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A unified approach to Bermudan and Barrier options
under stochastic volatility models with jumps
J. Lars Kirkby ∗Duy Nguyen†Zhenyu Cui ‡
May 4, 2017
Abstract
Many financial assets, such as currencies, commodities, and equity stocks, exhibit
both jumps and stochastic volatility, which are especially prominent in the market after
the financial crisis. Some strategic decision making problems also involve American-style
options. In this paper, we develop a novel, fast and accurate method for pricing Ameri-
can and barrier options in regime switching jump diffusion models. By blending regime
switching models and Markov chain approximation techniques in the Fourier domain,
we provide a unified approach to price Bermudan, American options and barrier options
under general stochastic volatility models with jumps. The models considered include He-
ston, Hull-White, Stein-Stein, Scott, the 3/2 model, and the recently proposed 4/2 model
and the α-Hypergeometric model with general jump amplitude distributions in the return
process. Applications include the valuation of discretely monitored contracts as well as
continuously monitored contracts common in the foreign exchange markets. Numerical
results are provided to demonstrate the accuracy and efficiency of the proposed method.
Keywords: American options, barrier options, stochastic volatility, regime switching,
jump diffusion, frame projection
AMS subject classifications: 91G80, 93E11, 93E20
∗School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30318, Email:
jkirkby3@gatech.edu
†Department of Mathematics, Marist College, 3399 North Road, Poughkeepsie, NY 12601, USA, Email:
nducduy@gmail.com
‡Corresponding author. School of Business, 1 Castle Point on Hudson, Stevens Institute of Technology,
Hoboken, NJ 07030. Email: zcui6@stevens.edu
1 Introduction
American call (put) options give the holder the right to buy (sell) an underlying asset at
a predetermined strike price at any point in time until the expiration date. According to
Barraclough and Whaley (2012) and Jensen and Pedersen (2016), a large portion of options
traded on the market are American-style options. Some strategic decision making problems
also involve American-style real options, see Copeland and Tufano (2004) and Battauz et al.
(2014). However, even under the simple Black and Scholes model based on geometric Brownian
motions (GBM), closed-form solutions do not exist for American options outside of the infinite
series expansions as in Zhu (2006). As a result, numerical methods must be employed to price
and hedge these financial derivatives. The computational approaches can be categorized as
Monte Carlo methods developed in Broadie and Glasserman (1997); Longstaff and Schwartz
(2001); Lindset and Lund (2007); Fujiwara and Kijima (2007), lattice method of Cox et al.
(1979), numerical partial differential equation (PDE) techniques in Ikonen and Toivanen (2004);
Muthuraman (2008), randomization techniques developed in Carr (1998), integral equation
methods of Myneni (1992); Carr et al. (1992); Jacka (1991), finite element method in Zhu
and Chen (2013), radial basis method in Ballestra and Pacelli (2013), expansion method in
Medvedev and Scaillet (2010), and the moving boundary approach in Muthuraman (2008). For
an overview of the literature on pricing American options, we refer the reader to Broadie and
Detemple (2004), Detemple (2005), Detemple and Emmerling (2009) and references therein.
The previous literature mentioned above considers American options under the constant
volatility model, i.e. the Black and Scholes model. However, empirical evidence reveals the
stochastic nature of the volatility. Earlier studies in Scott (1987) further confirm the random
change of volatility over time, and Rubinstein (1994) provides empirical evidence suggesting
that the constant volatility does not hold for options written on the S&P 500 index. Bates
(1996) provided further evidence of the existence of jumps in the underlying index. As a result,
several extensions have been proposed to account for more realistic descriptions for volatility
dynamics. Among the most popular extensions are stochastic volatility models of Hull and
White (1990), Heston (1993), Stein and Stein (1991), Scott (1987), the 3/2 model (see Carr
and Sun (2007)), and recently the 4/2 model of Grasselli (2016) and the α-Hypergeometric
model of Da Fonseca and Martini (2016).
While the pricing of European options under stochastic volatility models has been exten-
sively studied in the literature, relatively little attention has been paid to the problem of Amer-
ican options under stochastic volatility models. Approaches in the literature include the partial
differential equation (PDE) with penalty method of Zvan et al. (1998), the exercise-policy im-
provement procedure of Chockalingam and Muthuraman (2011), the sequential Monte Carlo
1
method developed in Rambharat and Brockwell (2010), and a nonparametric approach utilized
in Broadie et al. (2000). For Heston model, Ikonen and Toivanen (2008) consider the problem of
pricing American put options using five numerical methods, and Zhylyevskyy (2010) develops
a fast Fourier transform approach to price American options by extending the Geske-Johnson
compound option scheme.
So far the literature discussed above is limited to stochastic volatility models without jumps.
On the other hand, empirical studies reveal that there exist sudden rapid movements in the
mid quotes of stock prices, i.e. jumps during the trading period, as documented in empirical
studies by Bates (1996), Carr and Wu (2003), Lee and Hannig (2010), etc. The recent financial
crisis highlights that event risks such as a market crash may be significant, e.g. the flash crash
on May 6th, 2010. Motivated from the need to model random volatility fluctuations and the
existence of jumps, stochastic volatility models with jumps have been developed. They improve
the pricing and hedging performance of traditional jump diffusion models (see Merton (1976)),
and in particular, account for certain imperfections, such as the volatility smile. In contrast
to the case of diffusion models, very few studies are known for American options in stochastic
volatility models with jumps.
In a recent paper, Salmi et al. (2014) develop a (two-step implicit-explicit) IMEX scheme
to price American put options in Heston model with simultaneous jumps in both the stock and
the variance components. It is important to distinguish our paper from theirs: their method
is based on a discretization scheme of the partial integro-differential equation (PIDE) formu-
lations, while our method is focused on Markov chain approximations and Fourier techniques
using characteristic functions. Their method is based on numerical finite difference techniques,
while our method is of a probabilistic nature. Our method can handle a general class of
stochastic volatility models with general correlation structures between the two driving Brown-
ian motions.1Transform methods have also been considered in Fang and Oosterlee (2011) and
Zeng and Kwok (2014) for the Heston model, but require an explicit form for the transition
density of variance and the conditional characteristic function of integrated variance, which is
not generally available.
In this paper, by blending regime switching models, Markov chain approximation techniques
and Fourier methods, we develop a fast and accurate approach for pricing Bermudan and
American options and (European) barrier options in general stochastic volatility models with
jumps, which we refer to as the PROJ method. This includes discretely monitored contracts as
well as continuously monitored options common in the foreign exchange (Forex) market, and
encompasses regime switching jump diffusion models as a special case. The models considered
include those of Heston, Hull-White, Stein-Stein, the 3/2 model, the 4/2 model, and the α-
1Introducing jumps in the variance process, as in Salmi et al. (2014), is left as a topic for future research.
2
Hypergeometric model with any type of jump amplitude distribution in the return process.
We believe that the results presented in this paper are new, of interest to academics and
practitioners, and can be used as a comparison benchmark in future studies. Our paper makes
the following main contributions:
1. We extend the realm of transform-based pricing methods to incorporate barrier, Bermu-
dan and American options under very general regime switching jump diffusion models,
which permit different jump distributions in each regime. Capitalizing on the rapid con-
vergence and simplicity of frame projection with the linear basis, we devise an efficient
and accurate pricing algorithm.
2. By adopting a Markov chain approximation approach, we are able to reduce the problem
of pricing American options in stochastic volatility models with jumps to that of pricing
options in a regime switching jump diffusion model. The approach thus provides a unified
method for pricing American options in many well-known stochastic volatility models with
jumps, which include those of Heston, Hull-White, Scott, Stein-Stein, α-Hypergeometric
model, the 3/2 model, and the 4/2 model with virtually any jump amplitude distribution
in the asset price.
3. We provide numerical results for American options and barrier options under regime
switching jump diffusion models and under stochastic volatility models with jumps. For
some of these models, prices have never been reported in the literature. We verify the
accuracy of our method against benchmarks based on tree methods, and cases for which
prices exist in the literature. Due to the accuracy of our method, these results can be
used as reference benchmark values for future research.
The rest of the paper is structured as follows: Section 2 casts the probability settings, in-
troduces the regime switching jump diffusion framework and the weak approximation scheme
to stochastic volatility models with jumps using continuous time Markov chains (CTMCs).
Section 3 discusses the pricing algorithm for early-exercise contracts. Section 4 illustrates the
accuracy and efficiency of the method through numerical examples with Heston model, while
Section 5 considers additional stochastic volatility models. Section 6 concludes the paper with
future research directions.
2 Regime switching framework
This section introduces the main mathematical framework for pricing European and exotic
options under regime switching jump diffusions as well as stochastic volatility models with
jumps. Consider a complete probability space (Ω,F,Q) under which all processes are defined,
3
where Qis the risk-neutral probability measure. We assume the existence of a finite state
continuous-time process, denoted by α(t), which describes the current state of the economy at
time t, and resides in the set of possible states M:= {1,2, . . . , m0}. The transition dynamics
of α(t) are described by the generator matrix Q= [qij]m0×m0, whose elements qij satisfy (i)
qii ≤0, and qij ≥0, if i6=j, and (ii) Pjqij = 0, ∀i∈ M. In terms of qij , the current regime
makes transitions according to
Prob(α(t+ ∆t) = j|α(t) = i, α(t0),0≤t0≤t) = qij ∆t+o(∆t),∀j6=i. (1)
Let Btbe a standard Brownian motion independent of α(t). The underlying asset price St
evolves under the risk-neutral measure as
dSt
St−
= (r−q−λα(t)κα(t))dt +σα(t)dBt+ZE
[eγ(y,α(t−)) −1]πα(dy, dt),(2)
where κα(t)=RE[eγ(y,α(t−))−1]να(t)(y)dy is finite for each regime, with jump intensity λ(α(t), dy) =
λα(t)να(t)(y)dy, and πα(dy, dt) is the L´evy measure. Here ris the common risk-free interest rate,
and q≥0 the continuous dividend yield.
2.1 Characteristic functions
The methods developed in this paper rely on the characteristic function of the log return process
Xt= log(St/S0), t ∈[0, T ]. For a regime switching (RS) jump diffusion, Xtsatisfies
Xt=Zt
0
ζα(t)dt +Zt
0
σα(t)dB(t) + Zt
0ZE
γ(y, α(t−))πα(dy, dt),(3)
where ζα(t)=r−q−1
2σ2
α(t)−λα(t)κα(t)is the risk-neutral (convexity-adjusted) drift. Each state
j= 1, . . . , m0is characterized by a particular model of dynamics in that state, or equivalently
by its characteristic function. Given a time increment of size ∆t>0, and ξ∈R, define for
each state j= 1, ..., m0
e
φj
X∆t(ξ) := E[eiξX∆t|α(0 ≤s≤∆t) = j] := exp(ψj(ξ)∆t),(4)
where e
φj
X∆t(ξ) is the risk-neutral characteristic function (chf) of a process which spends the
entirety of ∆tin state j, and ψj(ξ) is its L´evy symbol. A full model of the process is specified
by describing each ψj(ξ) along with the generator Q. The characteristic function of a regime
switching jump diffusion is characterized by the set
ψj(ξ) = iξζj−1
2ξ2σ2
j+λj(φj(ξ)−1), j = 1, . . . , m0,(5)
where φj(ξ) = E[eiξγ(y,α(t))] is the characteristic function of the jump magnitude in state j. In
terms of φj(ξ),
ζj=r−q−1
2σ2
j−λj(φj(−i) −1).(6)
4
In general, if we are provided with a L´evy symbol ¯
ψj(ξ), then the risk-neutral model is obtained
by using ψj(ξ) := r−q−¯
ψj(−i)+ ¯
ψj(ξ), so ζj=r−q−¯
ψj(−i). Because our method incorporates
jumps by using only the characteristic function of the jump component, it can be applied with
general L´evy processes, including regime switching L´evy models, and stochastic volatility with
a L´evy jump component. Next, let’s recall a useful result of Buffington and Elliot (2002), which
plays a vital role later.
Corollary 1 (Buffington and Elliot (2002)) For ∆t>0, the characteristic function of X∆tis
given by the matrix exponential
E[exp(iX∆tξ)|α(0)] = 10·exp (∆t(Q0+diag(ψ1(ξ), . . . , ψm0(ξ))) I(0),(7)
where 1∈Rm0is a unit vector, and I(0) ∈Rm0is a vector of zeros, except for the value 1in
the position α(0).
2.2 Stochastic volatility
While regime switching models are an interesting class on their own, the main contribution
of this paper is to provide a framework for pricing exotic options under general stochastic
volatility models (with jumps). Consider the following two-dimensional system of correlated
time-homogeneous diffusions with independent jumps in the asset dynamics
(dSt=Stγ(vt)dt +m(vt)StdW 1
t+St(eJt−1)dNt,
dvt= ˆµ(vt)dt + ˆσ(vt)dW 2
t,(8)
where W1
t, W 2
tare Brownian motions with E[dW 1
tdW 2
t] = ρdt for ρ∈(−1,1). Here Ntis a
Poisson process,2and Jtis the jump amplitude, and they are independent of W1
tand W2
t. In
addition, we assume that there exists a constant C > 0 such that for all v1, v2in the state space
of vt
|ˆµ(v1)−ˆµ(v2)|+|ˆσ(v1)−ˆσ(v2)| ≤ C|v1−v2|,( ˆµ(v1))2+ (ˆσ(v1))2≤C2(1 + v2
1).(9)
The above conditions guarantee that there exists a unique solution vtpossessing the strong
Markov property (see Gihman and Skorohod (1979)). Moreover, we assume that ˆσ(.) and m(.)
are continuously differentiable, with ˆσ(·)>0 on the domain of vt.
Let Sc
tdenote the continuous part of the stock price, then from the Ito’s formula for jump
processes, we have
dlog(St) = 1
St
dSc
t−1
2S2
t
(dSc
t)2+dX
0<s≤t
[log(Ss)−log(Ss−)]
=γ(vt)−1
2m2(vt)dt +m(vt)dW (1)
t+dX
0<s≤t
[log(Ss)−log(Ss−)].(10)
2As mentioned in section 2.1, the method applies more generally to a stochastic volatility model with a L´evy
jump component. The special case of a stochastic volatility jump diffusion is considered here for simplicity.
5
Next, define ˆ
f(x) := Rx
c
m(u)
ˆσ(u)du with cbeing a constant, and h(x) := L(ˆ
f(x)) = ˆµ(x)ˆ
f0(x) +
1
2ˆσ2(x)ˆ
f00(x). Define f(vt, v0) := ρ(ˆ
f(vt)−ˆ
f(v0)), then we have
df(vt, v0) = ρd ˆ
f(vt) = ρh(vt)dt +ρm(vt)dW 2
t.(11)
Finally, define W∗
t:= W1
t−ρW 2
t
√1−ρ2, then 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,
we plug (11) into (10), and obtain
dlog(St) = γ(vt)−1
2m2(vt)dt +m(vt)(ρdW 2
t+p1−ρ2dW ∗
t) + dX
0<s≤t
[log(Ss)−log(Ss−)]
=γ(vt)−1
2m2(vt)dt +df(vt, v0)−ρh(vt)dt +p1−ρ2m(vt)dW ∗
t
+dX
0<s≤t
[log(Ss)−log(Ss−)].(12)
Let e
Xt:= log(St/S0)−f(vt, v0), then we can rewrite (8) as
de
Xt=γ(vt)−1
2m2(vt)−ρh(vt)dt +p1−ρ2m(vt)dW ∗
t+dX
0<s≤t
[log(Ss)−log(Ss−)],
dvt= ˆµ(vt)dt + ˆσ(vt)dW 2
t.
(13)
Our approach is as follows: we first approximate vtby a m0−state Markov chain with variance
states v={v1, v2, . . . , vm0}and with the corresponding generator Qgoverning the transition
between variance states (see Section 2.3 for details). We then specify the function f(·) as above
such that
log(St/S0) = e
Xt+f(vt, v0),(14)
and the dynamics are approximated by the regime switching jump diffusion process
de
Xt=µα(t)dt +σα(t)dW ∗
t+dX
0<s≤t
[log(Ss)−log(Ss−)],(15)
where (µα(t)=γ(vα(t))−1
2m2(vα(t))−ρh(vα(t)),
σα(t)=p1−ρ2m(vα(t)),(16)
and in particular W∗
tis a standard Brownian motion independent of W2
tthat drives the variance
process. The states {v1, . . . , vm0}are represented by the chain α(t) on j= 1, . . . , m0. Hence,
e
Xtis of the form of equation (3), and the pricing problem in a stochastic volatility model is
reduced to an equivalent pricing problem in a regime switching jump diffusion model.
Once e
Xthas been identified, the set of chfs describing the dynamics of e
Xtare
ψj(ξ) = iξµj−1
2ξ2σ2
j+λ(φ(ξ)−1), j = 1, . . . , m0,(17)
6
f(vk, vj)ρ
σv(vk−vj)
Heston µj(ρη
σv−1
2)vj+ (r−q−ρηθ
σv)
σjp(1 −ρ2)vj
f(vk, vj)ρ
σva(vk−vj) + b(log(vk)−log(vj))
4/2 µj(aρη
σv−a2
2)vj+ (ρbσv−b2
2−bρηθ
σv)1
vj+ (ρη
σv(b−aθ) + r−q−λκ −ab)
σj(a√vj+b
√vj)p(1 −ρ2)
f(vk, vj)1
2
ρ
σv(v2
k−v2
j)
Stein-Stein µj(ρη
σv−1
2)v2
j−ρηθ
σvvj+ (r−q−λκ −ρσv
2)
σjvjp(1 −ρ2)
f(vk, vj)ρ
bσv(log(bvk)−log(bvj))
3/2 µj1
2(ρbσv−2ρbη
b
θ
bσv−1) 1
bvj+ ( ρbη
bσv+r−q−λκ)
σjq(1−ρ2)
bvj
f(vk, vj)2ρ
σv(√vk−√vj)
Hull-White µj(ρσv
4−avρ
σv)√vj−1
2vj+ (r−q−λκ)
σjp(1 −ρ2)vj
f(vk, vj)ρ
σv(evk−evj)
Scott µjρ(η
σv(vj−θ)−σv
2)evj−e2vj
2+ (r−q−λκ)
σjevjp(1 −ρ2)
f(vk, vj)ρ
σv(evk−evj)
α-H µjρθ
σve(1+av)vj−ρ(η
σv+σv
2)evj−e2vj
2+ (r−q−λκ)
σjevjp(1 −ρ2)
Table 1: Transform for pricing with common stochastic volatility models with jumps.
where λ(φ(ξ)−1) is the state-independent characteristic exponent of the jump component. The
necessary transforms as well as the drift and volatility functions µα(t)and σα(t)for equation
(15) are provided in Table 1 for many popular stochastic volatility models, which we have
augmented with jumps.
For example, consider Heston stochastic volatility model with jumps:
Hes: (dSt=St(r−λκ)dt +St√vtdW 1
t+St(eJt−1)dNt,
dvt=η(θ−vt)dt +σv√vtdW 2
t,(18)
where ηis the mean reversion rate, θis the equilibrium level, and σv>0 is the volatility of
volatility. If we apply the change of variables
e
Xt= log St
S0−ρZvt
v0
m(u)
ˆσ(u)du = log St
S0−ρ
σv
(vt−v0),(19)
with f(vt, v0) = ρ
σv(vt−v0), then it follows that
de
Xt=h(ρη
σv−1
2)vt+ (r−ρηθ
σv−λκ)idt +p(1 −ρ2)vtdW ∗
t+dhNt
X
k=1
Yki,
dvt=η(θ−vt)dt +σv√vtdW 2
t.
(20)
Remark 1 An important property of this weak approximation approach is that the character-
istic function of the log stock price in the underlying stochastic volatility model is not even
required to employ a transform based approach, thus our method can be applied to a broader
class of stochastic volatility models where the characteristic functions either do not exist, or
7
have not been determined in the literature. This distinguishes our approach from the transform-
based analysis in the literature, e.g. Duffie et al. (2000). We also do not need the transition
density of the variance process to carry out our approach.
2.3 Markov chain approximation of variance
Suppose that under Qthe variance process follows a general diffusion process
(dvt= ˆµ(vt)dt + ˆσ(vt)dW 2
t,
v0∈R.(21)
Consider a m0-state Markov chain on v={v1, v2, . . . , vm0}with vi−1< viand k={k1, k2, . . . , km0−1},
and we denote the set of grid spacings with ki=vi−vi−1. Once v1and vm0have been deter-
mined (as illustrated in the following Remark 2), our approach is to first define a non-uniform
grid as in Tavella and Randall (2000):
vj=v0+ ¯αsinh c2
j
m0
+c1(1 −j
m0
), j = 2, . . . , m0−1,(22)
where
c1= arcsinh v1−v0
¯α, c2= arcsinh vm0−v0
¯α(23)
for ¯α < (vm0−v1), which we set to ¯α= (vm0−v1)/2.
Next we parameterize the generator Q(as in Lo and Skindilias (2014) for a general diffusion)
by
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,
(24)
where a±= max{±a, 0}. Note that if kis chosen such that
0<max
1≤i≤m0−1ki≤min
1≤i≤m0
ˆσ2(vi)
|ˆµ(vi)|,(25)
then we have
ˆσ2(vi)≥ki−1ˆµ−(vi) + kiˆµ+(vi),(26)
so the set of qij is well defined. That is, qij ≥0,∀1≤i6=j≤m, and Pm0
j=1 qij = 0, i = 1, . . . , m0.
Moreover, the first two instantaneous moments of the variance process are locally matched to
those of the Markov chain:
E[vt+∆t−vt] = ˆµ(vi)∆t, E[vt+∆t−vt]2= (ˆσ(vi))2∆t. (27)
Therefore the Markov chain satisfies the local consistency conditions, hence will converge weakly
to its continuous counterpart (see Kushner (1990)).
8
Remark 2 To determine the grid boundaries v1and vm0, we fix t=T/2and center the
grid about the mean of the variance process vtby: v1:= max{¯v, ¯µ(t)−γ¯σ(t)}and vm0:=
¯µ(t)+γ¯σ(t). Here we have defined ¯µ(t) = E[vt|v0]and ¯σ(t)is the standard deviation conditional
on v0. Further, we define the constants γ= 3 ∼4and ¯v= 0.0001 for models which require a
positive level of vt.3For Stein-Stein model, ¯v= 0.01 is recommended. For Scott model and the
α−Hypergeometric model, vtis allowed to become negative, so we take ¯v=−∞.
Remark 3 We note that for Heston model, 3/2 model, and 4/2 model (considered in Section
3.1) ,
¯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 and Scott models, we have
¯µ(t) = e−ηtv0+θ(1 −e−ηt),¯σ2(t) = σ2
v
2η(1 −e−2ηt).
For Hull-White model, we have
¯µ(t) = v0eavt,¯σ2(t) = v2
0e2avt(eσ2
vt−1).
2.4 Biorthogonal Projection
In order to price contracts in regime switching jump diffusion and stochastic volatility models,
we employ the option pricing method of frame projection introduced in Kirkby (2015a) that
orthogonally projects the probability density function of a random variable onto a Riesz basis,
which is a special class of frame. Let Xbe a random variable on Rwith unknown density
fX. Given a compactly supported symmetric function ϕ(ν) called the generator, a resolution
a > 0, and a reference point ν1, the infinite dimensional space Ma:= span{ϕa,n}n∈Zis formed
from the basis elements ϕa,n(ν) := a1/2ϕ(a(ν−νn)) centered over νn=ν1+ (n−1)/a. The
orthogonal projection PMafXof fXonto Mais determined by its dual basis {eϕa,n}n∈Z:
PMafX(ν) = X
n∈ZhfX,eϕa,niϕa,n(ν),
where the dual basis is biorthogonal, that is hϕa,k,eϕa,ni={k=n}(orthogonal bases are a special
case). The dual basis resides in the same space as ϕa,n, and can be characterized by the Fourier
transform of the generator, b
eϕ, which for the linear basis is given by
b
eϕ(ξ) = 12 sin2(ξ/2)
ξ2(2 + cos(ξ)),(28)
3If moments of the variance process are unknown, the grid can be fixed using v1=β1v0and vm0=β2v0.
For example, β1= 10−3and β2= 4.
9
and can be derived for higher order B-spline bases (see Kirkby (2015a) for further details, and
Kirkby (2016b) for an application of higher order B-splines to Asian options). The projection
coefficients satisfy a closed-form integral
hfX,eϕa,ni=a−1/2
π<Z∞
0
exp(−iνnξ)·φX(ξ)b
eϕξ
adξ,(29)
assuming the characteristic function φX(ξ) is known. In this work, we utilize the linear B-spline
generator ϕ(ν) = ϕ[1](ν) defined by
ϕ[1](ν) = (1 + ν)·[−1,0] (ν) + (1 −ν)·[0,1](ν).(30)
First we restrict Mato the finite set {νn}N
n=1, where νn= (1 −N/2)∆ + (n−1)/a, where the
choice of aand Nwill be discussed below. If we define an N-point frequency grid to satisfy
the Nyquist requirement
∆ξ= 2πa/N, ξn= (n−1)∆ξ, n = 1, ..., N, (31)
and constant Υa,N := 32a4/N, the coefficients a1/2Υa,N ·¯
βa,n ≈βa,n =hfX,eϕa,niare found
using the exponentially convergent discretization of (29)
¯
βa,nN
n=1 := <{D{Hj}},Dn{Hj}=
N
X
j=1
e−i2π
N(j−1)(n−1)Hj, n = 1, ..., N, (32)
where Dis the discrete Fourier transform (DFT). From (29) and the dual transform in (28),
the DFT input vector {Hj}N
j=1 is defined by
H1=1
24a2, Hn= exp(−iν1ξn)·φX(ξn)·(sin(ξn/(2a))/ξn)2
2 + cos(ξn/a), n ≥2,(33)
where ν1:= (1 −N/2)∆ centers the expansion around zero. Note that we have chosen the
linear basis in this work for its simplicity. However, higher order bases are likely to improve
accuracy, at the cost of a more complicated algorithm.
Remark 4 In light of Corollary 1, the PROJ method of Kirkby (2015a) can be directly applied
to price European call options with arbitrary payoffs g(S∆t)where the underlying model is a
regime switching L´evy process as in Section 2. Moreover, the Markov chain approximation of
Section 2.3 provides an alternative approach to price European options under stochastic volatility
models when the characteristic function of ln(St/S0)is unavailable or ill-behaved.
2.5 Barrier options
Without loss of generality4we consider knock-out barrier options, which are contracts that
become worthless if the underlying enters a prescribed knock-out region. Let Cdenote the
4Since knock-in options can be priced by parity.
10
continuation region, and Ccthe knock-out region for Xtwhere Xt= ln(St/S0). For a double
barrier option with knockout barriers Land Uin asset space, C= [lx, ux] where lx:= ln(L/S0)
and ux:= ln(U/S0) in log space. Define the terminal payoff GS0(X) = g(S0exp(X)) for some
g(·), and fix a set of monitoring dates tm=m∆t,m= 0, . . . , M , where ∆t=T/M. Let
V∗
m(Xtm, α(tm)) denote the value of a barrier option at time tmwhich has not previously been
knocked out, where Xtm= log(Stm/S0) and α(tm)∈ {1, . . . , m0}is the current state, and
represents the current variance level in the case of stochastic volatility. The barrier option
price can be found by backward recursion with
(V∗
M(XM, αM) = GS0(XM)
V∗
m(Xm, αm) = e−r∆tEV∗
m+1(Xm+1, αm+1){Xm∈C}|Xm, αmm=M−1, ..., 0,(34)
where5Xm:= Xtmand αm=α(tm). By definition, V∗
m+1(Xm+1, αm+1) = 0 for Xm+1 ∈ Cc. Let
Gm
j,k(x) := {Xm+1 =x, αm=j, αm+1 =k}, and define the state transition probabilities
P∆t
jk =Q[α(t+ ∆t) = k|α(t) = j], j, k = 1, . . . , m0.
Then with αm=jand Xm=x∈ C, we have
V∗
m(x, j) = e−r∆tEV∗
m+1(Xm+1, αm+1)|Xm=x, αm=j
=e−r∆tX
k=1,..,m0
P∆t
j,k EV∗
m+1(Xm+1, k)|Gm
j,k(x)
=e−r∆tX
k=1,..,m0
P∆t
j,k ZCV∗
m+1(y, k)pj,k (y|x)dy,
where we have defined the set of transition densities for the log return process
pj,k(y|x) = Q[X(∆t)∈y+dy|X(0) = x, α(0) = j, α(∆t) = k], j, k = 1, . . . , m0.
For a regime switching model, the set of transition densities can be recovered from
E[eiξX∆t|Gj,k ] = 1
P∆t
j,k Ek,j (ξ),E(ξ) = exp (∆t(Q0+ diag(ψ1(ξ), . . . , ψm0(ξ))) .
For a stochastic volatility model with log return process given by X∆t=e
X∆t+f(v∆t, v0), we
have E[eiξX∆t|Gj,k] = Ek,j (ξ)·exp(iξf (vk, vj))/P ∆t
j,k , which leads us to define
e
Ek,j (ξ) := (Ek,j (ξ),regime-switching model
Ek,j (ξ)·exp(iξf(vk, vj)),stochastic volatility model.(35)
This allows a unified treatment of regime-switching and stochastic volatility models in the
pricing algorithms that follow. Once the generator Qhas been determined6, the different
5A European option can be priced recursively by setting C= (−∞,∞).
6In the regime-switching case Qis provided as a model parameter, while for a stochastic volatility model it
is determined as in section 2.3.
11
definitions of e
Ek,j (ξ) are the main difference between SV and RS models in the algorithm.
Noting that pj,k(y|x) = pj,k (ν) where ν=y−x, we have
pj,k(ν)≈X
n∈ZZ∞
−∞
pj,k(η)eϕa,n (η)dηϕa,n (ν)
=a−1/2
π
1
P∆t
j,k X
n∈Z<Z∞
0
exp(−iνnξ)·e
Ek,j (ξ)·b
eϕ(ξ/a)dξϕa,n(ν).
Hence,
P∆t
j,k ·pj,k(ν)≈Υa,N X
1≤n≤N
¯
βj,k
a,n ·a1/2ϕa,n(ν) := ¯pj,k(ν),
where the set of {¯
βj,k
a,n}N
n=1 is computed7as in Section 2.4, for each j, k = 1, . . . , m0. In this
case, the DFT input in equation (33) is defined by setting φX(ξn) = e
Ek,j (ξn):
Hj,k
1=e
Ek,j (ξ1)
24a2, Hj,k
n= exp(−iν1ξn)·e
Ek,j (ξn)·(sin(ξn/(2a))/ξn)2
2 + cos(ξn/a), n ≥2.(36)
Moreover, the basis elements are centered over a grid {νn}N
n=1 defined in Section 2.4.
Now define a grid {xn}N/2
n=1 for the continuation region Cin log asset space as in Section 2.5.1
below. We approximate the value at node (Xm, αm)=(xn, j) by replacing the true (unknown)
density with its orthogonal projection
V∗
m(xn, j) = e−r∆tX
k=1,..,m0ZCV∗
m+1(y, k)·P∆t
j,k ·pj,k(y|xn)dy
=e−r∆tX
k=1,..,m0ZCnV∗
m+1(xn+ν, k)·P∆t
j,k ·pj,k(ν)dν
≈e−r∆tX
k=1,..,m0ZCnV∗
m+1(xn+ν, k)·¯pj,k(ν)dν, (37)
where Cn:= {ν∈R:xn+ν∈ C}. For each k= 1, . . . , m0, we can represent the convolution
from (38) in the following form which facilitates an efficient Toeplitz-based algorithm:
ZCnV∗
m+1(xn+ν, k)·¯pj,k(ν)dν = Υa,N
N
X
l=1
¯
βj,k
l·a1/2ZCnV∗
m+1(xn+ν, k)ϕa,l(ν)dν
≈Υa,N
N/2
X
l=1
¯
βj,k
N/2+(l−n)·a1/2ZCnV∗
m+1(xn+ν, k)ϕa,N/2+(l−n)(ν)dν,
(38)
where ¯
βj,k
l≡¯
βj,k
a,l are the projection coefficients of ¯pj,k(ν).
Remark 5 The approximation (38) holds exactly for a double barrier option with C= [lx, ux] =
[x1, xn](since the value is zero outside of C), but introduces a truncation error in the case of an
7As discussed below, the set of {¯
βj,k
a,n}N
n=1 are not stored explicitly. We instead store a set of vectors µj,k
defined in equation (39)
12
unbounded domain, particularly at the boundaries. For example, at the leftmost grid point x1,
only the terms ¯
βj,k
N/2,¯
βj,k
N/2+1,..., ¯
βj,k
N−1corresponding to the right half of the transition density
are present in the summation in (38), as seen in the first row of T∆t
j,k below. However, the
error is easily controlled by choosing ˆα= (xn−x1)/2sufficiently large, as in the experiments
below. Alternatively, Kirkby (2015b) proposes a grid extension method which approximates the
convolution outside of [x1, xn](resulting in accurate approximations for reduced ˆα), that can be
adapted to the present case. To simplify the presentation, we consider the case without boundary
modification.
To efficiently determine values from (38) along the full grid, we introduce a set of Toeplitz
matrices, where Υ∆t
a,N = exp(−r∆t)·Υa,N :
T∆t
j,k = Υ∆t
a,N ·
¯
βj,k
N/2¯
βj,k
N/2+1 ¯
βj,k
N/2+2 ··· ¯
βj,k
N−1
¯
βj,k
N/2−1¯
βj,k
N/2¯
βj,k
N/2+1 ··· ¯
βj,k
N−2
¯
βj,k
N/2−2¯
βj,k
N/2−1¯
βj,k
N/2¯
βj,k
N−3
.
.
..
.
..
.
.....
.
.
¯
βj,k
1¯
βj,k
2¯
βj,k
3··· ¯
βj,k
N/2
, j, k = 1, . . . , m0.
In particular, to calculate T∆t
j,k u, for u= (u1, ..., uN/2)>, we define
µj,k := DnΥ∆t
a,N ·(¯
βj,k
N/2, ..., ¯
βj,k
1,0,¯
βj,k
N−1, ..., ¯
βj,k
N/2+1)>o, j, k = 1, . . . , m0,(39)
and T∆t
j,k uis given by the first N/2 elements of
p=D−1µj,k ◦ D (u1, ..., uN/2,0, ..., 0)>,(40)
where v◦uis the element-wise (Hadamard) product of vwith u, and the vector of zeros is
of length N/2. Since a computation of the form T∆t
j,k uis required at each time step of the
algorithm, we will store the set of µj,k at initialization for future use. In particular, T∆t
j,k is
never explicitly constructed in the algorithm, and µj,k is stored instead. The full initialization
routine is summarized by Algorithm 1 in Appendix A, which applies to barrier as well as
Bermudan options. Note that given the algorithm’s structure, a single vector {Hn}N
n=1 can be
used to store Hj,k
nat each iteration, and similarly for ¯
βj,k
n.
Given the Topelitz representation T∆t
j,k of the set of transition densities with jfixed, the
values V∗
m(xn, j) in equation (38) are computed for the full set of {xn}N/2
n=1 with complexity
O(m0·Nlog2(N)) using
{Vm(xn, j)}N/2
n=1 := X
k=1,..,m0
T∆t
j,k θm+1
k, m =M−1,...,0,
where for each m= 1, . . . , M,θm
kis a vector with8
θm
n,k =a1/2ZCnVm(xn+ν, k)ϕa,n (ν)dν, k = 1,...m0, n = 1, . . . , N/2,
8Note that in this equation, ϕa,n(ν) is centered over xn.
13
where Cn:= {ν∈R:xn+ν∈ C} and Vm(xn, j) is used to denote the approximation of the
true value, V∗
m(xn, j). The set of dual or theta coefficients,{θm
n,j}N/2
n=1 for each j= 1, . . . , m0are
required at each stage m. We use the following closed-form approximation:
θm
n,j :=
[13Vm(x1, j) + 15Vm(x2, j )−5Vm(x3, j) + Vm(x4, j)] /48 n= 1
[Vm(xn−1, j) + 10Vm(xn, j ) + Vm(xn+1, j)] /12 n= 2, ..., N/2−1
13Vm(xN/2, j) + 15Vm(xN/2−1, j)
−5Vm(xN/2−2, j) + Vm(xN/2−3, j)/48 n=N/2
(41)
which follows by a local quadratic approximation of each Vm(xn, j) (see (55) for details). As
shown in the error analysis, the error introduced by this approximation is O(∆3), and will not
affect the rate of convergence with the linear basis.
Thus, after initialization, in order to find Vm(xn, j) for all j∈ {1, . . . , m0}, it requires
O(m2
0·Nlog2(N)). The algorithm complexity is thus O(M·m2
0·Nlog2(N) + N·m3
0), where
the final terms derives from populating the matrix exponentials e
E(ξn) for n= 1, . . . , N, each
at a cost of O(m3
0).
2.5.1 Barrier Grid Selection and Terminal Payoff
The log-asset grid, {xn}N/2
n=1, is adapted to the continuation region, C, for various contracts. We
begin with a choice of N∈N+, the number of basis elements, and the grid width parameter
ˆα > 0, which determines the truncated density support on (−ˆα, ˆα). For each contract type we
initially define
e
∆ := 2ˆα/(N−1),ea:= 1/e
∆.
For down-and-out options with C= [lx,∞), x1:= lxcoincides with the knock-out barrier, and
the grid xn=x1+ (n−1)∆, n = 1, ..., N/2 is defined by9
x1:= lx, n0:= b1−lx·eac,∆ := lx/(1 −n0), a := 1/∆,(42)
which satisfies xn0= 0. Once Nand ∆ have been fixed, the frequency grid is generated as in
equation (31).
For one-sided barrier options our grid choice aligns the barrier and initial price (both in log
space) with the grid, while the strike might not belong. Define the nearest grid point left of
log(K/S0) (strike in log space)
¯n:= b(a·(log(K/S0)−x1)+1c.(43)
9We use a log asset grid of size N/2 to employ the Toeplitz-based convolution represented by (39)-(40).
14
Puts ¯
δput
jδput
j
j= 0 ζ1−1
2ζζ
18 4(2 −ζ)e¯ρ/2+ 5 ·((1 −ζ−)eρ−+ (1 −ζ+)eρ+)
j= 1 ζ2
2
ζ
18 ·e−∆·4ζ·e¯ρ/2+ 5 (ζ−eρ−+ζ+eρ+)
Calls ¯
δcall
jδcall
j
j= 0 1
2+ζζ
2−1e(¯ρ+∆)/2σ2
18 5·(1 −q−)e∆σ−+ (1 −q+)e∆σ++ 4
j= 1 σ−1
2σ2e(¯ρ−∆)/2σ
18 ·4(ζ+ 1) + 5 ζ+1
2+σ−e∆σ−+ζ+1
2+σ+e∆σ+
Table 2: Terminal payoff coefficients derived from a three point Gaussian quadrature, where q±:=
(1 ±p3/5)/2, ζ±:= ζ·q±, and ρ±:= ¯ρ·q±.
With the formulae defined in Table 2, we can derive the DOP payoff coefficients :
θM
n=
K
2−L·ϑ[0,1], n = 1
K−exn·S0·(ϑ[−1,0] +ϑ[0,1]), n = 2, ..., ¯n−1
K1
2+¯
δput
0−e−¯ρϑ[−1,0] +δput
0, n = ¯n
K¯
δput
1−e∆−¯ρ·δput
1, n = ¯n+ 1
0, n = ¯n+ 2, ..., N/2
(44)
where θM
n,j ≡θM
nare identical for j= 1, . . . , m0. We have also defined the constants
ϑ[−1,0] := e−∆/2
18 4 + 5 cosh(√15∆/10) + √15 sinh(√15∆/10)
ϑ[0,1] := e∆/2
18 4 + 5 cosh(√15∆/10) −√15 sinh(√15∆/10).(45)
Similarly, we define the DOC coefficients
θM
n=
0n≤¯n−1
Kδcall
0e−¯ρ−¯
δcall
0n= ¯n
Ke∆−¯ρϑ[0,1] +δcall
1−1/2 + ¯
δcall
1 n= ¯n+ 1
S0exn·(ϑ[−1,0] +ϑ[0,1])−K n = ¯n+ 2, ..., N/2
(46)
For up-and-out contracts, double barriers and rebates, terminal coefficients and the grid choice
are provided in Kirkby (2015b). Algorithm 2 in Appendix A summarizes the procedure for
knock-out barrier options, while knock-in options are easily priced using parity relations.
3 Early exercise contracts
Early-exercise contracts are priced in a similar manner to barrier options, with a few mod-
ifications. In general, the Bermudan option price can be found by backward recursion with
15
V∗
M(XM, αM) = GS0(XM)
C∗
m(Xm, αm) = e−r∆tEV∗
m+1(Xm+1, αm+1)|Xm, αmm=M−1, ..., 0,
V∗
m(Xm, αm) = max{C∗
m(Xm, αm), GS0(Xm)},
(47)
where G(Xm) is the early exercise payoff at time tm, and C∗
m(Xm, αm) is the continuation value
as of tm. Theoretical values are denoted with a superscript star to distinguish them from the
approximations to follow.
Let {xn}N/2
n=1 define a uniformly spaced grid in log asset space (see Remark 10 for details).
For a Bermudan option, the continuation value satisfies
C∗
m(xn, j) = e−r∆tEV∗
m+1(Xm+1, αm+1)|Xm=xn, αm=j
≈e−r∆tX
k=1,..,m0Z∞
−∞ Vm+1(xn+ν, k)·¯pj,k(ν)dν := Cm(xn, j),
which requires the set of value coefficients,θm+1
j, defined at time tm+1 by
θm+1
n,j := a1/2Zxn+1
xn−1Vm+1(x, j)ϕa,n(x)dx =a1/2Zxn+1
xn−1
max{Cm+1(x, j), GS0(x)}ϕa,n(x)dx, (48)
where we note that here ϕa,n(x) is centered over the grid point xn, and is supported on
[xn−1, xn+1]. The integral is evaluated upon determining the set of early-exercise points, which
separate log asset space into a continuation and an exercise region. At time tm, we define for each
j= 1, . . . , m0the early-exercise point10 bxm(j), which satisfies GS0(bxm(j)) = Cm(bxm(j), j). For
concreteness, we consider the case of a Bermudan put option, with GS0(x)=(K−S0exp(x))+
(calls are priced similarly). The point bxm(j) is found by locating the bracketing left index and
grid point
bn(j) = max{1≤n≤¯n:GS0(xn)− Cm(xn, j)≥0}, xbn(j)=x1+ (bn(j)−1)∆,(49)
where ¯nis defined in equation (43), so the early exercise point satisfies xbn(j)≤bxm(j)< xbn(j)+1.
Once bn(j) is found, we use the linear approximation
bxm(j)≈xbn(j)+ ∆ GS0(xbn(j))− Cm(xbn(j), j)
GS0(xbn(j))− Cm(xbn(j), j)−GS0(xbn(j)+1)− Cm(xbn(j)+1, j),(50)
which is found by equating the linear approximations of GS0(x) and Cm(x, j), and solving for
the point of intersection, bxm(j). Starting with the set θM
kdefined in equation (56), the value
10These early exercise points are unique because we can think of them as taken from the corresponding
continuous early exercise boundary (for the continuous finite-maturity American option). Thus if this continuous
early exercise boundary is uniquely defined, then these early exercise points will also be uniquely defined due to
the convergence relation between Bermudan options and American options. Note that the stochastic volatility
dynamics considered in our paper is a special case of the model (6) and (7) in Detemple and Tian (2002),
then from Proposition 2 on page 922 of Detemple and Tian (2002), we have that the continuous early exercise
boundary is unique. We would like to thank an anonymous referee for raising this point.
16
recursion becomes
{Cm(xn, j)}N/2
n=1 := X
k=1,..,m0
T∆t
j,k θm+1
k, m =M−1,...,0,
where θm+1
kis defined in equation (48).
Corollary 2 Suppose for j∈ {1, . . . , m0}that we have identified an index bn(j)such that
xbn(j)≤bxm(j)< xbn(j)+1 . The value coefficients for n= 1, ..., N/2of the Bermudan put can
be evaluated by a closed-form three point Gaussian quadrature approximation, where Cn,j :=
Cm(xn, j):
θm
n,j :=
θM
nn= 1, ..., bn(j)−1
K1
2+¯
δput
0−S0exp(xbn(j))ϑ[−1,0] +δput
0
+γm
1Cbn(j)−1,j +γm
2Cbn(j),j +γm
3Cbn(j)+1,j n=bn(j)
K¯
δput
1−S0exp(xbn(j)+1)δput
1
+γm
4Cbn(j),j +γm
5Cbn(j)+1,j +γm
6Cbn(j)+2,j n=bn(j)+1
1
12 (Cn−1,j + 10Cn,j +Cn+1,j)n=bn(j)+2, ..., N/2−1
1
48 13CN/2,j + 15CN/2−1,j −5CN/2−2,j +CN/2−3,j n=N/2,
(51)
where in stage mwe define γm
l=γm
l(j)by
γm
1=−1
24 +ζ4
8−ζ3
3+ζ2
4, γm
2=5
12 −ζ4
4+ζ3
3+ζ2
2−ζ, γm
3=1
8+ζ4
8−ζ2
4(52)
γm
4=1
12 −ζ4
8+1
2ζ3−ζ2, γm
5=5
6+ζ4
4−2ζ3
3, γm
6=1
12 −ζ4
8+ζ3
6
and (with time superscripts and dependence on jsuppressed to ease notation)
¯ρ:= bxm(j)−xbn(j), ζ := a¯ρ. (53)
We have also defined in Table 2 a set of coefficients δput
l=δput
l(ζ, ¯ρ)and ¯
δput
l=¯
δput
l(¯ρ).
Proof: From the definition of bxm(j) and θm
n,j, it follows that
θm
n,j =a1/2
Zxn+1
xn−1
GS0(x)ϕa,n(x)dx, n ≤bn(j)−1
Zbxm(j)
xn−1
GS0(x)ϕa,n(x)dx +Zxn+1
bxm(j)Cm(x, j )ϕa,n(x)dx, n =bn(j),bn(j)+1
Zxn+1
xn−1Cm(x, j)ϕa,n (x)dx n ≥bn(j)+2,
(54)
17
where θm
n,j =θM
n,j ≡θM
nfor n≤bn(j)−1. We apply a local quadratic interpolation of the
value function to evaluate θm
n,j for n= 1,...,bn(j)−1. For each interval [xn−1, xn+1], n=
bn(j), . . . , N/2−1, define
e
Cm(x, j) = Cn−1,j
(x−xn)(x−xn+1)
2∆2− Cn,j
(x−xn−1)(x−xn+1)
∆2
+Cn+1,j
(x−xn−1)(x−xn)
2∆2,(55)
and integrate exactly. Given that ϕa,n(y)e
Cm(y, j) is piecewise cubic on [xn−1, xn+1], by splitting
the interval in half, integration by Simpson’s rule is exact. To evaluate the coefficients of bn(j)
and bn(j) + 1, the integral is similarly divided to account for the early exercise point, and a
three point Gaussian quadrature is applied to obtain the coefficients in (52).
Remark 6 The coefficient recursion for the Bermudan put is initialized by the set terminal
coefficients θM
k=θMsatisfying
θM
n=(K−S0exn·(ϑ[−1,0] +ϑ[0,1])n= 1, ..., ¯n−1
K·1
2−ϑ[−1,0]n= ¯n(56)
where ¯nis defined in equation (43), and θM
k,n = 0 for n= ¯n+ 1, ..., N/2. This initial vector is
the same for all k= 1, . . . , m0.
3.1 Stochastic volatility models
In addition to Heston model given in equation (18), we consider the following group of stochastic
volatility models, for which we provide the dynamics. Augmenting the model of Hull and White
(1990) with jumps, we obtain
HW: (dSt=St(r−λκ)dt +St√vtdW 1
t+St(eJt−1)dNt,
dvt=avvtdt +σvvtdW 2
t.(57)
Extending the 4/2 model introduced in Grasselli (2016) to include jumps, we have
4/2 Model: (dSt=St(r−λκ)dt +St[a√vt+b
√vt]dW 1
t+St(eJt−1)dNt,
dvt=η(θ−vt)dt +σv√vtdW 2
t.(58)
The model of Stein and Stein (1991) is extended to include jumps and to allow correlation
ρ∈(−1,1) between W1
tand W2
t:
SS: (dSt=St(r−λκ)dt +StvtdW 1
t+St(eJt−1)dNt,
dvt=η(θ−vt)dt +σvdW 2
t.(59)
We also consider the 3/2 model with jumps, which can be formulated as
(dSt=St(r−λκ)dt +St√vtdW 1
t+St(eJt−1)dNt,
dvt=vt[η(θ−vt)dt +σv√vtdW 2
t].(60)
18
where σv>0, θ∈Ris the mean reversion level, ηis given such that ηvt-a stochastic volatility
quantity- is the speed of mean reversion. However, we prefer the alternative form which results
from a change of variables
bvt=1
vt
,bη=ηθ, b
θ=η+σ2
v
ηθ ,bσv=−σv.(61)
In particular, we use the alternative formulation
3/2 Model:
dSt=St(r−λκ)dt +St
√bvt
dW 1
t+St(eJt−1)dNt,
dbvt=bη[b
θ−bvt]dt +bσv√bvtdW 2
t,bv0= 1/v0.
(62)
The model of Scott (1987) is extended to include jumps and to allow correlation ρ∈(−1,1)
between W1
tand W2
t:
Scott: (dSt=St(r−λκ)dt +StevtdW 1
t+St(eJt−1)dNt,
dvt=η(θ−vt)dt +σvdW 2
t.(63)
Given this formulation, vtis interpreted as a stochastic driver, which determines instantaneous
volatility through evt. Unlike previous models, vtis permitted to become negative.11 A similar
observation holds for the α-Hypergeometric model discussed next.
The α-Hypergeometric model (without jumps) was recently proposed by Da Fonseca and
Martini (2016). Unlike Heston model, for α-Hypergeometric model the strict positivity of
volatility is guaranteed. We extend their model by adding a jump component which results in
the dynamics
α−H: (dSt=St(r−λκ)dt +StevtdW 1
t+St(eJt−1)dNt,
dvt= (η−θeavvt)dt +σvdW 2
t,(64)
where η, v0∈(−∞,+∞), θ > 0, σv>0, av>0. For Scott model and the α-Hypergeometric
model, we allow the underlying driver vtto take negative values in the “variance” grid (see
Remark 2).
Remark 7 While the mean and variance of vtare unknown in the α-Hypergeometric model,
we can estimate them using a simple approximation.12 From the first order expansion
eavt≈eav0+aeav0(vt−v0) = eav0(1 −av0) + aeav0vt
the dynamics of the variance process in the α−Hypergeometric model are approximated by
dvt≈θaeav0η−θeav0(1 −av0)
θaeav0−vtdt +σvdW 2
t:= ¯η(¯
θ−vt)dt +σvdW 2
t,
so we can estimate the mean and variance using Scott model as in Remark 2 and 3 with t:= T/2.
11The alternative is to consider an instantaneous volatility of the form τevtfor τ∈(0,1), and restrict vt.
12The accuracy of this approach has been verified with Monte Carlo simulation.
19
3.2 Error Analysis
We now show that the PROJ algorithm (with the linear basis) converges at a rate of O(∆2)
in the pure regime switching case. For concreteness, the result is demonstrated for a double
barrier knock-out option, but extends to general barrier and Bermudan contracts, as discussed
in Remark 8 below. Consider the case of a double barrier knock-out option (with knock-out
region Cc), where C= [lx, ux] with ux−lx= ˆα, and recall Cn={ν∈R:xn+ν∈ C} for grid
point xn. Denote the valuation error by
Em(xn, j) := V∗
m(xn, j)− Vm(xn, j ),
where V∗
m(xn, j) is the true value, and Vm(xn, j) is the approximation in the absence of projec-
tion coefficient error.13 In particular
Vm(xn, j) = e−r∆tX
k=1,..,m0
P∆t
j,k ZCne
Vm+1(xn+ν, k)ˆpj,k(ν)dν
where Cn={ν∈R:xn+ν∈ C}, and
ˆpj,k(ν) :=
N
X
n=1 Z∞
−∞
pj,k(η)eϕa,n (η)dηϕa,n (ν) :=
N
X
n=1
βj,k
a,n ·ϕa,n(ν).
Here e
Vm+1(xn+ν, k) is taken to be the quadratic interpolate value function.14 Hence
|Em(xn, j)|=e−r∆tX
k=1,..,m0
P∆t
j,k ZCnV∗
m+1(xn+ν, k)pj,k(ν)−e
Vm+1(xn+ν, k)ˆpj,k(ν)dν
≤e−r∆tX
k=1,..,m0
P∆t
j,k ZCnV∗
m+1(xn+ν, k)pj,k(ν)−e
Vm+1(xn+ν, k)ˆpj,k(ν)dν
≤e−r∆tX
k=1,..,m0
P∆t
j,k ·(|E1
m+1(xn, j, k)|+|E2
m+1(xn, j, k)|),(65)
where
E1
m+1(xn, j, k) := ZCnV∗
m+1(xn+ν, k)pj,k(ν)− V∗
m+1(xn+ν, k)ˆpj,k(ν)dν
and
E2
m+1(xn, j, k) := ZCnV∗
m+1(xn+ν, k)ˆpj,k(ν)−e
Vm+1(xn+ν, k)ˆpj,k(ν)dν.
13For simplicity, we omit the error in calculating the projection coefficients, as they converge at a much
faster rate than the other sources, and does not affect the overall convergence rate (see Kirkby (2015b), which
considers the influence of this error).
14To be more precise, the coefficients θn,k are determined by the quadratic interpolation e
Vm+1(xn+ν, k) over
overlapping intervals [xn−1, xn+1]. However, the error bound is unaffected if we treat e
Vm+1(xn+ν, k) as a global
quadratic interpolation (rather than a set of overlapping local interpolations), which simplifies the analysis. In
fact, in the computation of θn,k we could alternatively use a global interpolation, but the local approach leads
to closed-form coefficient formulas.
20
From the Cauchy-Schwartz inequality,
|E1
m+1(xn, j, k)| ≤ ZCn|V∗
m+1(xn+ν, k)||pj,k(ν)−ˆpj,k(ν)|dν
≤BV,1kpj,k −ˆpj,kk2,(66)
where RCn|V∗
m+1(xn+ν, k)|2dν =RC|V∗
m+1(y, k)|2dν ≤B2
V,1independent of mfor a double
barrier contract (in general, we can take the finite bound over the grid [x1, xn]). Similarly,
|E2
m+1(xn, j, k)| ≤ ZCn
pj,k(ν)|V∗
m+1(xn+ν, k)−e
Vm+1(xn+ν, k)|dν
+ZCn|ˆpj,k(ν)−pj,k (ν)||V∗
m+1(xn+ν, k)−e
Vm+1(xn+ν, k)|dν
≤
V∗
m+1(·, k)−e
Vm+1(·, k)
C
∞ZCn
pj,k(ν)dν +ZCn|ˆpj,k (ν)−pj,k (ν)|dν
≤
V∗
m+1(·, k)−e
Vm+1(·, k)
C
∞1 + √ˆαkpj,k −ˆpj,kk2,(67)
where kgkC
∞:= sup{g(x) : x∈ C}. Let V∗
m+1,k(x) := V∗
m+1(x, k) and e
Vm+1,k(x) := e
Vm+1(x, k).
It follows that
V∗
m+1,k −e
Vm+1,k
C
∞≤
V∗
m+1,k −e
V∗
m+1,k
C
∞+
e
V∗
m+1,k −e
Vm+1,k
C
∞
where e
V∗
m+1,k is the quadratic interpolation taken from the true values of e
V∗
m+1,k. The first term
satisfies
V∗
m+1,k −e
V∗
m+1,k
C
∞≤∆3
12 ·
V∗(3)
m+1,k
C
∞
≤∆3·e−r∆t
12 max
j,k
p(3)
k,j
(−ˆα, ˆα)
1max
jn
V∗
m+2,j
C
∞o,(68)
where the second inequality follows from
dn
dxnV∗
m+1,k(x)=e−r∆tX
j=1,..,m0
P∆t
k,j ZCV∗
m+2,j(y)·p(n)
k,j (y−x)dy
≤e−r∆tX
j=1,..,m0
P∆t
k,j
p(n)
k,j
(−ˆα, ˆα)
1
V∗
m+2,j
C
∞
≤e−r∆tmax
j,k
p(n)
k,j
(−ˆα, ˆα)
1
V∗
m+2,j
C
∞,
where Pj=1,..,m0P∆t
k,j = 1. From (68), we obtain
max
k
V∗
m+1,k −e
V∗
m+1,k
C
∞≤ζ1∆3,(69)
where
ζ1:= e−r∆t
12 BV,2·max
j,k
p(3)
j,k
(−ˆα, ˆα)
1, ζ2(∆) := max
j,k kpj,k −ˆpj,kk2
21
and BV,2satisfies maxjn
V∗
m,j
C
∞o≤BV,2for all 0 ≤m≤Msince the true value function is
finite on C(bounded by the maximum discounted payoff). For the second term, we have
e
V∗
m+1,k −e
Vm+1,k
C
∞≤max
jnmax
nV∗
m+1,j(xn)− Vm+1,j (xn)o:= E∗
m+1.(70)
Hence, with E2
m+1 := maxn,j,k |E2
m+1(xn, j, k)|, we have from (67), (69) and (70) that
E2
m+1 ≤(1 + √ˆα·ζ2(∆)) ·(E∗
m+1 +ζ1∆3).(71)
Define E1
m+1 := maxn,j,k |E1
m+1(xn, j, k)|, which from (66) satisfies
E1
m+1 ≤BV,1·ζ2(∆).(72)
Combining (65) with (71) and (72), we have for 0 ≤m≤M−2:
E∗
m≤e−r∆tX
k=1,..,m0
P∆t
j,k ·(E1
m+1 +E2
m+1)
≤e−r∆t(E1
m+1 +E2
m+1)
≤e−r∆t(1 + √ˆα·ζ2(∆)) ·(E∗
m+1 +ζ1∆3) + e−r∆tBV,1·ζ2(∆)
=ζ3(∆)E∗
m+1 +ζ4(∆),(73)
where
ζ3(∆) := e−r∆t(1 + √ˆα·ζ2(∆)), ζ4(∆) := ζ3(∆) ·ζ1∆3+e−r∆tBV,1·ζ2(∆).
At the first recursive pricing date, M−1, we have
|EM−1(xn, j)| ≤ e−r∆tX
k=1,..,m0
P∆t
j,k |E1
M(xn, j, k)|,(74)
since the terminal payoff is known, from which E∗
M−1≤e−r∆tBV,1·ζ2(∆). Iterating backwards
using (73), we obtain
E∗
0≤(ζ3(∆))M−1E∗
M−1+ζ4(∆) ·
M−2
X
m=0
(ζ3(∆))m
≤e−r∆tBV,1·ζ2(∆) ·(ζ3(∆))M−1+ζ4(∆) ·
M−2
X
m=0
(ζ3(∆))m.(75)
The dominant term in ζ4(∆) is ζ2(∆) = maxj,k kpj,k −ˆpj,kk2=O(∆2) from Kirkby (2015a)
(the theoretical convergence rate of frame projection), and for ∆ sufficiently small, ζ3(∆) ≈1,
from which the second term in (75) decays like O(∆2). The same holds for the first term, and
we conclude that the overall rate of convergence is O(∆2).
22
Remark 8 In practice, option price approximations using frame projection converge at a much
faster rate than predicted, as illustrated in Table 3. By increasing the number of basis elements
N, the projection error can be controlled. For Bermudan options, there is also the error in
locating bxm(j), which is again O(∆2)using linear interpolation,15 and does not affect the overall
rate of convergence. While the proof above was presented for a double barrier option, the more
general case of unbounded continuation region Ccan be proved using the grid extension developed
in Kirkby (2015b). The method uses an approximation of the value function at the boundaries,
and allows for a smaller value of ˆαwhen truncating C(recall Remark 5). We refer the reader
to the previous reference for details, which can be readily adapted to the present case.
Remark 9 For stochastic volatility models, there is the additional error induced by a Markov
chain approximation of the underlying variance process. We find empirically that an algebraic
order of convergence holds with respect to m0, the number of variance states. Numerical exper-
iments demonstrating the convergence across a wide set of models are provided in Sections 4.3
and 5, where we use benchmark prices from Monte Carlo as well as three additional numerical
methods.
4 Numerical Examples
4.1 Some jump distributions
Recall from (2) that the quantity eγ(y,i)represents the jump amplitude associated with νi(y).
In the remainder of the paper, as a specific example, we take γ(y, i) = yfor all i∈ M. We list
below several popular models for the distribution of the random jump size Y.
Normal distribution: In this case, νj(y) is the density of the Normal(aj, b2
j) distribution, i.e.
eγ(Y,j)is log-normal. This jump model is first considered in Merton (1976). Under this model,
κj= exp(aj+1
2b2
j)−1. We then have φj(z) = eizaj−1
2z2b2
jand with the jump rate λjfor
j= 1,2, . . . , m0,
ψj(z)=izζj−1
2z2σ2
j+λjeizaj−1
2z2b2
j−1, j = 1,2, . . . , m0.(76)
However, this distribution has several major drawbacks: (i) the jumps cannot be large, hence
most of the jumps are not easily detected, and (ii) the return process log(St+h/St) lacks the
observed leptokurtic properties, i.e. fat tails, in the empirical data.
Double-exponential distribution: This model is proposed in Kou (2002). In this case,
νj(y) = pjη1,je−η1,j yI{y≥0}+ (1 −pj)η2,j eη2,j yI{y<0}, i = 1, . . . , m0.(77)
15Higher order schemes may be used, but are not found to improve convergence.
23
Here pj∈(0,1), η1,j >1, η2,j >0 are assumed to ensure the finiteness of E(eY) and E(St).
Then we have φj(z) = pjη1,j
η1,j −iz+ (1 −pj)η2,j
η2,j +izand with the jump rate λjfor j= 1,2, . . . , m0,
ψj(z) = izζj−1
2z2σ2
j+λjpj
η1,j
η1,j −iz+ (1 −pj)η2,j
η2,j + iz−1, j = 1,2, . . . , m0.(78)
This model can better explain the asymmetric leptokurtic feature and volatility smile ob-
served from empirical data. Under this model κi=pi
η1,i
η1,i −1+ (1 −pi)η2,i
η2,i + 1 −1.
Mixture of two normal distributions: In Florescu et al. (2013), a jump distribution as a mixture
of two normal distribution is proposed. In this case, νi(y) has the following form:
νj(y) = pj
1
√2πb1,j
e−(y−a1,j )2
2b2
1,j + (1 −pj)1
√2πb2,j
e−(y−a2,j )2
2b2
2,j , i = 1, . . . , m0,(79)
where a1,j >0, a2,j <0, and 0 < pj<1. Under this model, κj=piea1,j +1
2b2
1,j +(1−pj)ea2,j +1
2b2
2,j −
1,where pj∈(0,1) and (1−pj) represent the upward and downward jumps, respectively. Then
we have
φj(z) = pjeiza1,j −1
2z2b2
1,j + (1 −pj)eiza2,j −1
2z2b2
2,j ,
and with the jump rate λjfor j= 1,2, . . . , m0,
ψj(z) = izζj−1
2z2σ2
j+λj(φj(z)−1) , j = 1,2, . . . , m0.(80)
This model can be considered as a distribution having probability pjto jump up with the
jump size drawn from a normal distribution with parameters a1,j and b2
1,j, and with probability
(1 −pj) to jump down with the jump size drawn from a normal distribution with parameters
a2,j and b2
2,j in regime j.
4.2 American put options in a regime switching model
In this section, we consider Bermudan and American option pricing within regime switching
jump diffusion models. For pricing American options with PROJ, we use a two point Richardson
extrapolated price: 2·P(2·M1)−P(M1) where P(M1) is the Bermudan price with M1monitoring
dates. Bermudan options are priced with a fixed number of monitoring dates M.
To assess the convergence rate of PROJ for regime switching models, we use the regime
switching PROJ algorithm to price (single state) L´evy models for which reference prices are
available in the literature. In this case, we set each regime to be consistent with the single state
of the L´evy process. The errors are reported in Table 3 as a function of monitoring dates M,
and log return grid size N, for each of three test sets. The first set of tests are for the BSM
model:
Test 1 (BSM): S0= 100, K= 110, r= 0.1, q= 0.0, T= 1, σ= 0.20
24
with reference price for M= 10 obtained by the COS method of Fang and Oosterlee (2009),
and for M= 50,250 prices are obtained by the PROJ method for L´evy processes of Kirkby
(2016a). Errors are reported for PROJ by pricing the two state RS model with Qdefined by
q12 = 6, q21 = 9, as in Khaliq and Liu (2009). The second set of tests are for the BSM model:
Test 2 (BSM): S0= 100, K= 110, r= 0.1, q= 0.0, T= 1, σ= 0.25
with reference price for M= 10 obtained by the CONV method of Lord et al. (2008), and for
M= 50,250 prices are obtained by the PROJ method. For Test 2 and Test 3 we use the three
state RS model with Qdefined in equation (81), as considered by Rambeericha and Pantelous
(2016). The third set of tests are for the Normal Inverse Gaussian model (NIG):
Test 3 (NIG): S0= 100, K= 100, r= 0.05, q= 0.02, T= 1, α= 15, β=−5, δ= 0.5
which has a risk neutral L´evy symbol (in each state) defined by
ψ(ξ) = −δpα2−(β+ iξ)2−pα2−β2+ iξr−q+δpα2−(β+ 1)2−pα2−β2.
Reference prices for M= 12,52,252 are obtained by the Hilbert transform (HT) method of
Feng and Lin (2013). From Table 3 we see that the regime switching PROJ algorithm achieves
very high accuracy for pricing Bermudan options, and convergence is easily obtained. For the
NIG model with large M, a larger value of Nis required than for the BSM cases, due to its
highly peaked density for small ∆t. In addition to N, the only remaining parameter to apply
the PROJ method is ˆα(which determines log asset grid width), for which we set ˆα= 4 for
M >= 50 and ˆα= 6 for M < 50. When obtaining references prices, a larger value of ˆαmay
be set.
log2(N) 9 10 11 12 13 Ref. Source M
1.70e-05 9.46e-07 8.47e-08 3.31e-09 1.77e-11 10.479520123 COS 10
Test 1 7.37e-05 1.54e-07 3.36e-07 1.89e-08 4.12e-10 10.678044064 PROJ 50
5.99e-05 1.30e-05 1.65e-06 4.08e-08 3.32e-09 10.710838317 PROJ 250
7.69e-06 4.54e-07 1.66e-08 1.58e-09 1.42e-10 11.98745352 CONV 10
Test 2 9.64e-07 4.97e-07 7.74e-08 3.03e-09 9.90e-12 12.132564348 PROJ 50
8.33e-06 2.47e-06 2.56e-07 1.90e-08 1.00e-09 12.162005425 PROJ 250
2.30e-05 5.73e-07 3.36e-07 1.48e-08 7.28e-10 6.4574297377 HT 12
Test 3 4.34e-04 7.94e-06 1.63e-07 2.85e-09 3.88e-09 6.4833874148 HT 52
6.49e+00 1.24e-02 3.15e-04 5.00e-06 8.87e-09 6.489580997740 HT 252
Table 3: Regime Switching Convergence. Bermudan options pricing in standard (non-regime switch-
ing) L´evy models using the PROJ regime switching algorithm. Reference price sources are listed in
the “Source” column.
25
In Table 4, we consider a two-state regime switching diffusion model with no jumps. We
choose the parameters as: Q= −6 6
9−9!,K= 9, r= 0.05, σ1= 0.80 σ2= 0.30. For
comparison, we report the results using the finite difference scheme of Khaliq and Liu (2009) (in
“PDE” columns) and the tree method of Jiang et al. (2016) (in “Tree” columns). The results
obtained by our method are reported in PROJ columns. In particular, all three methods
produce very similar results, and the main source of error in this case is the observed slow
convergence of the Bermudan approximations to the American price, given the high volatility
in each state.
V1V2
S0PROJ PDE Tree PROJ PDE Tree
6.0 3.5286 3.5358 3.5325 3.4505 3.4583 3.4534
7.5 2.7084 2.7143 2.7124 2.6133 2.6196 2.6159
8.5 2.2773 2.2825 2.2815 2.1759 2.1812 2.1784
9.0 2.0906 2.0954 2.0950 1.9873 1.922 1.9900
9.5 1.9208 1.9252 1.9253 1.8165 1.820 1.8193
10.5 1.6257 1.6295 1.6306 1.5214 1.5253 1.5245
12.0 1.2745 1.2776 1.2803 1.1742 1.1773 1.1782
Table 4: American puts in a 2-state regime switching diffusion model. Parameters: K= 9, r =
0.05, σ1= 0.80, σ2= 0.30, T = 1, q12 = 6, q21 = 9. Vjdenotes the value of the put option in the regime
j. PROJ: N= 212,ˆα= 6, M1= 300 with Richardson extrapolation.
Next, we consider American put options in a 2-state regime switching jump diffusion model,
with jumps modeled by a mixture of normal and by a double exponential distribution. We
choose parameters as in Jiang et al. (2016): the generator Q= −0.5 0.5
0.5−0.5!,K= 100,
r= 0.05, σ1= 0.15 σ2= 0.25. For comparison, we report in Table 5 the results obtained by
the tree method of Jiang et al. (2016) (in Tree columns) using 100 time step size.
26
Mixture of Normal Jumps Double Exponential Jumps
Regime 1 Regime 2 Regime 1 Regime 2
S0PROJ Tree PROJ Tree PROJ Tree PROJ Tree
92 47.4172 47.5846 36.7820 36.8430 36.0193 36.0820 28.0313 28.0308
96 46.4487 46.6169 35.6286 35.6893 34.5986 34.6621 26.4319 26.4363
100 45.5286 45.6973 34.5529 34.6133 33.2572 33.3213 24.9477 24.9504
104 44.6528 44.8220 33.5478 33.6084 31.9913 32.0561 23.5727 23.5747
108 43.8178 43.9874 32.6068 32.6679 30.7967 30.8622 22.3001 22.3021
Table 5: American puts in regime switching jump diffusion model. Parameters: T= 1, K = 100, r =
0.05, σ1= 0.15, σ2= 0.25, p1=p2= 0.3445, λ1= 5, λ2= 2. Mixture of two normal jumps: a1,1=
a1,2= 0.3753, a2,1=a2,2=−0.5503, b1,1=b1,2= 0.18, b2,1=b2,2= 0.6944. DE jumps: η1,1=η1,2=
3.0465, η2,1=η2,2= 3.0775.PROJ: N= 212, ˆα= 8, M1= 300 with Richardson Extrapolation
Next, we compare our method with the finite scheme of Rambeericha and Pantelous (2016), in
a case of a three regime model with normally distributed jumps (i.e., in each state the process
evolves following the Merton’s jump diffusion). The generator of the Markov chain α(t) is given
by
Q=
−0.8 0.6 0.2
0.2−1.0 0.8
0.1 0.3−0.4
,(81)
and the arrival rate of jumps differs in each state.
In Table 6, we report the results obtained in Rambeericha and Pantelous (2016) with dif-
ferent sets of space step (M) and time step (N) in RP columns. The results obtained by our
method are reported in the PROJ row, where Richardson extrapolation is used to obtain the
American option price. In particular, the results agree in each regime to within 10−5∼10−6.
Regime 1 Regime 2 Regime 3
M N RP M N RP M N RP
64 128 11.1261351 64 128 13.8328230 64 128 15.7532644
128 256 11.1250693 128 256 13.8314890 128 256 15.7516936
256 512 11.1250406 256 512 13.8313990 256 512 15.7515986
PROJ - - 11.1250534 - - 13.8314058 - - 15.7516115
Diff - - 1.2800e-05 - - 6.8000e-06 - - 1.2900e-05
Table 6: American put option under a 3-regime jump diffusion model. Parameters: K=S0=
100, T = 1.0, r = 0.05, σ1=σ2=σ3= 0.15. Normal jumps for 3 regimes with ai=−0.5, bi= 0.45, i =
1,3, λ1= 0.3, λ2= 0.5, λ3= 0.7. PROJ with Richardson extrapolation: M1= 128, N= 212.
Table 7 reports the computational cost of our proposed algorithm in seconds, taken as an
average over 100 trials for each parameter setting, where all experiments are conducted in Mat-
lab 8.5 on a personal computer with Intel(R) Core(TM) i7-6700 CPU @3.40GHz. In particular,
27
we consider the cost as a function of the key parameters: m0,M, and N. For comparison,
m0= 1 illustrates the cost of the (single state) L´evy pricing algorithm of Kirkby (2016a). The
additional initialization cost of the matrix exponential e
Eis clear for m0= 2,3. With m0fixed,
the cost grows linearly with respect to M, and at the rate Nlog2(N) as we increase N. For
stochastic volatility models, considered next, the main cost becomes initialization of the matrix
exponentials, which grows at a rate of m3
0. However, accurate valuations are obtained with a
relatively small value of m0.
M= 12 M= 52 M= 252
m0123123123
9 0.0004 0.0698 0.0768 0.0017 0.0679 0.0797 0.0056 0.0768 0.1080
log2(N) 10 0.0005 0.1408 0.1555 0.0017 0.1384 0.1591 0.0076 0.1541 0.2038
11 0.0008 0.2894 0.3392 0.0027 0.2904 0.3480 0.0121 0.3200 0.4248
Table 7: CPU times in seconds for Bermudan option pricing in regime switching model averaged
over 100 trials each, with total log return space grid budget N.
From Tables 3 to Table 6, it can be seen that the method we propose can be used to compute
the value of Bermudan and American options within regime switching models with/without
jumps with very high accuracy. In the following sections, we will consider pricing American
put options and barrier options in stochastic volatility models. Moreover, the method requires
just milliseconds, as illustrated in Table 7.
4.3 Heston model
In this section, we consider the problem of pricing options in Heston model using the proposed
method. First, we consider the price of European call options for which reference prices are
available. The results obtained by our method, the COS method of Fang and Oosterlee (2008),
and the tree method of Liu and Nguyen (2015) are reported as PROJ, COS, and Tree columns in
Table 8 respectively, where European option prices for PROJ are computed using the recursive
barrier algorithm with an appropriate barrier. While COS and PROJ agree in all cases within
10−6or better, the tree method is less accurate, especially for a large v0.
28
v0= 0.04 v0= 0.09
S0PROJ COS Tree PROJ COS Tree
90 0.8852 0.8852 0.8852 1.9024 1.9024 1.9017
95 2.2588 2.2588 2.2587 3.6147 3.6147 3.6140
100 4.6105 4.6105 4.6106 6.0703 6.0703 6.0695
105 7.9290 7.9290 7.9290 9.2385 9.2385 9.2372
110 12.0006 12.0006 12.0007 13.0088 13.0088 13.0061
Table 8: European call options under Heston model (without jumps). Parameters: K= 100, r =
0.05, η = 3, θ = 0.04, ρ =−0.1, σv= 0.1, T = 0.25. PROJ values obtained using DOC barrier
algorithm with small L= 40: m0= 80, γ = 3.3, N = 210,ˆα= 4.
In Figure 1, we consider two sets of experiments using the PROJ barrier algorithm to price
European options under Heston model. In general, the same convergence behavior holds for
European, Bermudan, and barrier options. Focusing here on the European case, we can obtain
precise benchmark prices for comparison using the COS method of Fang and Oosterlee (2008)
for European options. The prices are verified up to 14 decimals by the PROJ method for
European options.
5 10 20 30 40 50 60
m0
-6
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
log10 |err|
Model 1
Model 2
Model 3
Model 4
Model 5
5 10 20 30 40 50 60
m0
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
log10 |err|
Model 6
Model 7
Model 8
Model 9
Model 10
Figure 1: Challenging parameter sets: convergence of DOC barrier algorithm for European calls
in Heston model as function of m0. Parameters: S0=W= 50, r= 0.05, T=.25. PROJ with
N= 212, M = 12,ˆα= 6, γ = 3.3.
For Models 1-5, η= 3, ρ =−0.6. We then set v0=θfor each, where respectively
v0= [0.01,0.02,0.03,0.04,0.05] and σv= [0.02,0.08,0.14,0.20,0.26]. For Models 6-10, we
fix σv= 0.15, θ= 0.02, v0= 0.04, and respectively η= [1,0.5,0.25,0.1,0.05] and ρ=
[−0.9,−0.8,−0.7,−0.6,−0.5]. The left panel in Figure 1 aims to capture the effect of increasing
variance level on obtained accuracy. For Models 1-5, initial and long term variance (v0, θ) are
increasing, as well as the volatility of variance, σv. Overall, with m0fixed accuracy is greater
for a smaller, more contained variance process, although practical accuracy of 10−3∼10−4is
easily obtained in general.
29
The right panel of Figure 1 illustrates accuracy as a function of the quantity q:= 2ηθ/(σ2
v)−
1, which is found to be [0.78,−0.11,−0.56,−0.82,−0.91] respectively for Models 6 −10. It is
well known that q∈[−1,0] results in a very difficult parameter set numerically, especially
as qapproaches −1. However, the method easily obtains practical accuracy in each of these
cases, which are chosen for their difficulty. A detailed investigation of near-singular behavior of
variance as qapproaches −1 can be found in Fang and Oosterlee (2011), for which they propose
a novel solution. As illustrated in Figure 1, our method is also able to handle challenging
parameter sets, despite its generality.
S08 9 10 11 12 CPU (sec)
ref.val. 2.000000 1.107621 0.520030 0.213677 0.082044 -
PROJ 0.00e+00 4.77e-03 3.20e-03 1.39e-03 4.37e-04 1.02
M= 10 COS 1.80e-02 4.79e-03 2.85e-03 1.31e-03 5.18e-04 6.9
HT 1.80e-02 4.75e-03 2.81e-03 1.29e-03 5.14e-04 6.6
PROJ 0.00e+00 2.43e-03 1.84e-03 7.84e-04 2.04e-04 1.20
M= 20 COS 9.54e-03 2.39e-03 1.40e-03 6.65e-04 2.78e-04 7.5
HT 9.53e-03 2.31e-03 1.31e-03 6.07e-04 2.53e-04 7.2
PROJ 0.00e+00 3.08e-03 2.84e-03 8.53e-04 1.19e-04 1.55
M= 40 COS 5.14e-03 1.07e-03 5.50e-04 2.54e-04 1.22e-04 8.9
HT 5.12e-03 9.11e-04 3.60e-04 1.27e-04 5.40e-05 8.4
M1= 8 PROJ 0.00e+00 1.11e-04 5.03e-04 2.09e-04 8.40e-06 2.11
Table 9: American option value convergence from Bermudan estimates as a function of M. Richard-
son extrapolation estimate with M1= 8. Parameters: K= 10, T = 0.25, r = 0.1, η = 5, σv= 0.9, θ =
0.16, v0= 0.0625. PROJ: N= 210, ˆα= 4, m0= 30. Ref. values from COS.
In Table 9, we replicate an American option pricing experiment in Heston model from
Fang and Oosterlee (2011) for which American reference values are provided.16 The rows
M= 10,20,40 illustrate the price discrepancy for a Bermudan approximation using the PROJ
algorithm, as compared with the COS method and the Hilbert Transform (HT) method of Zeng
and Kwok (2014). The row M1= 8 illustrates the price error for the Richardson extrapolated
value with M1= 8. We observe that practical accuracy is attained in each case and for all S0,
at a negligible cost. It is also interesting to note that the pricing error for each S0is nearly
identical to the errors reported by Fang and Oosterlee (2011) and(indicating that the dominant
source of error here is likely time discretization), with the exception S0= 8, for which we match
the reference value exactly. In this case, the optimal policy is to exercise immediately (ie at
t0), and the (intrinsic) value is just the payoff K−S0.
16CPU times for the COS method are reported by the authors with an Intel(R) 2.2GHz CPU and 4-GB
memory in MATLAB.
30
Next, we consider American put options under Heston model with jumps. We compare our
results with those obtained by using IMEX scheme of Salmi et al. (2014) and the tree method
in Table 10. For S0= 90, the prices obtained are similar for each, while for S0= 100,110
the tree method diverges from the other two. This divergence was also observed for the tree
method when pricing regime switching models in the presence of jumps.
S0PROJ IMEX Tree
90 11.6171 11.6199 11.6104
100 6.7295 6.7142 6.5267
110 4.2192 4.2616 4.3866
Table 10: American put options under Heston model. Parameters: K= 100, r = 0.03, η = 2, θ =
0.04, ρ =−0.50, σv= 0.25, v0= 0.04, T = 0.5. Normal jumps: a1=−0.5, b1= 0.04, λ = 0.20. PROJ:
N= 211, m0= 40,ˆα= 4, M1= 30 with Richardson Extrapolation.
An additional set of reference prices is provided in Table 11 for Heston model without
jumps, as well as for three jump distributions. For comparison, we also supply prices for the
Tree method. In the absence of jumps, the prices are closely matched, often to four decimals.
Prices also match closely in the case of jumps, but to a lesser extent.
No Jumps Normal Jumps DE Jumps Mixture NJ
KPROJ Tree PROJ Tree PROJ Tree PROJ Tree
40 0.0160 0.0160 0.5851 0.5826 0.3529 0.3497 0.0338 0.0311
45 0.2872 0.2872 1.1998 1.1970 0.8367 0.8311 0.3605 0.3538
50 1.7376 1.7376 2.7374 2.7524 2.3039 2.3082 1.8434 1.8433
55 5.1605 5.1604 5.8099 5.8413 5.4481 5.4572 5.2095 5.2116
60 10.0000 10.0000 10.1026 10.1137 10.0000 10.0000 10.0000 10.0000
Table 11: American puts in Heston model. Parameters: S0= 50, r = 0.05, T = 0.25, ρ =−0.10, v0=
0.04, σv= 0.10, η = 3, θ = 0.04. Normal Jumps a1=−0.10, b1= 0.30, λ = 1. Mixture of two normal
jumps: a1=−0.05, b1= 0.07, a2= 0.02, b2= 0.03, p = 0.6, λ = 1. DE jumps: η1= 10, η2= 5, p =
0.40, λ = 1.PROJ: N= 211, m0= 60,ˆα= 6, M1= 30 with Richardson Extrapolation.
5 Other stochastic volatility models
In this section we consider several additional stochastic volatility models and provide a large
set of reference prices for Bermudan and barrier options. For many of these models (and their
jump extensions), Bermudan/American and barrier option prices have not yet been reported
in the existing literature. Table 12 provides two test parameter settings for each stochastic
31
volatility model, while Table 13 provides a set of jump distributions that will be considered for
each set of parameters.
Test Set 1 Test Set 2
Model σvη ρ θ v0avσvη ρ θ v0av
SS 0.15 3 -0.6 0.18 0.15 - 0.18 2 -0.5 0.18 0.22 -
HW 0.15 - -0.6 - 0.04 0.01 0.10 - -0.7 - 0.03 0.03
3/2 0.15 4 -0.6 0.03 0.03 - 0.12 2 -0.3 0.04 0.035 -
4/2 0.15 0.5 -0.6 0.035 0.04 - 0.10 1.8 -0.7 0.04 0.04 -
Hes 0.15 4 -0.6 0.035 0.04 - 0.12 1.5 -0.8 0.035 0.04 -
Scott 0.15 3 -0.6 log(0.15) log(0.15) - 0.20 2 -0.9 log(0.16) log(0.18) -
α−H 0.25 0.01 -0.6 0.04 log(0.19) 0.01 0.20 0.05 -0.9 0.2 log(0.17) 0.03
Table 12: 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.
Jump Model Parameters
Normal a1=−0.10, b1= 0.30, λ = 1
Mixed Normals a1=−0.10, b1= 0.30, a2= 0.2, b2= 0.15, p = 0.6, λ = 1
Double Exponential η1= 10, η2= 5, p = 0.40, λ = 1
Table 13: Jump distribution parameters for numerical experiments
The first set of tests considers down-and-out barrier put options in stochastic volatility
models without jumps. To provide a benchmark for models (given a lack of prices reported in
the literature for most models), we calculate prices in Table 14 and 15 using a simple Euler
Monte Carlo (MC) scheme with 5,000,000 sample paths, and time partitioning to reduce bias.
The estimates from Monte Carlo are generally within a couple of cents of PROJ.17 In Table
16, we calculate prices in the two test sets with the addition of a double exponential jump
component.
17The choice of Monte Carlo scheme here is simply to provide a benchmark, rather than as a performance
comparison, in which case more sophisticated techniques could be employed.
32
Test Set 1 (PROJ) Test Set 1 (MC)
K 90 95 100 105 110 90 95 100 105 110
SS 0.3065 0.9559 2.1872 4.1594 6.9423 0.3067 0.9569 2.1915 4.1715 6.9663
HW 0.4380 1.2896 2.7593 4.8973 7.6707 0.4332 1.2800 2.7466 4.8851 7.6620
3/2 0.4134 1.2798 2.8455 5.1852 8.2507 0.4125 1.2794 2.8475 5.1929 8.2654
4/2 0.4550 1.3388 2.8562 5.0459 7.8611 0.4569 1.3409 2.8588 5.0476 7.8608
Hes 0.3847 1.1607 2.5430 4.6104 7.3524 0.3840 1.1605 2.5443 4.6158 7.3651
Scott 0.3329 1.1052 2.6125 4.9931 8.2130 0.3303 1.1010 2.6088 4.9927 8.2188
α-H 0.3830 1.1528 2.5275 4.5923 7.3405 0.3816 1.1512 2.5283 4.5988 7.3546
Table 14: Down-and-out barrier puts under Stochastic Volatility (No Jumps). Parameters are Test
Set 1 as in Table 12. T= 0.5, M= 30. PROJ: N= 213 , ˆα= 6, m0= 120. Knock out L= 80.
Test Set 2 (PROJ) Test Set 2 (MC)
K 90 95 100 105 110 90 95 100 105 110
SS 0.3096 0.9312 2.0578 3.7923 6.1750 0.3107 0.9351 2.0668 3.8108 6.2064
HW 0.4128 1.2708 2.8170 5.1299 8.1698 0.4072 1.2574 2.7965 5.1044 8.1422
3/2 0.4409 1.3241 2.8719 5.1382 8.0774 0.4410 1.3259 2.8773 5.1505 8.0991
4/2 0.4525 1.3326 2.8456 5.0325 7.8496 0.4543 1.3359 2.8484 5.0330 7.8458
Hes 0.3688 1.1058 2.4195 4.3950 7.0373 0.3701 1.1092 2.4278 4.4116 7.0650
Scott 0.3705 1.1510 2.5839 4.7725 7.7061 0.3710 1.1538 2.5916 4.7873 7.7295
α-H 0.3473 1.0917 2.4888 4.6674 7.6334 0.3461 1.0918 2.4923 4.6767 7.6514
Table 15: Down-and-out barrier puts under Stochastic Volatility (No Jumps). Parameters are Test
Set 2 as in Table 12. T= 0.5, M= 30. PROJ: N= 213 , ˆα= 6, m0= 120. Knock out L= 80.
Test Set 1 Test Set 2
K 90 95 100 105 110 90 95 100 105 110
SS 0.2576 0.7802 1.7411 3.2543 5.3920 0.2582 0.7637 1.6648 3.0410 4.9357
HW 0.3324 0.9787 2.1046 3.7684 5.9707 0.3101 0.9460 2.0969 3.8478 6.2076
3/2 0.3095 0.9478 2.1078 3.8744 6.2523 0.3289 0.9858 2.1493 3.8862 6.1944
4/2 0.3411 1.0052 2.1603 3.8597 6.0955 0.3394 1.0013 2.1535 3.8507 6.0867
Hes 0.3001 0.8971 1.9595 3.5625 5.7241 0.2926 0.8685 1.8903 3.4339 5.5247
Scott 0.2638 0.8383 1.9386 3.6949 6.1397 0.2897 0.8837 1.9661 3.6319 5.9081
α-H 0.3008 0.8952 1.9530 3.5527 5.7161 0.2794 0.8533 1.9110 3.5618 5.8474
Table 16: Down-and-out barrier puts under Stochastic Volatility with double exponential jumps.
Parameters as in Table 12. T= 0.5, M= 30. PROJ: N= 213 , ˆα= 6, m0= 120. Knock out L= 80.
To assess the efficiency of our approach, we compute the CPU times averaged over 20 trials
of each parameter setting in Table 17, as a function of the key parameters: m0,N, and M
which drive the computational cost. These time estimates are provided for Bermudan options,
while barrier options are somewhat less expensive to compute. Since the times are essentially
identical for each different model, we use Heston model (with normal jumps) as an example.
A value of m0= 10 ∼20 is often sufficient to obtain a quick price estimate, valid to within
several cents of the true price. Greater accuracy of about 10−3∼10−5is obtained by setting
m0= 30 ∼60, together with a value of N= 210. For short monitoring frequency ∆t<1/200,
we recommend a value of N= 211 to obtain practical accuracy of several decimals, given
33
that the transition densities become more peaked with decreasing ∆t. Finally, a large set of
Bermudan reference prices are recorded in Table 18 for each model across a set of strikes. Note
that time to maturity Tand Mvaries for each set of Bermudan prices.
M= 12 M= 52 M= 252
m010 20 40 60 10 20 40 60 10 20 40 60
9 0.11 0.30 0.85 1.53 0.16 0.54 1.68 3.35 0.44 1.61 5.96 12.56
log2(N) 10 0.22 0.58 1.54 2.88 0.31 0.91 2.82 5.74 0.74 2.60 9.34 20.31
11 0.52 1.47 4.13 8.21 0.58 1.70 5.18 10.57 1.30 4.51 16.10 34.92
Table 17: CPU times in seconds for Bermudan option pricing in stochastic volatility models averaged
over 20 trials each. Total log return space grid budget is N.
6 Conclusion
In this paper, we consider the problem of pricing American and barrier options under regime
switching jump diffusion models and general stochastic volatility models with jumps. By work-
ing in the Fourier domain, we develop a novel, fast and accurate method for pricing gener-
alized American and barrier options, including both discretely and continuously monitored
contracts.Numerical experiments confirm the accuracy and efficiency of the proposed method.
It would be interesting to extend the method to the case of American-style Asian options. An-
other future research direction is an extension to multi-factor stochastic volatility models (e.g.
see Pun et al. (2015)) as well as the case of jumps in the variance process as in Salmi et al. (2014).
Acknowledgements. We would like to thank the anonymous referees and the editor for
stimulating remarks, and comments which significantly help to improve the paper. The usual
disclaimer applies.
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34
Test Set 1 Test Set 2
K 90 95 100 105 110 90 95 100 105 110
No Jumps: T= 0.5, M = 50
SS 1.1225 2.1520 3.8382 6.4099 10.1181 1.8752 3.1442 5.0023 7.5775 10.9557
HW 1.3635 2.6666 4.6546 7.4028 10.9155 0.9106 2.0446 3.9392 6.7229 10.4221
3/2 0.8844 2.0153 3.9216 6.7248 10.4363 1.1072 2.3386 4.3043 7.0918 10.6981
4/2 1.3647 2.6951 4.7245 7.5124 11.0396 1.3346 2.6513 4.6671 7.4462 10.9756
Hes 1.3270 2.5582 4.4700 7.1732 10.7062 1.4404 2.6739 4.5584 7.2115 10.6960
Scott 0.5990 1.5434 3.3137 6.1317 10.0860 0.9989 2.1084 3.9496 6.6782 10.3618
α-H 1.3081 2.5165 4.4025 7.0880 10.6271 0.9140 1.9557 3.7322 6.4411 10.2023
Normal Jumps: T= 0.5, M = 20
SS 4.2406 5.6759 7.5326 9.9051 12.8623 4.7924 6.3802 8.3711 10.8225 13.7713
HW 4.4259 6.0044 8.0462 10.5965 13.6585 4.1042 5.5674 7.5166 10.0239 13.1030
3/2 4.0942 5.5517 7.5010 10.0153 13.1047 4.2487 5.7723 7.7782 10.3218 13.4055
4/2 4.4374 6.0284 8.0927 10.6716 13.7601 4.4112 5.9933 8.0475 10.6165 13.6975
Hes 4.3932 5.9315 7.9210 10.4213 13.4515 4.4724 6.0213 8.0071 10.4881 13.4887
Scott 3.8976 5.2448 7.0820 9.5251 12.6114 4.1619 5.6194 7.5412 10.0079 13.0494
α-H 4.3831 5.9124 7.8903 10.3794 13.4018 4.1025 5.5251 7.4053 9.8372 12.8647
Double Exponential Jumps: T= 0.5, M = 50
SS 3.0018 4.2673 6.0129 8.3615 11.4197 3.6068 5.0492 6.9472 9.3777 12.3975
HW 3.1938 4.6279 6.5867 9.1325 12.2811 2.8448 4.1468 6.0022 8.5090 11.7034
3/2 2.8300 4.1256 5.9825 8.4995 11.7075 2.9982 4.3698 6.2900 8.8343 12.0201
4/2 3.2013 4.6502 6.6350 9.2127 12.3881 3.1747 4.6135 6.5873 9.1552 12.3250
Hes 3.1639 4.5522 6.4534 8.9475 12.0714 3.2521 4.6507 6.5446 9.0126 12.0984
Scott 2.6177 3.7861 5.5173 7.9635 11.2008 2.9105 4.2057 6.0314 8.4952 11.6527
α-H 3.1512 4.5270 6.4117 8.8895 12.0029 2.8457 4.0989 5.8757 8.3008 11.4521
Mixture of Normal Jumps: T= 0.25, M = 20
SS 1.7562 2.8493 4.7188 7.5933 11.3881 2.2495 3.5833 5.5984 8.4000 11.9422
HW 1.9435 3.2686 5.3670 8.2836 11.9000 1.6981 2.8876 4.9264 7.8909 11.6158
3/2 1.6946 2.8823 4.9267 7.8989 11.6267 1.8115 3.0762 5.1579 8.1063 11.7756
4/2 1.9414 3.2768 5.3950 8.3292 11.9505 1.9330 3.2638 5.3776 8.3092 11.9311
Hes 1.9526 3.2343 5.2759 8.1592 11.7836 1.9948 3.2846 5.3176 8.1805 11.7870
Scott 1.5452 2.5923 4.5478 7.5553 11.3950 1.7704 2.9617 4.9652 7.8870 11.5926
α-H 1.8982 3.1610 5.1998 8.0977 11.7429 1.7094 2.8539 4.8264 7.7581 11.5041
Table 18: Bermudan options with stochastic volatility parameters from Table 12 and jump distri-
bution parameters from Table 13. Other parameters: S0= 100, r= 0.05, and q= 0.0. Prices found
with N= 213,m0= 120, and ˆα= 8 for jump models, ˆα= 6 for no jumps.
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A Algorithms
This section provides algorithms for pricing Bermudan and barrier options under stochastic
volatility (SV) and regime-switching (RS) jump diffusions. The algorithms are presented for
the SV case, while the RS case follows by simply supplying the RS model’s generator Qand
defining ¯
Λj,k ≡1 in initialization Algorithm 1. The only other difference is that the final
interpolation step in Algorithms 2 and 4 is not required for RS, since there is no volatility grid.
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Algorithm 1 Stochastic Volatility Initialization
Fix ∆, N
a= 1/∆; ∆ξ= 2πa/N;ξn= (n−1)∆ξ, n = 1, . . . , N
Fix a variance grid {vj}m0
j=1 and generator Qas in Section 2.3
e
E(ξn)=∆t(Q0+ diag(ψ1(ξn), . . . , ψm0(ξn)) , n = 1, . . . , N
e
E(ξ1)←exp( e
E(ξ1)) (matrix exponential)
¯
Λj,k = exp(i∆ξf(vk, vj)), j, k = 1 ...,m0; Λ ←¯
Λ (standard exponential)
for n= 2, . . . , N do
e
E(ξn)←exp( e
E(ξn)) ◦Λ (matrix exponential)
Λ←Λ◦¯
Λ
end for
ζn= exp(−i(1 −N/2)∆ ·ξn)) ·(sin(ξn/(2a))/ξn)2/(2 + cos(ξn/a)), n = 2, . . . , N
Υ∆t
a,N = exp(−r∆t)32a4/N
for j= 1, . . . , m0do
for k= 1, . . . , m0do
Hj,k
1←e
Ek,j (ξ1)/(24a2)
Hj,k
n←e
Ek,j (ξn)·ζn, n = 2, . . . , N
{¯
βj,k
n}N
n=1 ← <{FFT({Hj,k
n}N
n=1)}
µj,k = FFT nΥ∆t
a,N ·(¯
βj,k
N/2, ..., ¯
βj,k
1,0,¯
βj,k
N−1, ..., ¯
βj,k
N/2+1)>o
end for
end for
Algorithm 2 Down-and-out Barrier Option (put/call) - Stochastic Volatility
Fix ˆα, N ; determine x1,∆, n0as in Section 2.5.1
Call initialization Algorithm 1
Coefficient Matrix: θn,j ←0, n = 1, . . . , N/2, j = 1, . . . , m0
Initialize first column θn,1with θM
ndefined in equation (44) or (46) (put/call)
Cn,j ←0, n = 1, . . . , N/2, j = 1, . . . , m0
e
θ←FFT{(θ1,1, ..., θN/2,1,0, ..., 0)}}
for j= 1, . . . , m0do
for k= 1, . . . , m0do
p←iFFT{µj,k ◦e
θ};{Cn,j}N/2
n=1 ← {Cn,j}N/2
n=1 +{p}N/2
n=1
end for
end for
for m=M−2,...,0do
for k= 1, . . . , m0do
θ1,k = (13C1,k + 15C2,k −5C3,k +C4,k)/48
θn,k = (Cn−1,k + 10Cn,k +Cn+1,k)/12, n= 2, . . . , N/2−1
θN/2,k = (13CN/2,k + 15CN/2−1,k −5CN/2−2,k +CN/2−3,k )/48
end for
Call Algorithm 3 to update {Cn,j }
end for
Find k0∈ {1, . . . , m0}such that vk0≤v0< vk0+1
Price(v0) := Cn0,k0+ (Cn0,k0+1 −Cn0,k0)·(v0−vk0)/(vk0+1 −vk0)
39
Algorithm 3 Update Continuation Value
Cn,j ←0, n = 1, . . . , N/2, j = 1, . . . , m0
for k= 1, . . . , m0do
e
θ←FFT{(θ1,k, ..., θN/2,k,0, ..., 0)}}
for j= 1, . . . , m0do
p←iFFT{µj,k ◦e
θ}
{Cn,j}N/2
n=1 ← {Cn,j}N/2
n=1 +{p}N/2
n=1
end for
end for
Remark 10 Our grid choice for Bermudan options aligns both ln(K/S0)and ln(S0/S0) = 0
with grid points. With grid-width parameter ˆα > 0chosen, we initially define e
∆ = 2ˆα/(N−1),
and set n0=N/4. There are four cases to consider to determine ∆. First, if S0=K, then ∆ :=
e
∆. Second, if 0<|ln(K/S0)|<e
∆,then ∆ := e
∆and cubic spline interpolation is used at the end
of the algorithm to determine the price from values at grid points {n0−2, n0−1, n0, n0+1, n0+2}.
This is done for each of the two states which bracket the initial regime, from which the price
for v0is found by a second interpolation. In the next two cases, we assume |ln(K/S0)|>e
∆,
and define ¯n=bln(K/S0)/e
∆ + N/4c. Third, if K < S0,∆ := ln(K/S0)/(1 + ¯n−N/4), and
redefine ¯n←¯n+ 1. Fourth, if K > S0,∆ := ln(K/S0)/(¯n−N/4). After ∆is found, we set
a:= 1/∆, x1:= (1 −N/4)∆, and determine the frequency grid as in equation (31).
Algorithm 4 Bermudan Put - Stochastic Volatility
Fix ˆα, N ; determine x1,∆ as in Remark 10; n0=N/4; ¯n:= b(a·(log(K/S0)−x1)+1c
Call initialization Algorithm 1
Coefficient Matrix: θn,j ←0, n = 1, . . . , N/2, j = 1, . . . , m0
Initialize θM
ndefined in equation (56), and Gn=GS0(xn), n = 1,...,¯n
Cn,j ←0, n = 1, . . . , N/2, j = 1, . . . , m0
e
θ←FFT{(θM
1, ..., θM
N/2,0, ..., 0)}}
for j= 1, . . . , m0do
for k= 1, . . . , m0do
p←iFFT{µj,k ◦e
θ};{Cn,j}N/2
n=1 ← {Cn,j}N/2
n=1 +{p}N/2
n=1
end for
end for
for m=M−2,...,0do
for j= 1, . . . , m0do
Find bn(j), xbn(j)in (49), and bx(j) in (50) using {Cn,j }N/2
n=1 and {Gn}¯n
n=1
¯ρ←bx(j)−xbn(j), ζ ←a¯ρ
Find δput
0,¯
δput
0, δput
1,¯
δput
1by Table 2, and γ1, ..., γ6by equation (52)
Update {θn,j }N/2
n=1 with definition of θm
n,j in equation (51)
end for
Call Algorithm 3 to update {Cn,j }N/2
n=1
end for
Find k0∈ {1, . . . , m0}such that vk0≤v0< vk0+1;τ:= (v0−vk0)/(vk0+1 −vk0)
Price(v0) := max{Cn0,k0, Gn0}+ (max{Cn0,k0+1, Gn0} − max{Cn0,k0, Gn0})·τ
40
