- Access to this full-text is provided by Hindawi.
- Learn more
Download available
Content available from Discrete Dynamics in Nature and Society
This content is subject to copyright. Terms and conditions apply.
Research Article
Variance Swap Pricing under Markov-Modulated
Jump-Diffusion Model
Shican Liu ,
1
Yu Yang ,
2
Hu Zhang ,
1
and Yonghong Wu
2
1
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
2
School of Electrical Engineering Computing and Mathematical Sciences, Curtin University, Perth 6845, Australia
Correspondence should be addressed to Yu Yang; yu.yang9@postgrad.curtin.edu.au
Received 27 May 2020; Revised 8 December 2020; Accepted 16 December 2020; Published 8 January 2021
Academic Editor: Giancarlo Consolo
Copyright ©2021 Shican Liu et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
is paper investigates the pricing of discretely sampled variance swaps under a Markov regime-switching jump-diffusion model.
e jump diffusion, as well as other parameters of the underlying stock’s dynamics, is modulated by a Markov chain representing
different states of the market. A semi-closed-form pricing formula is derived by applying the generalized Fourier transform
method. e counterpart pricing formula for a variance swap with continuous sampling times is also derived and compared with
the discrete price to show the improvement of accuracy in our solution. Moreover, a semi-Monte-Carlo simulation is also
presented in comparison with the two semi-closed-form pricing formulas. Finally, the effect of incorporating jump and regime
switching on the strike price is investigated via numerical analysis.
1. Introduction
Risk, often measured by the variance (volatility) of a specific
underlying asset’s return, has always been a major concern
for the investors in financial markets. e variance (vola-
tility) changes sarcastically over the investment period,
providing practitioners opportunities to speculate on the
spread between the realized variance (volatility) and the
implied variance (volatility), as well as the motivation to
hedge against the variance risk. As a consequence, variance
(volatility) related derivative products have emerged and
become increasingly important in the financial market,
which has been witnessed by the dramatic rise of the yearly
trading volume in VIX futures (VIX futures provide fi-
nancial practitioners with the ability to trade a liquid var-
iance/volatility product based on the VIX index
methodology) over the past few years (see Figure 1) (the
popularity of VIX futures can be seen from the annual
volume in http://www.cboe.com/data/historical-options-
data/volume-put-call-ratios).
Among all the variance (volatility) related derivative
products, variance (volatility) swaps have drawn much at-
tention from the practitioners and researchers. A variance
(volatility) swap is not a swap in a traditional sense, but a
forward contract whose payoff at expiry is determined by the
difference between the realized variance (volatility) and a
preset fixed strike price. e realized variance (volatility) is
usually calculated according to a prespecified formula. For
details regarding calculation of the realized variance (vol-
atility), one can refer to the references: Lewis and Weithers
[1], Bossu [2], and Bossu et al. [3]. A long position in a
variance (volatility) swap generates profit if the realized
variance exceeds the preset strike price. One significant
feature of the variance swap useful in the valuation process is
that it requires zero initial cost since it is essentially a for-
ward contract.
Numerous research has been carried out on variance
(volatility) swap pricing over the last decades. e price of a
volatility swap is closely related to the price of the corre-
sponding variance swap with the same contract details by the
square-root relationship between volatility and variance.
erefore, we only focus on the valuation of variance swaps
in this paper.
e valuation approaches can be categorized into two
types: the model-independent approach and the stochastic
approach. e main idea of the model-free pricing technique
Hindawi
Discrete Dynamics in Nature and Society
Volume 2021, Article ID 9814605, 16 pages
https://doi.org/10.1155/2021/9814605
is to replicate the variance swap with a portfolio composed of
a call option, a put option, and a forward and follow the
routine of calculating the VIX index (see Martin [4]). It is
less complicated and easier to apply since it does not involve
any assumption of the specific form of the dynamics of the
asset’s price process. However, this technique has a draw-
back that it assumes continuous sampling time and con-
tinuous exercise of the options, which is unrealistic. On the
other hand, the stochastic approach is based on the as-
sumption that the underlying asset’s price process is driven
by a specific stochastic process (see Broadie and Jain [5],
Elliott et al. [6], Elliott and Lian [7], Little and Pant [8], and
Song-Ping and Guang-Hua [9]). Even though the early
studies of the stochastic approach also consider the con-
tinuously sampled variance swap as an approximation, an
increasing number of models are put forward to price the
variance swap on yearly, quarterly, monthly, and daily base
to investigate the discretely sampled variance swap (see
Broadie and Jain [5], Little and Pant [8], and Song-Ping and
Guang-Hua [9]). Both the discrete model and continuous
model are investigated in the paper Broadie and Jain [5]
where the authors prove that the discretely sampled variance
swap price converges to the continuously sampled variance
swap price as the observation frequency approaches infinity.
Little and Pant [8] assumed that the local volatility varies
with both time and the stock price according to a known
function and studied the discretely sampled variance swap
via a finite difference method. ey reduced the dimension
of the pricing problem by introducing a two-stage approach,
which greatly improved the efficiency and accuracy of their
pricing formula.
Since the variance swap is designed to hedge against the
risk from the volatility, it would be meaningless and un-
realistic to assume the constant volatility in the stochastic
process of the underlying asset. Consequently, stochastic
volatility models have been incorporated into the valuation
of the variance swaps (see Heston [10] and Sch¨
obel and Zhu
[11]). Song-Ping and Guang-Hua [9] priced a variance swap
with discrete sampling times based on the Heston stochastic
volatility model (see Heston [10]), assuming that the un-
derlying stock price is driven by a Cox–Ingersoll–Ross (CIR)
type stochastic process. An explicit solution with high ef-
ficiency and accuracy is derived in their paper by adopting
the dimension reduction technique from Little and Pant [8]
and using the generalized Fourier transform. Broadie and
Jain [5] considered both stochastic volatility and jump
diffusion in the dynamics of the underlying asset price
process, and the pricing formula was established by utilizing
the characteristic function method. Cheng and Zhao [12]
proposed a tractable approach to price volatility derivative
under a general stochastic volatility model. In their work, the
underlying volatility is assumed to comply with a beta prime
distribution, which is flexible and consistent with the feature
of the market data, and the jump diffusion process is also
captured by extending the model through stochastic time
changes. As the stochastic volatility is always investigated
together with a stochastic interest rate, a model incorpo-
rating these two factors is investigated in the paper Cao et al.
[13] where it is proved that the effect of the interest rate is
not as vital as that of the volatility on the variance swap
price.
Along another line, the regime-switching model, where
the parameters for the dynamics of the underlying asset’s
price are modulated by an observable Markov chain that
represents the general varying market states, is widely ac-
cepted as economically reasonable and has been applied to a
large number of financial models such as those related to
financial time series and derivative pricing (see Buffington
and Elliott [14], Liu et al. [15], Yao et al. [16], Boyle and
Draviam [17], Yuen and Yang [18], and Costabile et al. [19]).
Elliott et al. [6] first considered the regime-switching model
in a variance swap pricing problem where the quadratic
return related to the variance is analysed by combining the
probabilistic approach with the partial differential approach.
However, their pricing formula is based on continuous
sampling times. erefore, Elliott and Guang-Hua [7] im-
proved the accuracy of their former pricing formula and
derived a semiclosed solution of the variance and volatility
swap in a discretely sampled case via the characteristic
function method. However, in the literature mentioned
above, the price processes of the underlying asset are as-
sumed to be continuous under each fixed market state which
in the real world are often discontinuous and have jumps
even under the given market state. As a matter of fact, the
Markov regime-switching model also describes jumps, but
of a different type from the jumps depicted in jump-dif-
fusion models. e general jumps may be caused by some
unexpected financial events, which may have short-term and
temporal effects on the prices of the assets and liabilities. On
the other hand, the Markovian-type jump may result from
the changes in the entire economic environment, which may
affect the prices in the long run. erefore, to formulate a
suitable pricing model for the variance swaps over a time
period of any length and incorporate both types of jumps, it
is reasonable to consider the Markov-modulated jump
diffusion model. In fact, the Markov regime-switching
jump-diffusion model has been applied to solve financial
problems such as portfolio selection and option pricing (see
Elliott et al. [6], Yu [20], Ramponi [21], and Weron et al.
[22]). Moreover, Zhang et al. [23] investigated a stochastic
control problem in a regime-switching jump-diffusion
market and established a sufficient stochastic maximum
1e8
0.00
0.03
0.05
0.08
0.10
0.13
0.15
0.18
Percentage (%)
2008 2010 2012 2014 2016 20182006
Year
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Trading volume
Figure 1: Trading volume of VIX future.
2Discrete Dynamics in Nature and Society
principle. However, to the best of our knowledge, no work
has been done before to consider the Markov-modulated
jump-diffusion model in the variance swap pricing problem,
which motivates the work of this paper.
e aim of this paper is to price a discretely sampled
variance swap under Heston’s stochastic volatility model
with Markov-modulated jump diffusion. is will extend
the work of Elliott and Lian [7] with further consideration
of jump diffusion in the regime-switching model. Dif-
ferent from the characteristic function method used by
Elliott and Lian [7], we apply the generalized Fourier
transform method and the two-stage approach to obtain a
semi-closed-form pricing formula. To illustrate the ac-
curacy and efficiency of our discrete solution, we present a
semi-Monte-Carlo simulation and derive the pricing
formula of a variance swap with continuous sampling
times and compare the prices from the three methods
under a range of different observation frequencies. Since
the main contribution of this paper is integrating both the
regime switching and jump diffusion in the variance swap
pricing problem, we examine the effect of both the
Merton-type and Kou-type jumps under the regime-
switching model in numerical analysis. We also investi-
gate the effect of ignoring regime switching and the effect
of the model parameters such as the transition rate on the
swap price.
e rest of the paper is organized as follows. In Section 2,
our Heston’s stochastic volatility model with Markov-
modulated jump diffusion, including a measure change, is
described in detail. In Section 3, we derive our pricing
formula via the generalized Fourier transform under a two-
stage framework. In Section 4, several numerical examples
are given to demonstrate the efficiency and accuracy of our
pricing formula. Section 5 presents a summary of our paper.
2. Model Description
Before we start formulating our model, we first introduce
our basic idea of pricing a variance swap.
A variance swap is defined as a forward contract on the
future realized variance of the return from the specified
underlying financial asset. Generally, the payoff function of a
long position in a variance swap at expiry takes the form
V(T) � (σ2
R−Kvar) × G, where σ2
Rdenotes the realized
variance, Kvar is the strike price of the variance swap, and G
denotes the notional amount of the swap in dollars per
volatility point squared. Usually, the values are all consid-
ered on an annualized basis.
Furthermore, the value of a variance swap at time t,
which equals the expected present value of the payoff under
the risk-neutral measurement of Q, can be expressed as
follows:
V(t) � EQ
te−T
trtdtσ2
R−Kvar
G
,(1)
where rtis the related interest rate, σRdenotes the realized
volatility, and EQ
tdenotes the conditional expectation at
time t.
e nature of a forward contract indicates that the value
of a variance swap at entry is equal to zero. us, by setting
V(0) � 0, we can easily have the following fair strike price:
Kvar �EQ
0σ2
R
.(2)
e pricing of a variance swap problem is then reduced
to calculate the expectation in equation (2).
e realized variance σ2
Ris obtained by discretely
sampling over the contract lifetime period [0, Te], which is
also referred to as the total sampling period. Let Tebe the life
time of the contract; AF �N/Teis the annualized factor
converting this expression to an annualized variance, which
is assumed to be within a wide range from 5 to 252 according
to the sampling frequency.
e specific calculation of the realized variance σ2
Rdiffers
from contract to contract. Usually, the details of the cal-
culation would be specified in the contract initially. In this
paper, we use a typical formula which is also used by many
other researchers as follows:
σ2
R�AF
N
N
k�1
Stk−Stk−1
Stk−1
2
×1002,(3)
where Stkdenotes the underlying stock price at the k-th
observation time and Ndenotes the total number of the
observations.
us, our pricing of a variance swap problem is reduced
to the calculation of the conditional expectation of the re-
alized variance defined by (3) under the risk-neutral mea-
surement of Qat time 0. Next, we start formulating our
model.
2.1. Heston Model with Markov-Modulated Jump Diffusion.
In this paper, we use a complete probability space (Ω,F,P)
with Pbeing the real-world probability measure. e market
regime is divided into ndifferent states described by the
states of a Markov chain α(t). Following Elliott et al. [6], α(t)
is a continuous-time finite-state observable Markov chain
whose value can be selected from the state space
E�e1, e2,. . . , en
, where ei� (0,...,1,...,0)′∈Rnis a
canonical unit vector. Moreover, the semimartingale rep-
resentation theorem for the process α(t)can be obtained as
follows:
dα(t) � Q(t)α(t)dt+dM(t),(4)
where M(t), t ∈[0,∞)is a Rn-valued martingale incre-
ment process with respect to the natural filtration generated
by α(t), and
Q(t) �
q11 ··· q1n
⋮... ⋮
qn1··· qnn
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(5)
is the generator matrix of α(t), where qij denotes the in-
tensity of transition from state ito state jsatisfying iqij �0.
Furthermore, let wsand wybe two Wiener processes.
For consideration of the skew effect, we assume that wsand
wyare correlated with a constant correlation coefficient ρ.
Discrete Dynamics in Nature and Society 3
e stochastic process α(t)is assumed to be independent
with wsand wy.
For simplicity, we consider a financial market with only
two assets: a risk-less bond B(t)and a risky stock S(t). e
price of the bond is driven by the following deterministic
process:
dB(t) � rα(t)B(t)dt, (6)
where rα(t)�<r, α(t)>is the interest rate process which
depends on the market state. 〈·,·〉 denotes the inner product
in Rn, and r� (r1,. . . , rn)′is a vector representing different
interest rates under different market states. To be specific, ri
is the interest rate corresponding to the state ifor each
i�1,. . . , n. Note that the subsequent parameters of the risky
stock price process are defined in a similar way.
e price of stock is assumed to be driven by the fol-
lowing Markov-modulated jump diffusion process:
dS(t) � μα(t)dt+σ(y)dws+R
βα(t)(t, z)
Nα(t)(dz, dt)
S(t),
(7)
where μα(t)� 〈μ,α(t)〉,μ� (μ1,. . . ,μn)′denotes the ap-
preciation rate of the stock process, and
βα(t)(t, z) � β(t, z, α(t)) � 〈β(t, z),α(t)〉,β(t, z) � (β1
(t, z),...,βn(t, z))′is a generalized form of the jump-size z
(we consider an exponential form of β(t, z)in the following
sections).
Nα(t)(dz, dt) � 〈
N(dz, dt),α(t)〉, and
N(dz,
dt) � (
N1(dz, dt),. . . ,
Nn(dz, dt))′is a compensated
Poisson random measure which is also determined by the
Markov chain α(t)and can be rewritten as follows:
Nα(t)(dz, dt) � Nα(t)(dz, dt) − λα(t)vα(t)(dz)η(dt),(8)
where vα(t)(dz) � <v(dz),α(t)>,v(dz) � (v1(dz),. . . ,
vn(dz))′denotes the jump size distribution and
λα(t)�<λ,α(t)>,λ� (λ1,. . . ,λn)′is the jump intensity
which describes the expected number of jumps.
Nα(t)(dz, dt) � <N(dz, dt),α(t)>,N(dz, dt) � (N1(dz,
dt),. . . , Nn(dz, dt))′is the Markov-modulated Poisson
random measure. η(dt)is a generalized form of dt, and for
simplicity, we take η(dt) � dtin this paper. σ(y)denotes the
volatility rate of the stock process and is assumed to be a
function of ywhich is driven by the following stochastic
process:
dyt�a bα(t)−yt
dt+σv��
yt
√dwyt,(9)
where ais corresponding to the speed of mean reversion
adjustment, bα(t)� 〈b, α(t)〉, b � (b1,. . . , bn)′is the mean,
and σvdenotes the so-called volatility of volatility.
2.2. Change of Measure. A market described by the Mar-
kov-modulated jump-diffusion is not complete; thus,
there exists more than one equivalent martingale measure
[24]. In this paper, we apply the regime-switching gen-
eralized Esscher transform to determine a specific
equivalent martingale measure. is method was firstly
proposed by Elliott and applied widely in the pricing of
the currency and variance swap [6]. Elliott divided the risk
premium into the diffusion term and the jump term, and
the diffusion component and the jump component are
assumed to represent the systematic risk and the unsys-
tematic risk, respectively. e jump risk is unpriced. us,
the parameters of jump component are invariant under
the change of measure from Pto Q. Bo et al. took the jump
components more seriously and derived a closed-form
solution for the risk premium [25]. Different from Elliott’s
approach, Siu and Shen divided the risk premium into
three terms, including the diffusion term, the jump term,
and the Markov switching term, and the authors divided
each risk price by a risk-minimizing approach [26]. For
simplification, we adopt a two-step procedure like Elliott
et al. [27]; in the first step, we divide the asset price into a
continuous diffusive part and jump part and calculate the
risk premiums separately. In the second step, we consider
the regime effects into each coefficients. As our focus here
is the variance swap pricing, we omit the rigorous proof
details and give a simplified expression under the mea-
surement of Q.
Let
dw∗
s�dws+θα(t)dt,
dw∗
y�dwy+cdt, (10)
where θα(t)�<θ,α(t)>,θ�[μ1−r1/σ(y),.. . ,μn−rn/σ(y)]′,
and c�− (Λ/σv)��y
√.Λis the market price of the volatility
risk (risk premium).
Substituting (8) and (10) into (7) and (9), we obtain the
dynamics of the price process of the stock S(t)and yunder
the risk-neutral assumption as follows:
dS(t)
S(t)�rα(t)−λα(t)mα(t)
dt+σ(y)dw∗
s
+R
βα(t)(t, z)Nα(t)(dz, dt),(11)
where mα(t)�Rβα(t)(t, z)vα(t)(dz)and
dy�a∗b∗
α(t)−y
dt+σv��y
√dw∗
y,(12)
in which a∗�a−Λ,b∗
α(t)�abα(t)/a−Λ.
In the subsequent sections, we will only use the risk-
neutral probability measure Q.
Before we move on to the next section, we define three
natural filtration generated by the two wiener processes w∗
s,
w∗
yand the Markov chain α(t)up to time tas follows:
Fs(t) � σw∗
s(u):u≤t
,
Fy(t) � σw∗
y(u):u≤t
,
Fα(t) � σ α(u):u≤t
{ }.
(13)
3. Pricing Variance Swap
As we have mentioned in Section 2, to price a variance swap,
we are concerned with the calculation of the conditional
expectation as follows:
4Discrete Dynamics in Nature and Society
Kvar �EQ
0σ2
R
�EQ
0
AF
N
N
k�1
Stk−Stk−1
Stk−1
2
×1002
⎡⎣⎤⎦
�AF
N
N
k�1
EQ
0
Stk−Stk−1
Stk−1
2
⎡⎣⎤⎦×1002.
(14)
Our pricing problem can be further reduced to the
calculation of the Nconditional expectations of the same
form as
EQ
0
Stk−Stk−1
Stk−1
2
⎡⎣⎤⎦�EQStk−Stk−1
Stk−1
2
|Fs(0)∨Fy(0)∨Fα(0)
⎡⎣⎤⎦,
(15)
for some fixed equal time period Δt�Te/N�tk−tk−1, k �
1,..., N, which is referred to as the sampling period defined
as the time span between two observation points.
We consider the calculation under two cases: k�1 and
k>1. When k�1, we have only one unknown variable St1in
the expectation to be calculated since St0�S0is the current
stock price which is a known constant. We will discuss this
case later.
Now we investigate the latter case where k>1. In this
case, both Stkand Stk−1are unknown variables at initial time
which makes the calculation of the expectation rather
complicated and difficult to work out. erefore, to reduce
the dimension as well as the difficulty in computation, we
utilize the work of Little and Pant (see Little and Pant [8],
Song-Ping and Guang-Hua [9], and Cao et al. [13]) and
introduce a new variable Itdriven by the underlying process:
dIt�δtk−1−t
Stdt, (16)
where δ(·) is a step function with the following definition:
δ(tk−1−t) � δtk−1�0, t ≠tk−1,
1, t �tk−1,
(17)
and the property
R
δaF(t)dt�F(a),for any a∈Rand any integrable function F(t).(18)
us, Itis a new process only related to the previous
observation Stk−1, which can be written as follows:
It�0, t <tk−1,
Stk−1, t ≥tk−1.
⎧
⎨
⎩(19)
With the new defined variable It, we then employ the
two-stage approach from Little and Pant [8] to calculate the
expectation in (15).
To illustrate this approach, we first consider a contingent
claim Uk�Uk(t, St, It, yt,α(t)) defined over the period
[0, tk]with a future payoff function at expiry as
Uk(tk, Stk, Itk, ytk,α(tk)) � ((Stk/Itk) − 1)2. e value of this
claim at time tcould be written as Uk(t, St, It, yt,α(t)) �
EQ[e−r(tk−t)((Stk/Itk) − 1)2|Fs(t)∨Fy(t)∨Fα(t)].
Similar to that in Elliott and Lian [7], we first consider the
conditional expectation given the information about the
sample path of the Markov chain α(t)from time 0 to the expiry
time T,Fα(T), where T�tkin this case. For a given realized
path of α(t), the parameters such as rα(t),λα(t),mα(t), and bα(t)
are all deterministic functions. Under this assumption, we
denote the value of the contingent claim as Wk(t, St, It, yt) �
Uk(t, St, It, yt,α(t)|Fαt(T)) � EQ[e−r(tk−t)((Stk/Itk) − 1)2|F
(t)∨Fα(T)], where F(t) � Fs(t)∨Fy(t).
en, we can easily obtain the corresponding PIDE for
Wkin the following theorem by using the Feynman–Kac
theorem (some subscripts have been omitted without
ambiguity).
Theorem 1. Let Wk(t, St, It, yt) � Uk(t, S, I, y, α(t)|
Fα(T)) � EQ[e−r(tk−t)((Stk/Itk) − 1)2|F(t)∨Fα(T)], and S
is driven by the dynamics of (11). en, Wkis governed by the
following PIDE:
zWk
zt+rα(t)−λα(t)mα(t)
SzWk
zS+1
2σ2S2z2Wk
zS2
+a∗b∗
α(t)−y
zWk
zy+1
2σ2
vyz2Wk
zy2+ρσvyS z2Wk
zSzy
+δtk−1−t
SzWk
zI
+λα(t)R
Wk(t, Sβ(t, z, α(t)))vα(t)(dz) − rα(t)+λα(t)
Wk�0,
(20)
subject to the terminal condition:
Wktk, S, I, y
�S
I−1
2
.(21)
Proof. Let ϕ�Wk(t, St, It, yt); according to It o’s formula
with jump, we first obtain
dϕ�ϕtdt+ϕSdSc(t) + ϕIdI+1
2ϕSS dSc(t)
2+ϕydy
+1
2ϕyy(dy)2+ϕSy dSc(t)dy
+ϕ(t, S(t), I, y) − ϕ(t, S(t− ), I, y),
(22)
Discrete Dynamics in Nature and Society 5
where dSc(t) � [rα(t)−λα(t)mα(t)]dt+σα(t)dw∗
s
Sc(t)de-
notes the continuous part of the stock price process, and the
discrete part of Ito’s formula can be written as
ϕ(t, S(t), I, y) − ϕ(t, S(t− ), I, y ) � R[ϕ(t, Sβ(t, z, α(t) ) − ϕ(t, S )]Nα(t)(dz, dt)
�R[ϕ(t, Sβ(t, z, α(t) )) − ϕ(t, S )]
Nα(t)(dz, dt)
+R[ϕ(t, Sβ(t, z, α(t) )) − ϕ(t, S )]λα(t)vα(t)(dz)dt
�R[ϕ(t, Sβ(t, z, α(t) )) − ϕ(t, S )]
Nα(t)(dz, dt)
+λα(t)R
ϕ(t, Sβ(t, z, α(t))vα(t)(dz)dt−λα(t)R
ϕ(t, S )vα(t)(dz)dt
�R[ϕ(t, Sβ(t, z, α(t) )) − ϕ(t, S )]
Nα(t)(dz, dt)
+λα(t)R
ϕ(t, Sβ(t, z, α(t))vα(t)(dz)dt−λα(t)ϕ(t, S )dt.
(23)
Substituting (23) into (22) and extracting the coefficient
of the dtterm, we can prove eorem 1.
Due to the definition of the function δ(·), the PIDE at
any time other than tk−1could be reduced to
zWk
zt+rα(t)−λα(t)mα(t)
SzWk
zS+1
2σ2S2z2Wk
zS2
+a∗b∗
α(t)−y
zWk
zy+1
2σ2
vyz2Wk
zy2+ρσvyS z2Wk
zSzy
+λα(t)R
Wk(t, Sβ(t, z, α(t)))vα(t)(dz) − rα(t)+λα(t)
Wk�0.
(24)
us, the term related to the variable Itis not considered in
the PIDE anymore except at the time tk−1, which seemingly
indicates the success in dimension reduction. However, we still
have to consider the time point tk−1, where the variable I
experience a jump. Furthermore, the variable Iis still present in
the terminal condition. To handle this and ensure that the
claim’s value remains continuous (which is required by the no-
arbitrary pricing theory), we utilize Little and Pant’s approach
and consider an additional jump condition at time tk−1:
lim
t↑tk−1
Wk(t, S, y, I) � lim
t↓tk−1
Wk(t, S, y, I).(25)
According to Little and Pant’s two-stage approach, we
divide the time period into two parts [0, tk−1]and [tk−1, tk],
during each of which the variable Icould be treated as
constant. us, we have completed the dimension reduction
for the PIDE during each of the time spans. en, we solve
the PIDE in (24) backwards through two stages. We first
derive the solution of the PIDE in the first stage in [tk−1, tk],
which will provide the terminal condition for the PIDE in
the second stage in [0, tk−1]through the jump condition.
en, we further solve the PIDE in the second stage and
obtain the analytical solution for the PIDE within the whole
time period.
After the analytical solution Wk(·) is obtained, we can
further solve for Uk(·) while taking into account the path
change of the Markov chain α(t)as
Ukt, St, It, yt,α(t)
�EQUkt, St, It, yt,α(t)|Fα(T))
Fα(t)
�EQWkt, St, It, yt
|Fαt(t)
.
(26)
en, we can eventually calculate the conditional ex-
pectation that we are concerned with in (15) according to the
Feynman–Kac theorem as follows:
EQ
0
Stk−Stk−1
Stk−1
2
⎡⎣⎤⎦�ertkUk0, S0, I0, y0,α(0)
.(27)
us, we can obtain the fair strike price Kvar in (14).
Now we have illustrated our basic idea for solving this
variance swap pricing problem; we then first start by solving
the PIDE in (24) by the two-stage approach and the gen-
eralized Fourier transform method. □
3.1. Variance Swap Pricing by the Two-Stage Process. As we
have stated before, we divide the time period [0, tk]into two
parts. Let T�tk�kΔt,Δt�Te/N�tk−tk−1. en, the two
time spans are denoted by [0, T −Δt]and [T−Δt, T]. We
first solve the PIDE in the first stage.
3.1.1. Stage I Algorithm. Let T−Δt≤t≤T, and x�lnS;
then, the equation in (24) can be easily converted to the
following PIDE:
6Discrete Dynamics in Nature and Society
zWk
zt+rα(t)−λα(t)mα(t)−1
2y
zWk
zx
+1
2yz2Wk
zx2a∗b∗
α(t)−y
zWk
zy+1
2σ2
vyz2Wk
zy2+ρσvxy z2Wk
zxzy
−rα(t)+λα(t)
Wk+λα(t)R
Wk(t, x +z)vα(t)(dz) � 0,
(28)
subject to the terminal condition
Wk(T, x, I, y) � ex
I−1
2
.(29)
Next we apply the generalized Fourier transform method
to solve the above equation. Let V(w) � F(Wk) �
RWk(x)e−jwxdx, and let Φz(w)denote the characteristic
function of the jump size distribution. en, (28) can be
converted to the following partial differential equation
(PDE):
zV
zt+rα(t)−λα(t)mα(t)−1
2y
(jwV) + 1
2y(jw)2V
+a∗b∗
α(t)−y
zV
zy+1
2σ2
vyz2V
zy2+ρσvyzV
zy(jw)
−rα(t)+λα(t)
V+λα(t)Φz(w)V�0,
(30)
with the transformed terminal condition
VT�F(Wk(T)) � Fex
I−1
2
⎡⎣⎤⎦.(31)
e conversion from (28) to (30) is quite simple
according to the table of Fourier transform pairs. e only
process we have to specify is the conversion of the term
λα(t)RWk(t, x +z)vα(t)(dz)to λα(t)Φz(w)V.
Let v(dz) � p(z)dzin (28) (the subscript is omitted here
for convenience), and p(z)is the density function of the jump
size distribution. en, we have the Fourier transform of the
integral term arising from the jump diffusion as follows:
FR
Wk(t, x +z) ]p(z)dz
�RR
Wk(t, x +z)p(z)e−jwx dzdx
�R
p(z)dzR
Wk(x+z)e−jwxdx
�R
p(z)dzR
Wk(y)e−jw(y−z)dy
�R
p(z)ejwzdzR
Wk(y)e−jwydy
�Φz(w)V.
(32)
Moreover, we specify the Fourier transform of the ter-
minal condition as follows:
VT�FWk(T)
�Fex
I−1
2
�Fe2x
I2−2ex
I+1
�2πδ−2j(w)
I2−2δ−j(w)
I+δ0(w)
,
(33)
where δa(·), for any complex number a, is the generalized
delta function with the same definition and property as in
(17) and (18).
Now we are concerned with the solution of the PDE (30)
with the above terminal condition, which can be assumed to
be of the following form by applying Heston’s solution
scheme:
V�eL(w,t)+M(w,t)yVT.(34)
Substituting the above expression into (30), we obtain
the following ordinary differential equations (ODEs) with
the corresponding terminal conditions, respectively:
−_
M� − 1
2(jw )(1−jw ) +( ρv(jw ) − a∗)M+1
2σ2
vM2,
M(w, T ) � 0,
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩(35)
−_
L�rα(t)−λα(t)mα(t)
(jw) − rα(t)+λα(t)
+λα(t)Φz(w) + a∗b∗
α(t)M,
L(w, T) � 0.
(36)
Discrete Dynamics in Nature and Society 7
e ODE (35) can be solved explicitly as
M(w, t ) � A+B
σ2
v
1−eB(T−t)
1−CeB(T−t),
A�ρσv(jw ) − a∗,
B��������������������
A2+σ2
v(jw )(jw −1)
,
C�A+B
A−B.
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(37)
As for the ODE (36), let ϕl(w) � [rα(t)−λα(t)mα(t)]
(jw) − (rα(t)+λα(t)) + λα(t)Φz(w) + a∗b∗
α(t)M; then, the
equation can be solved numerically as
L(w, t) � T
t〈ϕl(w),α(s)〉ds. (38)
Finally, according to the inverse Fourier transform, we
obtain the solution to the PIDE in (28) as
Wk(t, x, I, y) � F−1(V) � F−1eL(w,t)+M(w,t)yVT
�1
2πR
VTeL(w,t)+M(w,t)yejwxdw
�e2x
I2eL(− 2j,t)+M(− 2j,t)y−2ex
IeL(− j,t)+M(− j,t)y
+eL(0,t)+M(0,t)y.
(39)
From (40), we could easily have M(− j, t) � M(0, t) � 0
for ∀t∈[T−Δt, T]; therefore, we can further obtain
Wk(t, x, I, y) � e2x
I2eL(− 2j,t)+M(− 2j,t)y−2ex
IeL(− j,t)+eL(0,t).
(40)
en, as we have stated before, we can obtain the ter-
minal condition for stage two, which is based on the solution
Wk(t, x, I, y)in the first stage, through the jump condition
in (25):
Wk(T−Δt, x, I, y) � eL(− 2j,T−Δt)+M(− 2j,T−Δt)y
−2eL(− j,T−Δt)+eL(0,T−Δt).(41)
Note that here we are making use of the fact that
limt↓tk−1lnSt�lnItaccording to the definition of It, and the
terminal condition above only contains one stochastic
variable, y.
With the terminal condition, we can now move on to
solving the PIDE (24) in the second stage.
3.1.2. Algorithm of Stage II. Let 0 ≤t≤T−Δt. In this stage,
based on the terminal condition in (41), we calculate
Wk(t, x, I, y) � EQe−r(T−Δt−t)Wk(T−Δt, x, I, y)|Ft(t)
�EQe−r(T−Δt−t)eL(− 2j,T−Δt)+M(− 2j,T−Δt)y
−2eL(− j,T−Δt)+eL(0,T−Δt)|Ft(t)
�e−r(T−Δt−t)eL(− 2j,T−Δt)G(t, y)
−2eL(− j,T−Δt)+eL(0,T−Δt),
(42)
where G(t, y) � EQ(eM(− 2j,t)y|Ft(t)).
Since here we only have one unknown stochastic variable
ywhich is contained in the term G(t, y), we have to solve for
G(t, y)to finally obtain the closed-form expression of Wk.
According to the Feynman–Kac formula, G(t, y)should
satisfy the following PDE with the corresponding terminal
condition:
Gt+1
2σ2
vyGyy +(a∗(b∗
α(t)−y))Gy�0,
G(T−Δt, y ) � eM(− 2j,T−Δt)y.
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩(43)
We assume that Gis of the following affine form:
G(t, y) � eR(t)+H(t)y.(44)
Substituting (44) into (43), we obtain the following
ODEs:
−zR
zt�a∗bα(t)H,
−zH
zt� − a∗H+1
2σ2
vH2
i,
(45)
with the terminal conditions R(T−Δt) � 0 and
H(T−Δt) � M(− 2j, T −Δt). After some simple derivation,
we obtain
H(t) � 2a∗
σ2
v
e−a∗(T−Δt)
e−a∗(T−Δt)−c0
,(46)
R(t) � T−Δt
t〈a∗bα(t)H, α(t)〉dt, (47)
where c0�1− (2a∗/σ2
vM(− 2j, T −Δt)).
Substituting (43) into (42), we can finally obtain the
solution to the PIDE in eorem 1 through the two-stage
approach as follows:
Wk(t, x, I, y) � e−r(T−Δt−t)eL(− 2j,T−Δt)+R(t)+H(t)y
−2eL(− j,T−Δt)+eL(0,T−Δt).(48)
8Discrete Dynamics in Nature and Society
3.2. Variance Swap Pricing under Regime-Switching Markov
Chain. Now we have obtained Wkas the value of Ukbased
on a given realized path of the Markov chain α(t), and we
calculate Ukby taking into account the change of the sample
path of the Markov chain.
Combining (26) and (49), we obtain
Ukt, St, It, yt,α(t)
�EQWkt, St, It, yt
|Fαt(t)
�EQe−r(T−Δt−t)eL(− 2j,T−Δt)+R(t)+H(t)y
−2eL(− j,T−Δt)+eL(0,T−Δt)|Fα(t)
�e−r(T−Δt−t)Φk1(− 2j, t)Φk2(t)eH(t)y
−2Φk1(− j, t) + Φk1(0, t),
(49)
where
Φk1(w, t) � EQeT
T−Δt〈ϕl(w),α(s)〉ds|Fα(t)
�EQetk
tk−1〈ϕl(w),α(s)〉ds|Fα(t)
,
Φk2(t) � EQeT−Δt
t〈a∗b∗
α(s)H,α(s)〉ds|Fα(t)
�EQetk−1
t〈a∗b∗
α(s)H,α(s)〉ds|Fα(t)
.
(50)
Φk1(w, t)and Φk2(t)can be calculated by utilizing the
following formula in the Proposition 3.2 of Elliott and Lian
[7]:
EQeT
t〈v,α(t)〉ds|Fαt(t)
�〈exp T
t
Q′+diag[v]ds
α(t), E〉,
(51)
where E� (1,1,. . . ,1)T∈Rnand Q′denotes the transpose
of the transition matrix Q.
According to (27), we have
EQ
0
Stk−Stk−1
Stk−1
2
⎡⎣⎤⎦�ertkUk0, S0, I0, y0,α(0)
�erΔtΦk1(− 2j, 0)Φk2(0)eH(0)y0
−2Φ1(− j, 0) + Φ1(0,0).
(52)
Since we have only considered the case where k>1 as we
have stated before, we have to further work on the case
where k�1 to obtain the summation in (14) fully.
For k�1, we have tk�t1�Tand tk−1�t0�0�
T−Δt, which indicates that [0, T] � [T−Δt, T]. erefore,
we could derive the term EQ
0[(St1−St0/St0)2]by making use
of the result of the first stage in (41) as follows:
EQ
0
St1−St0
St0
2
⎡⎣⎤⎦�ert1U10, S0, I0, y0,α(0)
�ert1EQU10, S0, I0, y0,α(0)|Fα(T))
Fα(0)
�ert1EQW10, S0, I0, y0
|Fαt(0)
�ert1EQeL(− 2j,0)+M(− 2j,0)y0−2eL(− j,0)
+eL(0,0)|Fα(0)
�erΔtΦ11(− 2j, 0)eM(− 2j,0)y0
−2Φ11(− j, 0) + Φ11 (0,0).
(53)
Let N�AF ∗Teand Δt�Te/N�1/AF. Combining
(14), (52), and (53), we can eventually obtain our fair strike
price as follows:
Kvar �AF
N
N
k�1
EQ
0
Stk−Stk−1
Stk−1
2
⎡⎣⎤⎦×1002
�erΔt
Te
f1y0
+
N
k�2
fky0
⎡⎣⎤⎦×1002,
(54)
where
f1y0
�Φ11 (− 2j, 0)eM(− 2j,0)y0−2Φ11 (− j, 0) + Φ11(0,0),(55)
fky0
�Φk1(− 2j, 0)Φk2(0)eH(0)y0−2Φk1(− j, 0)
+Φk1(0,0), k �2,. . . , N. (56)
Comments: we conclude our calculation algorithm of the
discretely sampled variance swap price in Appendix B. Also,
the integrations of (47) and (51) are evaluated numerically
by Simpson rule using Python, with the accuracy of o(h4)(h
denotes the sliced step size).
4. Numerical Examples
In this section, we present several numerical examples for
illustrating our semiclosed pricing formula for a variance
swap in a Heston model with Markov-modulated jump
diffusion. e effect of incorporating regime switching and
jump diffusion would be investigated. We will also derive the
counterpart pricing formula for a continuously sampled
model and compare the two different prices under varying
observation frequency, which will be helpful for readers to
understand the improvement in accuracy of our discrete
sampling solution.
For simplicity, we assume that there are two market
regimes: regime 1 and regime 2, which can be interpreted as
the “bullish” and the “bearish” market, respectively. In this
case, we have n�2 and the state space of the Markov chain is
reduced to E�e1, e2
; we assume the generator matrix Qas
Discrete Dynamics in Nature and Society 9
Q�−0.1 0.1
0.4−0.4
.(57)
A basic set of parameters applied in this section is
shown in Table 1. We would change some of the pa-
rameters while keeping others fixed to investigate the
effect of the change of the particular parameter. As we can
see from Table 1, the interest rate rand the mean reversion
value bin “bullish” regime I (r�0.06, b �0.09) are higher
than those in “bearish” regime II (r�0.03, b �0.04), and
the jump in the “bullish” market is more oscillatory with
higher jump intensity, which is economically reasonable.
Note that the same parameters except the jump intensity
are also adopted in Elliott and Lian [7] in a Heston
stochastic volatility model with only regime switching. In
addition, we consider both the Merton-type jump and the
Kou-type jump, whose characteristic functions and
density functions are listed in Table 2. e parameters
corresponding to the jump process are also adopted in
Feng and Linetsky [28]. e lifetime of the variance swap
contract is assumed to be Te�1.
4.1. Model Validation. To show the improvement of accu-
racy of our solution, we first compare our pricing formula
with the semi-Monte-Carlo simulation and the continuously
sampled counterpart.
Different from the traditional Monte-Carlo simulation,
the semi-Monte-Carlo simulation is only required to sim-
ulate the sample path of the Markov chain. e simulation
process is discussed in detail in Elliott and Lian [7] and Liu
et al. [15]. We will implement the semi-Monte-Carlo sim-
ulation through the following procedures:
(1) Simulate 10000 sample paths for the Markov chain
α(t)through the generator matrix Qfollowing the
method of Liu et al. [15].
(2) For the ith sample path obtained, we calculate fi
k
according to the equation (55) and (56).
(3) Calculate the fair strike price Kvarifor the ith sample
path using the results from the previous step and the
formula (54).
(4) Obtain the final fair strike price Kvar by taking ex-
pectation of Kvari, i �1,...,100000.
e continuous pricing formula can be obtained as
follows:
Kvar �1−e−aTe
aTe
V0+a
TeTe
0eQ′tdiag b1−e−at
a
⎡⎣⎤⎦α(t), Edt
+eQ′Tediag λjμ2
j+σ2
j
α(t), E.
(58)
e derivation details can be found in Appendix B.
Figure 2 compares the fair strike prices obtained from
our discretely sampled pricing formula (54), the continu-
ously sampled counterpart (58), and the semi-Monte-Carlo
simulation. As we can see from the figure, the results from
our discrete pricing formula match the results from the
semi-Monte-Carlo simulation, which provides verification
for our solution. In fact, our pricing formula is more effi-
ciently ideal with an elegant closed form. On the other hand,
one can also see from the figure that the fair strike price of
the discrete model can be extremely high and deviate
drastically from the continuous counterpart when the ob-
servation frequency is low. erefore, it would be very in-
appropriate to use the continuous price as an approximation
of the discrete one in this situation. However, as the sam-
pling period narrows, the discrete Kvar declines rapidly and
asymptotically approaches the continuous Kvar.
4.2. Regime-Switching Effect. Next, we examine the effect of
incorporating regime-switching in our variance swap
pricing model. We assume that the model without regime
switching coincides with the “bullish” case of the model
with regime switching, but with zero probability of
switching to the other regime. Moreover, to get rid of the
impact of the jump-diffusion part, we add the same jumps
to both of the models with and without regime switching.
We have documented different prices of both models under
a range of observation frequencies in Table 3 where we
could find that Kvar with regime switching is smaller than
that without regime switching. is is reasonable due to the
possibility of switching to the “bearish” regime where Kvar
calculated from our pricing formula is smaller because of
the smaller values of the related parameters. As a matter of
fact, things would be the opposite if we assume that the
model without regime switching coincides with the
“bearish” case. To be more specific, comparisons between
the pricing model with and without regime switching when
entering markets with different initial regimes are shown in
Figures 3 and 4.
In Figure 3, we add the Merton-type jump to both of the
models with the parameters specified in Table 1, while in
Figure 4, we consider the Kou-type jump with parameters
specified in Table 1. We can conclude from Figures 3(a) and
4(a) that with the consideration of regime-switching pos-
sibility, Kvar under “bullish” market will be dragged down by
the “bearish” market. Inversely, from Figures 3(b) and 4(b),
we can see that Kvar under the “bearish” economy will be
pulled up by the effect of the “bullish” market. Moreover, the
difference between the two models seems wider in
Figures 3(b) and 4(b) than that in Figures 3(a) and 4(a). is
can be explained by the different transition rates defined in
the generator matrix Q. e transition rate from regime 1
(bullish) to regime 2 (bearish) is four times the rate from
regime 2 (bearish) to regime 1 (bullish). Consequently, the
difference between Kvar of the model with and without
regime switching in Figures 3(b) and 4(b) is approximately
four times bigger than that in Figures 3(a) and 4(a).
To verify the influence of the transition rates on the fair
strike price, we apply another set of generator matrix:
Q2�−0.4 0.4
0.1−0.1
,(59)
10 Discrete Dynamics in Nature and Society
where the transition rate from regime 1 to regime 2 is now
bigger than that from regime 2 to regime 1. e numerical
result is shown in Figure 5 where we can see that the
difference of the price between the model with and without
regime switching is wider when entering the “bullish”
market (regime 1) than the “bearish” market (regime 2),
Kvar
120
110
100
90
80
70
60
50
50 100 150 200 250
Observation frequency
Explicit method
Continuous
Semi_MC
Figure 2: Calculated Kvar from the discrete model, the continuous model, and the semi-Monte-Carlo simulation.
Table 3: Prices of variance swap with regime switching and without regime switching.
Observation frequency N Kvar (regime switching) Kvar (non-regime switching)
Quarterly 4 210.20 211.83
Monthly 12 127.32 128.22
Fortnightly 26 107.68 108.48
Weekly 52 99.54 100.30
Daily 252 93.17 93.92
Continuously ∞90.62 89.25
Table 1: Model parameters.
Notations Parameters Regime I Regime II
rInterest rate 0.06 0.03
μAppreciation rate 0.08 0.04
aMean reversion rate 3.46 3.46
bMean reversion value 0.009 0.004
σvVolatility of volatility 0.14 0.14
ρCorrelation coefficient −0.82 −0.82
λJump intensity 0.2 0.1
Merton jump
μMean of jump size 0.05 0.04
σJump size volatility 0.086 0.078
Kou jump
η1Inverse mean one 25 20
η2Inverse mean two 50 45
pExponential occurrences 0.2 0.16
Table 2: Jump model parameters.
Model Γ(dz)ϕz(w)
Merton (e− (z−μ)2/���
2π
√δ)dz ejμw− (w2/2)δ2
Kou pλ1e−λ1zIz>0+ (1−p)λ2eλ2zIz<0dz(pλ1/λ1−jw) + ((1−p)λ2/λ2+jw)
Discrete Dynamics in Nature and Society 11
100
95
90
85
80
75
70
50 100 150 200 250
Observation frequency
Kvar
Regime-switching model (bull)
Model without regime switching (bull)
(a)
50 100 150 200 250
Observation frequency
Regime switching (bear)
Model without regime switching (bear)
50
60
70
80
90
Kvar
(b)
Figure 4: Comparison of Kvar with and without regime switching: Kou-type jump. (a) Regime 1. (b) Regime 2.
150
140
130
120
110
100
90
Kvar
50 100 150 200 250
Observation frequency
Regime-switching model (bull)
Model without regime switching (bull)
(a)
Kvar
90
85
80
75
70
65
60
55
50
50 100 150 200 250
Observation frequency
Regime switching (bear)
Model without regime switching (bear)
(b)
Figure 5: Comparison of Kvar with and without regime switching with generator Q2. (a) Regime 1. (b) Regime 2.
Kvar
110.0
107.5
105.0
102.5
100.0
97.5
95.0
92.5
90.0
50 100 150 200 250
Observation frequency
Regime-switching model (bull)
Model without regime switching (bull)
(a)
Kvar
90
85
80
75
70
65
60
55
50
50 100 150 200 250
Observation frequency
Regime switching (bear)
Model without regime switching (bear)
(b)
Figure 3: Comparison of Kvar with and without regime switching: Merton-type jump. (a) State 1. (b) State 2.
12 Discrete Dynamics in Nature and Society
(1) Initialization
(2) for AF in [1,252]do
(3) Δt�Te/AF
(4) for kin[1,AF]do (the loop here is for calculating fk)
(5) T�kΔt
(6) if k�� 1then
(7) Calculate f1(y0)by (55)
(8) else
(9) Calculate fk(y0)by (56)
(10) end
(11) f+ � fk
(12) Kvar �10000 ∗erΔtf/Te
(13) end
(14) end
A
LGORITHM
1:
Algorithm of computing the fair strike price using our pricing formula.
180
160
140
120
100
80
60
40
Kvar
25 50 75 100 125 150 175 200
Observation frequency
λ = 0
λ = 0.05
λ = 0.1
λ = 0.15
λ = 0.2
λ = 0.25
(a)
Kvar
25 50 75 100 125 150 175 200
Observation frequency
100
100
90
80
70
60
50
µ = 0.01
µ = 0.03
µ = 0.05
µ = 0.07
µ = 0.09
(b)
Figure 6: Comparison of Kvar with the Merton-type jump. (a) Comparison of Kvar with varying jump intensity. (b) Comparison of Kvar
with different mean.
Kvar
90
80
70
60
50
25 50 75 100 125 150 175 200
Observation frequency
λ = 0
λ = 0.1
λ = 0.2
λ = 0.3
λ = 0.4
λ = 0.5
(a)
Kvar
90
80
70
60
50
25 50 75 100 125 150 175 200
Observation frequency
p = 0
p = 0.2
p = 0.4
p = 0.6
p = 0.8
p = 1.0
(b)
Figure 7: Comparison of Kvar with and without regime switching. (a) Comparison of Kvar with varying jump intensity. (b) Comparison of
Kvar with different probabilities.
Discrete Dynamics in Nature and Society 13
which implies that the transition rates do have a great in-
fluence on Kvar in our model.
4.3. Jump-Diffusion Effect. Finally, we investigate the impact
of the jump diffusion. Here, we also consider Merton-type
jump and Kou-type jump in Figures 6 and 7, respectively.
Note that we only focus on the case of a “bullish” market
since the regime switching is not the major concern now.
In Figure 6, we assume that the jump size is driven by a
normal distribution. e influence of the jump intensity is
investigated in Figure 6(a) by changing the value of λwhile
keeping other parameters fixed. Note that by setting λ�0,
we consider the pricing model without a jump. As the figure
depicts, Kvar is positively correlated to the jump intensity
and the jump diffusion does have a great influence on the
price. Figure 6(b) shows the different prices with a varying
mean of the jump size, from which we can also conclude that
Kvar increases with the mean. ese results are in line with
Broadie and Jain [5] where a similar conclusion about the
effect of jumps is drawn.
Figure 7 depicts the jump effects under Kou’s model with
the jump size being driven by a double exponential distri-
bution. Same result is obtained from Figure 7(a) about the
positive correlation between the jump intensity and the Kvar.
Also, Figure 7(b) shows the positive effect of the weight we
assign on each exponential distribution.
ree main conclusions drawn from the numerical
analysis are summarized as follows:
(i) e discretely sampled variance swap price con-
verges to that of the continuously sampled coun-
terpart as the observation frequency approaches
infinity.
(ii) Interactions of different market states resulting
from the regime-switching probability are evident.
For instance, the fair strike price under the “bullish”
market will be smaller compared to the non-regime-
switching market with same model parameters due
to the possible transition to the “bearish” market.
(iii) Jumps have a significant effect on the variance swap
price. More specifically, the fair strike price will
increase as the jump intensity increases.
5. Conclusion
In this paper, we have investigated the pricing of a discretely
sampled variance swap under the framework of Heston’s
stochastic volatility model with Markov regime-switching
jump diffusion. e model parameters, including those
related to the jump diffusion, are modulated by a Markov
chain which is used to represent different market states. e
pricing problem is reduced to the calculation of a series of
conditional expectations based on the fact that a variance
swap is essentially a forward contract that requires zero
initial cost. By utilizing the two-stage approach and the
generalized Fourier transform method, we have obtained a
semiclosed pricing formula for the fair strike price Kvar. To
show the improvement of the accuracy of our solution, we
have also derived the price of a continuously sampled
variance swap and compared the results from the two
pricing formulas with a semi-Monte-Carlo simulation. We
found that the discrete price asymptotically approaches the
continuous price as the observation frequency increases,
though the differences between the two prices could be
drastic when the frequency is low. We have presented nu-
merical analysis where a market with two regimes is as-
sumed. We concluded that the possibility of regime
switching could have a significant effect on the price, which
is primarily related to the value of the transition rates.
Moreover, we have examined the effect of incorporating the
jump diffusion into the pricing model by considering both
the Merton-type and the Kou-type jump. By changing the
value of related jump parameters, we find that either jump
has a significant effect on the price, which is positively
correlated to the jump intensity.
Appendix
A. Derivation of the Continuous Model
Our derivation of the continuous strike price is based on the
results of Broadie and Jain [5], Elliott et al. [6, 24], Elliott and
Guang-Hua [7], and Buffington and Elliott [14].
According to the work of Broadie and Jain [5], two terms
will contribute to the continuous fair strike price, including
the stochastic volatility and the jump diffusion term.
Let
Kvar �Kvarvol +Kvarjump,(A.1)
where Kvarvol is the part of the fair strike price calculated by
taking expectation of the cumulative volatility term:
Kvarvol �EQ1
TeTe
0
ytdt|Fα(0)∨Fy(0)
,(A.2)
and Kvarjump is the component of Kvar derived from the
expectation of jump diffusion process:
Kvarjump �E
M
j�1
1
Tj
N Tj
i�1
ln zi
2
⎛
⎜
⎜
⎜
⎜
⎜
⎝⎞
⎟
⎟
⎟
⎟
⎟
⎠|Fα(0)∨Fs(0)
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.
(A.3)
Let Te�M
i�1Ti, and Ti�T
0mα(t)�idsdenotes the
occupation time of the Markov chain and let Mbe the total
occupation of Markov chain. For each Ti, the parameters of
the model stick to a specific status.
For Kvarvol, we can utilize the result derived in Elliott
et al. [6] and Elliott and Guang-Hua [7]:
Kvarvol �1−e−aT
aT V0+a
TT
0〈exp Q′t
diag b1−e−at
a
⎡⎣⎤⎦α(t), E〉.
(A.4)
Here, we mainly focus on the proof of Kvarjump from
jump diffusion terms, utilizing the work of Ramponi [21], in
which the author investigated the start forward option with
regime switching jump diffusion by Fourier transform.
14 Discrete Dynamics in Nature and Society
N(Ti)is orthogonal to N(Tj)when i≠j,i, j �1. . . M.
Based on the work of Broadie and Jain [5], we obtain the
characteristic function of (A.3) for a given sample path of the
Markov chain from time 0 to time Te:
Φ�E[e−jwM
j�11/Tj
(N(Tj)
i�1ln(Yi)
( )2)|Fα(T)]
�
M
i�1
E[e−jw 1/Tj
(N(Tj)
i�1(ln(Yi))2)|Fα(T)]
�
M
i�1
eλjTj(ϕi(w)− 1),
(A.5)
where
ϕi(w) � �����������
Tj
−2jwσ2
j+Tj
eμjjw/−2jwσ2
j+Tj.(A.6)
en, we take the path change of the Markov chain into
consideration and let φ�E[Φ|Fαt(0)]; we obtain
φ�E
M
i�1
eλiTiϕi(w)− 1
( )
⎡⎣⎤⎦�E eM
i�1λiTiϕi(w)− 1
( )
�E ejM
i�1λiTiϑ(w)
�ejTϑM(w)E ejM−1
i�1Tiϑi(w)
�〈eQ′+jdiag(θ(w))α(0), E〉,
(A.7)
where ϑi(w) � ϑi(w) − ϑM(w),ϑi(w) � λi(ϕi(w) − 1), and
� (ϑ1(w),ϑ2(w),...,ϑM(w))′. Here we assume TM�T−
M−1
i�1Ti. Here we utilize the result from Buffington and
Elliott [14]:
E ejM−1
i�1λiTiϑi(w)
�〈eQ′+jdiag(θ(w))α(0), E〉,(A.8)
where E� (1,1,...,1)′∈RM, and θ� (ϑ1(w),ϑ2(w),...,
ϑM−1(w),0)′.
us,
Kvarjump �zΦ
zw|w�0�〈eQ′diag(κ)α(0), E〉,(A.9)
where κ� 〈λj(μ2
j+σ2
j)〉N
i�1. Consequently, we obtain the fair
strike price for a continuously sampled variance swap as
Kvar �1−e−aT
aT V0+a
TT
0〈eQ′tdiag b(1−e−at )
a
α(t), E〉
+〈(eQ′diag(κ)α(t), E〉,
(A.10)
where κ� 〈λj(μ2
j+σ2
j)〉N
i�1.
B. Algorithm of Our Pricing Formula
In this appendix, we conclude our algorithm for computing
the discrete variance swap price. For the initialization part,
we simply write a class to include all the formulas obtained in
our derivation. e benefit of using class is that parameters
in different status can inherit our class method easily without
redefinition (Algorithm 1).
Data Availability
e data and codes used to support the findings of this study
are available from the corresponding author upon request.
Conflicts of Interest
e authors declare that they have no conflicts of interest.
Acknowledgments
e authors are particularly grateful to A/Prof. Benchawan
Wiwatanapataphee for valuable suggestions and technical
support. is study was supported by the Curtin Interna-
tional Postgraduate Research Scholarship (CIPRS) and
Chinese Scholarship Council (CSC).
References
[1] B. Lewis and T. Weithers, Variance Swaps and Cboe S&p 500
Variance Futures, Chicago Trading Company, LLC, Chicago,
IL, USA, 2007.
[2] S. Bossu, Introduction to Variance Swaps, pp. 50–55, Wilmott
Magazine, London, UK, 2006.
[3] S. Bossu, E. Strasser, and R. Guichard, “Just what you need to
know about variance swaps,” JPMorgan Equity Derivatives
Report, vol. 4, 2005.
[4] I. Martin, “Simple variance swaps,” Technical Report, Na-
tional Bureau of Economic Research, Cambridge, MA, USA,
2011.
[5] M. Broadie and A. Jain, “e effect of jumps and discrete
sampling on volatility and variance swaps,” International
Journal of eoretical and Applied Finance, vol. 11, no. 8,
pp. 761–797, 2008.
[6] R. J. Elliott, T. Kuen Siu, and L. Chan, “Pricing volatility swaps
under heston’s stochastic volatility model with regime
switching,” Applied Mathematical Finance, vol. 14, no. 1,
pp. 41–62, 2007.
[7] R. J. Elliott and L. Guang-Hua, “Pricing variance and volatility
swaps in a stochastic volatility model with regime switching:
discrete observations case,” Quantitative Finance, vol. 13,
no. 5, pp. 687–698, 2013.
[8] T. Little and V. Pant, “A finite-difference method for the
valuation of variance swaps,” in Quantitative Analysis In
Financial Markets: Collected Papers of the New York University
Mathematical Finance Seminar, pp. 275–295, New York, NY,
USA, 2001.
[9] Z. Song-Ping and L. Guang-Hua, “A closed form exact so-
lution for pricing variance swaps with stochastic volatility,”
Mathematical Finance, vol. 21, no. 2, pp. 233–256, 2011.
[10] S. L. Heston, “A closed-form solution for options with sto-
chastic volatility with applications to bond and currency
options,” Review of Financial Studies, vol. 6, no. 2,
pp. 327–343, 1993.
[11] R. Sch¨
obel and J. Zhu, “Stochastic volatility with an ornstein-
uhlenbeck process: an extension,” Review of Finance, vol. 3,
no. 1, pp. 23–46, 1999.
[12] Y. Cheng and Bo Zhao, “A general jump-diffusion process to
price volatility derivatives,” Journal of Futures Markets,
vol. 39, no. 1, pp. 15–37, 2019.
Discrete Dynamics in Nature and Society 15
[13] J. Cao, G. Lian, T. R. N. Roslan, and R. N. R. Teh, “Pricing
variance swaps under stochastic volatility and stochastic in-
terest rate,” Applied Mathematics and Computation, vol. 277,
pp. 72–81, 2016.
[14] J. Buffington and R. J. Elliott, “American options with regime
switching,” International Journal of eoretical and Applied
Finance, vol. 5, no. 5, pp. 497–514, 2002.
[15] R. H. Liu, Q. Zhang, and G. Yin, “Option pricing in a regime-
switching model using the fast fourier transform,” Interna-
tional Journal of Stochastic Analysis, vol. 2006, Article ID
018109, 22 pages, 2006.
[16] D. D. Yao, Q. Zhang, and X. Yu Zhou, “A regime-switching
model for european options,” in Stochastic Processes, Opti-
mization, and Control eory: Applications in Financial En-
gineering, Queueing Networks, and Manufacturing Systems,
pp. 281–300, Springer, Berlin, Germany, 2006.
[17] P. Boyle and T. Draviam, “Pricing exotic options under re-
gime switching,” Insurance: Mathematics and Economics,
vol. 40, no. 2, pp. 267–282, 2007.
[18] F. L. Yuen and H. Yang, “Option pricing with regime
switching by trinomial tree method,” Journal of Computa-
tional and Applied Mathematics, vol. 233, no. 8, pp. 1821–
1833, 2010.
[19] M. Costabile, A. Leccadito, I. Massab´
o, and E. Russo, “Option
pricing under regime-switching jump-diffusion models,”
Journal of Computational and Applied Mathematics, vol. 256,
pp. 152–167, 2014.
[20] J. Yu, “Optimal asset-liability management for an insurer
under markov regime switching jump-diffusion market,”
Asia-Pacific Financial Markets, vol. 21, no. 4, pp. 317–330,
2014.
[21] A. Ramponi, “Fourier transform methods for regime-
switching jump-diffusions and the pricing of forward starting
options,” International Journal of eoretical and Applied
Finance, vol. 15, no. 5, p. 1250037, 2012.
[22] R. Weron, M. Bierbrauer, and S. Tr¨uck, “Modeling electricity
prices: jump diffusion and regime switching,” Physica A:
Statistical Mechanics and Its Applications, vol. 336, no. 1-2,
pp. 39–48, 2004.
[23] X. Zhang, R. J. Elliott, and T. K. Siu, “A stochastic maximum
principle for a markov regime-switching jump-diffusion
model and its application to finance,” SIAM Journal on
Control and Optimization, vol. 50, no. 2, pp. 964–990, 2012.
[24] R. J. Elliott, L. Chan, and T. K. Siu, “Option pricing and
esscher transform under regime switching,” Annals of Fi-
nance, vol. 1, no. 4, pp. 423–432, 2005.
[25] L. Bo, Y. Wang, and X. Yang, “Markov-modulated jump-
diffusions for currency option pricing,” Insurance: Mathe-
matics and Economics, vol. 46, no. 3, pp. 461–469, 2010.
[26] T. K. Siu and Y. Shen, “Risk-minimizing pricing and esscher
transform in a general non-markovian regime-switching
jump-diffusion model,” Discrete & Continuous Dynamical
Systems-B, vol. 22, no. 7, p. 2595, 2017.
[27] R. J. Elliott, C.-J Osakwe, and U. Osakwe, “Option pricing for
pure jump processes with markov switching compensators,”
Finance and Stochastics, vol. 10, no. 2, p. 250, 2006.
[28] L. Feng and V. Linetsky, “Pricing options in jump-diffusion
models: an extrapolation approach,” Operations Research,
vol. 56, no. 2, pp. 304–325, 2008.
16 Discrete Dynamics in Nature and Society
Available via license: CC BY 4.0
Content may be subject to copyright.
