This paper focuses on the pricing of variance and volatility swaps under Heston model (1993). To this end, we apply this model to the empirical financial data: CAC 40 French Index. More precisely, we make an application example for stock market forecast: CAC 40 French Index to price swap on the volatility using GARCH(1,1) model.
1. Introduction
Black and Scholes’ model [1] is one of the most significant models discovered in finance in XXe century for valuing liquid European style vanilla option. Black-Scholes model assumes that the volatility is constant but this assumption is not always true. This model is not good for derivatives prices founded in finance and businesses market (see [2]).
“The volatility of asset prices is an indispensable input in both pricing options and in risk management. Through the introduction of volatility derivatives, volatility is now, in effect, a tradable market instrument” Broadie and Jain [3].
Volatility is one of the principal parameters employed to describe and measure the fluctuations of asset prices. It plays a crucial role in the modern financial analysis concerning risk management, option valuation, and asset allocation. There are different types of volatilities: implied volatility, local volatility, and stochastic volatility (see Baili [4]).
To this end, the new financial products are variance and volatility swaps, which play a decisive role in volatility hedging and speculation. Investment banks, currencies, stock indexes, finance, and businesses markets are useful for variance and volatility swaps.
Volatility swaps allow investors to trade and to control the volatility of an asset directly. Moreover, they would trade a price index. The underlying is usually a foreign exchange rate (very liquid market) but could be as well a single name equity or index. However, the variance swap is reliable in the index market because it can be replicated with a linear combination of options and a dynamic position in futures. Also, volatility swaps are not used only in finance and businesses but in energy markets and industry too.
The variance swap contract contains two legs: fixed leg (variance strike) and floating leg (realized variance). There are several works which studied the variance swap portfolio theory and optimal portfolio of variance swaps based on a variance Gamma correlated (VGC) model (see Cao and Guo [5]).
The goal of this paper is the valuation and hedging of volatility swaps within the frame of a GARCH(1,1) stochastic volatility model under Heston model [6]. The Heston asset process has a variance that follows a Cox et al. [7] process. Also, we make an application by using CAC 40 French Index.
The structure of the paper is as follows. Section 2 considers representing the volatility swap and the variance swap. Section 3 describes the volatility swaps for Heston model, gives explicit expression of σt2, and discusses the relationship between GARCH and volatility swaps. Finally, we make an application example for stock market forecast: CAC 40 French Index using GARCH/ARCH models.
2. Volatility Swaps
In this section we give some definitions and notations of swap, stock’s volatility, stock’s volatility swap, and variance swap.
Definition 1.
Swaps were introduced in the 1980s and there is an agreement between two parties to exchange cash flows at one or several future dates as defined by Bruce [8]. In this contract one party agrees to pay a fixed amount to a counterpart which in turn honors the agreement by paying a floating amount, which depends on the level of some specific underlying. By entering a swap, a market participant can therefore exchange the exposure from the varying underlying by paying a fixed amount at certain future time points.
Definition 2.
A stock’s volatility is the simplest measure of its risk less or uncertainty. Formally, the volatility σR(S) is the annualized standard deviation of the stock’s returns during the period of interest, where the subscript R denotes the observed or “realized” volatility for the stock S.
Definition 3 (see [<xref ref-type="bibr" rid="B6">9</xref>]).
A stock volatility swap is a forward contract on the annualized volatility. Its payoff at expiration is equal to
(1)NσRS-Kvol,
where σR(S)≔(1/T)∫0Tσs2ds,σt is a stochastic stock volatility, Kvol is the annualized volatility delivery price, and N is the notional amount of the swap in Euro annualized volatility point.
Definition 4 (see [<xref ref-type="bibr" rid="B6">9</xref>]).
A variance swap is a forward contract on annualized variance, the square of the realized volatility. Its payoff at expiration is equal to
(2)NσR2S-Kvar,
where Kvar is the delivery price for variance and N is the notional amount of the swap in Euros per annualized volatility point squared.
Notation 1.
We note that σR2(S)=V.
Using the Brockhaus and Long [10] and Javaheri [11] approximation which is used in the second order Taylor formula for x, we have
(3)EV≈EV-VarV8E3/2V,
where Var(V)/8E3/2(V) is the convexity adjustment. Thus, to calculate volatility swaps we need both E(V) and Var(V).
The realized discrete sampled variance is defined as follows:
(4)VnS≔nn-1T∑i=1nln2StiSti-1,V≔limn→∞VnS,
where T is the maturity (years or days).
3. Volatility Swaps for Heston Model3.1. Stochastic Volatility Model
Let (Ω,F,Ft,P) be probability space with filtration Ft, t∈[0;T]. We consider the risk-neutral Heston stochastic volatility model for the price St and variance follows the following model:
S1dSt=rtStdt+σtStdw1t,dσt2=kθ2-σt2dt+ξσtdw2t,
where rt is deterministic interest rate, σ0>0 and θ>0 are short and long volatility, k>0 is a reversion speed, ξ>0 is a volatility of volatility parameter, and w1(t) and w2(t) are independent standard Brownian motions.
We can rewrite the system S1 as follows:
(S2)dSt=rtStdt+σtStdw1(t)dσt2=kθ2-σt2dt+ρξσtdw1(t)+ξ1-ρσtdwt,
where w(t) is standard Brownian motion which is independent of w1(t) and the indicator economic X. Let cov(dw1(t),dw2(t))=ρdt, and we can transform the system (S2) to S1 if we replace ρdw1(t)+1-ρdw(t) by dw2(t).
3.2. Explicit Expression and Properties of <inline-formula>
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In this section we reformulated the results obtained in [12], which are needed for study of variance and volatility swaps, and price of pseudovariance, pseudovolatility, and the problems proposed by He and Wang [13] for financial markets with deterministic volatility as a function of time. This approach was first applied to the study of stochastic stability of Cox-Ingersoll-Ross process in Swishchuk and Kalemanova [14]. The Heston asset process has a variance σt2 that follows Cox et al. [7] process, described by the second equation in S1. If the volatility σt follows Ornstein-Uhlenbeck process (see, e.g., Oksendal [15]), then Ito’s lemma shows that the variance σt2 follows the process described exactly by the second equation in S1.
We start to define the following process and function:
(5)vt≔ektσt2-θ2,Φt≔ξ-2∫0tekΦsσ02-θ2+w~2s+θ2e2kΦs-1ds.
Definition 5.
We define B(t)≔w~2(Φt-1), where w~2 is an Ft-measurable one-dimensional Wiener process, F~t≔FΦt-1, and t∧s≔min(t,s), where Φt-1 is an inverse function of Φt. The properties of B(t) are as follows:
(a) We obtain mean value for V(11)E(V)=1T∫0TEσt2dt
using Lemma 6, and we find
(12)E(V)=1-e-kTkTσ02-θ2+θ2.
(b) Variance for V equals Var(V)=E(V2)-E2(V), and the second moment may be found as follows: using formula (8) of Lemma 6: E(V2)=(1/T2)∬0TE(σt2σs2)dtds,
(13)EV2=ξ2T2∬0Te-kt+sekt∧s-1kσ02-θ2+e2kt∧s-12kθ2dtds+E2(V)
and taking (13) and variance formula we find
(14)VarV=ξ2T2∬0Te-kt+sekt∧s-1kσ02-θ2+e2kt∧s-12kθ2dtds;
after calculations we obtain
(15)VarV=ξ2e-2kT2k3T22e2kT-4kTekT-2σ02-θ2+2kTe2kT-3e2kT+4ekT-1θ2
which achieves the proof.
Corollary 8.
If k is large enough, we find
(16)EV=θ2,Var(V)=0.
Proof.
The idea is the limit passage k→∞.
Remark 9.
In this case a swap maturity T does not influence E(V) and Var(V).
3.3. GARCH(1,1) and Volatility Swaps
GARCH model is needed for both the variance swap and the volatility swap. The model for the variance in a continuous version for Heston model is
(17)dσt2=kθ2-σt2dt+ξσtdw2t.
The discrete version of the GARCH(1,1) process is described by Engle and Mezrich [16]:
(18)νn+1=1-α-βV+αun2+βνn,
where V is the long-term variance, un is the drift-adjusted stock return at time n, α is the weight assigned to un2, and β is the weight assigned to νn. Further we use the following relationship (19) to calculate the discrete GARCH(1,1) parameters:
(19)V=C1-α-βθ=VΔtL,σ0=VΔtSk=1-α-βΔtξ2=α2K-1Δt,
where ΔtL=1/252, 252 trading days in any given year, and ΔtS=1/63, 63 trading days in any given three months.
Now, 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(1,1) process. It is well known that this limit has the property that the increment in instantaneous variance is conditionally uncorrelated with the return of the underlying asset. This unfortunately implies that, at each maturity T, the implied volatility is symmetric. Hence, for assets whose options are priced consistently with a symmetric smile, these observations can be used either to initially calibrate the model or as a test of the model’s validity. It is worth mentioning that it is not suitable to use at-the-money implied volatilities in general to price a seasoned volatility swap. However, our GARCH(1,1) approximation should still be pretty robust.
4. Application
In this section, we apply the analytical solutions from Section 3 to price a swap on the volatility of the CAC 40 French Index for five years (October 2009–April 2013).
The first step of this application is to study the stationarity of the series. To this end, we used the unit root test of Dickey-Fuller (ADF) and Philips Péron test (PP).
4.1. Unit Root Tests and Descriptive Analysis
In this section, we summarized unit root tests and descriptive analysis results of Scac (see Table 1).
Unit root test.
Test
ADF
PP
Scac
−34.16458
−35.01017
Unit root test confirms the stationarity of the series.
In Table 2 all statistic parameters of CAC 40 French Index are shown. For the analysis 1155 observations were taken. Mean of time series is 0.0000528, median 0, and standard deviation 0.014589. Skewness of CAC 40 French Index is -0.078899, so it is negative and the mean is larger than the median, and there is left-skewed distribution. Kurtosis is 7.255109, large than 3, so we called leptokurtic, indicating higher peak and fatter tails than the normal distribution. Jarque-Bera is 809.0892. So we can forecast an uptrend.
Mean
Median
Std. Dev.
Skewness
Kurtosis
Jarque-B
Scac
5.28E-5
0.0000
0.014589
−0.078899
7.255109
809.0892
GARCH(1,1) models are clearly the best performing models as they receive the lowest score on fitting metrics whilst representing the lowest MAE, RMSE, MAPE, SEE, and BIC among all models. They are closely followed by GARCH(2,1) which is placed comfortably lower than both ARCH(2) and ARCH(4). However the GARCH(1,1) model is simple and easy to handle. The results also show that GARCH(1,1) model improves the forecasting performance (see Table 3).
Models
Adju R2
SEE
BIC
RMSE
MAE
MAPE
ARCH(2)
0.989953
0.007369
−2.620676
0.013674
0.009786
3.612218
ARCH(4)
0.989971
0.007062
−2.801014
0.010689
0.007441
3.469134
GARCH(2,1)
0.992352
0.003072
−7.893673
0.002668
0.002835
2.946543
GARCH(1,1)
0.999122
0.002672
−8.993776
0.002668
0.001983
2.743416
Numerical Applications. We have used Eviews software, and we found C = 2.03 × 10^{−7}, α=-0.008411, β=0.980310, and K=7.255109. To this end, we find the following: V = 72.23942208 × 10^{−7}; θ=0.00182043; σ0=0.0004551; k=7.081452; ξ2=0.11151.
We use the relations (9) and (10) for a swap maturity T=0.9 years, and we find
(20)EV=2.8273×10-6,VarV=5.0873×10-9.
The convexity adjustment is Var(V)/8E3/2(V)=0.13376 and E(V)≈-0.13208.
Remark 10.
If the nonadjusted strike is equal to 0.23456, then the adjusted strike is equal to 0.23456-0.13376=0.1008.
According to Figure 3E(V) is increasing exponentially and converges when T→∞ towards 3.3140 × 10^{−6}. But Var(V) is increasing linearly during the first year and is decreasing exponentially during 1,∞ years when Var(V)→0, if T→∞.
4.2. Conclusions
According to results founded, the GARCH(1,1) is a very good model for modeling the volatility swaps for stock market. Also, we remark the influence of the French financial crisis (2009) on CAC 40 French Index.
Moreover, we presented a probabilistic approach, based on changing of time method, to study variance and volatility swaps for stock market with underlying asset and variance that follow the Heston model. We obtained the formulas for variance and volatility swaps but with another structure and another application to those in the papers by Brockhaus and Long [10] and Swishchuk [12]. As an application of our analytical solutions, we provided a numerical example using CAC 40 French Index to price swap on the volatility (Figure 1).
GARCH(1,1) CAC 40 French Index forecasting.
Also, we compared the forecasting performance of several GARCH models using different distributions for CAC 40 French Index. We found that the GARCH(1,1) skewed Student t model is the most promising for characterizing the dynamic behaviour of these returns as it reflects their underlying process in terms of serial correlation, asymmetric volatility clustering, and leptokurtic innovation. The results also show that GARCH(1,1) model improves the forecasting performance. This result later further implies that the GARCH(1,1) model might be more useful than the other three models (ARCH(2), ARCH(4), and GARCH(2,1)) when implementing risk management strategies for CAC 40 French Index (Figure 2).
CAC 40 French Index conditional variance.
CAC 40 French Index E(V) and Var(V).
Appendix
We give a reminder for each parameter.
(1) Std. Dev. (standard deviation) is a measure of dispersion or spread in the series. The standard deviation is given by
(A.1)s=1N-1∑i=1Nyi-y-2,
where N is the number of observations in the current sample and y- is the mean of the series.
(2) Skewness is a measure of asymmetry of the distribution of the series around its mean. Skewness is computed as
(A.2)S=1N∑i=1Nyi-y-σ^3,
where σ^ is an estimator for the standard deviation that is based on the biased estimator for the variance (σ^=s(N-1)/N).
(3) Kurtosis measures the peakedness or flatness of the distribution of the series. Kurtosis is computed as
(A.3)K=1N∑i=1Nyi-y-σ^4,
where σ^ is again based on the biased estimator for the variance.
(4) Jarque-Bera is a test statistic for testing whether the series is normally distributed. The statistic is computed as
(A.4)Jarque-Bera=N6S2+K-324,
where S is the skewness and K is the kurtosis.
(5) Mean absolute error (MAE) is as follows: MAE=(1/N)∑i=1Nyi-y^i.
(6) Mean absolute percentage error (MAPE) is as follows: MAPE=∑i=1N(yi-y^i)/yi.
(7) Root mean squared error (RMSE) is as follows: RMSE=(1/N)∑i=1Nyi-y^i2.
(8) Adjusted R-squared (adjust R2) is considered.
(9) Sum error of regression (SEE) is considered.
(10) Schwartz criterion (BIC) is measured by nlnSEE+klnn.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was given ATRST (ex: ANDRU) financing within the framework of the PNR Project (Number 8/u23/1050) and Averroès Program.
BlackF.ScholesM.The pricing of option and corporate liabilitiesHullJ.BroadieM.JainA.The effect of jumps and discrete sampling on volatility and variance swapsBailiH.Stochastic analysis and particle filtering of the volatilityCaoL.GuoZ.-F.Optimal variance swaps investmentsHestonS.A closed-form solution for options with stochastic volatility with applications to bond and currency optionsCoxJ. C.IngersollJ.RossS.A theory of the term structure of interest ratesBruceT.DemeterfiK.DermanE.KamalM.ZouJ.A guide to volatility and variance swapsBrockhausO.LongD.Volatility swaps made simpleJavaheriA.SwishchukA.Variance and volatility swaps in energy marketsHeR.WangY.Price pseudo-variance, pseudo covariance, pseudo-volatility, and pseudo-correlation swaps-in analytical close formsProceedings of the 6th PIMS Industrial Problems Solving Workshop (PIMS IPSW '02)2002Vancouver, CanadaUniversity of British Columbia2737SwishchukA.KalemanovaA.The stochastic stability of interest rates with jump changesOksendalB.EngleR. F.MezrichJ.Grappling with GARCH