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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).

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