
105 4.8623 4.8677 [4.8431,4.8924] 2.9415 2.9305 [2.9172,2.9439]
110 3.4313 3.4358 [3.4270, 3.4446] 1.5839 1.5875 [1.5764,1.5986]
Table 7: Options under Heston’s model (with jumps): S0= 100, r = 0.05, M = 200, T = 0.25.Heston
parameters: η= 3, θ= 0.04, ρ=−0.10, σv= 0.10, v0= 0.04. Normal jump parameters: λ= 5, a =−0.10, b =
0.30. DE jump parameters: λ= 5, η1= 10, η2= 5, p = 0.40.
22

Normal Jumps Double Exponential Jumps
KAPROJ MC 95CI APROJ MC 95CI
90 20.8970 20.8939 [20.8357, 20.9521] 17.7766 17.7753 [17.7439, 17.8066]
95 18.3265 18.3523 [18.2893, 18.4152] 14.7980 14.7978 [14.7654, 14.8302]
100 16.0255 16.0399 [15.9989, 16.0810] 12.1503 12.1326 [12.1034, 12.1617]
105 13.9850 13.9930 [13.9328, 14.0531] 9.8440 9.8460 [9.8228, 9.8691]
110 12.1915 12.2145 [12.1662, 12.2628] 7.8779 7.8765 [7.8499, 7.9032]
Table 8: Options under Heston’s model (with jumps): S0= 100, r = 0.05, M = 200, T = 1. Heston Parameters:
η= 2, θ = 0.04, ρ =−0.50, σ = 0.20, v0= 0.04. Normal jump parameters: λ= 5, a =−0.10, b = 0.30. DE jump
parameters: λ= 5, η1= 10, η2= 5, p = 0.40.
In Table 9, we report the price of Asian call options with varying Mand Kobtained by our method
and the method of lower bound of Fusai and Kyriakou (2016). Both maximal lower bound (MLB)
and suboptimal lower bound (SLB) are reported, as well as the Monte Carlo results from that work.
A close match is obtained. While both methods apply to general L´evy models and Heston (Bate’s)
model, our approach applies to general stochastic volatility as well as regime switching models, and
characteristic functions of the underlying process (which may be unknown) are not required.
MK MLB SLB MC APROJ
12 90 11.74225 11.74164 11.74271 11.74261
12 100 3.69266 3.69250 3.69286 3.69283
12 110 0.17619 0.17570 0.17629 0.17688
50 90 11.75670 11.75616 11.75717 11.75708
50 100 3.72152 3.72137 3.72173 3.72175
50 110 0.18445 0.18405 0.18455 0.18513
250 90 11.76070 11.76018 11.76115 11.76099
250 100 3.72956 3.72940 3.72976 3.73002
250 110 0.18683 0.18645 0.18693 0.18786
Table 9: Call options under Heston’s model with normal jumps (Bate’s Model): S0= 100, r = 0.0367, T = 1.0.
Heston parameters: η= 3.99, θ= 0.014, ρ=−0.79, σv= 0.27, √v0= 0.094. Normal jump: λ= 0.11, a =
−0.1391, b = 0.15.
5.3 Additional Stochastic Volatility Models
This section investigates several additional stochastic volatility models. We compare two base
parameter sets (Test Set 1 and 2) for the stochastic volatility component, provided in Table 10. For
the 3/2 model, these are the parameters before the reparameterization in equation (58). In Table
11 prices are calculated for each test and five models as a function of strike, which are confirmed by
23

Test Set 1 Test Set 2
Model σvη ρ θ v0avσvη ρ θ v0av
Stein-Stein 0.15 3 -0.6 0.18 0.15 - 0.18 2 -0.5 0.18 0.22 -
Hull-White 0.15 - -0.6 - 0.04 0.01 0.10 - -0.7 - 0.03 0.03
3/2 Model 0.15 4 -0.6 0.03 0.03 - 0.12 2 -0.3 0.04 0.035 -
4/2 Model 0.15 0.5 -0.6 0.035 0.04 - 0.10 1.8 -0.7 0.04 0.04 -
Heston 0.15 4 -0.6 0.035 0.04 - 0.12 1.5 -0.8 0.035 0.04 -
Table 10: Parameters for stochastic volatility tests. In all cases, r= 0.05, S0= 100. For the 4/2 Model:
a= 0.5, b = 0.5·v0.
Monte Carlo. Table 12 considers the same set of models, but with an additional double exponential
(DE) jump component added to each. Comparing Table 11 to Table 12, we see that the addition of
jumps produces a large increase in price for each model at all strikes. To obtain benchmark prices to
four decimals, we set P= 5 and ¯
P= 3, and fix a large value of m0= 200 ∼300. The variance grid is
determined by setting grid parameter γ= 4.5. Notice that the more deeply the option is in-the-money,
the less important the underlying model becomes, where in the limit (OTM) the price is governed by
the common growth rate of r. For OTM and ATM options, the prices vary greatly.
Test Set 1 Test Set 2
K 90 95 100 105 110 90 95 100 105 110
SS 10.5287 5.8467 2.1231 0.3377 0.0157 10.6398 6.2254 2.8007 0.8571 0.1620
HW 10.5505 6.0225 2.5973 0.7779 0.1549 10.5163 5.8550 2.2962 0.5525 0.0759
3/2 10.5147 5.8491 2.2935 0.5573 0.0793 10.5284 5.9304 2.4506 0.6749 0.1191
4/2 10.5473 6.0168 2.6033 0.7972 0.1700 10.5464 6.0141 2.5982 0.7913 0.1660
Hes 10.5619 6.0350 2.5674 0.7171 0.1184 10.5681 6.0526 2.5866 0.7217 0.1154
Table 11: Call options under Stochastic Volatility (No Jumps). Parameters as in Table 10. T=.25, M= 50.
Test Set 1 Test Set 2
K 90 95 100 105 110 90 95 100 105 110
SS 11.7194 7.5241 3.9750 1.6747 0.7332 11.8281 7.7802 4.4450 2.1778 0.9983
HW 11.7637 7.6429 4.2709 2.0650 0.9599 11.7251 7.5360 4.0593 1.8463 0.8366
3/2 11.7243 7.5329 4.0557 1.8477 0.8398 11.7421 7.5839 4.1640 1.9620 0.9031
4/2 11.7627 7.6402 4.2723 2.0768 0.9740 11.7619 7.6381 4.2686 2.0720 0.9698
Hes 11.7674 7.6498 4.2598 2.0244 0.9209 11.7719 7.6609 4.2756 2.0338 0.9217
Table 12: Call Options under Stochastic Volatility (With DE Jumps). Stochastic volatility parameters as in
Table 10. T=.25, M= 50. Common DE Jump Parameters: λ= 3, p = 0.4, η1= 10, η2= 5.
24

10 50 100 150 200 250
m0
-7
-6
-5
-4
-3
-2
-1
log10 |err|
Nonuniform
SS
HW
3/2
4/2
Hes
10 50 100 150 200 250
m0
-7
-6
-5
-4
-3
-2
-1
log10 |err|
Uniform
SS
HW
3/2
4/2
Hes
Figure 2: Convergence in m0for Test Set 2 with DE jumps: λ= 3, p = 0.4, η1= 10, η2= 5. Common
parameters: S0=K= 100, r= 0.05, T=.25, M= 50. Comparison of nonuniform (left) and uniform (right)
variance grid, both using width parameter γ= 4.5.
In Figure 2, value convergence is illustrated with respect to the variance grid size m0for each
model, where benchmark prices are respectively [4.44504,4.05935,4.16396,4.26858,4.27560] for Stein-
Stein, Hull-White, 3/2 model, 4/2 model, and Heston’s model. Comparing the value approximations
using a uniform grid with the nonuniform grid proposed in Section A.1 with α= (vm0−v1)/5, we
observe a considerable benefit when grid points are clustered more densely around the initial variance,
v0, than when they are spread uniformly about [v1, vm0]. In either case, fewer than 100 grid points
are typically required to obtain basis point accuracy.
To further illustrate the effect of the Markov chain approximation for the continuous state space
of stochastic volatility, Table 13 presents ATM prices as a function of increasing m0, given for Test
Set 1 with DE jumps. In particular, values as small as m0= 10 are often sufficient to obtain penny
accuracy. For the many of the cases we tested (figures not reported), m0= 100 was sufficient for
four correct decimals. As one can see from Table 13 however, the 3/2 model with Test Set 1 was an
exception. In practice, a value of m0= 30 ∼60 is recommended for practical accuracy, together with
a value of P= 4 ∼5 and ¯
P= 2 ∼3.
m010 20 40 60 80 100 200 300
Stein-Stein 3.97841 3.97564 3.97518 3.97509 3.97506 3.97504 3.97502 3.97502
Hull-White 4.26737 4.27005 4.27069 4.27081 4.27085 4.27087 4.27089 4.27090
3/2 Model 4.01312 4.04385 4.05277 4.05447 4.05506 4.05534 4.05571 4.05578
4/2 Model 4.40002 4.27423 4.27194 4.27218 4.27220 4.27223 4.27226 4.27226
Heston 4.25786 4.25929 4.25967 4.25973 4.25975 4.25976 4.25978 4.25978
Table 13: Option values with respect to m0on non-uniform grid. T= 0.25, M= 50, K= 100. Test set 1
with common DE jump parameters: λ= 3, p = 0.4, η1= 10, η2= 5.
25

M= 12 M= 52 M= 252
m010 20 40 80 10 20 40 80 10 20 40 80
6 0.04 0.13 0.44 1.32 0.14 0.53 1.84 5.58 0.64 2.48 8.73 26.75
log2(N) 7 0.08 0.27 0.83 2.38 0.29 1.06 3.46 10.05 1.32 5.12 16.60 52.01
8 0.18 0.54 1.63 4.58 0.60 2.07 6.73 18.99 2.58 9.60 33.07 95.51
Table 14: Cpu times in seconds for Heston’s model averaged over 10 trials each. Total asset space grid budget
N= 2P+¯
P.
Table 14 reports the cpu times in seconds for the stochastic volatility algorithm (presented for
Heston’s but the times across models are nearly identical). The times are presented for varying levels
of N= 2P+¯
Pand m0which control the accuracy. The top row with ¯
P= 2 and P= 4 represents
times required for accuracy of about e−02 ∼e−04, while the bottom row with ¯
P= 3 and P= 5
corresponds to accuracy of about e−03 ∼e−05, assuming we fix m0≥40. Considering the difficulty
of pricing Asian options under stochastic volatility (especially with jumps), and the very high cost of
Monte Carlo, the cost of APROJ is negligible by comparison. An additional advantage of APROJ
over MC and PIDE approaches is that it can be used to price contracts with small values of Mwith-
out incurring discretization error. This also enables more efficient pricing of continuously monitored
contracts, as illustrated in Appendix A.3.
The results in Tables 5-13 indicate that the APROJ performs well for volatility models with an
additional jump component. Extensive numerical experiments reveal that option values are stable as
a function of m0, and a very small value of m0= 10 ∼40 is often sufficient to obtain e-03 accuracy
or better. However, it would be interesting to find the optimal m0and establish an error bound for
the approximated option values as a function of m0. We plan to investigate these problems in future
work.
6 Conclusion
In this paper, we consider the problem of pricing Asian options under stochastic volatility models and
regime switching jump diffusions. By working in the Fourier domain, we develop a fast and highly
accurate method for pricing Asian options, including discretely monitored contracts as well as the
continuously monitored options common in foreign exchange markets. The main contribution of this
work is the development of a transform based regime-switching approximation framework for pricing
Asian options under general stochastic volatility models, which is also very promising for alternative
exotic contracts. Numerical experiments confirm the effectiveness of the proposed method. This uni-
fied framework has proven effective at solving a wide variety of problems in option pricing, including
barrier and American options Kirkby et al. (2017), realized variance derivatives Cui et al. (2017b);
Leitao Rodriguez et al. (2019), and cliquet-style equity-linked annuities Cui et al. (2017a). For a
26

review of recent developments in this area, see Cui et al. (2019a). Interesting extensions of this work
include American-style Asian options and floating strikes, stochastic interest rates (as in Boyarchenko
and Levendorskii (2013)), as well as models with jumps in volatility or time-changed processes (as in
Zeng and Kwok (2016); Cui et al. (2019b)), which we leave as topics for future research.
Acknowledgements The usual disclaimer applies. The research of Duy Nguyen is partially sup-
ported by a Marist College summer research grant.
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A Implementation Details
Cubic ¯
ϑ[3]
kϑ[3]
k
k=∗11
90 14
3(2 + cosh(∆)) + 1
2(cosh(3∆/2) + 9 cosh(5∆/4) + 23cosh(∆/2))
+1
6(cosh(7∆/4) + 121 cosh(3∆/4) + 235cosh(∆/4))
k=−123
24
1
90 1
328 + 7e−∆+1
12 14e∆+e−7∆/4+ 242 cosh(3∆/4) + 470 cosh(∆/4)
+1
4e−3∆/2+ 9e−5∆/4+ 46 cosh(∆/2)
k= 0 1
2
1
20 28
27 +1
54 e−7∆/4+ 121e−3∆/4+ 235e−∆/4
+1
18 e−3∆/2+ 23e−∆/2+1
2e−5∆/4+14
27 e−∆
k= 1 1
24
1
20 1
54 e−7∆/4+1
18 e−3∆/2+1
2e−5∆/4+7
27 e−∆
Table 15: Stable coefficient approximations derived for the cubic basis using a five point Newton-Cotes
quadrature (Boole’s rule).
A.1 Nonuniform Variance Grid
To apply the Markov chain approximation, we determine the variance grid {vj}m0
j=1 as follows. First, we fix
t=T/2 and center the grid about the mean of the variance process vtby: v1:= max{¯v, ¯µ(t)−γ¯σ(t)}if the
domain of vtis positive; otherwise v1= ¯µ(t)−γ¯σ(t). We next choose vm0:= ¯µ(t) + γ¯σ(t). Here we have defined
¯µ(t) = E[vt|v0] and ¯σ(t) the standard deviation conditional on v0. Further, we define the constant7γ= 4.5 and
¯v= 0.00001.8Finally, we generate v2, v3, . . . , vm0−1using the procedure
vi=v0+ ¯αsinh c2
i
m0
+c11−i
m0
where
c1= arcsinh v1−v0
¯α, c2= arcsinh vm0−v0
¯α
for ¯α < (vm0−v1). This transformation concentrates more grid points near the critical point v0, where the
non-uniformity of the grid is determined by the parameter ¯α: smaller ¯αresults in a more nonuniform grid.
For computations in this paper we choose ¯α= (vm0−v1)/5. Since v0is not likely a member of the variance
grid, find the bracketing index j0such that vj0≤v0< vj0+1. Holding the points v1, v2constant,9we shift the
remaining points vj, j ≥2 by v0−vj0so that vj0=v0is a member of the adjusted grid.
7We can increase γto sufficiently cover the domain of vt. From numerical experimentation, we find that γ= 4.5 is
sufficient for the models considered in this work.
8If moments of the variance process are unknown, the grid can be fixed using v1=β1v0and vm0=β2v0. For example,
β1= 10−3and β2= 4.
9This keeps an ”anchor” at the boundary in the case where v0≈0.
31

Remark 3 We note that for Heston, 3/2 (with reformulated parameters), and 4/2 models
¯µ(t) = e−ηtv0+θ(1 −e−ηt ),¯σ2(t) := σ2
v
ηv0(e−ηt −e−2ηt ) + θσ2
v
2η(1 −e−ηt +e−2ηt ).
In the Stein-Stein’s model, we have
¯µ(t) = e−ηtv0+θ(1 −e−ηt ),¯σ2(t) := σ2
v
2η(1 −e−2ηt).
For Hull-White’s model, the variance process is described by the moments
¯µ(t) = v0eavt,¯σ2(t) = v2
0e2avt(eσ2
vt−1).
A.2 Approximation of Ψ
To approximate ¯
Ψ = {¯
Ψ}n,k for n= 1, . . . , N ,k= 1, . . . , N +NM−1, we apply Gaussian quadrature over
each interval Im
k= [x∗
k−2∆, x∗
k+ 2∆], where x∗
kis defined in Section 4.2. After a change of variables, this is
accomplished by applying an Nq-point quadrature applied to each subinterval of [−2,2] = [−2,−1] ∪[−1,0] ∪
[0,1] ∪[1,2] as follows. Specifically,
Ψ(n, k) := a1/2ZIm
k
(ey+ 1)iξna1/2ϕ(a(y−x∗
k))dy
=Z[−2,2]
exp iξnln 1 + exp x∗
k+y
aϕ(y)dy
≈
4·Nq
X
l=1
ωl·exp (iξnln (1 + exp (x∗
k+γl))) ϕ(γl),
where {(γl, ωl)}4·Nq
l=1 are the nodes and weights. If we define the sample grid
ηj, j = 1, ..., Nη, Nη:= (N+NM−1−1) ·Nq+ 4 ·Nq,(49)
the Gaussian approximation of Ψ(n, k), for n= 1, . . . , N and k= 1, . . . , N +NM−1, is given by
¯
Ψ(n, k) :=
4·Nq
X
l=1
θn
Nq(k−1)+l·σl=
2·Nq
X
l=1 θn
Nq(k−1)+l+θn
Nq(k+3)+1−l·σl,(50)
where σl:= ϕ[3](γl)·ωl, l = 1, ..., 4·Nq, and
θn
j:= exp (iξnln (1 + exp (ηj))) , n = 1, ..., N, j = 1, ..., Nη.
To populate ¯
Ψ(n, k) efficiently, note that θn
j=θn−1
j·exp(i∆ξln(1+exp(ηj))). In this work, we utilize a five-point
Gaussian quadrature, detailed in Appendix A.2.1.
A.2.1 Five-Point Gaussian Quadrature
A composite five point Gaussian quadrature is implemented by applying a five point rule over each interval
[r, r + 1], for r=−2,−1,0,1. By symmetry we only evaluate for r=−2,−1. On [−2,−1], we obtain the nodes
and weights
{γl}5
l=1 =n−3
2−g3,−3
2−g2,−3
2,−3
2+g2,−3
2+g3o,{ωl}5
l=1 =1
2{ˆv3,ˆv2,ˆv1,ˆv2,ˆv3},
while on [−1,0] we have
{γl}10
l=6 =n−1
2−g3,−1
2−g2,−1
2,−1
2+g2,−1
2+g3o,{ωl}10
l=6 =1
2{ˆv3,ˆv2,ˆv1,ˆv2,ˆv3},
where we define the constants
g2:= 1
6q5−2p10/7, g3:= 1
6q5+2p10/7
ˆv1:= 128/225,ˆv2:= (322 + 13√70)/900,ˆv3:= (322 −13√70)/900.
The final weights σl=ϕ[3](γl)·wlare found by evaluating the cubic generator defined in equation (38) at each
of {γl}10
l=1, and σlis stored for repeated use.
32

A.3 Continuous Monitoring
Rather than set Mextremely large to estimate the option value with continuous monitoring, we can utilize the
now standard technique of Richardson extrapolation from values computed with modest levels of M. Let VN(M)
denote the discretely monitored value approximation with Mmonitoring dates, and with Nfixed. By choosing
d∈N+, we can approximate the continuously monitored value with a four-point Richardson extrapolation:
V∞
N(d) := 1
21 64VN(2d+3)−56VN(2d+2 ) + 14VN(2d+1)− VN(2d).
as applied in Zhang and Oosterlee (2013). An efficient APROJ extrapolation algorithm can be devised by
reusing the matrix ¯
Ψ for each d.
d/K 90 95 100 105 110
1 12.10018 8.05890 4.63329 2.20531 0.89033
2 12.10042 8.05935 4.63364 2.20555 0.89056
3 12.10045 8.05938 4.63367 2.20558 0.89058
4 12.10045 8.05938 4.63367 2.20558 0.89059
5 12.10045 8.05938 4.63367 2.20558 0.89059
Table 16: Continuously monitored Asian option values by Richardson Extrapolation. Heston’s model
under Test Set 2 from Table 10 with DE jumps: λ= 4, p= 0.40, η1= 15, η2= 5.
B Error Analysis
This section demonstrates stability of the characteristic function recursion, and provides a bound on the rate
of convergence of the APROJ value error.10 Returning shortly to the case of m= 2, we have for m≥3
(¯
φj
Zm−1(ξ)) := φj
Zm−1(ξ)−¯
φj
Zm−1(ξ) = ZR
(ey+ 1)iξ(fj
Ym−1(y)−¯
fj
Ym−1(y))dy
=ZR/Gm
(ey+ 1)iξfj
Ym−1(y)dy
+ ZGm
(ey+ 1)iξfj
Ym−1(y)dy −Υa,N
N
X
k=1
βj
Ym−1,kΨm−1(ξ , k)!
+ Υa,N
N
X
k=1
βj
Ym−1,k(Ψm−1(ξ , k)−¯
Ψm−1(ξ, k)) + Υa,N
N
X
k=1
¯
Ψm−1(ξ, k)(βj
Ym−1,k −¯
βj
Ym−1,k)
:= τ(Gm−1) + Ej
m−1,1(ξ) + Ej
m−1,2(ξ)+Ej
m−1(ξ),
where Gm= [¯µm−¯a/2,¯µm+ ¯a/2], and the final term will be further split into two components.11 The term
τ(Gm−1) captures density truncation error and can be bounded by τ(G) which converges exponentially in ¯a= 2 ¯
P
for most processes of interest. If we define PMafj
Ym−1to be the true (untruncated) orthogonal projection of
fj
Ym−1onto the space Ma, the second term satisfies
Ej
m−1,1(ξ) := ZGm
(ey+ 1)iξfj
Ym−1(y)dy −Υa,N
N
X
k=1
βj
Ym−1,kΨm−1(ξ , k)
≤
(ey+ 1)iξ
Gm−1
2·

fj
Ym−1−PMafj
Ym−1


R
2
≤√¯a·C1(fj
Ym−1)·∆4,
10To avoid excessive notation, we continue to suppress the superscript on {ϕm−1
a,k }={ϕa,k}, where the grid shift is
understood.
11Note that for the purpose of this proof we have defined βj
Ym−1,k so that a1/2Υa,N βj
Ym−1,k =hfj
Ym−1,eϕa,ki.
33

where ∆4is the theoretical convergence rate of cubic projection (in practice, the rate is much faster). The
constant C1(fj
Ym−1) is bounded by a constant multiple of 

(−iξ)4φYj
m−1(ξ)

2≤

ξ4φ¯
k
X∆t(ξ)

2for some 1 ≤¯
k≤
m0, where the inequality follows by Lemma 1. Hence, for a constant C1we have the bound
Ej
m−1,1(ξ)≤C1·√¯a·∆4(51)
which is independent of ξ, j, m. This will govern the overall rate of convergence. Next define the numerical
integration error
(¯
Ψ) := sup{|¯
Ψ(ξn, k)−Ψ(ξn, k)|: 1 ≤n≤N , 1≤k≤N+NM−1}.
For a constant C(ϕ), it follows from Lemma 5.2 of Kirkby (2016) that
Ej
m−1,2(ξ)=Υa,N
N
X
k=1
βj
Ym−1,k(Ψm−1(ξ , k)−¯
Ψm−1(ξ, k))
≤√¯a·(¯
Ψ) ·C(ϕ)·

fj
Ym−1

2≤C2·√¯a·(¯
Ψ)
for some C2, where the second inequality again follows from Lemma 1.
The remaining term, which provides the link between errors though time, is
Ej
m−1(ξ) := Υa,N
N
X
k=1
¯
Ψm−1(ξ, k)(βj
Ym−1,k −¯
βj
Ym−1,k)
=a−1/2
N
X
k=1
¯
Ψm−1(ξ, k)·(¯
βj
Ym−1,k),
where (¯
βj
Ym−1,k) := a1/2Υa,N (βj
Ym−1,k −¯
βj
Ym−1,k). Since ¯
βj
Ym−1,k contains several sources of error, we define
˘
βj
Ym−1,k to be the approximation using the true φj
Ym−1rather than ¯
φj
Ym−1. Hence
(¯
βj
Ym−1,k) = hfj
Ym−1,eϕa,k i − a1/2Υa,N ˘
βj
Ym−1,k+a1/2Υa,N ˘
βj
Ym−1,k −¯
βj
Ym−1,k
:= 1(¯
βj
Ym−1,k) + 2(¯
βj
Ym−1,k),
which yields Ej
m−1(ξ) = Ej
m−1,3(ξ) + Ej
m−1,4(ξ).
The first term can be captured by the Corollary 3.2 of Kirkby (2016), modified slightly to the present
case. We assume that e
φj
X∆t,j= 1, . . . , m0are analytic within a strip Dd:= {z∈C:=(z)∈(−d, d)}, for
some d > 0, and satisfy equation (18). We further define a constant CM, which is bounded by a multiple of
max1≤m≤M,1≤j≤m0

φj
Ym


Hd, where kfkHdis the Hardy norm of fon Dd.
Corollary 3 Fix a= 2Pand N=a·¯a, where ¯a= 2 ¯
Pfor ¯
P > 1 + log2|¯µM|. Assume Lemma 1 holds for some
c¯
k, κ¯
k>0and ν¯
k∈(0,2]. Then for some 0< γ ≤d
sup
1≤k≤Na1/2Υa,N ·˘
βj
Ym,k − hfj
Ym,eϕa,k i≤a−1/2
πCM
e−(¯a−2|¯µM|)γ/2
1−e−¯aγ +τae
φ¯
k
X∆t
:= a−1/2
πM(a, ¯a) (52)
independently of 1≤m≤Mand j= 1, . . . , m0where τa(e
φ¯
k
X∆t) = O(aexp(−∆tc¯
k·(2πa)ν¯
k)). For large enough
a > 0, and d < ∞,γwill approach d.
By Corollary 3 and |¯
Ψm−1(ξ, k)| ≤ 1 we have
|Ej
m−1,3(ξ)| ≤ a−1/2
N
X
k=1 |¯
Ψm−1(ξ, k)|·|1(¯
βj
Ym−1,k)| ≤ ¯a
πM(a, ¯a).
The discretization error represented by the first term in M(a, ¯a) decays exponentially in ¯a, while the truncation
error decays exponentially in a(for fixed ¯a).
The final term to estimate is
Ej
m−1,4(ξ) := a−1/2
N
X
k=1
¯
Ψm−1(ξ, k)·2(¯
βj
Ym−1,k)
34

where, with hm−1
a,k (ξ) := b
eϕ(ξ/a) exp(iξxm−1
k),
2(¯
βj
Ym−1,k) = a−1/2
π< ∆ξ
N
X
n=1
0φj
Ym−1(ξn)−¯
φj
Ym−1(ξn)hm−1
a,k (ξn)!
=a−1/2
π< ∆ξ
N
X
n=1
0 m0
X
l=1 φl
Zm−2(ξn)−¯
φl
Zm−2(ξn)El,j (ξn)!hm−1
a,k (ξn)!
=a−1/2
π< ∆ξ
N
X
n=1
0
m0
X
l=1
(¯
φl
Zm−2(ξn))El,j (ξn)·hm−1
a,k (ξn)!
where P0indicates that the first and last terms in the sum are halved.
Denoting (¯
φZm−2) := max1≤n≤N,1≤k≤m0|(¯
φk
Zm−2(ξn))|, and ˜(¯
φk
Zm−2(ξn)) ·(¯
φZm−2) = (¯
φk
Zm−2(ξn)),
2(¯
βj
Ym−1,k) = (¯
φZm−2)a−1/2
π< ∆ξ
N
X
n=1
0hm−1
a,k (ξn)
m0
X
l=1
˜(¯
φl
Zm−2(ξn))El,j (ξn)!
≤C·(¯
φZm−2)a−1/2
π,
for Nsufficiently large and some 0 < C<∞, since we can majorize the error term by an L2function which
admits an upper frame bound. Thus,
Ej
m−1,4(ξ) := a−1/2
N
X
k=1
¯
Ψm−1(ξ, k)·2(¯
βj
Ym−1,k)
=a−1/2(¯
φZm−2)
N
X
k=1
¯
Ψm−1(ξ, k)a−1/2
π< ∆ξ
N
X
n=1
0hm−1
a,k (ξn)
m0
X
l=1
˜(¯
φl
Zm−2(ξn))El,j (ξn)!
=O (¯
φZm−2)
a1/2
N
X
k=1
¯
Ψm−1(ξ, k)a−1/2
π< ∆ξ
N
X
n=1
0hm−1
a,k (ξn)
m0
X
l=1 El,j (ξn)!!
=O (¯
φZm−2)
a1/2
N
X
k=1
¯
Ψm−1(ξ, k)a−1/2
π< ∆ξ
N
X
n=1
0hm−1
a,k (ξn)φj
Y1(ξn)!!
=O (¯
φZm−2)
a1/2
N
X
k=1
¯
Ψm−1(ξ, k)·a1/2Υa,N ¯
βj
Y1,Nm−1+k!.
Given the exponential decay of ¯
βj
Y1,n for processes satisfying the assumptions above, we have for some constants
Cj:
Ej
m−1,4(ξ) = O (¯
φZm−2)
a1/2
N
X
k=1
¯
Ψm−1(ξ, k)·a1/2Υa,N ¯
βj
Y1,k!
≤Cj(¯
φZm−2)a−1/2|¯
φj
Z1(ξ)|.
Summarizing,
(¯
φj
Zm−1(ξ)) ≤τ(G) + |Ej
m−1,1(ξ)|+|Ej
m−1,2(ξ)|+|Ej
m−1,3(ξ)|+|Ej
m−1,4(ξ)|
≤τ(G) + C1·√¯a·∆4+C2·√¯a·(¯
Ψ) + ¯a
πM(a, ¯a)+a−1/2Cj|¯
φj
Z1(ξ)|(¯
φZm−2)
≤γM(a, ¯a) + Ω(a, ξ)(¯
φZm−2),
where γM(a, ¯a) is the term in parentheses, and Ω(a, ξ) := a−1/2max1≤j≤m0Cj|¯
φj
Z1(ξ)|. Using the same logic
as above, we can show that (¯
φZ1)≤γM(a, ¯a). Iterating from M−1 we obtain
(¯
φZM−1(ξ)) ≤γM(a, ¯a)
M−3
X
m=0
Ω(a, ξ)m+ Ω(a, ξ)M−2(¯
φZ1)
=γM(a, ¯a)1−Ω(a, ξ)M−2
1−Ω(a, ξ)+ Ω(a, ξ)M−2(¯
φZ1)
≤γM(a, ¯a)1−Ω(a, ξ)M−1
1−Ω(a, ξ)≤2γM(a, ¯a),
35

where the final inequality holds for asufficiently large. By the definition of φj
YM(ξ), and (¯
φj
YM(ξ)) := φj
YM(ξ)−
¯
φj
YM(ξ) we have with (¯
φk
ZM−1(ξ)) = (¯
φZM−1(ξ)) ·˜(¯
φk
ZM−1(ξ))
|(¯
φj
YM(ξ))|=X
k=1,...,m0
(¯
φk
ZM−1(ξ))Ek,j (ξ)=|(¯
φZM−1(ξ))|X
k=1,...,m0
˜(¯
φk
ZM−1(ξ))Ek,j (ξ)
≤¯
Cγ M(a, ¯a),
for some ¯
Cindependent of ξ. Hence, the characteristic function recursion is stable, and the error in ¯
φj
Mcan
be made arbitrarily small. In particular, with the truncation error τ(G) (which converges exponentially for
processes of interest) is controlled by the choice of ¯a,(¯
Ψ) sufficiently small by Gaussian quadrature, and
M(a, ¯a) exponentially convergent by Corollary 3, the error is dominated by that of cubic projection, which
is O(∆4). We now state the main result, whose proof is analogous to Section 5.2 in Kirkby (2016), which
characterizes the final valuation error.
Proposition 4 Given a European-style payoff g(AM)on the arithmetic average AM, suppose that the as-
sumptions of Lemma 1 hold, and that ¯ahas been fixed sufficiently large to control truncation error. De-
fine the value approximation VN◦g(S0, α0)as in equation (46), where N:= a¯a. Then the value error,
E(VN) = |V ◦ g(S0, α0)− VN◦g(S0, α0)|decays as E(VN) = O(a−4)as the resolution parameter ais in-
creased.
In practice, we typically observe a faster rate of error decay than this result would suggest, but it serves as
a conservative estimate.
C Additional stochastic volatility models
C.1 Heston and Bate’s models
Consider the Heston stochastic volatility model Heston (1993) with jumps (also known as the Bates model Bates
(1996) in the literature):
Hes: 


dSt
St−= (r−q−λκ)dt +√vtdW 1
t+dPN(t)
i=1 (eJi−1),
dvt=η(θ−vt)dt +σv√vtdW 2
t,
(53)
where ηis the mean reversion rate, θis the equilibrium level, and σv>0 is the volatility of volatility. Note
that to ensure vt>0, the Feller’s condition 2ηθ > σ2
vis imposed (see Heston (1993)). Applying the transform
in equation (21) yields
e
Xt= log St
S0−ρZvt
v0
κ(u)
ˆσ(u)du = log St
S0−ρ
σv
(vt−v0),(54)
with f(vt, v0) = ρ
σv(vt−v0). It follows that



de
Xt=h(ρη
σv−1
2)vt+ (r−q−ρηθ
σv−λκ)idt +p(1 −ρ2)vtdW ∗
t+dPN(t)
i=1 Ji,
dvt=η(θ−vt)dt +σv√vtdW 2
t.
(55)
C.2 3/2 Model with jumps
Next consider the dynamics of the 3/2 stochastic volatility model with jumps
3/2: 


dSt
St−= (r−q−λκ)dt +√vtdW 1
t+dPN(t)
i=1 (eJi−1),
dvt=vt[η(θ−vt)dt +σv√vtdW 2
t], v(0) = v0,
(56)
where in (56) vtis the variance of the asset St,ris the risk-free interest rate, σv>0, θ∈Ris the mean reversion
level, ηis given such that ηvt-a stochastic volatility quantity-is the speed of mean reversion.
36

While this form can be applied directly, we prefer an alternative formulation which nests the 3/2 model
within the 4/2 model introduced in Section 3.1.1. Applying Ito’s formula we have
d1
vt=ηθ η+σ2
v
ηθ −1
vtdt −σv
√vt
dW 2
t(57)
bvt:= 1
vt
,bη:= ηθ, b
θ:= η+σ2
v
ηθ ,bσv:= −σv,(58)
from which (56) is reduced to









dSt
St−= (r−q−λκ)dt +1
√bvt
dW 1
t+d

N(t)
X
i=1
(eJi−1)
,
dbvt=bη[b
θ−bvt]dt +bσv√bvtdW 2
t,bv0= 1/v0.
(59)
Equation equation (21) prescribes the change of variables
e
Xt= log St
S0−ρ
bσv
log bvt
bv0,(60)
which results in the decorrelated dynamics









de
Xt=h1
2ρbσv−2ρbηb
θ
bσv−11
bvt
+ρbη
bσv
+ (r−q−λκ)idt +s(1 −ρ2)
bvt
dW ∗
t+d

N(t)
X
i=1
Ji
,
dbvt=bη[b
θ−bvt]dt +bσv√bvtdW 2
t,bv0= 1/v0.
(61)
We see that the 4/2 model in (27) indeed contains the 3/2 model as a special case where a= 0, b= 1, and the
parameters for 3/2 are selected using the re-parameterization in equation (58).
C.3 Hull-White’s model with jumps
Augmenting the traditional Hull and White (1990) model with jumps produces
HW: 


dSt
St−= (r−q−λκ)dt +√vtdW 1
t+dPN(t)
i=1 (eJi−1),
dvt=avvtdt +σvvtdW 2
t.
(62)
The change of variable that will help us to remove the correlation, ρ, between the two stochastic processes W1
t,
W2
tin (62) is given by
e
Xt= log St
S0−2ρ
σv
(√vt−√v0).(63)
The decorrelated dynamics satsify









de
Xt=hρσv
4−avρ
σv√vt−1
2vt+ (r−q−λκ)idt +p(1 −ρ2)vtdW ∗
t+d

N(t)
X
i=1
Ji
,
dvt=avvtdt +σvvtdW 2
t.
(64)
C.4 Stein-Stein’s model with jumps
Stein and Stein (1991) consider a stochastic volatility model where the two Brownian motions are independent.
In this section, we extend their model by allowing for correlation and add a jump component. More specifically,
the dynamics of the model is specified as follow:
SS: 


dSt
St−= (r−q−λκ)dt +vtdW 1
t+dPN(t)
i=1 (eJi−1),
dvt=η(θ−vt)dt +σvdW 2
t,
(65)
The change of variable that will help us to remove the correlation between the two stochastic processes W1
t,
W2
tin (65) is given by
e
Xt= log St
S0−1
2
ρ
σv
(v2
t−v2
0).(66)
37

Then, we have









de
Xt=hρη
σv−1
2v2
t−ρηθ
σv
vt+r−q−λκ −ρσv
2idt +vtp(1 −ρ2)dW ∗
t+d

N(t)
X
i=1
Ji
,
dvt=η(θ−vt)dt +σvdW 2
t.
(67)
C.5 α-Hypergeometric model
The α-Hypergeometric model was recently proposed by Da Fonseca and Martini (2016). Unlike Heston model,
for α-Hypergeometric model the strict positivity of volatility is guaranteed. The dynamics of the stock price is
given by 


dSt
St−= (r−q−λκ)dt +evtdW 1
t+dPN(t)
i=1 (eJi−1),
dvt= (η−θeavvt)dt +σvdW 2
t, v(0) = v0,
(68)
where η, v0∈(−∞,+∞), θ > 0, σv>0, av>0. Let
Xt= log( St
S0
)−ρ
σv
(evt−ev0)−(r−q−λκ)t, (69)
then we have



dXt=hρθ
σve(1+av)vt−ρ(η
σv+σv
2)evt−e2vt
2idt +evtp1−ρ2dW ∗
t+dPN(t)
i=1 Ji,
dvt= (η−θeavvt)dt +σvdW 2
t.
(70)
C.6 Jacobi model
An interesting recent SV model in the literature is the Jacobi model (without jump) of Ackerer et al. (2016),
which specifies a bounded variance process Q(v), where v∈[vmin, vmax ] for 0 ≤vmin < vmax , defined by the
quadratic function
Q(v) = (v−vmin )(vmax −v)
(√vmax −√vmin)2,0≤Q(v)≤v, v ∈[vmin , vmax].
With Zt:= log(St), the dynamics under the Jacobi model with jumps are



dZt= (r−q−λκ −vt/2)dt +pvt−ρ2Q(vt)dW ∗
t+ρpQ(vt)dW (2)
t+dPN(t)
i=1 Ji,
dvt=η(θ−vt)dt +αpQ(vt)dW (2)
t.
(71)
where η≥0, θ∈[vmin, vmax ], and α > 0. Note here that E[dW (2)
tdW ∗
t] = 0, as the correlation structure is
already incorporated with correlation parameter ρ. From the equality
ρZt
0pQ(vs)dW (2)
s=ρ
α(vt−v0)−ρ
αZt
0
η(θ−vs)ds,
we can derive the auxiliary process ˜
Xt:= Zt−ρ
αvtwith
d˜
Xt=r−q−λκ −vt
2−ρ
αη(θ−vt)dt +pvt−ρ2Q(vt)dW ∗
t+d

N(t)
X
i=1
Ji

=r−q−λκ −ρ
αηθ+vtρ
αη−1
2dt +pvt−ρ2Q(vt)dW ∗
t+d

N(t)
X
i=1
Ji
.
38