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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: [email protected];

Fax: 403-770-8104; Tel: 403-220-5540Lecturer, Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edin-

burgh, United Kingdom; Email: [email protected]. candidate, Department of Mathematics and Statistics, University of Calgary, Calgary,

Alberta, Canada; Email: [email protected]

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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

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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 .

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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

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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.

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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

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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


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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

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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

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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

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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

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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 ]

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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

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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.

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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

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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

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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)

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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)

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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


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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


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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).

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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

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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)

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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 .

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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

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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.

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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)

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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


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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 )

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= 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)

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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 .


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 .


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 .

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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 .

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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

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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

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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

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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

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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.

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