research article on volatility swaps for stock market...

Research ArticleOn Volatility Swaps for Stock Market ForecastApplication Example CAC 40 French Index

Halim Zeghdoudi12 Abdellah Lallouche3 and Mohamed Riad Remita1

1 LaPS Laboratory Badji-Mokhtar University BP 12 23000 Annaba Algeria2 Department of Computing Mathematics and Physics Waterford Institute of Technology Waterford Ireland3Universite 20 Aout 1955 Skikda Algeria

Correspondence should be addressed to Halim Zeghdoudi hzeghdoudiyahoofr

Received 3 August 2014 Revised 21 September 2014 Accepted 29 September 2014 Published 9 November 2014

Academic Editor Chin-Shang Li

Copyright copy 2014 Halim Zeghdoudi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper focuses on the pricing of variance and volatility swaps under Heston model (1993) To this end we apply this model tothe empirical financial data CAC 40 French Index More precisely we make an application example for stockmarket forecast CAC40 French Index to price swap on the volatility using GARCH(11) model

1 Introduction

Black and Scholesrsquo model [1] is one of the most significantmodels discovered in finance in XXe century for valuingliquid European style vanilla option Black-Scholes modelassumes that the volatility is constant but this assumption isnot always true This model is not good for derivatives pricesfounded in finance and businesses market (see [2])

ldquoThe volatility of asset prices is an indispensable input inboth pricing options and in risk management Through theintroduction of volatility derivatives volatility is now in effecta tradable market instrumentrdquo Broadie and Jain [3]

Volatility is one of the principal parameters employed todescribe and measure the fluctuations of asset prices It playsa crucial role in the modern financial analysis concerningrisk management option valuation and asset allocationThere are different types of volatilities implied volatility localvolatility and stochastic volatility (see Baili [4])

To this end the new financial products are varianceand volatility swaps which play a decisive role in volatilityhedging and speculation Investment banks currencies stockindexes finance and businesses markets are useful for vari-ance and volatility swaps

Volatility swaps allow investors to trade and to controlthe volatility of an asset directly Moreover they would trade

a price index The underlying is usually a foreign exchangerate (very liquid market) but could be as well a single nameequity or index However the variance swap is reliable inthe index market because it can be replicated with a linearcombination of options and a dynamic position in futuresAlso volatility swaps are not used only in finance andbusinesses but in energy markets and industry too

The variance swap contract contains two legs fixed leg(variance strike) and floating leg (realized variance) Thereare several works which studied the variance swap portfoliotheory and optimal portfolio of variance swaps based on avariance Gamma correlated (VGC) model (see Cao and Guo[5])

The goal of this paper is the valuation and hedging ofvolatility swaps within the frame of a GARCH(11) stochasticvolatility model under Heston model [6] The Heston assetprocess has a variance that follows a Cox et al [7] processAlso we make an application by using CAC 40 French Index

The structure of the paper is as follows Section 2considers representing the volatility swap and the varianceswap Section 3 describes the volatility swaps for Hestonmodel gives explicit expression of 120590

2

119905 and discusses the

relationship between GARCH and volatility swaps Finallywe make an application example for stock market forecastCAC 40 French Index using GARCHARCH models

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 854578 6 pageshttpdxdoiorg1011552014854578

2 Journal of Probability and Statistics

2 Volatility Swaps

In this sectionwe give somedefinitions andnotations of swapstockrsquos volatility stockrsquos volatility swap and variance swap

Definition 1 Swaps were introduced in the 1980s and thereis an agreement between two parties to exchange cash flowsat one or several future dates as defined by Bruce [8] Inthis contract one party agrees to pay a fixed amount to acounterpart which in turn honors the agreement by paying afloating amount which depends on the level of some specificunderlying By entering a swap a market participant cantherefore exchange the exposure from the varying underlyingby paying a fixed amount at certain future time points

Definition 2 A stockrsquos volatility is the simplest measure of itsrisk less or uncertainty Formally the volatility 120590

119877(119878) is the

annualized standard deviation of the stockrsquos returns duringthe period of interest where the subscript 119877 denotes theobserved or ldquorealizedrdquo volatility for the stock 119878

Definition 3 (see [9]) A stock volatility swap is a forwardcontract on the annualized volatility Its payoff at expirationis equal to

119873(120590119877(119878) minus 119870vol) (1)

where 120590119877(119878) = radic(1119879) int

119879

01205902

119904119889119904 120590119905is a stochastic stock

volatility 119870vol is the annualized volatility delivery price and119873 is the notional amount of the swap in Euro annualizedvolatility point

Definition 4 (see [9]) A variance swap is a forward contracton annualized variance the square of the realized volatilityIts payoff at expiration is equal to

119873(1205902

119877(119878) minus 119870var) (2)

where 119870var is the delivery price for variance and 119873 isthe notional amount of the swap in Euros per annualizedvolatility point squared

Notation 1 We note that 1205902119877(119878) = 119881

Using the Brockhaus and Long [10] and Javaheri [11]approximation which is used in the second order Taylorformula forradic119909 we have

119864 (radic119881) asymp radic119864 (119881) minus

Var (119881)811986432

(119881)

(3)

where Var(119881)811986432(119881) is the convexity adjustment Thus tocalculate volatility swaps we need both 119864(119881) and Var(119881)

The realized discrete sampled variance is defined asfollows

119881119899(119878) =

119899

(119899 minus 1) 119879

119899

sum

119894=1

ln2 (119878119905119894

119878119905119894minus1

) 119881 = lim119899rarrinfin

119881119899(119878)

(4)

where 119879 is the maturity (years or days)

3 Volatility Swaps for Heston Model

31 Stochastic VolatilityModel Let (Ω 119865 119865119905P) be probability

space with filtration 119865119905 119905 isin [0 119879] We consider the risk-

neutral Heston stochastic volatility model for the price 119878119905and

variance follows the following model

119889119878119905= 119903119905119878119905119889119905 + 120590

1199051198781199051198891199081(119905)

1198891205902

119905= 119896 (120579

2minus 1205902

119905) 119889119905 + 120585120590

1199051198891199082(119905)

(1198781)

where 119903119905is deterministic interest rate 120590

0gt 0 and 120579 gt 0 are

short and long volatility 119896 gt 0 is a reversion speed 120585 gt 0

is a volatility of volatility parameter and 1199081(119905) and 119908

2(119905) are

independent standard Brownian motionsWe can rewrite the system (1198781) as follows

119889119878119905= 119903119905119878119905119889119905 + 120590

1199051198781199051198891199081(119905)

1198891205902

119905= 119896 (120579

2minus 1205902

119905) 119889119905 + 120588120585120590

1199051198891199081(119905)

+ 120585radic1 minus 120588120590119905119889119908 (119905)

(1198782)

where 119908(119905) is standard Brownian motion which is inde-pendent of 119908

1(119905) and the indicator economic 119883 Let

cov(1198891199081(119905) 119889119908

2(119905)) = 120588119889119905 and we can transform the system

(1198782) to (1198781) if we replace 1205881198891199081(119905) + radic1 minus 120588119889119908(119905) by 119889119908

2(119905)

32 Explicit Expression andProperties of1205902119905 In this sectionwe

reformulated the results obtained in [12] which are neededfor study of variance and volatility swaps and price ofpseudovariance pseudovolatility and the problems proposedby He andWang [13] for financial markets with deterministicvolatility as a function of timeThis approachwas first appliedto the study of stochastic stability of Cox-Ingersoll-Rossprocess in Swishchuk and Kalemanova [14]TheHeston assetprocess has a variance 1205902

119905that follows Cox et al [7] process

described by the second equation in (1198781) If the volatility120590119905follows Ornstein-Uhlenbeck process (see eg Oksendal

[15]) then Itorsquos lemma shows that the variance 1205902119905follows the

process described exactly by the second equation in (1198781)We start to define the following process and function

V119905= 119890119896119905(1205902

119905minus 1205792)

Φ (119905) = 120585minus2int

119905

0

119890119896Φ(119904)

(1205902

0minus 1205792+ 1199082(119904) + 120579

21198902119896Φ(119904)

)

minus1

119889119904

(5)

Definition 5 We define 119861(119905) = 1199082(Φminus1

119905) where 119908

2is anF

119905-

measurable one-dimensional Wiener process F119905= FΦminus1

119905

and 119905 and 119904 = min(119905 119904) whereΦminus1

119905is an inverse function ofΦ

119905

The properties of 119861(119905) are as follows

(a) F119905-martingale and 119864(119861(119905)) = 0

(b) 119864(1198612(119905)) = 1205852(((119890119896119905minus1)119896)(120590

2

0minus1205792)+((1198902119896119905

minus1)2119896)1205792)

(c) 119864(119861(119904)119861(119905)) = 1205852(((119890119896(119905and119904)

minus1)119896)(1205902

0minus1205792)+ ((119890

2119896(119905and119904)minus

1)2119896)1205792)

Journal of Probability and Statistics 3

Lemma 6(a) Consider the following

1205902

119905= 119890minus119896119905

(1205902

0minus 1205792+ 119861 (119905)) + 120579

2 (6)

(b)

119864 (1205902

119905) = 119890minus119896119905

(1205902

0minus 1205792) + 1205792 (7)

(c)

119864 (1205902

1199041205902

119905) = 1205852119890minus119896(119905+119904)

(

119890119896(119905and119904)

minus 1

119896

(1205902

0minus 1205792) +

1198902119896(119905and119904)

minus 1

2119896

1205792)

+ 119890minus119896(119905+119904)

(1205902

0minus 1205792)

2

+ 119890minus119896119905

(1205902

0minus 1205792) 1205792

+ 119890minus119896119904

(1205902

0minus 1205792) 1205792+ 1205794

(8)

Proof See [12]

Theorem 7 One has

(a)

119864 (119881) =

1 minus 119890minus119896119879

119896119879

(1205902

0minus 1205792) + 1205792 (9)

(b)

Var (119881) = 1205852119890minus2119896119879

211989631198792[(21198902119896119879

minus 4119896119879119890119896119879

minus 2) (1205902

0minus 1205792)

+ (21198961198791198902119896119879

minus 31198902119896119879

+ 4119890119896119879

minus 1) 1205792]

(10)

Proof (a) We obtain mean value for 119881

119864 (119881) =

1

119879

int

119879

0

119864 (1205902

119905) 119889119905 (11)

using Lemma 6 and we find

119864 (119881) =

1 minus 119890minus119896119879

119896119879

(1205902

0minus 1205792) + 1205792 (12)

(b) Variance for 119881 equals Var(119881) = 119864(1198812) minus 1198642(119881) and

the second moment may be found as follows using formula(8) of Lemma 6 119864(1198812) = (1119879

2)∬

119879

0119864(1205902

1199051205902

119904)119889119905119889119904

119864 (1198812)

=

1205852

1198792∬

119879

0

[119890minus119896(119905+119904)

(

119890119896(119905and119904)

minus 1

119896

(1205902

0minus 1205792)

+

1198902119896(119905and119904)

minus 1

2119896

1205792)]119889119905119889119904

+ 1198642(119881)

(13)

and taking (13) and variance formula we find

Var (119881)

=

1205852

1198792∬

119879

0

[119890minus119896(119905+119904)

(

119890119896(119905and119904)

minus 1

119896

(1205902

0minus 1205792)

+

1198902119896(119905and119904)

minus 1

2119896

1205792)]119889119905119889119904

(14)

after calculations we obtain

Var (119881) = 1205852119890minus2119896119879

211989631198792[(21198902119896119879

minus 4119896119879119890119896119879

minus 2) (1205902

0minus 1205792)

+ (21198961198791198902119896119879

minus 31198902119896119879

+ 4119890119896119879

minus 1) 1205792]

(15)

which achieves the proof

Corollary 8 If 119896 is large enough we find

119864 (119881) = 1205792 Var (119881) = 0 (16)

Proof The idea is the limit passage 119896 rarr infin

Remark 9 In this case a swap maturity 119879 does not influence119864(119881) and Var(119881)

33 GARCH(11) and Volatility Swaps GARCH model isneeded for both the variance swap and the volatility swapThemodel for the variance in a continuous version forHestonmodel is

1198891205902

119905= 119896 (120579

2minus 1205902

119905) 119889119905 + 120585120590

1199051198891199082(119905) (17)

The discrete version of the GARCH(11) process is describedby Engle and Mezrich [16]

]119899+1

= (1 minus 120572 minus 120573)119881 + 1205721199062

119899+ 120573]119899 (18)

where 119881 is the long-term variance 119906119899is the drift-adjusted

stock return at time 119899 120572 is the weight assigned to 1199062

119899 and

120573 is the weight assigned to ]119899 Further we use the following

relationship (19) to calculate the discrete GARCH(11) param-eters

119881 =

119862

1 minus 120572 minus 120573

120579 =

119881

Δ119905119871

1205900=

119881

Δ119905119878

119896 =

1 minus 120572 minus 120573

Δ119905

1205852=

1205722(119870 minus 1)

Δ119905

(19)

where Δ119905119871= 1252 252 trading days in any given year and

Δ119905119878= 163 63 trading days in any given three monthsNow we will briefly discuss the validity of the assumption

that the risk-neutral process for the instantaneous variance is


a continuous time limit of a GARCH(11) process It is wellknown that this limit has the property that the incrementin instantaneous variance is conditionally uncorrelated withthe return of the underlying asset This unfortunately impliesthat at each maturity 119879 the implied volatility is symmetricHence for assets whose options are priced consistently witha symmetric smile these observations can be used eitherto initially calibrate the model or as a test of the modelrsquosvalidity It is worthmentioning that it is not suitable to use at-the-money implied volatilities in general to price a seasonedvolatility swap However our GARCH(11) approximationshould still be pretty robust

4 Application

In this section we apply the analytical solutions from Sec-tion 3 to price a swap on the volatility of the CAC 40 FrenchIndex for five years (October 2009ndashApril 2013)

The first step of this application is to study the stationarityof the series To this end we used the unit root test of Dickey-Fuller (ADF) and Philips Peron test (PP)

41 Unit Root Tests and Descriptive Analysis In this sectionwe summarized unit root tests and descriptive analysis resultsof 119878cac (see Table 1)

Unit root test confirms the stationarity of the seriesIn Table 2 all statistic parameters of CAC 40 French Index

are shown For the analysis 1155 observations were takenMean of time series is 00000528 median 0 and standarddeviation 0014589 Skewness of CAC 40 French Index isminus0078899 so it is negative and the mean is larger than themedian and there is left-skewed distribution Kurtosis is7255109 large than 3 so we called leptokurtic indicatinghigher peak and fatter tails than the normal distributionJarque-Bera is 8090892 So we can forecast an uptrend

GARCH(11) models are clearly the best performingmodels as they receive the lowest score on fitting metricswhilst representing the lowest MAE RMSE MAPE SEEand BIC among all models They are closely followed byGARCH(21) which is placed comfortably lower than bothARCH(2) and ARCH(4) However the GARCH(11) modelis simple and easy to handle The results also show thatGARCH(11) model improves the forecasting performance(see Table 3)

Numerical Applications We have used Eviews software andwe found 119862 = 203 times 10minus7 120572 = minus0008411 120573 = 0980310and 119870 = 7255109 To this end we find the following 119881 =7223942208 times 10minus7 120579 = 000182043 120590

0= 00004551 119896 =

7081452 1205852 = 0111 51We use the relations (9) and (10) for a swap maturity 119879 =

09 years and we find

119864 (119881) = 28273 times 10minus6

Var (119881) = 50873 times 10minus9

(20)

The convexity adjustment is Var(119881)811986432(119881) = 013376 and119864(radic119881) asymp minus013208

Table 1 Unit root test

Test ADF PP119878cac minus3416458 minus3501017

04

03

02

01

00

minus01

minus02

minus03

minus04

250 500 750 1000

YYF

Figure 1 GARCH(11) CAC 40 French Index forecasting

Remark 10 If the nonadjusted strike is equal to 023456 thenthe adjusted strike is equal to 023456 minus 013376 = 01008

According to Figure 3 119864(119881) is increasing exponentiallyand converges when 119879 rarr infin towards 33140 times 10minus6But Var(119881) is increasing linearly during the first yearand is decreasing exponentially during [1infin[ years whenVar(119881) rarr 0 if 119879 rarr infin

42 Conclusions According to results founded theGARCH(11) is a very good model for modeling the volatilityswaps for stock market Also we remark the influence of theFrench financial crisis (2009) on CAC 40 French Index

Moreover we presented a probabilistic approach basedon changing of time method to study variance and volatilityswaps for stock market with underlying asset and variancethat follow the Heston model We obtained the formulasfor variance and volatility swaps but with another structureand another application to those in the papers by Brockhausand Long [10] and Swishchuk [12] As an application ofour analytical solutions we provided a numerical exampleusing CAC 40 French Index to price swap on the volatility(Figure 1)

Also we compared the forecasting performance of sev-eral GARCH models using different distributions for CAC40 French Index We found that the GARCH(11) skewedStudent 119905 model is the most promising for characterizingthe dynamic behaviour of these returns as it reflects theirunderlying process in terms of serial correlation asymmetricvolatility clustering and leptokurtic innovation The resultsalso show that GARCH(11) model improves the forecast-ing performance This result later further implies that theGARCH(11) model might be more useful than the otherthree models (ARCH(2) ARCH(4) and GARCH(21)) whenimplementing riskmanagement strategies for CAC40 FrenchIndex (Figure 2)


Table 2

Mean Median Std Dev Skewness Kurtosis Jarque-B119878cac 528119864 minus 5 00000 0014589 minus0078899 7255109 8090892

Table 3

Models Adju 1198772 SEE BIC RMSE MAE MAPE

ARCH(2) 0989953 0007369 minus2620676 0013674 0009786 3612218ARCH(4) 0989971 0007062 minus2801014 0010689 0007441 3469134GARCH(21) 0992352 0003072 minus7893673 0002668 0002835 2946543GARCH(11) 0999122 0002672 minus8993776 0002668 0001983 2743416

250 500 750 1000

times10minus2

030

028

026

024

022

020

018

Conditional standard deviation

Figure 2 CAC 40 French Index conditional variance

Appendix

We give a reminder for each parameter(1) Std Dev (standard deviation) is a measure of disper-

sion or spread in the series The standard deviation is givenby

119904 = radic1

119873 minus 1

119873

sum

119894=1

(119910119894minus 119910)2

(A1)

where119873 is the number of observations in the current sampleand 119910 is the mean of the series

(2) Skewness is a measure of asymmetry of the distribu-tion of the series around its mean Skewness is computed as

119878 =

1

119873

119873

sum

119894=1

(

119910119894minus 119910

)

3

(A2)

where is an estimator for the standard deviation thatis based on the biased estimator for the variance ( =

119904radic(119873 minus 1)119873)(3) Kurtosis measures the peakedness or flatness of the

distribution of the series Kurtosis is computed as

119870 =

1

119873

119873

sum

119894=1

(

119910119894minus 119910

)

4

(A3)

where is again based on the biased estimator for the vari-ance

30

25

20

15

10

05

00

minus050 1 2 3 4 5 6 7 8 9

Maturity (years)

Var(V

)

Line plot of Var(V) donn ees volatility 3vlowast8c

(a)

020

018

016

012

014

010

008

0 1 2 3 4 5 6 7 8 9

Maturity (years)

Line plot of E(V) donn ees volatility 3v lowast8c

E(V

)

(b)

Figure 3 CAC 40 French Index 119864(119881) and Var(119881)

(4) Jarque-Bera is a test statistic for testing whether theseries is normally distributed The statistic is computed as

Jarque-Bera = 119873

6

(1198782+

(119870 minus 3)2

4

) (A4)

where 119878 is the skewness and 119870 is the kurtosis

(5) Mean absolute error (MAE) is as follows MAE =

(1119873)sum119873

119894=1

1003816100381610038161003816119910119894minus 119910119894

1003816100381610038161003816

(6)Mean absolute percentage error (MAPE) is as followsMAPE = sum

119873

119894=1

1003816100381610038161003816(119910119894minus 119910119894)119910119894

1003816100381610038161003816

(7) Root mean squared error (RMSE) is as followsRMSE = radic(1119873)sum

119873

119894=1(119910119894minus 119910119894)2

(8) Adjusted R-squared (adjust 1198772) is considered(9) Sum error of regression (SEE) is considered


(10) Schwartz criterion (BIC) is measured by 119899 ln (SEE) +119896 ln (119899)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was given ATRST (ex ANDRU) financing withinthe framework of the PNR Project (Number 8u231050) andAverroes Program

References

[1] F Black and M Scholes ldquoThe pricing of option and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3 pp 637ndash659 1973

[2] J Hull Options Futures and Other Derivatives Prentice HallNew York NY USA 4th edition 2000

[3] M Broadie and A Jain ldquoThe effect of jumps and discrete sam-pling on volatility and variance swapsrdquo International Journal ofTheoretical and Applied Finance vol 11 no 8 pp 761ndash797 2008

[4] H Baili ldquoStochastic analysis and particle filtering of the volatil-ityrdquo IAENG International Journal of Applied Mathematics vol41 no 1 article 09 2011

[5] L Cao and Z-F Guo ldquoOptimal variance swaps investmentsrdquoIAENG International Journal of AppliedMathematics vol 41 no4 pp 334ndash338 2011

[6] S Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 pp 327ndash343 1993

[7] J C Cox J Ingersoll and S Ross ldquoA theory of the term structureof interest ratesrdquo Econometrica Journal of the EconometricSociety vol 53 no 2 pp 385ndash407 1985

[8] T Bruce Fixed Income Securities Tools for TodayrsquosMarkets JohnWiley amp Sons New York NY USA 1996

[9] K Demeterfi E Derman M Kamal and J Zou ldquoA guide tovolatility and variance swapsrdquoThe Journal of Derivatives vol 6no 4 pp 9ndash32 1999

[10] O Brockhaus and D Long ldquoVolatility swaps made simplerdquo RiskMagazine vol 2 no 1 pp 92ndash96 2000

[11] A JavaheriThe volatility process [PhD thesis] Ecole des Minesde Paris Paris France 2004

[12] A Swishchuk ldquoVariance and volatility swaps in energy mar-ketsrdquo Journal of Energy Markets vol 6 no 1 pp 33ndash49 2013

[13] R He and YWang ldquoPrice pseudo-variance pseudo covariancepseudo-volatility and pseudo-correlation swaps-in analyticalclose formsrdquo in Proceedings of the 6th PIMS Industrial ProblemsSolving Workshop (PIMS IPSW rsquo02) pp 27ndash37 University ofBritish Columbia Vancouver Canada 2002

[14] A Swishchuk and A Kalemanova ldquoThe stochastic stabilityof interest rates with jump changesrdquo Theory Probability andMathematical Statistics vol 61 pp 161ndash172 2000

[15] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 1998

[16] R F Engle and J Mezrich ldquoGrappling with GARCHrdquo RiskMagazine vol 8 no 9 pp 112ndash117 1995

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Stochastic AnalysisInternational Journal of


2 Volatility Swaps

In this sectionwe give somedefinitions andnotations of swapstockrsquos volatility stockrsquos volatility swap and variance swap

Definition 1 Swaps were introduced in the 1980s and thereis an agreement between two parties to exchange cash flowsat one or several future dates as defined by Bruce [8] Inthis contract one party agrees to pay a fixed amount to acounterpart which in turn honors the agreement by paying afloating amount which depends on the level of some specificunderlying By entering a swap a market participant cantherefore exchange the exposure from the varying underlyingby paying a fixed amount at certain future time points

Definition 2 A stockrsquos volatility is the simplest measure of itsrisk less or uncertainty Formally the volatility 120590

119877(119878) is the

annualized standard deviation of the stockrsquos returns duringthe period of interest where the subscript 119877 denotes theobserved or ldquorealizedrdquo volatility for the stock 119878

Definition 3 (see [9]) A stock volatility swap is a forwardcontract on the annualized volatility Its payoff at expirationis equal to

119873(120590119877(119878) minus 119870vol) (1)

where 120590119877(119878) = radic(1119879) int

119879

01205902

119904119889119904 120590119905is a stochastic stock

volatility 119870vol is the annualized volatility delivery price and119873 is the notional amount of the swap in Euro annualizedvolatility point

Definition 4 (see [9]) A variance swap is a forward contracton annualized variance the square of the realized volatilityIts payoff at expiration is equal to

119873(1205902

119877(119878) minus 119870var) (2)

where 119870var is the delivery price for variance and 119873 isthe notional amount of the swap in Euros per annualizedvolatility point squared

Notation 1 We note that 1205902119877(119878) = 119881

Using the Brockhaus and Long [10] and Javaheri [11]approximation which is used in the second order Taylorformula forradic119909 we have

119864 (radic119881) asymp radic119864 (119881) minus

Var (119881)811986432

(119881)

(3)

where Var(119881)811986432(119881) is the convexity adjustment Thus tocalculate volatility swaps we need both 119864(119881) and Var(119881)

The realized discrete sampled variance is defined asfollows

119881119899(119878) =

119899

(119899 minus 1) 119879

119899

sum

119894=1

ln2 (119878119905119894

119878119905119894minus1

) 119881 = lim119899rarrinfin

119881119899(119878)

(4)

where 119879 is the maturity (years or days)

3 Volatility Swaps for Heston Model

31 Stochastic VolatilityModel Let (Ω 119865 119865119905P) be probability

space with filtration 119865119905 119905 isin [0 119879] We consider the risk-

neutral Heston stochastic volatility model for the price 119878119905and

variance follows the following model

119889119878119905= 119903119905119878119905119889119905 + 120590

1199051198781199051198891199081(119905)

1198891205902

119905= 119896 (120579

2minus 1205902

119905) 119889119905 + 120585120590

1199051198891199082(119905)

(1198781)

where 119903119905is deterministic interest rate 120590

0gt 0 and 120579 gt 0 are

short and long volatility 119896 gt 0 is a reversion speed 120585 gt 0

is a volatility of volatility parameter and 1199081(119905) and 119908

2(119905) are

independent standard Brownian motionsWe can rewrite the system (1198781) as follows

119889119878119905= 119903119905119878119905119889119905 + 120590

1199051198781199051198891199081(119905)

1198891205902

119905= 119896 (120579

2minus 1205902

119905) 119889119905 + 120588120585120590

1199051198891199081(119905)

+ 120585radic1 minus 120588120590119905119889119908 (119905)

(1198782)

where 119908(119905) is standard Brownian motion which is inde-pendent of 119908

1(119905) and the indicator economic 119883 Let

cov(1198891199081(119905) 119889119908

2(119905)) = 120588119889119905 and we can transform the system

(1198782) to (1198781) if we replace 1205881198891199081(119905) + radic1 minus 120588119889119908(119905) by 119889119908

2(119905)

32 Explicit Expression andProperties of1205902119905 In this sectionwe

reformulated the results obtained in [12] which are neededfor study of variance and volatility swaps and price ofpseudovariance pseudovolatility and the problems proposedby He andWang [13] for financial markets with deterministicvolatility as a function of timeThis approachwas first appliedto the study of stochastic stability of Cox-Ingersoll-Rossprocess in Swishchuk and Kalemanova [14]TheHeston assetprocess has a variance 1205902

119905that follows Cox et al [7] process

described by the second equation in (1198781) If the volatility120590119905follows Ornstein-Uhlenbeck process (see eg Oksendal

[15]) then Itorsquos lemma shows that the variance 1205902119905follows the

process described exactly by the second equation in (1198781)We start to define the following process and function

V119905= 119890119896119905(1205902

119905minus 1205792)

Φ (119905) = 120585minus2int

119905

0

119890119896Φ(119904)

(1205902

0minus 1205792+ 1199082(119904) + 120579

21198902119896Φ(119904)

)

minus1

119889119904

(5)

Definition 5 We define 119861(119905) = 1199082(Φminus1

119905) where 119908

2is anF

119905-

measurable one-dimensional Wiener process F119905= FΦminus1

119905

and 119905 and 119904 = min(119905 119904) whereΦminus1

119905is an inverse function ofΦ

119905

The properties of 119861(119905) are as follows

(a) F119905-martingale and 119864(119861(119905)) = 0

(b) 119864(1198612(119905)) = 1205852(((119890119896119905minus1)119896)(120590

2

0minus1205792)+((1198902119896119905

minus1)2119896)1205792)

(c) 119864(119861(119904)119861(119905)) = 1205852(((119890119896(119905and119904)

minus1)119896)(1205902

0minus1205792)+ ((119890

2119896(119905and119904)minus

1)2119896)1205792)



1205902

119905= 119890minus119896119905

(1205902

0minus 1205792+ 119861 (119905)) + 120579

2 (6)

(b)

119864 (1205902

119905) = 119890minus119896119905

(1205902

0minus 1205792) + 1205792 (7)

(c)

119864 (1205902

1199041205902

119905) = 1205852119890minus119896(119905+119904)

(

119890119896(119905and119904)

minus 1

119896

(1205902

0minus 1205792) +

1198902119896(119905and119904)

minus 1

2119896

1205792)

+ 119890minus119896(119905+119904)

(1205902

0minus 1205792)

2

+ 119890minus119896119905

(1205902

0minus 1205792) 1205792

+ 119890minus119896119904

(1205902

0minus 1205792) 1205792+ 1205794

(8)

Proof See [12]

Theorem 7 One has

(a)

119864 (119881) =

1 minus 119890minus119896119879

119896119879

(1205902

0minus 1205792) + 1205792 (9)

(b)

Var (119881) = 1205852119890minus2119896119879

211989631198792[(21198902119896119879

minus 4119896119879119890119896119879

minus 2) (1205902

0minus 1205792)

+ (21198961198791198902119896119879

minus 31198902119896119879

+ 4119890119896119879

minus 1) 1205792]

(10)


119864 (119881) =

1

119879

int

119879

0

119864 (1205902

119905) 119889119905 (11)


119864 (119881) =

1 minus 119890minus119896119879

119896119879

(1205902

0minus 1205792) + 1205792 (12)



2)∬

119879

0119864(1205902

1199051205902

119904)119889119905119889119904

119864 (1198812)

=

1205852

1198792∬

119879

0

[119890minus119896(119905+119904)

(

119890119896(119905and119904)

minus 1

119896

(1205902

0minus 1205792)

+

1198902119896(119905and119904)

minus 1

2119896

1205792)]119889119905119889119904

+ 1198642(119881)

(13)


Var (119881)

=

1205852

1198792∬

119879

0

[119890minus119896(119905+119904)

(

119890119896(119905and119904)

minus 1

119896

(1205902

0minus 1205792)

+

1198902119896(119905and119904)

minus 1

2119896

1205792)]119889119905119889119904

(14)


Var (119881) = 1205852119890minus2119896119879

211989631198792[(21198902119896119879

minus 4119896119879119890119896119879

minus 2) (1205902

0minus 1205792)

+ (21198961198791198902119896119879

minus 31198902119896119879

+ 4119890119896119879

minus 1) 1205792]

(15)



119864 (119881) = 1205792 Var (119881) = 0 (16)




1198891205902

119905= 119896 (120579

2minus 1205902

119905) 119889119905 + 120585120590

1199051198891199082(119905) (17)


]119899+1

= (1 minus 120572 minus 120573)119881 + 1205721199062

119899+ 120573]119899 (18)



119899 and



119881 =

119862

1 minus 120572 minus 120573

120579 =

119881

Δ119905119871

1205900=

119881

Δ119905119878

119896 =

1 minus 120572 minus 120573

Δ119905

1205852=

1205722(119870 minus 1)

Δ119905

(19)






4 Application








0= 00004551 119896 =



119864 (119881) = 28273 times 10minus6

Var (119881) = 50873 times 10minus9

(20)




04

03

02

01

00

minus01

minus02

minus03

minus04

250 500 750 1000

YYF








Table 2


Table 3



250 500 750 1000

times10minus2

030

028

026

024

022

020

018



Appendix



119904 = radic1

119873 minus 1

119873

sum

119894=1

(119910119894minus 119910)2

(A1)



119878 =

1

119873

119873

sum

119894=1

(

119910119894minus 119910

)

3

(A2)




119870 =

1

119873

119873

sum

119894=1

(

119910119894minus 119910

)

4

(A3)


30

25

20

15

10

05

00

minus050 1 2 3 4 5 6 7 8 9

Maturity (years)

Var(V

)


(a)

020

018

016

012

014

010

008

0 1 2 3 4 5 6 7 8 9

Maturity (years)


E(V

)

(b)




6

(1198782+

(119870 minus 3)2

4

) (A4)



(1119873)sum119873

119894=1

1003816100381610038161003816119910119894minus 119910119894

1003816100381610038161003816


119873

119894=1

1003816100381610038161003816(119910119894minus 119910119894)119910119894

1003816100381610038161003816


119873

119894=1(119910119894minus 119910119894)2






Acknowledgment


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205902

119905= 119890minus119896119905

(1205902

0minus 1205792+ 119861 (119905)) + 120579

2 (6)

(b)

119864 (1205902

119905) = 119890minus119896119905

(1205902

0minus 1205792) + 1205792 (7)

(c)

119864 (1205902

1199041205902

119905) = 1205852119890minus119896(119905+119904)

(

119890119896(119905and119904)

minus 1

119896

(1205902

0minus 1205792) +

1198902119896(119905and119904)

minus 1

2119896

1205792)

+ 119890minus119896(119905+119904)

(1205902

0minus 1205792)

2

+ 119890minus119896119905

(1205902

0minus 1205792) 1205792

+ 119890minus119896119904

(1205902

0minus 1205792) 1205792+ 1205794

(8)

Proof See [12]

Theorem 7 One has

(a)

119864 (119881) =

1 minus 119890minus119896119879

119896119879

(1205902

0minus 1205792) + 1205792 (9)

(b)

Var (119881) = 1205852119890minus2119896119879

211989631198792[(21198902119896119879

minus 4119896119879119890119896119879

minus 2) (1205902

0minus 1205792)

+ (21198961198791198902119896119879

minus 31198902119896119879

+ 4119890119896119879

minus 1) 1205792]

(10)


119864 (119881) =

1

119879

int

119879

0

119864 (1205902

119905) 119889119905 (11)


119864 (119881) =

1 minus 119890minus119896119879

119896119879

(1205902

0minus 1205792) + 1205792 (12)



2)∬

119879

0119864(1205902

1199051205902

119904)119889119905119889119904

119864 (1198812)

=

1205852

1198792∬

119879

0

[119890minus119896(119905+119904)

(

119890119896(119905and119904)

minus 1

119896

(1205902

0minus 1205792)

+

1198902119896(119905and119904)

minus 1

2119896

1205792)]119889119905119889119904

+ 1198642(119881)

(13)


Var (119881)

=

1205852

1198792∬

119879

0

[119890minus119896(119905+119904)

(

119890119896(119905and119904)

minus 1

119896

(1205902

0minus 1205792)

+

1198902119896(119905and119904)

minus 1

2119896

1205792)]119889119905119889119904

(14)


Var (119881) = 1205852119890minus2119896119879

211989631198792[(21198902119896119879

minus 4119896119879119890119896119879

minus 2) (1205902

0minus 1205792)

+ (21198961198791198902119896119879

minus 31198902119896119879

+ 4119890119896119879

minus 1) 1205792]

(15)



119864 (119881) = 1205792 Var (119881) = 0 (16)




1198891205902

119905= 119896 (120579

2minus 1205902

119905) 119889119905 + 120585120590

1199051198891199082(119905) (17)


]119899+1

= (1 minus 120572 minus 120573)119881 + 1205721199062

119899+ 120573]119899 (18)



119899 and



119881 =

119862

1 minus 120572 minus 120573

120579 =

119881

Δ119905119871

1205900=

119881

Δ119905119878

119896 =

1 minus 120572 minus 120573

Δ119905

1205852=

1205722(119870 minus 1)

Δ119905

(19)






4 Application








0= 00004551 119896 =



119864 (119881) = 28273 times 10minus6

Var (119881) = 50873 times 10minus9

(20)




04

03

02

01

00

minus01

minus02

minus03

minus04

250 500 750 1000

YYF








Table 2


Table 3



250 500 750 1000

times10minus2

030

028

026

024

022

020

018



Appendix



119904 = radic1

119873 minus 1

119873

sum

119894=1

(119910119894minus 119910)2

(A1)



119878 =

1

119873

119873

sum

119894=1

(

119910119894minus 119910

)

3

(A2)




119870 =

1

119873

119873

sum

119894=1

(

119910119894minus 119910

)

4

(A3)


30

25

20

15

10

05

00

minus050 1 2 3 4 5 6 7 8 9

Maturity (years)

Var(V

)


(a)

020

018

016

012

014

010

008

0 1 2 3 4 5 6 7 8 9

Maturity (years)


E(V

)

(b)




6

(1198782+

(119870 minus 3)2

4

) (A4)



(1119873)sum119873

119894=1

1003816100381610038161003816119910119894minus 119910119894

1003816100381610038161003816


119873

119894=1

1003816100381610038161003816(119910119894minus 119910119894)119910119894

1003816100381610038161003816


119873

119894=1(119910119894minus 119910119894)2






Acknowledgment


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











4 Application








0= 00004551 119896 =



119864 (119881) = 28273 times 10minus6

Var (119881) = 50873 times 10minus9

(20)




04

03

02

01

00

minus01

minus02

minus03

minus04

250 500 750 1000

YYF








Table 2


Table 3



250 500 750 1000

times10minus2

030

028

026

024

022

020

018



Appendix



119904 = radic1

119873 minus 1

119873

sum

119894=1

(119910119894minus 119910)2

(A1)



119878 =

1

119873

119873

sum

119894=1

(

119910119894minus 119910

)

3

(A2)




119870 =

1

119873

119873

sum

119894=1

(

119910119894minus 119910

)

4

(A3)


30

25

20

15

10

05

00

minus050 1 2 3 4 5 6 7 8 9

Maturity (years)

Var(V

)


(a)

020

018

016

012

014

010

008

0 1 2 3 4 5 6 7 8 9

Maturity (years)


E(V

)

(b)




6

(1198782+

(119870 minus 3)2

4

) (A4)



(1119873)sum119873

119894=1

1003816100381610038161003816119910119894minus 119910119894

1003816100381610038161003816


119873

119894=1

1003816100381610038161003816(119910119894minus 119910119894)119910119894

1003816100381610038161003816


119873

119894=1(119910119894minus 119910119894)2






Acknowledgment


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 2


Table 3



250 500 750 1000

times10minus2

030

028

026

024

022

020

018



Appendix



119904 = radic1

119873 minus 1

119873

sum

119894=1

(119910119894minus 119910)2

(A1)



119878 =

1

119873

119873

sum

119894=1

(

119910119894minus 119910

)

3

(A2)




119870 =

1

119873

119873

sum

119894=1

(

119910119894minus 119910

)

4

(A3)


30

25

20

15

10

05

00

minus050 1 2 3 4 5 6 7 8 9

Maturity (years)

Var(V

)


(a)

020

018

016

012

014

010

008

0 1 2 3 4 5 6 7 8 9

Maturity (years)


E(V

)

(b)




6

(1198782+

(119870 minus 3)2

4

) (A4)



(1119873)sum119873

119894=1

1003816100381610038161003816119910119894minus 119910119894

1003816100381610038161003816


119873

119894=1

1003816100381610038161003816(119910119894minus 119910119894)119910119894

1003816100381610038161003816


119873

119894=1(119910119894minus 119910119894)2






Acknowledgment


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgment


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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