Variance and Volatility Swaps in Energy Markets

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Journal of Energy Markets Volume 6/Number 1, Spring 2013 (3349)

Variance and volatility swaps

in energy markets

Anatoliy Swishchuk

Mathematical and Computational Finance Laboratory,

Department of Mathematics and Statistics, University of Calgary,

2500 University Drive NM, Calgary, AB T2N 1N4, Canada;

email: [email protected]

(Received July 30, 2010; accepted June 8, 2011)

This paper focuses on the pricing of variance and volatility swaps in the energy

market. An explicit variance swap formula and a closed-form volatility swap for-

mula (using the BrockhausLong approximation) for energy assets with stochas-

tic volatility are found. These formulas follow the continuous-time generalized

autoregressive conditional heteroskedasticity (1,1) .GARCH.1;1// model (mean-

reverting) or the Pilipovic one-factor model. A numerical example is presented

for the AECO natural gas index.

1 INTRODUCTION

Variance swaps are quite common in commodity markets such as the energy market,

and they are commonly traded.We consider the OrnsteinUhlenbeck (OU) processforcommodity assetswith stochastic volatilityfollowing the continuous-time generalized

autoregressive conditionalheteroskedasticity (GARCH)modelor the Pilipovic (1998)

one-factor model.The classical stochastic processfor the spot dynamics of commodity

prices is given by the Schwartz model (Schwartz 1997). It is defined as the exponential

of an OU process, and has become the standard model for energy prices with mean-

reverting features.

We consider a risky asset in the energy market with stochastic volatility following

a mean-reverting stochastic process satisfying the following stochastic differential

equation (continuous-time GARCH(1,1) model):

d2.t / D a.L 2.t// dt C 2.t/ dWt ;

The authors research was supported by the Natural Sciences and Engineering Research Council of

Canada.

33


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34 A.Swishchuk

where a is the speed of mean reversion, L is the mean-reverting level (or equilibrium

level), is the volatility of volatility .t/, and Wt is a standard Wiener process. Using

a change-of-time method we find an explicit solution of this equation and, using this

solution, we are able to find the variance and volatility swaps pricing formula under

the physical measure. Then, using the same argument, we find the option pricing

formula under the risk-neutral measure. We applied the approximation of Brockhaus

and Long (2000) to find the value of the volatility swap. A numerical example for the

Alberta Energy Company (AECO) natural gas index for the period from May 1, 1998

to April 30, 1999 is presented.

Commodities are emerging as an asset class in their own right.The array of products

offered to investors ranges from exchange-traded funds to sophisticated products

including principal-protected structured notes on individual commodities or baskets

of commodities and commodity range-accrual or variance swaps.

More and more institutional investors are including commodities in their asset

allocation mix, and hedge funds are also increasingly active players in commodities.

For example, Amaranth Advisors lost US$6 billion during September 2006 from

trading natural gas futures contracts, leading to the funds demise.

Concurrent with these developments, a number of recent papers have examined

the risk and return characteristics of investments in individual commodity futures or

commodity indices composed of baskets of commodity futures (see, for example, Erb

and Harvey 2006; Gorton and Rouwenhorst 2006; Ibbotson 2006; Kan and Oomen

2007a,b).

However, since all but the most plain vanilla of investments contain an exposure

to volatility, it is equally important for investors to understand the risk-and-return

characteristics of commodity volatilities.We focus on energy commodities for the following reasons.

(1) Energy is the most important commodity sector, and crude oil and natural gas

constitute the largest components of the two most widely tracked commodity

indices: the Standard & Poors Goldman Sachs Commodity Index (S&P GSCI)

and the Dow JonesAIG Commodity Index (DJ AIGCI).

(2) The existence of a liquid options market: crude oil and natural gas have the

deepest and most liquid options markets among all commodities.

The idea is to use variance (or volatility) swaps on futures contracts.

At maturity, a variance swap pays off the difference between the realized variance of

the futures contract over the life of the swap and the fixed variance swap rate. Since a

variance swap has zero net market value at initiation, the absence of arbitrage implies

that the fixed variance swap rate is equal to the conditional risk-neutral expectation

of the realized variance over the life of the swap.

Journal of Energy Markets Volume 6/Number 1, Spring 2013


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Variance and volatility swaps in energy markets 35

Therefore, for example, the time-series average of the payoff and/or excess return

on a variance swap is a measure of the variance risk premium.

Variance risk premiums in energy commodities, specifically crude oil and natural

gas, have been considered by Trolle and Schwartz (2010). The same methodology

was used by Carr and Wu (2009) in their study of equity variance risk premiums. The

idea was to use variance swaps on futures contracts.

The study in Trolle and Schwartz (2010) is based on daily data from January 2,

1996 to November 30, 2006, a total of 2750 business days. The source of the data is

NYMEX.

Trolle and Schwartz (2010) found that:

(1) the average variance risk premiums are negative for both energy commodities

but more strongly statistically significant for crude oil than for natural gas;

(2) the natural gas variance risk premium (defined in US dollar terms or in returnterms) is higher during the cold months of the year (seasonality and peaks for

natural gas variance during the cold months of the year);

(3) energy risk premiums in dollar terms are time varying and correlated with the

level of the variance swap rate; in contrast, energy variance risk premiums

in return terms, particularly in the case of natural gas, are much less closely

correlated with the variance swap rate.

The S&P GSCI is composed of twenty-four commodities with the weight of each

commodity determined by their relative levels of world production over the past

five years. The DJ AIGCI comprises nineteen commodities with the weight of each

component determined by liquidity and world production values, with liquidity being

the dominant factor. Crude oil and natural gas are the largest components in both

indices. In 2007, their weights were 51.30% and 6.71%, respectively, in the S&P

GSCI, and 13.88% and 11.03%, respectively, in the DJ AIGCI.

The Chicago Board Options Exchange (CBOE) recently introduced a crude oil

volatility index (ticker symbol OVX). This index also measures the conditional risk-

neutral expectation of crude oil variance, but is computed from a cross-section of

listed options on the United States Oil Fund (USO), which tracks the price of West

Texas Intermediate as closely as possible. The CBOE crude oil exchange-traded fund

volatility index (Oil VIX, with ticker symbol OVX) measures the markets expec-

tation of thirty-day volatility of crude oil prices by applying the VIX methodology to

the United States Oil Fund, LP (ticker symbol USO) options spanning a wide range of

strike prices. We note that crude oil and natural gas trade in units of 1000 barrels and

10 000 British thermal units (mmBtu), respectively. Unless stated otherwise, prices

are quoted as US dollars and cents per barrel or mmBtu.

Research Paper www.risk.net/journal


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36 A.Swishchuk

The continuous-time GARCH model has also been exploited by Javaheri et al

(2002) to calculate the volatility swap for the S&P500 index. They used a partial

differential equation approach and mentioned that it would be interesting to use an

alternative method to calculate F.v;t/ and the other above quantities (Javaheri et al

2002, p. 8, Section 3.3). This paper uses the alternative method, namely the change-

of-time method, to obtain variance and volatility swaps. The change-of-time method

was also applied in Swishchuk (2004) for pricing variance, volatility, covariance and

correlation swaps for the Heston model. The first paper on pricing of commodity

contracts was published by Black (1976).

2 THE MEAN-REVERTING STOCHASTIC VOLATILITY MODEL

In this section we introduce the mean-reverting stochastic volatility model (MRSVM)

and study some properties of this model that we can use later for calculating varianceand volatility swaps.

Let .; F; Ft ; P / be a probability space with a sample space , -algebra of

Borel sets F and probability P. The filtration Ft , t 2 0;T, is the natural filtrationof a standard Brownian motion Wt , t 2 0;T, such that FT D F. We consider arisky asset in the energy market with stochastic volatility following a mean-reverting

stochastic process in the following stochastic differential equation:

d2.t / D a.L 2.t// dt C 2.t/ dWt ; (2.1)

where a > 0 is a speed (or strength) of mean reversion, L > 0 is the mean-reverting

level (or equilibrium level, or long-term mean), > 0 is the volatility of volatility.t/ and Wt is a standard Wiener process.

2.1 An explicit solution for the MRSVM

Let:

Vt WD eat .2.t/ L/: (2.2)

Then, from (2.1) and (2.2), we obtain

dVt D aeat .2.t / L/ dt C eat d2.t/ D .Vt C eatL/ dWt : (2.3)

Using a change-of-time approach to Equation (2.3) (see Ikeda and Watanabe 1981;

Elliott 1982) we obtain the following solution of Equation (2.3):

Vt D 2.0/ L C QW .1t /



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or (see (2.2))

2.t / D eat 2.0/ L C QW .1t / C L; (2.4)

where QW .t/ is an Ft -measurable standard one-dimensional Wiener process and 1tis an inverse function to t :

t D 2Zt0

.2.0/ L C QW .s/ C easL/2 ds (2.5)

We note that

1t D 2Zt0

.2.0/ L C QW .1t / C easL/2 ds; (2.6)

which follows from (2.5).

2.2 Some properties of the process QW .1t

/

We note that the process QW .1t / is QFt WD F1t

-measurable and is an QFt -martingale.We then have

E QW .1t / D 0: (2.7)

We calculate the second moment of QW .1t / (see (2.6))

E QW2.

1t / D Eh QW .

1t /iD E1t

D 2Zt0

E.2.0/ L C QW .1s / C easL/2 ds

D 2

.2.0/ L/2t C 2L.2.0/ L/.eat 1/

a

C L2.e2at 1/

2aCZt0

E QW2.1s / ds

: (2.8)

From (2.8), solving this linear ordinary nonhomogeneous differential equation withrespect to E QW2.1t /:

E QW2.1t /dt

D 2.2.0/ L/2 C 2L.2.0/ L/eat C L2e2at C E QW2.1t /;



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38 A.Swishchuk

we obtain:

EQ

W2.1t /

D 2

.2.0/ L/2 e2t 1

2C 2L.

2.0/ L/.eat e2t/a 2 C

L2.e2at e2t/2a 2

:

We note that:

E QW .1s / QW .1t /

D 2

.2.0/ L/2 e2.t^s/ 1

2C 2L.

2.0/ L/.ea.t^s/ e2.t^s//a 2

C L2.e2a.t^/ e2.t^s//

2a

2

(2.9)

and the second moment for QW2.1t / above follows from (2.9).

2.3 An explicit expression for the process QW .1t

/

It turns out that we can find the explicit expression for the process QW .1t /. From theexpression (see Section 2.1)

Vt D 2.0/ L C QW .1t /

we have the following relationship between W.t/ and QW .1t /:

d QW .1t / D Zt0

S.0/ L C Leat C QW .1s / dW.t/:

It is a linear stochastic differential equation with respect to QW .1t / and we cansolve it explicitly. The solution is as follows:

QW .1t / D 2.0/.expfW.t/ 122tg 1/ C L.1 eat /

C aL expfW.t/ 12

2tgZt0

eas expfW.s/ C 12

2sg ds: (2.10)

It is easy to see from (2.10) that QW .1t / can be presented in the form of a linearcombination of two zero-mean martingales m1.t / and m2.t /:

QW .1t / D m1.t/ C Lm2.t/;



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where:

m1.t / WD 2.0/.expfW.t/ 122tg 1/

and:

m2.t / D .1 eat / C a expfW.t/ 122tgZt0

eas expfW.s/ C 12

2sg ds:

Indeed, the process QW .1t / is a martingale (see Section 2.2). Furthermore, it iswell-known that the process expfW.t/ 1

22tg and, hence, the process m1.t / are

martingales. Then the process m2.t/, being the difference between two martingales,

is also martingale.

2.4 Some properties of the mean-reverting stochastic volatility

2.t/: the first two moments, the variance and the covariation

From (2.4) we obtain the mean value of the first moment for mean-reverting stochastic

volatility 2.t/:

E2.t / D eat 2.0/ L C L: (2.11)

This means that E2.t / ! L when t ! C1. We need this moment to value thevariance swap.

Using formulas (2.4) and (2.9), we can calculate the second moment of 2.t /:

E2.t / D .eat .2.0/ L/ C L/2

C 2e2at

.2.0/ L/2 e2t 1

2C 2L.

2.0/ L/.eat e2t/a 2

C L2.e2at e2t /

2a 2

:

Combining the first and second moments we have the variance of 2.t/:

var.2.t// D E2.t/2 .E2.t//2

D 2e2at

.2.0/ L/2 e2

t 12

C 2L.2.0/ L/.eat e2

t /a 2

C L2.e2at e2t/

2a 2

:



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40 A.Swishchuk

From the expression for QW .1t / (see (2.10)) and for 2.t / in (2.4) we can findthe explicit expression for 2.t / through W.t/:

2.t / D eat 2.0/ L C QW .1t / C LD eat 2.0/ L C m1.t / C Lm2.t/ C LD 2.0/eat expfW.t/ 1

22tg

C aLeat expfW.t/ 12

2tgZt0

eas expfW.s/ C 12

2sg ds;(2.12)

where m1.t/ and m2.t/ are defined as in Section 2.3.

From (2.12) it follows that 2.t/ > 0 as long as 2.0/ > 0. The covariation for

2.t / may be obtained from (2.4), (2.7) and (2.9):

E2.t/2.s/ D ea.tCs/.2.0/ L/2

C ea.tCs/

2

.2.0/ L/2 e2.t^s/ 1

2

C 2L.2.0/ L/.ea.t^s/ e2.t^s//

a 2

C L2.e2a.t^s/ e2.t^s//

2a 2

C eat .2.0/ L/L C eas.2.0/ L/L C L2: (2.13)

We need this covariance to value the volatility swap.

3 A VARIANCE SWAP FOR THE MEAN-REVERTING STOCHASTIC

VOLATILITY MODEL

To calculate the variance swap for 2.t / we need E2.t/. From (2.11) it follows that:

E2.t / D eat 2.0/ L C L:

Then E2R WD EV takes the following form:

E2R WD EV WD 1TZ

T

0

E2.t / dt

D 20 LaT

.1 eaT/ C L: (3.1)



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FIGURE 1 A variance swap.

0 1 2 3 4 50

0.015

0.005

0.010

T

For the AECO natural gas index (May 1, 1998April 30, 1999), using formula (3.1).

Figure 1 depicts a variance swap (price versus maturity) for the AECO natural gas

index (May 1, 1998April 30, 1999), using formula (3.1) and parameters from Table 1

on page 47. Recall that:

V WD 1T

ZT0

2.t / dt:

4 A VOLATILITY SWAP FOR THE MEAN-REVERTING STOCHASTIC

VOLATILITY MODEL

To calculate the volatility swap for 2.t/ we need Ep

V D EpR and it means thatwe need more than just E2.t /, because the realized volatility R WD

pV D

q2R.

Using the BrockhausLong approximation we then get

E

pV

pEV

var.V /

8.EV /3=2 : (4.1)

We have calculated EV in Equation (3.1). We need

var.V / D EV2 .EV /2 (4.2)



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42 A.Swishchuk

From (3.1) it follows that .EV /2 has the form

.EV /2

D.2.0/

L/2

a2T2 .1 eaT

/2

C 22.0/

L

aT .1 eaT

/L C L2

: (4.3)

Let us calculate EV2 using (2.9) and (2.13):

EV2 D 1T2

ZT0

ZT0

E2.t/2.s/ dt ds

D 1T2

ZT0

ZT0

ea.tCs/.2.0/ L/2

C ea.tCs/

2

.2.0/ L/2 e2.t^s/ 1

2

C 2L.2.0/ L/.ea.t^s/ e2.t^s//

a 2


2a 2

C eat .2.0/ L/L C eas.2.0/ L/L C L2

dt ds:

(4.4)

After calculating the integrals in the second, fourth and sixth lines in (4.4), we have

EV2 D 1T2

ZT0

ZT0

E2.t/2.s/ dt ds

D .2.0/ L/2

a2T2.1 eaT/2

C 1T2

ZT0

ZT0

ea.tCs/

2

.2.0/ L/2 e2.t^s/ 1

2

C 2L.2.0/ L/.ea.t^s/ e2.t^s//

a 2

C L2.e2a.t^s/ e2

.t^s//2a 2

dt ds

C .2.0/ L/L

aT.1 eaT/ C .

2.0/ L/LaT

.1 eaT/ C L2: (4.5)



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Taking into account (4.2), (4.3) and (4.5), we arrive at the following expression for

var.V /:

var.V / D EV2 .EV /2

D 1T2

ZT0

ZT0

ea.tCs/

2

.2.0/ L/2 e2.t^s/ 1

2

C 2L.2.0/ L/.ea.t^s/ e2.t^s//

a 2


2a 2

dt ds

D 2.0/ L

T2

ZT0

ZT0

ea.tCs/.e2.t^s/ 1/ dt ds

C 2L2.2.0/ L/

.a2 2/T2ZT0

ZT0

ea.tCs/.ea.t^s/ e2.t^s// dt ds

C 2L2

.2a 2/T2ZT0

ZT0

ea.tCs/.e2a.t^s/ e2.t^s// dt ds: (4.6)

After calculating the three integrals in (4.6) we obtain

var.V / D EV2 .EV /2

D1

T2Z

T

0Z

T

0

ea.tCs/2.2.0/ L/2 e

2.t^s/ 12

C 2L.2.0/ L/.ea.t^s/ e2.t^s//

a 2


2a 2

dt ds:

(4.7)

From (4.1) and (4.7) we get the volatility swap:

Ep

V

pEV

var.V /

8.EV /3=2

: (4.8)

Figure 2 on the next page depicts a volatility swap (price versus maturity) for the

AECO natural gas index (May 1, 1998April 30, 1999), using formula (4.8) and

parameters from Table 1 on page 47.



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44 A.Swishchuk

FIGURE 2 A volatility swap.

0 1 2 3 4 5

T

0

0.01

0.03

0.02


5 A MEAN-REVERTING RISK-NEUTRAL STOCHASTIC VOLATILITY

MODEL

In this section, we are going to obtain the values of variance and volatility swaps under

risk-neutral measure P, using the same arguments as in Sections 3 and 4, where, in

place ofa and L, we are going to take a and L:

a ! a WD a C ; L ! L WD aLa C ;

where is a market price of risk (see Section 3).

5.1 A risk-neutral stochastic volatility model

Consider our model (2.1):

d2.t/ D a.L 2.t// dt C 2.t / dWt : (5.1)Let be market price of risk and define the following constants:

a WD a C ; L WD aL=a:Then, in the risk-neutral world, the drift parameter in (5.1) has the following form:

a.L 2.t// D a.L 2.t// 2.t/: (5.2)



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If we define the process Wt as

Wt WD Wt C t; (5.3)

where Wt is a standard Brownian motion, then the risk-neutral stochastic volatility

model has the following form:

d2.t / D .aL .a C /2.t// dt C 2.t/ dWtor, equivalently

d2.t/ D a.L 2.t// dt C 2.t / dWt ; (5.4)

where

a WD a C ; L WD aLa

C

(5.5)

and Wt is as defined in (5.3).

Now we have the same model in (5.4) as in (2.1), and we are going to apply

our change-of-time method to this model (5.4) to obtain the values of variance and

volatility swaps.

5.2 Variance and volatility swaps for the risk-neutral SVM

Using the same arguments as in the previous section (where in place of (2.4) we have

to take (5.4)) we get the following expressions for variance and volatility swaps taking

into account (5.5).

For the variance swaps we have (see (3.1) and (5.5)):

E2R WD EV WD1

T

ZT0

E2.t/ dt

D 2.0/ L

aT.1 eaT/ C L: (5.6)

Figure 3 on the next page depicts a variance swap with risk-adjusted parameters (price

versus maturity) for the AECO natural gas index (May 1, 1998April 30, 1999), using

formula (5.6) and parameters from Section 5.3.

For the volatility swap we obtain (see (4.8) and (5.5))

Ep

V

pEV

var.V /

8.EV /3=2: (5.7)

Figure 4 on the next page depicts a volatility swap with risk-adjusted parameters (price

versus maturity) for the AECO natural gas index (May 1, 1998April 30, 1999), using

formula (5.7) and parameters from Section 5.3.



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46 A.Swishchuk

FIGURE 3 A variance swap (risk-adjusted parameters).

0 1 2 3 4 5

T

6

0.002

0.004

0.006

0.008

0.010

0.012

0


FIGURE 4 A volatility swap (risk-adjusted parameters).

0 1 2 3 4 5

T

6

0

0.025

0.020

0.015

0.010

0.005




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TABLE 1 Parameters.

a L

4.6488 1.5116 2.7264 0.18

FIGURE 5 Comparison: adjusted (lower line) and nonadjusted (upper line) price.

0 1 2 3 4 5

T

1.52

1.54

1.56

1.58

1.60

1.62

1.64

5.3 A numerical example: the AECO natural gas index

We calculate the value of variance and volatility swap prices of a daily natural gas

contract. To apply our formula for calculating these values we need to calibrate the

parameters a, L, 20 and (T is monthly). These parameters can be obtained from

futures prices for the AECO natural gas index for the period from May 1, 1998 to

April 30, 1999 (see Bos et al 2002). The parameters are given in Table 1.

For the variance swap we use (3.1), and for the volatility swap we use (4.8).

From Table 1 we can calculate the values for risk-adjusted parameters a and L:

a D a C D 4:9337

andL D aL

a C D 2:5690:

For the value of 2.0/ we can take 2.0/ D 2:25. For a variance swap and volatil-ity swap with risk-adjusted parameters we use formulas (5.6) and (5.7), respectively.



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48 A.Swishchuk

FIGURE 6 Convexity adjustment.

0 1 2 3 4 5

0.05

0.10

0.15

0.20

0

T

Figure 5 on the preceding page depicts a comparison of adjusted (lower line) and non-

adjusted (upper line) prices (naive strike versus adjusted strike). Figure 6 depicts the

convexity adjustment, which decreases with swap maturity (the volatility of volatility

over a long period of time is low).

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Variance and Volatility Swaps in Energy Markets