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On Volatility Swaps for Stock Market Forecast: Application Example CAC 40 French Index Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2014, Article ID 854578, 6 pages http://dx.doi.org/10.1155/2014/854578 Research Article On Volatility Swaps for Stock Market Forecast: Application Example CAC 40 French Index Halim Zeghdoudi,1,2 Abdellah Lallouche,3 and Mohamed Riad Remita1 1 LaPSLaboratory,Badji-MokhtarUniversity,BP12,23000Annaba,Algeria 2 Department of Computing Mathematics and Physics, Waterford Institute of Technology, Waterford, Ireland 3 Universite´ 20 Aout, 1955 Skikda, Algeria Correspondence should be addressed to Halim Zeghdoudi; [email protected] Received 3 August 2014; Revised 21 September 2014; Accepted 29 September 2014; Published 9 November 2014 Academic Editor: Chin-Shang Li Copyright © 2014 Halim Zeghdoudi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper focuses on the pricing of variance and volatility swaps under Heston model (1993). To this end, we apply this modelto 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 a price index. The underlying is usually a foreign exchange rate (very liquid market) but could be as well a single name Black and Scholes’ model [1]isoneofthemostsignificant equity or index. However, the variance swap is reliable in models discovered in finance in XXe century for valuing the index market because it can be replicated with a linear liquid European style vanilla option. Black-Scholes model combination of options and a dynamic position in futures. assumes that the volatility is constant but this assumption is Also, volatility swaps are not used only in finance and not always true. This model is not good for derivatives prices businesses but in energy markets and industry too. founded in finance and businesses market (see [2]). The variance swap contract contains two legs: fixed leg “The volatility of asset prices is an indispensable input in (variance strike) and floating leg (realized variance). There both pricing options and in risk management. Through the are several works which studied the variance swap portfolio introduction of volatility derivatives, volatility is now, in effect, theoryandoptimalportfolioofvarianceswapsbasedona atradablemarketinstrument”Broadie and Jain [3]. variance Gamma correlated (VGC) model (see Cao and Guo Volatility is one of the principal parameters employed to [5]). describe and measure the fluctuations of asset prices. It plays The goal of this paper is the valuation and hedging of a crucial role in the modern financial analysis concerning volatility swaps within the frame of a GARCH(1,1) stochastic risk management, option valuation, and asset allocation. volatility model under Heston model [6]. The Heston asset There are different types of volatilities: implied volatility, local process has a variance that follows a Cox et al. [7]process. volatility, and stochastic volatility (see Baili [4]). Also, we make an application by using CAC 40 French Index. To this end, the new financial products are variance Thestructureofthepaperisasfollows.Section2 andvolatilityswaps,whichplayadecisiveroleinvolatility considers representing the volatility swap and the variance hedging and speculation. Investment banks, currencies, stock swap. Section 3 describes the volatility swaps for Heston 2 indexes, finance, and businesses markets are useful for vari- model, gives explicit expression of , and discusses the ance and volatility swaps. relationship between GARCH and volatility swaps. Finally, Volatility swaps allow investors to trade and to control we make an application example for stock market forecast: the volatility of an asset directly. Moreover, they would trade CAC 40 French Index using GARCH/ARCH models. 2 Journal of Probability and Statistics 2. Volatility Swaps 3. Volatility Swaps for Heston Model In this section we give some definitions and notations of swap, 3.1. Stochastic Volatility Model. Let (Ω, , , P) be probability stock’s volatility, stock’s volatility swap, and variance swap. space with filtration , ∈[0;]. We consider the risk- neutral Heston stochastic volatility model for the price and Definition 1. Swapswereintroducedinthe1980sandthere variance follows the following model: is an agreement between two parties to exchange cash flows at one or several future dates as defined by Bruce [8]. In = + 1 () , (1) this contract one party agrees to pay a fixed amount to a 2 2 2 counterpart which in turn honors the agreement by paying a =(− )+2 () , floating amount, which depends on the level of some specific underlying. By entering a swap, a market participant can where is deterministic interest rate, 0 >0and >0are therefore exchange the exposure from the varying underlying short and long volatility, >0is a reversion speed, >0 by paying a fixed amount at certain future time points. is a volatility of volatility parameter, and 1() and 2() are independent standard Brownian motions. Definition 2. Astock’svolatilityisthesimplestmeasureofits We can rewrite the system (1) as follows: risk less or uncertainty. Formally, the volatility () is the annualized standard deviation of the stock’s returns during = + 1 () the period of interest, where the subscript denotes the 2 =(2 −2)+ () 2 observed or “realized” volatility for the stock . 1 ( ) √ Definition 3 (see [9]). A stock volatility swap is a forward + 1− () , contract on the annualized volatility. Its payoff at expiration () is equal to where is standard Brownian motion which is inde- pendent of 1() and the indicator economic .Let ( () − ), ( (), ()) = vol (1) cov 1 2 ,andwecantransformthesystem (2) to (1) if we replace 1() + √1−()by 2(). () := √(1/) ∫ 2, where 0 is a stochastic stock 2 volatility, vol istheannualizedvolatilitydeliveryprice,and 3.2. Explicit Expression and Properties of . In this section we is the notional amount of the swap in Euro annualized reformulated the results obtained in [12], which are needed volatility point. for study of variance and volatility swaps, and price of pseudovariance, pseudovolatility, and the problems proposed Definition 4 (see [9]). A variance swap is a forward contract by He and Wang [13] for financial markets with deterministic on annualized variance, the square of the realized volatility. volatility as a function of time. This approach was first applied Its payoff at expiration is equal to to the study of stochastic stability of Cox-Ingersoll-Ross 2 processinSwishchukandKalemanova[14]. The Heston asset ( () − ), (2) 2 var process has a variance that follows Cox et al. [7]process, (1) describedbythesecondequationin .Ifthevolatility where var isthedeliverypriceforvarianceand is follows Ornstein-Uhlenbeck process (see, e.g., Oksendal thenotionalamountoftheswapinEurosperannualized 2 volatility point squared. [15]), then Ito’s lemma shows that the variance follows the process described exactly by the second equation in (1). 2 We start to define the following process and function: Notation 1. We note that () =. V := (2 −2), Using the Brockhaus and Long [10]andJavaheri[11] approximation which is used in the second order Taylor √ −2 Φ() 2 2 2 2Φ() −1 formula for ,wehave Φ () := ∫ (0 − + ̃2 () + ) . 0 () (√) ≈ √ () − Var , (5) 83/2 () (3) −1 Definition 5. We define () := ̃2(Φ ),wherẽ2 is an F- ()/83/2() where Var is the convexity adjustment. Thus, to F̃ := F −1 measurable one-dimensional Wiener process, Φ , calculate volatility swaps we need both () and Var(). −1 and ∧:=min(, ),whereΦ is an inverse function of Φ. The realized discrete sampled variance is defined as () follows: The properties of are as follows: F̃ (()) =0 2 (a) -martingale and ; () := ∑ln ( ), := lim () , (−1) →∞ (2()) = 2((( −1)/)(2 −2)+((2 −1)/2)2) =1 −1 (b) 0 ; (4) 2 (∧) 2 2 2(∧) (c) (()()) = ((( − 1)/)(0 − ) + (( − 2 where is the maturity (years or days). 1)/2) ). Journal of Probability and Statistics 3 Lemma 6. and taking (13) and variance formula we find (a) Consider the following: Var () 2 − 2 2 2 2 (∧) = (0 − +()) + , (6) −1 = ∬ [−(+) ( (2 −2) 2 0 (b) 0 (14) 2(∧) 2 − 2 2 2 −1 2 ( )= (0 − )+ , (7) + )] ; 2 (c) after calculations we obtain (∧) −1 2(∧) −1 (22)=2−(+) ( (2 −2)+ 2) 2−2 0 2 () = [(22 −4 −2)(2 −2) Var 232 0 (15) 2 −(+) 2 2 − 2 2 2 2 2 2 + (0 − ) + (0 − ) +(2 −3 +4 −1) ] − 2 2 2 4 + (0 − ) + . which achieves the proof. (8) Corollary 8. If is large enough, we find Proof. See [12]. 2 () =, Var () =0. (16) Theorem 7. One has Proof. The idea is the limit passage →∞. (a) Remark 9. In this case a swap maturity does not influence 1−− () () () = (2 −2)+2, (9) and Var . 0 3.3. GARCH(1,1) and Volatility Swaps. GARCH model is (b) needed for both the variance swap and the volatility swap. 2−2 The model for the variance in a continuous version for Heston () = [(22 −4 −2)(2 −2) model is Var 232 0 2 =(2 −2)+ () . + (22 −32 +4 −1)2]. 2 (17) (10) The discrete version of the GARCH(1,1) process is described by Engle and Mezrich [16]: Proof. (a) We obtain mean value for 2 ]+1 =(1−−)+ +], (18) 1 () = ∫ (2) (11) where is the long-term variance, is the drift-adjusted 0 2 stock return at time , is the weight assigned to ,and using Lemma 6,andwefind is the weight assigned to ].Furtherweusethefollowing − relationship (19) to calculate the discrete GARCH(1,1) param- 1− 2 2 2 eters: () = (0 − )+ . (12) = 2 2 (b) Variance for equals Var() = ( )−(),and 1−− the second moment may be found as follows: using formula 2 2 2 2 = ,= (8) of Lemma 6: ( )=(1/)∬ ( ), 0 0 Δ Δ (19) (2) 1−− = Δ 2 (∧) −1 = ∬ [−(+) ( (2 −2) 2 2 0 2 (−1) 0 = , (13) Δ 2(∧) −1 2 + )] where Δ = 1/252, 252 trading days in any given year, and 2 Δ =1/63,63tradingdaysinanygiventhreemonths. Now, we will briefly discuss the validity of the assumption +2 () that the risk-neutral process for the instantaneous variance is 4 Journal of Probability and Statistics a continuous time limit of a GARCH(1,1) process.
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