Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
Post on 04-Jun-2018
229 Views
Preview:
Transcript
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
1/46
Pricing Volatility Swaps Under Hestons Stochastic
Volatility Model with Regime Switching
Robert J. Elliott Tak Kuen Siu Leunglung Chan
January 16, 2006
Abstract
We develop a model for pricing volatility derivatives, such as variance swaps and
volatility swaps under a continuous-time Markov-modulated version of the stochasticThe Corresponding Author: RBC Financial Group Professor of Finance, Haskayne School of
Business, University of Calgary, Calgary, Alberta, Canada, T2N 1N4; Email: relliott@ucalgary.ca;
Fax: 403-770-8104; Tel: 403-220-5540Lecturer, Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edin-
burgh, United Kingdom; Email: T.K.Siu@ma.hw.ac.ukPh.D. candidate, Department of Mathematics and Statistics, University of Calgary, Calgary,
Alberta, Canada; Email: lchan@math.ucalgary.ca
1
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
2/46
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
3/46
1. Introduction and Summary
Volatility is one of the major features used to describe and measure the uctuations
of asset prices. It is popular as a measure of risk and uncertainty. It plays a signicant
role in three pillars of modern nancial analysis: risk management, option valuation
and asset allocation. There are different measures of volatility, including realised
volatility, implied volatility and model-based volatility. Realised volatility is thestandard deviation of the historical nancial returns; implied volatility is the volatility
inferred from the market option price data based on an assumed option pricing model,
such as the Black-Scholes model; model-based volatility includes both parametric and
non-parametric specications of the volatility dynamics, such as the ARCH models
introduced by Engle (1982), the GARCH models introduced by Bollerslev (1986) and
Taylor (1986), independently, and their variants. These models are introduced to
provide a better specication, measurement and forecasting of volatilities of various
nancial assets. Recent major nancial issues, such as the collapse of LTCM, the
Asian nancial crisis and the problems of Barings and Orange Country, reveal that
the global nancial markets have become more volatile. Due to large and frequent
shifts in the volatilities of various assets in the recent past, there has been a growing
and practical need to develop some models with related nancial instruments to
hedge volatility risk. On the other hand, market speculators may be interested in
3
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
4/46
guessing the direction that the volatility may take in the future. These practical needs
facilitate the growth of the market for derivative products related to volatility. The
initial suggestions for such products included options and futures on the volatility
index. Brenner and Galai (1989, 1993) proposed the idea of developing a volatility
index. The Chicago Board Options Exchange (CBOE) introduced a volatility index
in 1993, called VIX, which was based on implied volatilities from options on the S&P
500 index. The VIX Index reects the market expectations of near-term volatility
inferred from the S&P 500 stock index option prices. For a detailed discussion about
various volatility derivative products, see Brenner et al. (2001).
Variance swaps and volatility swaps are popular volatility derivative products
and they have been actively traded in over-the-counter markets since the collapse
of LTCM in late 1998. In particular, the variance swaps and the volatility swaps
on stock indices, currencies and commodities are quoted and traded actively. These
products are popular among market practitioners as a hedge for volatility risk. The
variance swap is a forward contract in which the long position pays a xed amount
K var / $1 nominal value at the maturity date and receives the oating amount ( 2)R / $1
nominal value, where K var is the strike price and ( 2)R is the realized variance.
The volatility swap is the same as the variance swap, except that the realized
variance (2)R is replaced by the realized volatility ()R .
4
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
5/46
Recently, there has been a considerable interest in pricing and hedging variance
swaps and volatility swaps. Grunbuchler and Longstaff (1996) developed pricing
models for options on variance based on Hestons stochastic volatility model. Carr
and Madan (1998) showed how derivatives on volatility can be valued. Demeter
et al. (1999) discussed the properties and valuation of volatility swaps. Heston
and Nandi (2000) discussed the valuation of options and swaps on variance and
volatility in the context of discrete-time GARCH models. Brockhaus and Long (2000)
discussed the pricing issues of volatility swaps based on Hestons stochastic volatility
model. Javaheri et al. (2002) developed a partial differential equation approach for
pricing volatility swaps under a continuous-time version of the GARCH(1, 1) model.
Brenner et al. (2001) considered the use of the Stein-Stein model to value volatility
swaps. Matytsin (2000) considered the valuation of volatility swaps and options on
variance for a stochastic volatility model with jump-diffusion dynamics. Howison et
al. (2004) also considered the valuation of volatility swaps for a stochastic volatility
model driven by a jump-diffusion. They used an asymptotic approximation for the
solution of the partial differential equation for pricing. Carr et al. (2005) valued
options on variance which were based on quadratic variation. Swishchuk (2004) used
an alternative probabilistic approach to value variance and volatility swaps under
the Heston (1993) stochastic volatility model. Elliott and Swishchuk (2004) valued
5
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
6/46
variance swaps in the context of a fractional Black-Scholes market. Swishchuk (2005)
developed a model for valuing variance swaps under a stochastic volatility model with
delay.
In this paper, we develop a model for pricing volatility derivatives, such as vari-
ance swaps and volatility swaps under a continuous-time Markov-modulated version
of Hestons stochastic volatility (SV) model, (Heston (1993)). In particular, the pa-
rameters of a continuous-time version of Hestons SV model depend on the states of
a continuous-time observable Markov chain process, which can be interpreted as the
states of an observable macroeconomic factor, such as an observable economic indica-
tor for business cycles and the sovereign ratings for the region. The market described
by the Markov-modulated model is incomplete in general, and hence, there is more
than one equivalent martingale pricing measure. We adopt the regime switching Es-
scher transform introduced by Elliott et al. (2005) to determine a martingale pricing
measure for the valuation of variance and volatility swaps. We consider both the
probabilistic approach and the PDE approach for the valuation of volatility deriva-
tives. We assume that the Markov chain process is observable in both the proba-
bilistic approach and the PDE approach. We shall document economic consequences
for the prices of the variance swaps and volatility swaps of the incorporation of the
regime-switching in the volatility dynamics by conducting a Monte Carlo experiment.
6
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
7/46
The next section describes the model. Section three demonstrates the use of the
probabilistic approach for pricing volatility and variance swaps under the continuous-
time Markov-modulated version of Hestons stochastic volatility model. Section four
considers the use of the PDE approach for the valuation. In Section ve, we shall con-
duct the Monte Carlo experiment for comparing the prices implied by the stochastic
volatility models with and without switching regimes. We shall document the eco-
nomic consequences for the prices of switching regimes. The nal section suggests
some possible topics for further investigation.
2. The Model
In this section, we adopt the probabilistic approach for the valuation of volatility
derivatives under a continuous-time Markov-modulated stochastic volatility model.
Our model can be considered the regime-switching augmentation of the
model by Swishchuk (2004) for pricing and hedging volatility swaps. The
Markov-modulated version of Hestons stochastic volatility model can describe the
consequences for the asset price and volatility dynamics of the transitions of the
states of an observable macroeconomic factor, which affects the asset prices and
volatility dynamics, such as an observable economic indicator of business cycles, or
the sovereign ratings of the region by some international rating agencies, such as
7
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
8/46
the Standard & Poors, Moodys and Fitch, etc. In particular, the parameters in the
asset price dynamics and the stochastic volatility dynamics depend on the state of
an economic indicator, which is described by the observable Markov chain.
First, consider a continuous-time nancial model with two primary securities,
namely a risk-free bond B and a risky stock S . Fix a complete probability space
(,
F ,
P ) with
P being the real-world probability measure. Let
T be the time index
set [0, ). On (, F , P ) we consider a continuous-time nite-state observable Markovchain X := {X t}tT with state space S which might be the set {s1, s2, . . . , s N }, wheres i RN , for i = 1, 2, . . . , N . The states of the Markov chain process X describe thestates of an observable economic indicator. Without loss of generality, the state space
of the chain can be identied with the set
{e1, e2, . . . , e N
} of unit vectors in
RN .
Write ( t) for the generator, or Q-matrix, [ ij (t)]i,j =1 ,2,...,N of X . Following Elliott
et al. (1994), the following semi-martingale representation theorem for the process
X can be obtained:
X t = X 0 + t
0(s)X s ds + M t . (2.1)
Here {M t}tT is an RN -valued martingale increment process with respect to thenatural ltration generated by X .
Let W 1 := {W 1t }tT and W 2 := {W 2t }tT denote two correlated standard8
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
9/46
Brownian Motions on (,
F ,
P ) with respect to the
P -augmentation of the ltra-
tion F W := {F W t }tT , where W t := ( W 1, W 2). We assume that
Cov(dW 1t , dW 2t ) = dt , (2.2)
where (1, 1) is the instantaneous correlation coefficient between W 1
and W 2.
We further suppose that the processes X is independent with ( W 1, W 2). Pan
(2002) documented from her empirical studies on stock indices that the
correlation coefficient between the diffusive shocks to the volatility level
and the level of the underlying price is signicantly negative. The model
considered here can incorporate this negative correlation coefficient.
Let {r (t, X t )}tT be the instantaneous market interest rate of the bond B, whichdepends on the state of the economic indicator described by X ; that is,
r (t, X t ) = < r, X t > , t T , (2.3)
where r := ( r 1, r 2, . . . , r N ) with ri > 0 for each i = 1, 2, . . . , N and < , > denotesthe inner product in RN .
To simplify the notation, write rt for r(t, X t ). Then, the dynamics of the price
9
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
10/46
process
{B t
}tT for the bond B are described by:
dB t = rt B t dt ,
B0 = 1 . (2.4)
Suppose the expected appreciation rate {t}tT of the risky stock S depends on thestate of the economic indicator X and is described by:
t := (t, X t ) = < , X t > , (2.5)
where := ( 1, 2, . . . , N ) with i R, for each i = 1, 2, . . . , N .
Let { t}tT denote the long-term volatility level. We assume that t depends onthe state of the economic indicator X and is given by:
t := (t, X t ) = < , X t > , (2.6)
where := ( 1, 2, . . . , N ) with i > 0, for each i = 1, 2, . . . , N .
Let and denote the speed of mean reversion and the volatility of volatility,
respectively. (In general, we could consider a more general case where both the
speed of mean reversion and the volatility of volatility depend on the states of
the economic indicator X .) However, in order to make our model more analytically
tractable, we suppose that both are constant. Suppose that the dynamics of the price
10
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
11/46
process
{S t
}tT and the short-term volatility process :=
{t
}tT of the risky stock
are governed by the following equations:
dS t = t S t dt + t S t dW 1t ,
d2t = (2t 2t )dt + t dW 2t , (2.7)
where Cov(dW 1t , dW 2t ) = dt.
Note that the variance process 2t follows a Cox, Ingersoll and Ross (1985) process.
Let := 1 ; Write W := {W t}tT for a standard Brownian motion, whichis independent of W 1 and X . Then, we can write the dynamics of {S t}tT and := {t}tT as:
dS t = t S t dt + t S t dW 1t ,
d2t = (2t 2t )dt + t dW 1t + t dW t . (2.8)
Let Y t denote the logarithmic return ln( S t /S 0) over the interval [0 , t ]. Then,
Y t = t
0u
12
2u du + t
0udW 1u . (2.9)
In our model, there are three sources of randomness: X , W 1 and W 2. Let F X :=
{F X t }tT , F W 1 := {F W 1t }tT and F W 2 := {F W 2t }tT be the P -augmentation of thenatural ltrations generated by {X t}tT , {W 1t }tT and {W 2t }tT , respectively. Let
11
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
12/46
F S :=
{F S t
}tT denote the
P -augmentation of the natural ltration generated by
{S t}tT .
Our model is a regime switching version of Hestons stochastic volatility model
and is in general incomplete. There are, therefore, innitely many equivalent martin-
gale pricing measures. In the sequel, we shall adopt the regime-switching Esscher
transform developed in Elliott et al. (2005) to determine an equivalent martin-
gale pricing measure for the volatility and variance swaps. The regime-switching
Esscher transform provides market practitioners with a convenient and
exible way to determine an equivalent martingale measure in the incom-
plete market. The choice of the equivalent martingale measure can be
justied by the regime-switching minimal entrophy equivalent martingale
(MEMM) measure (See Elliott et al. (2005)). One drawback of using the
Esscher transform is that the pricing rule by the Esscher transform is not
linear, which is considered by nancial economists as a desirable property
for a pricing rule. There are other possible approaches for determining an
equivalent martingale measure in an incomplete market, for instance, the
minimum variance hedging in Duffie and Richardson (1991) and Schweizer
(1992). In the minimum-variance hedging, the instrinsic value process
corresponding to a given contingent claim is used as the optimal tracking
12
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
13/46
process. The instrinsic value process is dened by the minimal martingale
measure introduced in F ollmer and Schweizer (1991) and Schweizer (1991)
and corresponds to the risk-neutral approach for valuing the claim. The
minimum-variance hedging can provide a pertinent solution to address
the pricing and hedging of a contingent claim while the Esscher transform
mainly deals with the valuation of the claim. The hedging strategies are
optimal in sense of minimizing the expected quadratic costs. The Esscher
transform can provide a convenient and intuitive way to price a claim.
Let Gt denote the right continuous completion of the -algebra F X t F W 1
t F W 2
t ,
for each t T . Let t := ( t, X t , t ) denote a regime switching Esscher process,which is written as follows:
t = ( t, X t , t ) = < (t, t ), X t > , (2.10)
where (t, t ) := (( t, t , e1), (t, t , e2), . . . , (t, t , eN )) and ( t, t , ei) is F W 2
t
measurable, for each i = 1, 2, . . . , N . So, (t, X t , t) is an N -dimensional F X t F W 2
t -
measurable random vector.
Following Elliott et al. (2005), for such a process we dene the regime switching
Esscher transform Q P on Gt with respect to a family of parameters {u}u[0,t ]
13
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
14/46
by:
dQdP Gt
=exp
t0 u dY u
E P exp t
0 u dY u F X t F W 2
t
, t T . (2.11)
Note that t
0 u dY u |F X t F W 2
t N ( t
0 u (u 12 2u )du, t
0 2u 2u du), that is a normal
distribution with mean t
0 u (u 12 2u )du and variance t
0 2u 2u du, under P . Then
given F X t F W 2
t , the Radon-Nikodym derivative of the regime switching Esscher
transform is given by:
dQdP Gt
= exp t
0u u dW 1u
12
t
02u
2u du . (2.12)
The central tenet of the fundamental theorem of asset pricing, which is also called
the fundamental theorem of nance in Ross (2005), established the equivalence be-
tween the absence of arbitrage opportunities and the existence of a positive linear
pricing operator (or positive state space prices). It was rst established by Ross
(1973) in a nite state space setting. Cox and Ross (1976) provided the rst state-
ment of risk-neutral pricing. Ross (1978) and Harrsion and Kreps (1979) then ex-
tended the fundamental theorem in a general probability space and characterized the
risk-neutral pricing as the expectation of the discounted assets payoff with respect
to an equivalent martingale measure. Harrison and Kreps (1979), and Harrison and
Pliska (1981, 1983) established the equivalence between the absence of arbitrage and
the existence of an equivalent martingale measure under which all discounted price
14
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
15/46
processes are martingale. The fundamental theorem was then extended by several
authors, including Dybvig and Ross (1987), Back and Pliska (1991), Schachermayer
(1992) and Delbaen and Schachermayer (1994). In our setting, let := {t}tT denote a process for the risk-neutral regime switching Esscher parameters. Due to
the presence of the uncertainty generated the processes X and W 2, the martingale
condition is characterised by considering an enlarged ltration and requiring:
S 0 = E Q exp t
0r s ds S t F X t F W
2
t , for any t T . (2.13)
One can interpret this condition as one when information about the Markov chain
and the stochastic volatility process are known to the markets agent in advance.
Give these arguments which are similar to those in Elliott et al. (2005), it can be
shown that the martingale condition (2.11) implies that t := (t, X t , t ) should be
given by
t = r(t, X t ) (t, X t )
2t=
(t, X t , t )t
, t T , (2.14)
where t := (t, X t , t ) Gt is the market price of risk at time t.
Then t = < (t, t ), X t > , where (t, t ) = ( r 1 1 t , r 2 2
t , . . . , rN N
t ). This is an
N -dimensional F W 2
t -measurable random vector.
15
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
16/46
Using (2.12), the Radon-Nikodym derivative of
Q with respect to
P is given by:
dQdP Gt
= exp t
0
r u uu
dW 1u 12
t
0
r u uu
2du . (2.15)
By Girsanovs theorem, W 1t = W 1t + t
0 (r s s
s )ds is a standard Brownian motion with
respect to {Gt}tT under Q . Since W and X are independent of W 1, W is astandard Brownian motion under Q and X remains unchanged under the
change of the probability measure from P to Q . Let 2t := 2t (r t t ).Then, the dynamics of S and under Q are
dS t = rt S t dt + t S t d W 1t ,
d2t = (2t 2t )dt + t d W 1t + t dW t . (2.16)
Let W 2t := W 1t + W t . Then, the dynamics of can be written as:
d2t = (2t 2t )dt + t d W 2t . (2.17)
When there is no regime switching, (i.e. the Markov chain X is degenerate), the
risk-neutral dynamics under Q reduce to the risk-neutral dynamics in Heston (1993).
3. The Probabilistic Approach
In this section, we assume the dynamics of the long-term volatility level { t}tT switchover time according to one of the regimes determined by the state of an observable
Markov chain. We may interpret the states of the observable Markov chain as those of
16
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
17/46
some observable economic indicator. First, we shall consider the valuation of variance
swaps, which is more simple than the valuation of volatility swaps. Then, we shall
discuss the hedging of variance swaps and volatility swaps.
3.1. Valuation
A variance swap is a forward contract on annualized variance, which is the square
of the realized annual volatility. Let 2R (S ) denote the realized annual stock variance
over the life of the contract. Then,
2R (S ) := 1T
T
02u du . (3.1)
In practice, variance swaps are written on the realized variance evaluated
based on daily closing prices with the integral in (3.1) replaced by a dis-
crete sum. Hence, variance swaps with payoffs depending on the realized
variance dened in (3.1) are only approximations to those of the actual
contracts. See Javaheri et al. (2002) for discussions on this point.
Let K v and N denote the delivery price for variance and the notional amount
of the swap in dollars per annualized volatility point squared. Then, the payoff of
the variance swap at expiration time T is given by N (2R (S ) K v). Intuitively, thebuyer of the variance swap will receive N dollars for each point by which the realized
annual variance 2R (S ) has exceeded the variance delivery price K v. We can adopt the
17
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
18/46
risk-neutral regime switching Esscher transform
Q for the valuation of the variance
swap. In fact, the value of the variance swap can be evaluated as the expectation of
its discounted payoff with respect to the measure Q , which is exactly the same asthe value of a forward contract on future realized variance with strike price K v.
As in Elliott, Sick and Stein (2003), we initially consider the evaluation of the
conditional value, or price, of a derivative given the information about the sample
path of the Markov chain process from time 0 to time T , say F X T . In particular, given
F X T , the conditional price of the variance swap P (X ) is given by:
P (X ) = E Q [e R T 0 ru du N (2R (S ) K v)|F X T ]
= e R T
0 ru du NE Q (2R (S )|F X T ) e R
T 0 ru du NK v . (3.2)
Hence, the valuation of the variance swap given F X T can be reduced to the problemof calculating the mean value of the underlying variance E Q (2R (S )|F X T ).
Note that under Q , the volatility dynamics can be written as:
2t = 20 +
t
0 ( 2s 2s )ds +
t
0 s d W 2s . (3.3)
Given F X T , 2t is a known function of time t. Hence,
E Q (2t |F X T ) = 20 +
t
0 [ 2s E Q (2s |F X T )]ds . (3.4)
18
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
19/46
This implies that
dE Q (2t |F X T )dt
= [2t E Q (2t |F X T )] . (3.5)
Solving (3.5) gives:
E Q (2t |F X T ) = 20e t +
t
0 2s e
(t s)ds . (3.6)
By It os lemma,
4t = 40 +
t
0[2 2s (
2s 2s ) + 22s ]ds +
t
02 3s d W
2s . (3.7)
Hence,
dE Q (4t |F X T )dt
= (2 2t + 2)E Q (
2t |F X T ) 2E Q (4t |F X T ) . (3.8)
Solving (3.8) gives:
E Q (4t |F X T ) = 40e 2t +
t
0e 2 (t s)(2 2s +
2) 20e s +
s
0 2u e
(s u)du ds . (3.9)
Hence,
V arQ (2t |F X T )
= E Q (4t |F X T ) E 2Q (2t |F X T )=
20 2
(e t e 2t ) +
t
0e 2 (t s) 2 2 2s +
2 s
0 2u e
(s u)du ds
2 t
0 2s e
(t s)ds2
. (3.10)
19
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
20/46
The results for E Q (2t
|F X T ) and V arQ (2t
|F X T ) are consistent with those in Shreve
(2004).
Write V := 2R (S ) to simplify the notation. Then, E Q (V |F X T ) can be calculatedas follows:
E Q (V |F X T ) = 1
T T
020e
t + t
0 2s e
(t s)ds dt
= 20
T (1 e T ) + T
T
0 t
0 2s e (t s)ds dt , (3.11)
We evaluate V arQ (V |F X T ) as follows:
V arQ (V |F X T )= CovQ (V, V |F X T )
= 1 /T 2 T
0 T
0CovQ (2t , 2s |F X T )dtds (3.12)
We rst derive an expression for CovQ (2t , 2s |F X T ). Without loss of generality, wesuppose that t > s . Then, we dene t,s as follows:
t,s := 2t E Q (2t |F W 2
s F X T )
= 2
t 2
s e (t s)
t
s 2
u e (t u)
du . (3.13)
Then, it is immediate that
E Q (t,s |F W 2
s F X T ) = 0 , (3.14)20
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
21/46
E Q (t,s 2s
|F X T ) = 0 , (3.15)
so
CovQ (t,s , 2s |F X T ) = 0 . (3.16)
Hence,
CovQ (2t ,
2s |F
X T )
= CovQ (t,s + 2s e
(t s) + t
s 2u e
(t u) , 2s |F X T )= e (t s)V arQ (
2s |F X T )
= e (t s)20 2
(e s e 2s ) +
s
0e 2 (s u) 2 2 2u +
2
u
0 2ze
(u z)dz du 2 s
0 2u e
(s u)du2
. (3.17)
Therefore,
V arQ (V |F X T )= 1 /T 2
T
0 T
0e (t s)
20 2
(e s e 2s ) +
s
0e 2 (s u) 2 2 2u +
2
u
0 2ze
(u z)dz du 2 s
0 2u e
(s u)du2
dtds
= 20
2
2T 2 (1 e T )T + 14 (1 e 2T )2 + 1 /T 2 T
0 T
0e (t s)
s
0e 2 (s u)
2 2 2u + 2
u
0 2ze
(u z)dz du 2 s
0 2u e
(s u)du2
dtds . (3.18)
For each i = 1, 2, . . . , N , let i := i (r i i); Write := ( 1, 2, . . . , N ). Given21
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
22/46
F X T , the conditional price of the variance swap is given by:
P (X ) = e R T
0 ru du N 20T
(1 e T ) + T
T
0 t
0< 2, X t > e (t s)ds dt K v .(3.19)
Consider now the valuation of the volatility swap given F X T . A stock volatilityswap is a forward contract on the annualized volatility. Let K s denote the annualized
volatility delivery price and N is the notational amount of the swap in dollar per
annualized volatility point. Then, the payoff function of the volatility swap is given
by N (R (S ) K s ), where R (S ) := 1T T
0 2u du. In other words, the payoff of the
volatility swap is equal to the payoff of the variance swap when 2R (S ) is replaced by
R (S ) and K v is replaced by K s . Given F X T , the conditional price of the volatilityswap is given by:
P s (X ) = E Q [e R T 0 ru du N (R (S ) K s )|F X T ]
= e R T
0 ru du NE Q (R (S )|F X T ) e R T
0 ru du NK s
= e R T
0 ru du NE Q ( V |F X T ) e R T
0 ru du NK s . (3.20)
For the valuation of the volatility swap, we need to evaluate E Q ( V |F
X T ). We
adopt the approximation for E Q ( V |F X T ) introduced by Brockhaus and Long (2000),based on the second-order Taylor expansion for the function V . This approxima-tion method has also been adopted in Javaheri et al. (2002) and Swishchuk (2004).
22
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
23/46
It gives
E Q ( V |F X T ) E Q (V |F X T ) V arQ (V |F X T )
8[E Q (V |F X T )]3/ 2 , (3.21)
where the term V ar Q (V |F
XT )
8[E Q (V |F XT )]
3 / 2 is the convexity adjustment.
Hence, given F X T , the conditional price of the volatility swap can be approximatedas:
P s (X ) e R T
0 ru du N E Q (V |F X T ) V arQ (V |F X T )
8[E Q (V |F X T )]3/ 2 K s . (3.22)
23
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
24/46
3.2. Hedging
Hedging variance swaps and volatility swaps is a challenging but practically im-
portant task. Javaheri et al. (2002) contended that hedging these products is difficult
in practice. Hence, they considered the pricing of a variance swap and a volatility
swap by the expectations of the discounted payoffs under the real-world probability
measure. This is an example of actuarial-based valuation method. In this section, we
shall discuss the hedging of variance swaps and volatility swaps. Different methods on
hedging variance swaps, have been proposed in the literature. These methods include
the simple delta hedging, the delta-gamma hedging, hedging using option portfolios,
hedging using a log contract and the vega hedging, etc. For a comprehensive overview
of various hedging strategies, see Demeter et al. (1999), Howison et al. (2004) and
Windcliff et al. (2003). Simple delta hedging does not work well since it is an implicit
linear approximation, which cannot incorporate the effects of large price movements
and the realized variance or volatility will increase substantially when the underlying
share prices move either up or down dramatically. This is the case even one considers
a Geometric Brownian Motion for the price dynamics of the underlying share. Simple
delta hedging is even more difficult to apply when one considers a stochastic volatility
model and a regime-switching stochastic volatility model, which is even more com-
plicated. Hedging using option portfolios and hedging using a log contract works
24
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
25/46
well when one considers an asset price model with non-stochastic volatility. The vega
hedging provides market practitioners with a convenient way to hedge variance swaps
and volatility swaps under a stochastic volatility model, in particular, the Heston SV
model (see Howison et al. (2004) and Carr (2005)). Howison et al. (2004) considered
the use of Vega to hedge volatililty derivatives and derived a general formula for the
Vega of a volatility derivative. Here, following Howison et al. (2004), we adopt the
Vega hedging for a variance swap and a volatility swap since it can provide a conve-
nient way to hedge these products under the regime-switching Hestons SV model. In
the case of the volatility swap, we shall derive an approximate formula for the Vega
of the contract based on the approximate price of the contract.
First, we consider the hedging of the variance swap. Let I t :=
t
0 2udu. Write
F W 2t for the P -augmentation of the -algebra generated by the values of W 2 up to
and including time t. Note that given F X T , F W 2t is equivalent to F W
2
t . Then, using
the results in Section 3.1, the price of the variance swap P (t, X ) at time t is given
by:
P (t, X ) = 1
T e R
T t ru du N I
t + E
Q
T
t2
udu
F X
T F W 2
t T K
v
= 1
T e R
T t ru du N I t +
2t
(e t e T ) + T
t s
t< 2, X u >
e (s u)du ds T K v . (3.23)
25
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
26/46
Let 2t := 2t , which represents the instantaneous volatility of the variance process
at time t. Then, the Vega of the variance swap is given by:
P (t, X )
= 2N t
T 2(e t e T )
= 2N t
T (e t e T ) , (3.24)
which can be evaluated given the current value of the volatility level t .
We shall consider the hedging of the volatility swap. First, we dene R1(t, 2t ),
R2(t, 2t ) and R3(t, 2t ) as follows:
R1(t, 2t ) = I t + 2t
(e t e T ) + T
t s
t< 2, X u > e (s u )du ds ,
R2(t, 2t) =
2t 2
T 2 3[
e (T t )
2
1
e 2 (T t)] +
1
T 2 T
t T
te (h s)
s
0
e 2 (s u) 2 2 2u + 2
u
0 2z e
(u z)dz du 2 s
0 2u
e (s u )du2
dhds ,
and
R3(t, 2t ) = 2t 2
2T
2 (1
e t )t +
1
4 (1
e 2t )2 + 1 /T 2
t
0 t
0
e (u s)
s
0e 2 (s u ) 2 2 2u +
2 u
0 2z e
(u z)dz du 2 s
0
2u e (s u)du
2duds .
26
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
27/46
From the results in Section 3.1, the price of the volatility swap P s (t, X ) at time t can
be approximated as:
P s (t, X ) e R T
t ru du N E Q (V |F X T F W 2t ) V arQ (V |F X T F W
2
t )8[E Q (V |F X T F W
2
t )]3/ 2 K s
= e R T
t ru du N R1(t, 2t ) R3(t, 2t ) + R2(t, 2t )
8R1(t, 2t )3/ 2 K s . (3.25)
Write Ri for Ri(t, 2t ) (i = 1, 2, 3). Then, the Vega of the volatility swap is approxi-
mated as:
P s (t, X )
= e R T
t ru du N t
R1 1/ 2(e t e T ) + 3t8
(R3 + R2)R1 5/ 2
(e t + e T ) + t 2R14T 2 3
e (T t)2
+ 1
e 2 (T t ) , (3.26)
which can be evaluated given the current value of the volatility level t .
4. The P.D.E. Approach
In this section, we adopt a partial differential equation (P.D.E.) approach for
evaluating the expectations of the discounted values of V and V 2, which are useful
for computing the prices of the variance swaps and volatility swaps. The P.D.E.
approach has been adopted by Javaheri, et al. (2002) for the valuation and hedging of
volatility swaps within the framework of a GARCH(1, 1) stochastic volatility model.
Here, we provide a regime switching modication of the problem and derive regime
switching P.D.E.s and the corresponding systems of coupled P.D.E.s satised by the
27
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
28/46
expectations of the discounted values of V and V 2. We adopt a regime switching
version of the Feyman-Kac formula to obtain the regime switching P.D.Es. The
derivation of the regime switching version of the Feyman-Kac formula follows from
the martingale approach and It os differentiation rule in Buffington and Elliott (2002).
First, let V t := 2R,t (S ) := 1T
t0
2u du. Then, given F W
2
t F X T , the price of thevariance swap is given by:
P (X, t ) = e R T
t ru du NE Q (2R,T (S )|F W
2
t F X T ) e R T
t ru du NK v , (4.1)
and the price of the volatility swap is given by:
P s (X, t )
= e R T
t ru du NE Q (R,T (S )|F W 2
t F X T ) e R T
t ru du NK s . (4.2)
Now, suppose 2t = , X t = X and V t = V are given at time t. Then, the price of
the variance swap is given by:
P (X, ,V, t ) = E Q (P (X, t )|2t = , V t = V, X t = X ) , (4.3)
and the price of the volatility swap is given by:
P s (X, ,V, t ) = E Q (P s (X, t )|2t = , V t = V, X t = X ) . (4.4)
Buffington and Elliott (2002a, b) adopted a similar method to determine the price of
a standard European call option.
28
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
29/46
By the double expectation formula,
P (X, ,V, t )
= NE Q e R T t ru du 2R,T (S ) e R
T t ru du K v t = , V t = V, X t = X , (4.5)
and
P s (X, ,V, t )
= NE Q e R T t ru du R,T (S ) e R
T t ru du K s 2t = , V t = V, X t = X . (4.6)
Hence, for the evaluation of P (X, ,V, t ) and P s (X, ,V, t ), we need to compute
1. E Q e R T
t ru du 2t = , V t = V, X t = X
2. E Q e R T
t ru du 2R,T (S ) 2t = , V t = V, X t = X
3. E Q e R T
t ru du R,T (S ) 2t = , V t = V, X t = X
The rst expectation is equal to the value of a zero-coupon bond at time t given that
2t = , V t = V, X t = X , which pays one unit of account at maturity time T . It is
given by:
E Q e R T t ru du 2t = , V t = V, X t = X
= E Q e R T t ru du X t = X
:= B(t ,T,X ) . (4.7)
29
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
30/46
Let B := diag r
, where diagr is the matrix with the vector r on its diagonal.
Write t (r ) := exp[B(T t)]I . Then, by Elliott and Kopp (2004), B(t ,T,X ) isgiven by:
B(t ,T,X ) = < t (r ), X > = < exp[B(T t)], X > I . (4.8)
The third expectation can be approximated by the using the formula in Section
2. More specically,
E Q e R T t ru du R,T (S ) 2t = , V t = V, X t = X
E Q e 2 R T t ru du 2R,T (S ) 2t = , V t = V, X t = X
V arQ e 2 R
T t ru du 2R,T (S ) 2t = , V t = V, X t = X
8 E Q e 2 R T
t ru du 2R,T
(S ) 2t = , V t = V, X t = X . (4.9)
Hence, in order to provide an approximation to the third expectation, we need to
evaluate
1. E Q (e 4 R T
t ru du 4R,T (S )|2t = , V t = V, X t = X )
2. E Q (e 2 R T
t ru du 2R,T (S )
|2t = , V t = V, X t = X ).
Suppose Ht := {W 2u , X u |u [0, t ]}. Since V t is a path integral of 2t and 2tis a Markov process given knowledge of X , (V t , 2t ) is a two-dimensional Markov
30
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
31/46
process given the knowledge of X . Since X is also a Markov process, (X t , 2t , V t ) is a
three-dimensional Markov process with respect to the information set Ht . Hence,
M 1(X, , V, t) := E Q (e R T t ru du 2R,T (S )|2t = , V t = V, X t = X )
= E Q (e R T t ru du 2R,T (S )|Ht ) , (4.10)
M 2(X, , V, t) := E Q (e 2 R T t ru du 2R,T (S )|2t = , V t = V, X t = X )
= E Q (e 2 R T t ru du 2R,T (S )|Ht ) , (4.11)
and
M 3(X, , V, t) := E Q (e 4 R T t ru du 4R,T (S )|2t = , V t = V, X t = X )
= E Q (e 4 R T t ru du 4R,T (S )|Ht ) , (4.12)
Now, write
M 1(X, , V, t) := e R t
0 r u du M 1(X, ,V, t )
= E Q (e R T 0 ru du 2R,T (S )|Ht ) , (4.13)
M 2(X, ,V, t ) := e 2 R t
0 r u du M 2(X, ,V, t )
= E Q (e 2 R T
0 ru du 2R,T (S )|Ht ) , (4.14)and
M 3(X, ,V, t ) := e 4 R t
0 r u du M 3(X, ,V, t )
31
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
32/46
= E Q (e 4 R T 0 ru du 4R,T (S )
|Ht ) . (4.15)
Then, it can be shown that M 1, M 2 and M 3 are Ht -martingales under Q .
In the sequel, we shall derive the P.D.E. for M 1, M 2 and M 3. For each i = 1, 2, 3,
let M i( ,V, t ) denote the N -dimensional vector ( M i(e1, ,V, t ), . . . , M i(eN , ,V, t )).
Then,
M i(X, ,V, t ) = < M i( , V, t), X t > . (4.16)
Then, by applying Itos differentiation rule to M i(X, ,V, t ),
M i(X, ,V, t ) = M i(X, , V, 0) + t
0
M iu
+ ( 2u 2u ) M i
+ 12
22u 2 M i 2
+ 2u M iV
du + t
0
M i
u d W 2u + t
0< M i , dX u > , (4.17)
and
dX t = ( t)X t dt + dM t . (4.18)
Due to the fact that M 1, M 2 and M 3 are Ht -martingale under Q , all terms withbounded variation in the above It os intergral representation for M i (i = 1, 2, 3) must
be identical to zero. Hence, for i = 1, 2, 3, M i satises the following P.D.E.:
M it
+ ( 2t 2t ) M i
+ 12
22t 2 M i 2
+ 2t M iV
+ < M i , X > = 0 . (4.19)
32
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
33/46
Now, let M i( ,V, t ) denote the N -dimensional vector ( M i(e1, ,V, t ), . . . , M i(eN , ,V, t )),
where i = 1, 2, 3. Then,
M i(X, ,V, t ) = < M i( , V, t), X t > . (4.20)
M 1 satises the following P.D.E.:
exp
t
0
ru du
r t M 1 +
M 1
t + ( 2t
2t )
M 1
+
1
2 22t
2M 1
2 +
2t
M 1
V + < M 1, X > = 0 , (4.21)
with terminal condition M 1(X, ,V,T ) = V T .
M 2 satises the following P.D.E.:
exp
2
t
0r u du
2r t M 2 +
M 2
t + ( 2t
2t )
M 2
+
1
2 22t
2M 2
2 + 2t
M 2
V + < M 2, X > = 0 , (4.22)
with terminal condition M 2(X, ,V,T ) = V T .
M 3 satises the following P.D.E.:
exp
4
t
0r u du
4r t M 3 +
M 3
t + ( 2
t 2
t)M 3
+
1
2 22
t
2M 3
2 + 2
t
M 3
V + < M 3, X > = 0 , (4.23)
with terminal condition M 3(X, ,V,T ) = V 2T .
33
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
34/46
Note that with X = e j ( j = 1, 2, . . . , N ),
r t = < r,X t > = r j ,
t = < , X t > = j . (4.24)
Let M ij := M i(e j , ,V,T ), where i = 1, 2, 3 and j = 1, 2, . . . , N . Then, M i =
(M i1, M i2, . . . M iN ). Hence, M 1 satises the following system of N coupled P.D.E.s:
r j M 1 j + M 1 j
t + ( 2 j 2t )
M 1 j
+ 12
22t 2M 1 j 2
+ 2tM 1 jV
+ < M 1, e j > = 0 , (4.25)
with terminal condition M 1(e j , ,V,T ) = V T .
M 2 satises the following system of N coupled P.D.E.s:
2r j M 2 j + M 2 jt + (2 j 2t ) M 2 j + 12
22t 2M 2 j 2
+ 2t M 2 j
V
+ < M 2, e j > = 0 , (4.26)
with terminal condition M 2(e j , ,V,T ) = V T .
M 3 satises the following system of N coupled P.D.E.s:
4r j M 3 j + M 3 jt + ( 2 j 2t ) M 3 j + 12 22t 2
M 3 j 2 + 2t M 3 jV
+ < M 3, e j > = 0 , (4.27)
with terminal condition M 3(e j , ,V,T ) = V 2T .
34
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
35/46
Once M 1, M 2 and M 3 are solved from the above systems of N coupled P.D.E.s,
we can use them to approximate the prices of the variance swap and the volatility
swap.
5. Monte Carlo Experiment
In this section, we shall perform a Monte Carlo Experiment for the prices of the
variance swaps and the volatility swaps implied by the regime-switching Hestons
stochastic volatility model. We shall document economic consequences for the prices
of the variance swaps and the volatility swaps of a regime-switching in Hestons
stochastic volatility model by comparing the prices with those obtained from Hestons
SV model without regime-switching. We shall compute the prices of the variance
swaps and the volatility swaps with various delivery prices under both the regime-
switching Hestons SV model and Hestons SV model without switching regimes by
Monte Carlo simulation. For illustration, we suppose that the number of regimes N =
2 throughout this section. The rst and second regimes, namely X t = 1 and X t = 2,
can be interpreted as the Good and Bad economic states, respectively. We also
assume that Hestons SV model without switching regimes coincides with the rst
regime of the regime-switching Hestons SV model. In this case, we can investigate
economic consequences for the prices of the variance swaps and the volatility swaps
35
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
36/46
when we allow the possibility that the dynamics of Hestons SV model switches over
time to the one corresponding to the Bad economic states. We generate 10,000
simulation runs for computing each price. All computations were done by C++
codes with GSL functions.
We shall assume some specimen values for the parameters of regime-switching
Hestons SV model and the one without switching regimes. When the economy is
good (bad), the interest rate is high (low). Let r1 and r2 denote the annual interest
rates for the Good state and the Bad state, respectively. Then, we suppose that
r 1 = 5% and r2 = 2%. The appreciation rate of the underlying risky asset is high
(low) when the economy is good (bad). In each case, the appreciation rate should
be higher than the corresponding interest rate. Hence, we suppose that the annual
appreciation rate 1 = 7% for the Good state and the annual appreciation rate
2 = 5% for the Bad state. When the economy is good (bad), the underlying risky
asset is less (more) volatile. Hence, we suppose that 1 = 0.12 and 2 = 0.24. The
speed of mean reversion = 0.2 and the volatility of volatility parameter = 0.08.
We also assume that the correlation coefficient is negative and is equal to 0.5.The transition probabilities of the Markov chain are 11 = 0.5, 12 = 0.5, 21 = 0.5,
22 = 0.5. The notational amount of the variance swap or the volatility swap is 1
million. We suppose that the current economic state X 0 = 1 and that the current
36
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
37/46
volatility level V 0 = 0.12. The delivery prices of the variance swap and the volatility
swap range from 80% to 125% of the current levels of the variance and the standard
deviation of the underlying risky asset, respectively. The time-to-expiry of both the
variance swap and the volatility swap is 1 year. Since the regime-switching Heston
stochastic volatility model is a continuous-time model, we need to discretize it when
we compute the prices of the variance swaps and volatility swaps by Monte Carlo
simulation. We suppose that the number of steps for the discretization is 20. Table
1 displays the prices of the variance swaps for various delivery prices implied by
the regime-switching Heston stochastic volatility model and its non-regime-switching
counterpart.
Table 1: Prices of Variance Swaps with and without Switching Regimes
Delivery Prices Prices with Switching Regimes Prices without Switching Regimes
in % in million in million
80 1.14556 0.847634
85 1.1431 0.845326
90 1.14165 0.843224
95 1.13823 0.840911
100 1.13607 0.838839
105 1.13167 0.836526
110 1.12934 0.834374
115 1.12657 0.832061
120 1.12196 0.829986
125 1.12082 0.82767
37
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
38/46
Table 2 displays the prices of the volatility swaps for various delivery prices im-
plied by the regime-switching Heston stochastic volatility model and its non-regime-
switching counterpart.
Table 2: Prices of Volatility Swaps with and without Switching Regimes Delivery Prices Prices with Switching Regimes Prices without Switching Regimes
in % in million in million
80 0.632453 0.467974
85 0.624144 0.461601
90 0.616364 0.455246
95 0.607529 0.448786
100 0.599301 0.442436
105 0.589936 0.436079
110 0.581685 0.429701
115 0.573195 0.423343
120 0.563777 0.416992
125 0.55599 0.410641
From Table 1 and Table 2, we see that the prices of the variance swaps and the
volatility swaps implied by the regime-switching Heston stochastic volatility model
are signicantly higher than the corresponding prices of the variance swaps and the
volatility swaps, respectively, implied by the standard Heston stochastic volatility
without switching regimes. This reveals that a higher risk premium is required to
compensate for the risk from the structural change of the volatility dynamics to the
one with higher long-term volatility level due to the possible transitions of the states
38
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
39/46
of the economy to the Bad state. This illustrates the economic signicance of
incorporating the switching regimes in the volatility dynamics for pricing variance
swaps and volatility swaps.
6. Further Research
For further investigation, it is interesting to explore and develop some criteria to de-
termine the number of states of the Markov chain in our framework which will incor-
porate important features of the volatility dynamics for different types of underlying
nancial instruments, such as commodities, currencies and xed income securities. It
is also interesting to explore the applications of our model to price various volatility
derivative products, such as options on volatilities and VIX futures, which are a listed
contract on the Chicago Board Options Exchange. It is also of practical interest to
investigate the calibration and estimation techniques of our model to volatility index
options. Empirical studies comparing the performance of models on volatility swaps
are interesting topics to be investigated further.
Acknowledgment
We would like to thank the referee for many valuable comments and suggestions.
39
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
40/46
References
1. Back, K. and Pliska, S. R. (1991) On the fundamental theorem of asset pricing
with an innite state space, Journal of Mathematical Economics , 20, pp. 1-18.
2. Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity,
Journal of Econometrics , 31, pp. 307-327.
3. Brenner, M., and Galai, D. (1989) New nancial instruments for hedging changes
in volatility, Financial Analyst Journal , July/August, pp. 61-65.
4. Brenner, M., and Galai, D. (1993) Hedging volatility in foreign currencies,
Journal of Derivatives , 1, pp. 53-59.
5. Brenner, M., Ou, E., and Zhang, J. (2001) Hedging volatility risk, Working
paper, Stern School of Business, New York University, United States.
6. Brockhaus, O., and Long, D. (2000) Volatility swaps made simple, Risk , 2(1),
pp. 92-95.
7. Buffington, J. and Elliott, R. J. (2002a) Regime switching and European op-
tions, In Stochastic Theory and Control, Proceedings of a Workshop, Lawrence,
K.S. , October, Springer Verlag, pp. 73-81.
40
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
41/46
8. Buffington, J. and Elliott, R. J. (2002b) American options with regime switch-
ing, International Journal of Theoretical and Applied Finance , 5, pp. 497-514.
9. Carr, P. (2005) Delta hedging with stochastic volatility, Working paper, Quan-
titative Financial Research, Bloomberg LP, New York.
10. Carr, P., and Madan, D. (1998) Towards a theory of volatility trading, In the
book: Volatility, Risk book publications.
11. Carr, P., Geman H., Madan, D., and Yor, M. (2005) Pricing options on realized
variance, Finance and Stochastics , 9(4), pp. 453-475.
12. Cox, J., Ingersoll, J. and Ross, S. (1985) A theory of the term structure of
interest rates, Econometrica , 53, pp. 385-407.
13. Cox, J. and Ross, S. (1976) The valuation of options for alternative stochastic
processes, Journal of Financial Economics , 3, pp. 145-166.
14. Delbaen, F. and Schachermayer, W. (1994) A general version of fundamental
theorem of asset pricing, Mathematische Annalen , 300, pp. 463-520.
15. Demeter, K., Derman, E., Kamal, M., and Zou, J. (1999) A guide to volatility
and variance swaps, The Journal of Derivatives , 6(4), pp. 9-32.
41
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
42/46
16. Duffie, D. and Richardson, H. R. (1991) Mean-variance hedging in continuous
time, The Annals of Applied Probability , 1, pp. 1-15.
17. Dybvig, Philip H., and Ross, S. (1987) Arbitrage, In Eatwell J., Milgate M.,
and Newman P. eds. the New Palsgrave: A dictionary of Economics , 1, pp.
100-106.
18. Elliott, R. J., Aggoun, L. and Moore, J. B. (1994) Hidden Markov Models:
Estimation and Control . Springer-Verlag: Berlin-Heidelberg-New York.
19. Elliott, R. J., Sick, G. A. and Stein, M. (2003) Modelling electricity price risk,
Working paper, Haskayne School of Business, University of Calgary.
20. Elliott, R. J., Chan, L. and Siu, T. K. (2005) Option pricing and Esscher
transform under regime switching, Annals of Finance , 1(4), pp. 423-432.
21. Elliott, R. J. and Kopp, E. P. (2004) Mathematics of Financial Markets . Springer-
Verlag: Berlin-Heidelberg-New York.
22. Elliott, R. J., and Swishchuk, A. (2004) Pricing options and variance swaps
in Brownian and fractional Brownian market, Working paper, University of
Calgary.
42
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
43/46
23. Engle, R. (1982) Autoregressive conditional heteroskedasticity with estimates
of the variance of U.K. ination, Econometrica , 50, pp. 987-1008.
24. Follmer, H. and Schweizer, M. (1991) Hedging of contingent claims under
incomplete information, In: M.H.A. Davis and R. J. Elliott (eds.), Applied
Stochastic Analysis, Stochastics Monographs, Vol.5, Gordon and Breach, Lon-
don/New York, pp. 389-414.
25. Grunbuchler, A., and Longstaff, F. (1996) Valuing futures and options on
volatility, Journal of Banking and Finance , 20, pp. 985-1001.
26. Harrison, J. M. and Kreps, D. M. (1979) Martingales and arbitrage in multi-
period securities markets, Journal of Economic Theory , 20, pp. 381-408.
27. Harrison, J. M. and Pliska, S. R. (1981) Martingales and stochastic integrals in
the theory of continuous trading, Stochastic Processes and Their Applications ,
11, pp. 215-280.
28. Harrison, J. M. and Pliska, S. R. (1983) A stochastic calculus model of contin-
uous trading: complete markets, Stochastic Processes and Their Applications ,15, pp. 313-316.
29. Heston, S. L. (1993) A closed-form solution for options with stochastic with
43
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
44/46
applications to bond and currency options, Review of Financial Studies , 6(2),
pp. 237-343.
30. Heston, S. L., and Nandi, S. (2000) Derivatives on volatility: some simple
solution based on observables, Working paper, Federal Reserve Bank of Atlanta.
31. Howison, S., Rafailidis A., and Rasmussen, H. (2004) On the pricings and
hedging of volatility derivatives, Applied Mathematical Finance , 11(4), pp. 317-
346.
32. Javaheri, A., Wilmott, P., and Haug, E. G. (2002) GARCH and volatility
swaps, Wilmott Magazine , January, pp. 1-17.
33. Matytsin, A. (2000) Modeling volatility and volatility derivatives, Working pa-
per, Columbia University.
34. Pan, J. (2002) The jump-risk premia implicit in options: evidence from an
integrated time-series study, Journal of Financial Economics , 63, pp. 3-50.
35. Ross, S. A. (1973) Return, risk and arbitrage, Wharton Discussion Paper pub-
lished in Risk and Return in Finance , edited by I. Friend and J. Bicksler, pp.
189-217. Cambridge: Ballinger, 1976.
44
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
45/46
36. Ross, S. A. (1978) A simple approach to the valuation of risky streams, Journal
of Business , 3, pp. 453-476.
37. Ross, S. A. (2005) Neoclassical Finance, Princeton and Oxford: Princeton
University Press.
38. Rubinstein, M. (1976) The valuation of uncertain income streams and the pric-
ing of options, Bell Journal of Economics and Management Science , 7, pp.
407-425.
39. Schachermayer, W. (1992) A Hilbert space proof of the fundamental theorem
of asset pricing in nite discrete time, Insurance: Mathematics and Economics ,
11, pp. 249-257.
40. Schweizer, M. (1991) Option hedging for semimartingales, Stochastic Processes
and Their Application , 37, pp. 339-363.
41. Schweizer, M. (1992) Mean-variance hedging for general claims, The Annals of
Applied Probability , 2, pp. 171-179.
42. Shreve, S. E. (2004) Stochastic Calculus for Finance, New York: Springer.
43. Swishchuk, A. (2004) Modeling of variance and volatility swaps for nancial
markets with stochastic volatilities, Wilmott magazine , 2, pp. 64-72.
45
8/14/2019 Robert Elliott - Pricing Volatility Swap and Variance Swap under Heston Model with Regime Switching.pdf
46/46
44. Swishchuk, A. (2005) Modeling and pricing of variance swaps for stochastic
volatilities with delay, Wilmott magazine , Forthcoming.
45. Taylor, S. J. (1986) Modelling Financial Time Series. New York: John Wiley
& Sons.
46. Windcliff, H., Forsyth, P. A. and Vetzal, K. R. (2003) Pricing methods and
hedging strategies for volatility derivatives, Working paper, University of Wa-
terloo.
46
top related