Is the variance swap rate affine in the spot variance? Evidence from S&P500 data
aa r X i v : . [ q -f i n . M F ] A p r Is the variance swap rate affine in the spot variance?Evidence from S&P500 data
M.E. MANCINO † , S. SCOTTI ‡ and G. TOSCANO ∗ § † Department of Economics and Management, University of Florence, Via delle Pandette 32,50127 Florence, Italy ‡ LPSM, Universit de Paris, rue Thomas Mann 5, 75205 Paris cedex 13, France § Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy ( submitted April 5, 2020 )We empirically investigate the functional link between the variance swap rate and the spot variance.Using S&P500 data over the period 2006-2018, we find overwhelming empirical evidence supporting theaffine link analytically found by Kallsen et al. (2011) in the context of exponentially affine stochasticvolatility models. Tests on yearly subsamples suggest that exponentially mean-reverting variance modelsprovide a good fit during periods of extreme volatility, while polynomial models, introduced in Cuchiero(2011), are suited for years characterized by more frequent price jumps. Keywords : Variance swap; Spot variance; Exponentially affine models; Exponentially mean-revertingvariance models; Polynomial models.
JEL Classification : C2, C12, C51, G12, G13.
1. Introduction
The class of the exponentially affine processes, introduced in the seminal paper by Duffie et al. (2000) and characterized by Filipovic (2001), has received large consensus in the quantitativefinance literature, based on its main advantages in terms of analytical tractability and empiricalflexibility. The classic example of an exponentially affine process, and the only one with continuouspaths, is the CIR diffusion, see Cox et al. (1985). The related stochastic volatility model, studied byHeston (1993), is considered as a reference model by scholars and practitioners. Kallsen et al. (2011)have studied the valuation of options written on the quadratic variation of the asset price within theexponentially affine stochastic volatility framework. In particular, they have proved, analytically,the existence of an affine link between the expected cumulated variance, i.e., the variance swap rate,and the spot variance. Note that the class of stochastic volatility models considered in Kallsen et al. (2011) allows for jumps and leverage effects, but fails to include some popular stochastic volatilitymodels, e.g., the models by Beckers (1980), Grasselli (2016), Hagan et al. (2002), Platen (1997). Thevariance swap is possibly the most plain vanilla contingent claim written on the realized variance.Indeed, it can be seen, to some extent, as the forward of the integrated variance of log-returns(see, for instance, Bernis et al. (2019), Carr and Sun (2007), Carr and Wu (2008), Filipovic et al. (2016), Jiao et al. (2019), Kallsen et al. (2011)). Volatility derivatives appear nowadays with avast demand, especially after the global financial crisis of 2008, which induced large fluctuationsin the volatility and other indicators of market stress. The large demand for volatility derivatives ∗ Corresponding author. Email: [email protected] et al. (2011), two natural questions arise: (i) could we analyticallyidentify a wider class of models which admits an affine link between the variance swap rates and thespot variance? (ii) is it possible to test if empirical data satisfy a given link (e.g., affine, quadratic)between the variance swap rate and the unobservable spot variance? This paper contributes toanswering both questions. With regard to question (i), we prove that a larger class of models ex-hibits a linear link between the variance swap rate and the spot variance, and we show that aquadratic (respectively, affine) link appears between the variance swap rate and the multidimen-sional stochastic process characterizing the model, in the presence (respectively, absence) of jumps,within the class of polynomial models (see Cuchiero (2011) and Cuchiero et al. (2012)). With regardto question (ii), we set up a simple testing procedure, based on Ordinary-Least-Squares (OLS), inwhich the unobservable spot variance is replaced with efficient Fourier estimates thereof. Then, weapply it to S&P500 empirical data over the period 2006-2018.In particular, our first result is showing that a model exhibits the affine link between the varianceswap rate and the spot variance if the stochastic differential equation satisfied by the latter is thesum of an affine drift and a zero-mean stochastic process. We term this class exponentially mean-reverting variance models. This class is fairly large. In fact, it contains not only exponentially affineprocesses with jumps (see, e.g., Bates (1996), Barndorff-Nielsen and Shephard (2001, 2002a), Duffie et al. (2003), Jiao et al. (2017, 2019)), but also, under suitable conditions (see Cuchiero et al. (2012),Ackerer et al. (2018)), polynomial processes. Moreover, it also contains some models based on thefractional Brownian motion, like the rough Heston model (see, for instance Bayer et al. (2016),El Euch et al. (2019), El Euch and Rosenbaum (2019), Gatheral et al. (2018)). However, it is worthnoting that many popular models, e.g., the CEV model (Beckers (1980)), the SABR model (Hagan et al. (2002)), the 3 / / et al. (2013) for the 3/2 model). Further,we consider the class of stochastic volatility models based on polynomial processes, introducedin Cuchiero (2011) and Cuchiero et al. (2012). The exponentially affine models by Kallsen et al. (2011), which exhibit an affine link between the variance swap rate and the spot variance, areincluded in the polynomial class, as a special case, see Example 3.1 in Cuchiero et al. (2012). In thepolynomial framework, we prove the existence, in the presence of jumps, of a quadratic correctionin the link between the theoretical variance swap rate and the spot variance.In the financial market, traded variance swaps are actually written on the realized variance, thatis the finite sum of squared log-returns sampled over a discrete grid. Instead, the correspondingtheoretical pricing formulae use the continuous time approximation given by the quadratic variationof the log-price, in virtue of higher mathematical tractability. Thus, we also study the case wherethe theoretical variance swap rate, i.e., the expected future quadratic variation, is replaced byits empirical counterpart, namely the expected future realized variance. In this regard, we showthat polynomial processes exhibit a quadratic link between the expected future realized varianceand the (multidimensional) stochastic process characterizing the model. The pricing error relatedto this approximation has been investigated by Broadie and Jain (2008), who conclude that theapproximation works quite well, based on simulated data obtained from four different models (theBlack-Scholes model, the Heston stochastic volatility model, the Merton jump-diffusion model andthe Bates stochastic volatility and jump model).Based on these results, our second contribution is testing, using OLS, if an affine or a quadraticlink is satisfied by actual financial data, namely S&P 500 daily data. This may allow us to determinewhich class of models, affine or polynomial, provides a better fit for empirical data. Clearly, such atest requires the availability of a the daily time-series of price, variance swap rate and spot varianceobservations. However, while S&P 500 prices and variance swap rates, in the form of the (squared)VIX index (see Carr and Wu (2008), CBOE (2019)), are quoted on the market, the spot varianceis a latent process. Thus, the main hurdle impeding the testing of the affine/quadratic link isthe latent nature of volatility process. To overcome this hurdle, the spot variance is estimated by2eans of the Fourier method proposed in Malliavin and Mancino (2009) and extended to jump-diffusions in Cuchiero and Teichmann (2015). The topic of the efficient estimation of the spotvariance is relatively recent, unlike that of the efficient estimation of the integrated variance. Anearly attempt goes through the following idea: first, the integrated variance is estimated over a localtime window, relying on the realized variance formula; then, a localized spot variance estimate isobtained through a numerical derivative. However, this differentiation-based estimation proceduregives rise to strong numerical instabilities, see Mykland and Zhang (2006), Foster and Nelson(1996), Comte and Renault (1998). Further, this procedure requires the use of high-frequencyprices to be efficient. However, it is well known that empirical high-frequency (tick-by-tick) pricesare contaminated by microstructure noise, preventing the realized variance from converging to theintegrated variance of the price process. The Fourier spot variance estimator allows to mitigatenumerical instabilities, by relying on the integration of the price observations rather than on adifferentiation procedure. Further, it results to be robust to different kinds of microstructure noisecontaminations, provided that the highest frequency to be included in the Fourier series is chosenappropriately (see Mancino and Sanfelici (2008)).The findings of our empirical tests are summarized as follows. First, we obtain overwhelmingempirical evidence supporting the use of exponentially affine models in financial applications. Ex-ponentially affine models imply the existence of an affine link between the variance swap rate andthe spot variance, with strictly positive coefficients. The test of the affine link over the period 2006-2018 is coherent with this prediction, in that it yields statistically significant positive coefficientsand an R larger than 0 .
95. Instead, the test of the quadratic link between the variance swap rateand the spot variance over the period 2006-2018 yields a non-significant quadratic coefficient. Thisresult may shed light on the negligibility of the discrete sampling effect affecting the variance-swappricing formula. In fact, the absence of a significant quadratic coefficient confirms that the dailysampling used to compute the VIX index is enough to match the continuous-time approximationof the latter, i.e., the expected future quadratic variation. This empirical finding, which is achievedin a non-parametric fashion, i.e., without assuming any parametric form for the price evolution,supports the numerical findings by Broadie and Jain (2008).The affine and quadratic tests are performed also on yearly subsamples, to investigate the sen-sitivity of the results to different economic scenarios. Test results on yearly subsamples are morenuanced. In particular, the intercept in the affine test is not significant in 2008 and 2011, two yearscharacterized by extreme volatility spikes. This suggests that S&P500 data in 2008 and 2011 areconsistent only with the broader assumption of an exponentially mean-reverting variance frame-work, which does not put any restrictions on the sign of the intercept (see, e.g., the rough-Bergomimodel in Bayer et al. (2016), Jacquier et al. (2018)). Moreover, the quadratic test yields significantquadratic corrections in the years characterized by a relatively high number of price jumps. Thisfindings support the use of polynomial models with jumps in periods when jumps are frequent. Ingeneral, our empirical analysis reveals that jumps play a non-negligible role, as we detect price-jumps in approximately 10% of days of our 13-year sample. This result is in accordance with a largeliterature, see, e.g., Bakshi et al. (1997), Barndorff-Nielsen and Shephard (2002b), Bates (1996),Eraker (2004). Perhaps surprisingly, high-volatility periods and periods with a larger number ofjumps do not necessarily coincide. For example, in 2007, 2010 and 2013, in spite of a relatively lowVIX index, the number of days with jumps is relatively large.The paper is organized as follows. In Section 2 we describe the analytical framework of the paper,illustrating the exponentially affine model, the exponentially mean-reverting variance model andthe polynomial model. In Section 3 we detail the spot variance estimation method and performempirical tests to investigate if S&P500 daily data over the period 2006-2018 are consistent withthe affine or the quadratic link. Section 4 concludes. The proofs are in the Appendix.3 . Variance swap rate and model set-up
In this section we introduce the problem of the variance swap valuation and investigate the typesof models under which an affine link between the variance swap rate and the spot variance exists.According to the fundamental theorem of asset pricing by Delbaen and Schachermayer (1994),the time evolution of the logarithm of the asset price follows a square-integrable semimartingalemodel, that is X t = A Xt + M Xt , (1)where M is a square-integrable martingale and A is a finite-variation process on a filtered space(Ω , F , P ). Being interested in the pricing problem, asset price dynamics are specified under a riskneutral measure along the paper. Moreover, in the paper we denote by [ X ] t the quadratic variationof the process X up to time t . The semimartingale hypothesis assures that the [ X ] t is finite forall times t and coincides with the quadratic variation of the martingale M X , if the finite-variationprocess A has continuous paths.A classical result proves that the quadratic variation can be obtained as the limit of the realizedvariance. More precisely, letting π m := { t < t < . . . < t m = τ } be a partition of a genericinterval [0 , τ ] and | π m | := sup k =1 ,...,m ( t k − t k − ) be the step of the partition, the realized variance isdefined as RV m [0 ,τ ] = m X k =1 ( X t k − X t k − ) . (2)Then, the following convergence holds in probability[ X ] τ = lim | π m |→ RV m [0 ,τ ] . (3)A financial product, called variance swap , was introduced to hedge volatility risk. Definition t, t + τ ], the other paying a fixedamount, generally called the rate or strike. Variance swap buyer pays the fix amount and receivesthe realized variance RV m [ t,t + τ ] , with the convention that t k − t k − is one day, t = t and t m = t + τ .The strike V S τt reads V S τt = τ − E h RV m [ t,t + τ ] | F t i . (4)Based on higher mathematical tractability, the finite-sample realized variance (2) is replaced, inthe theoretical variance swap pricing formula, by its continuous-time approximation, the quadraticvariation [ X ] τ . As a consequence, the strike of the variance swap (4), under the continuous-timelimit, reads V S τt = τ − E [[ X ] t + τ − [ X ] t | F t ] . (5)The simulation study by Broadie and Jain (2008), based on four different models (the Black-Scholes model, the Heston stochastic volatility model, the Merton jump-diffusion model and theBates stochastic volatility and jump model), suggests that the continuous-time approximation forthe variance swap pricing formula works quite well.4 model-free pricing method, used to compute the VIX index (see CBOE (2019)), has been alsoproposed by Carr and Lee (2008). This method exploits the fact that the variance swap can beperfectly statically replicated through vanilla Puts and Calls, as pointed by the next result (seeCarr and Wu (2006) for the proof). Proposition (Variance Swap rate) Assuming that the underlying asset price X t has continu-ous paths, then the variance swap can be statically replicated by a weighted position on vanilla Putsand Calls, that reads V S τ = 2 τ e rτ (cid:18)Z F K P ( K )d K + Z ∞ F K C ( K )d K (cid:19) , (6) where F , τ and r denote, respectively, the forward of the underlying, the maturity and the risk-freeinterest rate, which is assumed to be constant. The prices of the Call and Put options with strike K and maturity τ are denoted, respectively, by C ( K ) and P ( K ) .Moreover, in the presence of jumps in the price process X t , the formula (6) is subject to thecorrection ǫ J , which depends only on the jump measure and reads ǫ J = − τ E (cid:20)Z τ Z (cid:18) e x − − x − x (cid:19) ν (d t , d x ) (cid:21) , (7) where ν (d t , d x ) denotes the compensated Levy measure of the jump process. As far as equity models are concerned, in this work we focus on a two-dimensional framework,where the first process is the logarithm of asset price as in (1) and the second, called varianceprocess, is the variance of the martingale part in (1) or a function of the latter. More precisely, inthe rest of the paper we consider various model specifications within the following general class forthe price evolution ( d X t = µ t d t + √ V t d B t + d J t d V t = α t d t + d Z t (8)where B is a Brownian motion, J is a compensated jump process characterized by the Levy measure ν and Z is an integrable stochastic process with zero mean. Note that the process Z is not requiredto be a semimartingale. This allows us to include also the fractional Brownian motion case, see,for instance, Section 7 of Al`os et al. (2007). The class of models (8) and its extension to multi-dimensional volatility processes are extremely large and include almost all stochastic volatilitymodels commonly used in finance. Exponentially affine model
With pricing and forecasting applications in mind, researchers focus on some subclasses of (8),which are able to capture equity stylized facts while still remaining parsimonious. During the lasttwo decades, a large literature, started by Duffie et al. (2000), has focused on exponentially affinemodels, which are defined as follows, see Definition 2.1 in Duffie et al. (2003).
Definition
X, V ) is calledaffine if the characteristic function of the process has an exponential affine dependence on the initialcondition. 5hat is, for every 0 ≤ u < t , there exists functions ( ψ x ( a,b ) ( t, u ) , ψ v ( a,b ) ( t, u ) , φ ( a,b ) ( t, u )) such that E h e aX t + bV t |F u i = exp { X u ψ x ( a,b ) ( t − u ) + V u ψ v ( a,b ) ( t − u ) + φ ( a,b ) ( t, u ) } . Under natural financial hypotheses, we have ψ x ( a,b ) ( t − u ) = 1. Moreover, Duffie et al. (2000)show that ψ v ( a,b ) satisfies a generalised first order non-linear differential equation of Riccati typeand φ ( a,b ) is a primitive of a functional of ψ v ( a,b ) .The most popular exponentially affine model, and the only one with continuous paths, is themodel by Heston (1993), which reads ( d X t = (cid:0) r − V t (cid:1) d t + √ V t d B t d V t = κ ( θ − V t )d t + σ √ V t d W t (9)where B and W are correlated Brownian motions. Moreover, it is easy to verify that, under theHeston model, the variance swap strike (5) has the following expression: V S τt = θ + ( V t − θ ) 1 − e − κτ κτ . (10)The class of exponentially affine models is wide, including also jumps processes, and has beenextensively investigated, see for instance Bates (1996), Benth (2011), Bernis et al. (2019), Filipovicand Mayerhofer (2009), Hubalek et al. (2017), Horst and Xu (2019), Jiao et al. (2019), Keller-Ressel (2011). In this regard, we highlight the results by Keller-Ressel et al. (2011) and Cuchieroand Teichmann (2013), who show that exponentially affine processes are regular. Note that, in theexponentially affine framework, the variance process V needs to be driven by a martingale Z (see(8)) with finite quadratic variation. Moreover, the drift process α and the Levy measure ν of thejump process J in (8) need to be affine with respect to the variance process V .Kallsen et al. (2011) show that the affine link between the spot variance and the expectedintegrated variance holds for any exponentially affine stochastic volatility model. Their result ispresented in the following proposition. Proposition (Laplace transform of the quadratic variation) Let ( X, V ) be an exponentialaffine stochastic volatility model. Then, the triplet ( X t , V t , [ X ] t ) is a Markov exponentially affineprocess. Moreover, the process [ X ] t has the following characteristic function E h e u [ X ] t + τ |F t i = exp (cid:8) u [ X ] t + V t Ψ Vu ( τ ) + Φ Vu ( τ ) (cid:9) , where Ψ Vu satisfies a couple of first order non-linear differential equations of Riccati type and Φ Vu isa primitive of a functional of Ψ Vu . More precisely, using the parameter notation for the exponentiallyaffine model introduced in Lemma 4.2 of Kallsen et al. (2011), they satisfy ∂ Ψ Vu ∂ t ( t ) = 12 γ (cid:0) Ψ Vu ( t ) (cid:1) + β Ψ Vu ( t ) + γ u + Z R + × R (cid:16) e x Ψ Vu ( t )+ ux − − Ψ Vu ( t ) h ( x ) (cid:17) κ (d x ) , Ψ Vu (0) = 0 , Φ Vu ( t ) = Z t (cid:20) β Ψ Vu ( s ) + γ u + Z R + × R (cid:16) e x Ψ Vu ( s )+ ux − − Ψ Vu ( s ) h ( x ) (cid:17) κ (d x ) (cid:21) d s. oreover, assuming that R τ E [ V t ]d t < ∞ , then V S τt = V t Ψ( τ ) + Φ( τ ) , (11) where Ψ( τ ) (respectively, Φ( τ ) ) is the partial derivative of Ψ Vu ( τ ) (respectively, Φ Vu ( τ ) ) with respectto u , at u = 0 . Note that this affine link is not satisfied by all stochastic volatility models with an explicit Laplacetransform. For instance, it is not satisfied by the 3 / / et al. (2013).In the following proposition we complete the result by Kallsen et al. (2011), showing that thefunctions Ψ( τ ) and Φ( τ ) are strictly positive. This additional result is interesting in view of ourempirical study of section 3, where we test if S&P data are coherent with the exponential affineframework, based on the significance of the estimates of the coefficients in (11). The proof ofthis additional result crucially relies on the characterization of exponentially affine models byFilipovic (2001), who shows, under mild conditions (mainly the non-negativity of V ), that thevolatility process V has to be a continuous-state branching processes with immigration in theexponentially affine framework. Note that the explicit stochastic differential equation satisfied bya generic continuous-state branching process with immigration is provided by Dawson and Li(2006) and Li and Ma (2008), who also detail the conditions to have a stationary distribution forthe variance process. The existence of a stationary distribution is usually considered as a naturalproperty of the variance process. Proposition
Let ( X, V ) follow an exponentially affine stochastic volatility model and assumethat the variance process V admits a non-degenerate stationary distribution. Then Ψ( t ) > and Φ( t ) > for all t > . Based on Proposition 2.3, the exponential affine framework could be rejected by empirical dataif any of the coefficient estimates is not strictly positive. In that event, it could be worth inves-tigating the adequacy of the more general exponentially mean-reverting variance and polynomialframeworks, respectively detailed in Subsection 2.2 and 2.3.
Exponentially mean-reverting variance model
In this subsection we introduce a more general subclass of the stochastic volatility models includedin (8), which we name exponentially mean-reverting variance models . Moreover, we show that,under this paradigm, an affine relationship between the variance swap rate and the spot varianceholds.
Definition
X, V ) is anexponentially mean-reverting variance model if (
X, V ) satisfies ( d X t = µ t d t + √ V t d B t + d J t d V t = κ ( θ − V t )d t + d Z t (12)where κ >
0, the jump process J is square-integrable and its Levy measure is affine in the volatilityprocess.A relevant example inside this class is the rough-Heston model (see, e.g., El Euch et al. (2019),El Euch and Rosenbaum (2019), Gatheral et al. (2018)). In fact, we do not put any constrainton the process Z , except the fact that it has zero mean. Therefore, this class includes not onlyexponentially affine processes but also processes driven by a fractional Brownian motion.7he next result shows that the expected quadratic variation of an exponentially mean-revertingvariance model is affine in the spot variance. Proposition
Let ( X, V ) be an exponentially mean-reverting variance model, as defined in(12). Then the expectation of the quadratic variation [ X ] of the log-price is an affine function ofthe spot variance V , i.e., there exist deterministic functions Ψ and Φ such that E [[ X ] τ ] = V Ψ( τ ) + Φ( τ ) . (13)Differently from the case of the exponentially affine framework, in this case the coefficientsΨ( τ ) and Φ( τ ) are not strictly positive. A first example of a model satisfying Definition 3 butnot Definition 2 is the Hull-White stochastic volatility model, see Hull and White (1987), underwhich the volatility is log-normal. In particular, the Hull-White model fits Definition 3 for θ = 0and dZ t = ξV t dW t , where W is a Brownian motion and ξ >
0. A straightforward computationshows that the variance swap rate is linear with respect to the spot volatility. Moreover, note thatthis model only admits a degenerate steady-state distribution, namely a Dirac delta on zero. Amore interesting example is the rough-Bergomi volatility model, see Bayer et al. (2016). This casecould be seen as an extension of the Hull-White model, where the Brownian motion is replacedby a fractional Brownian motion with Hurst parameter smaller than 1 /
2. The main mathematicaldifficulty inherent to rough models is that the volatility is not a Markov process. This problemcould be overcome by taking an infinite-dimensional point of view (see, e.g., Jaber et al. (2019) andreferences therein). The initial value of the variance process V is then replaced by a function g that takes into account the initial conditions. Thus, under the infinite-dimensional viewpoint, thelink between the variance swap rate and the initial variance is the functional linear link betweenthe variance swap rate and the function g . The particular case of the rough-Bergomi volatilitymodel is studied in Jacquier et al. (2018), where a linear link is detailed. Finally, note that it isnot possible to work with the function g empirically, unless this function is assigned a parametricform.In Section 3.2.1, we show that empirical subsamples related to the years 2008 and 2011, where,respectively, the outbreak of the global financial crisis and the Euro-zone debt crisis took place,exhibit a non-significant intercept parameter. This result can tilt the balance in favor of log-normalmodels like Hull-White and rough-Bergomi during crisis periods. Polynomial model
In this section we consider the class of stochastic volatility models based on polynomial processes,introduced in Cuchiero (2011) and Cuchiero et al. (2012). As pointed out in Cuchiero et al. (2012),exponentially affine processes are polynomial processes. Moreover, under suitable restrictions, thepolynomial class could be considered as a sub-class of (8), see Cuchiero (2018).Let P k denote the vector space of polynomials up to degree k . In the bi-dimensional case, wehave the following definition of a polynomial process. Definition
X, V ) is said m -polynomial, if, for all k ∈ { , . . . , m } , f ∈ P k , ( x, v ) in the state space and t >
0, it holdsthat ( x, v ) → E ( x,v ) [ f ( X t , V t )] ∈ P k , (14)where, for any 0 ≤ u < t , we adopt the standard notation E ( x,v ) [ f ( X t , V t )] = E [ f ( X t , V t ) | X u = x, V u = v ]. Also, the semigroup is assumed to be strongly continuous. Moreover, if ( X, V ) is m -polynomial for all m ∈ R, then (
X, V ) is said polynomial.A relevant non-affine example in this class is the Jacobi stochastic volatility model, see Ackerer8 t al. (2018). Other examples and applications of polynomial process could be found in Ackerer andFilipovic (2020), Callegaro et al. (2017), Cuchiero (2011, 2018), Cuchiero et al. (2018), Filipovic et al. (2016).The following proposition allows us to investigate the existence of a quadratic correction in thelink between theoretical variance swap rates and spot variance in the polynomial framework.
Proposition
Let ( X, V ) be a -polynomial process describing a stochastic volatility model,then the expected quadratic variation of X belongs to P in ( x, v ) . Moreover, if ( X, V ) has contin-uous paths, then the expected quadratic variation of X is affine in ( x, v ) . This result suggests that the presence of a statistically significant quadratic correction could beexplained by the presence of jumps in the underlying. In fact, the empirical analysis in Section3.2 seems to support this finding. In particular, in Section 3.2.2, we point out that a quadraticcoefficient is statistically significant in the years with a higher frequency of price jumps.We conclude the section by discussing the effects of discrete sampling on the functional form ofthe variance swap rate. Indeed, the actual price of traded variance swaps relies on the computationof the realized variance in place of its asymptotic approximation, given by the quadratic variation(see Definition 1). In this regard, the following result holds.
Proposition If ( X, V ) is a -polynomial process describing a stochastic volatility model, thenthe expected realized variance of X belongs to P . Based on Proposition 2.6, the variance swap rate for a polynomial stochastic volatility model isat most quadratic in (
X, V ), that is there exist coefficients p i,j ( · ) , i, j = 0 , ,
2, such that
V S τt = p , ( τ ) + p , ( τ ) X t + p , ( τ ) V t + p , ( τ ) X t + p , ( τ ) X t V t + p , ( τ ) V t . (15)This result is interesting in that it may help collect empirical evidence supporting the resultby Broadie and Jain (2008). The authors show, for some well known models, that the expectedquadratic variation provides an efficient approximation of the actual VIX index, whose computa-tion is based on a daily sampling scheme (see Carr and Wu (2006), CBOE (2019)). In other words,non-significant estimates of the quadratic coefficients in (15) may represent empirical evidence thatthe continuous-time approximation works well enough.Finally, note that, based on Proposition 2.5, the expression (15) is also implied by the assumptionthat the data-generating process is a polynomial stochastic volatility model with jumps. Section3.2 analyzes if a second order correction fits S&P500 data better than the affine link implied bythe affine framework (11).
3. Empirical study
In this Section we perform an empirical study to investigate if S&P500 daily data over the period2006 − p i,j ( · ) , i, j = 0 , ,
2, in (15). To perform this study, we use the daily series of varianceswap rate and log-price observations, plus a daily series of estimates of the unobservable spotvariance. Accordingly, this Section begins with the description of the dataset used for the empiricalexercise, while Section 3.1 describes the method employed to reconstruct the spot variance pathon a daily grid from the series of high-frequency prices in the dataset.The dataset, ranging over the period 2006-2018, is composed of the series of S&P500 trade prices,recorded at the 1-minute sampling frequency (see panel a) in Figure 1), and the series of VIX indexvalues, recorded at the beginning of each trading day (see panel b) in Figure 1). The period 2006-2018 encompasses a number of volatility peaks, corresponding to historical financial events such as9
006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 time
S&P quo t e a) time V I X quo t e b) Figure 1. Panel a): 1-minute S&P500 trade prices over the period 2006-2018; panel b): Opening quotes of the VIXindex over the period 2006-2018. the global financial crisis of 2008, the flash-crash of May 2010, the Eurozone debt crisis of 2011,the Brexit events of 2016 and the US-China ‘trade war’ of 2018. For a detailed description of theevents that have affected the US stock market since the 1990s, see Horst and Xu (2019).
Spot variance estimation
The latent spot variance at the beginning of each trading day is reconstructed from 1-minuteprices using the Fourier methodology, according to Malliavin and Mancino (2009) and Cuchieroand Teichmann (2015). More precisely, we first detect the presence of jumps in our sample dataand, secondly, we use the the estimator by Cuchiero and Teichmann (2015), with sparse sampling,in days where jumps are detected. In the other days, the estimator by Malliavin and Mancino(2009) is employed with the entire high frequency data set. For the reader convenience, we brieflyrecall the definition and main properties of these spot volatility estimators.The Fourier estimator by Malliavin and Mancino (2009) is defined as follows. Let [0 , π ] be thetime horizon and consider the time grid { t < . . . < t m = 2 π } . For any integer k , | k | ≤ N ,define the discrete Fourier transform c k (d X m ) := 12 π m X j =1 e − i kt j − ( X t j − X t j − ) . (16)Then, for any integer k , | k | ≤ N , consider the following convolution formula c k ( V mN ) := 2 π N + 1 X | h |≤ N c h (d X m ) c k − h (d X m ) . (17)Formula (17) contains the identity relating the Fourier transform of the log-price process X t to theFourier transform of the variance V t . By (17) we gather all the Fourier coefficients of the variancefunction by means of the Fourier transform of the log-returns. Then, the reconstruction of thevariance function V t from its Fourier coefficients is obtained through the Fourier-Fej´er summation,10.e., the Fourier estimator of the spot variance is defined as follows: for any t ∈ (0 , π ), b V mN,M ( t ) = X | k | 4, fromthe 5-minute frequency and downwards.Secondly, after downsampling the log-price series at the 5-minute frequency, we apply the jumpdetection test by Corsi et al. (2010) for the null hypothesis that the price is a continuous semi-martingale. Test results at the 99 . 9% confidence level show that jumps are detected in 10 . 35% ofthe daily subsamples over the period 2006-2018. Figure 3 shows the values of the test statisticcomputed from daily subsamples (panel a)) and the ensuing percentage of days with jumps peryear (panel b)). The percentage of jumps detected per year is compatible with the empirical resultsin Corsi et al. (2010). Based on the results of the two tests, in order to obtain spot variance esti-11 min. 5 min. 10 min. 15 min. price sampling frequency t o t a l r ea li z ed v a r i an c e a) price sampling frequency R e j e c t i on r a t e o f t he nu ll hp . b) Figure 2. Panel a): rejection rate of the null hypothesis of the noise-detection test by A¨ıt-Sahalia and Xiu (2019),performed on daily subsamples of S&P500 prices over the period 2006-2018, for different sampling frequencies; panelb): volatility signature plot, i.e., total S&P500 realized variance for the period 2006-2018 as a function of the pricesampling frequency. time -4-2024681012 a) daily statistic99.9% critical values b) time pe r c en t age o f da ys w i t h j u m p s pe r y ea r Figure 3. Panel a): values of the statistic of the jump-detection test by Corsi et al. (2010) computed over the period2006-2018; panel b): ensuing percentage of days with jumps per year. mates, we proceed as follows. On the consecutive days in which the hypothesis of absence of jumpsis not rejected (amounting approximately to 90% of the sample), the Fourier estimator (18) isapplied with all prices recorded at 1-minute frequency. Instead, for the sparse days in which jumpsare detected (amounting approximately 10% of the sample) we use spot variance estimates obtainthrough the Fourier estimator (20), applied to sparsely sampled 5-minute prices. In the case of theestimator (18), the cutting frequencies have been selected as N = m / / M = m / / (16 π ),according to Mancino and Recchioni (2015), who find these cutting frequencies to be optimal inthe presence of different types of noise and noise intensities. For the estimator (20), instead, thefrequency M is selected as M = ( m/ / , in accordance to Cuchiero and Teichmann (2015). The12 006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 Time S po t v o l a t ili t y e s t i m a t ed t r a j e c t o r y Figure 4. Estimated S&P500 spot variance trajectory on a daily grid over the period 2006-2018. resulting spot variance estimates at the beginning of each trading day are shown in Figure 4. Empirical test results We now focus on testing the empirical link between the rate of the variance swap with time tomaturity equal to one month, i.e., the VIX index squared, and the couple spot variance - log return.If estimates of the unobservable spot variance are available, equation (15) can be rewritten, in thecase of the S&P500 index, as follows. Let L = 3267 denote the number of days in our sample andlet τ = 1 / 12. Then, for t i = i − , i = 1 , , ..., L , we writeVIX t i = p , ( τ ) + p , ( τ ) X t i + p , ( τ ) b V mt i + p , ( τ ) X t i + p , ( τ ) X t i b V mt i + p , ( τ ) ( b V mt i ) , (21)where:- VIX t i denotes the opening quote of the VIX index on i -th day;- X t i denotes the opening log-price of the S&P500 index on i -th day;- b V mt i denotes the estimated spot variance at the beginning of the i -th day, obtained from asample of size m using the estimator (18) or (20).Some comments are needed. First, based on the results of Propositions 2.2 and 2.5, the presence ofjumps does not spoil the affine/polynomial structure, thus the regression coefficients p i,j ( τ ) , i, j =0 , , 2, include the potential contribution of jumps. In the following, we drop the argument τ from p i,j ( τ ) as we always consider monthly coefficients. Secondly, the consistency of the spot varianceestimators (18) and (20) allows us to neglect the finite-sample error related to estimation of V mt i .We aim at testing the significance of the estimates of the coefficients in equation (21) within threeprogressively broader frameworks: the affine framework, introduced in Definition 2 and extendedin Definition 3 (hereafter, affine framework ); the polynomial framework of Definition 4, where thevariance swap rate is first assumed to be a quadratic function of V only (hereafter, quadraticframework ) and then is assumed to be a polynomial function of the couple ( X, V ) (hereafter, polynomial framework ). 13 .2.1. Affine framework. In this paragraph we consider the exponentially affine and expo-nentially mean-reverting variance models, which both imply the existence of an affine relationshipbetween the variance swap rate and the spot variance. Note that in the affine framework theequation (21) reduces to VIX t i = p , + p , b V mt i . (22)Recall that the main discriminant factor between the exponentially affine model and its extensionto the exponentially mean-reverting variance class is that the former implies the coefficients p , and p , in equation (22) are strictly positive. Thus, we are not only interested in testing if theaffine dependence between the variance swap rate and the spot variance is satisfied by empiricaldata, but also in verifying if both parameter estimates are significantly different from zero, as thiswould allow us to accept or reject the exponentially affine framework.The coefficients in (22) are estimated using OLS. Note that VIX t i is scaled as(VIX t i / / (30 / R larger than 0 . 95 and significant coefficients estimates, as shownin Table 1. In particular, the fact that both coefficient estimates are significant and positive sug-gests that an exponentially affine framework is a suitable fit for the S&P500 data over the period2006-2018. Note that the regression standard errors have been computed using the Newey-Westmethodology (see Newey and West (1987)), to account for the presence of heteroskedasticity andautocorrelations in the residuals. Table 1. Affine framework (22): estimation from S&P500 data over the period 2006-2018(p-values less than 10 − are reported as zero). framework coeff. estimate std. err. t stat. p value R affine p , p , et al. (2016), Jacquier et al. (2018) model (see the discussion in Section 2.2). Finally, notethat the empirical analysis suggests that the drift of the variance process is affine in the varianceitself. As a consequence, models with stronger mean reversion, e.g., the 3 / able 2. Affine framework (22): estimation from S&P500 data over yearly subsamples(p-values less than 10 − are reported as zero). Affine frameworkyear coeff. estimate std. err. t stat. p value R p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , Quadratic framework. In this paragraph we extend the analysis to take into accounta possible quadratic link between variance swap rates and spot variance. Through this section,the term quadratic is used to refer to polynomial models that are not exponentially affine, whereno risk of confusion exists. According to Propositions 2.5 and 2.6, when polynomial models withjumps are considered and/or discrete-sampling effects can not be neglected, the variance swap rateis a bivariate polynomial in X and V . Based on these results, the rest of our empirical study isaimed to investigating the possible existence of a quadratic link between the variance swap rateand the spot variance and/or the log-price. In particular, the analysis is split into two parts. In thisparagraph, devoted to what we have called the quadratic framework , we check if a quadratic formwith respect to the spot variance fits the data better than an affine form. In the next paragraphwe will deal with the general form as in (21). In the quadratic framework, equation (21) readsVIX t i = p , + p , b V mt i + p , (cid:16) b V mt i (cid:17) . (23)However, we measure the sample linear correlation between spot variance estimates and their squareand find it to be approximately equal to 0 . 84, thus signalling the presence of collinearity, whichrepresents a violation of the OLS hypotheses. The problem of collinearity is typical of polynomialregressions and can be solved by transforming the regressors in equation (23) through the useof orthogonal polynomials, i.e., by performing an orthogonal polynomial regression (see Narula151979)). This way we are able to isolate the actual additional contribution of the square of varianceestimates to the dynamics of the VIX index squared, if any. Accordingly, using the Gram-Schmidtalgorithm, we transform the vector of spot variance estimates and the vector of their squared valuesinto orthogonal vectors and estimate the following regression model:VIX t = q , + q , Z (1) t + q , Z (2) t , (24)where Z (1) and Z (2) denote, respectively, the orthogonal transformations of the vector of the spotvariance estimates and the vector of the squared spot variance estimates. Clearly, the coefficientsin equation (24) are not comparable to those those in (23). However, this is not relevant for ourstudy, as we aim only at assessing the significance of the additional contribution of the squaredvariance estimates to the dynamics of the VIX squared, not at making inference of the coefficientsin equation (23).The results of the OLS estimation of the coefficients in (24) over the period 2006-2018 are reportedin Table 3. These results point out that the additional contribution of the squared spot varianceis not statistically significant. In order to interpret these results, we first need to recall that, fromone side the class of polynomial models includes exponentially affine one as a subclass, from theother, in the presence of jumps, while the polynomial model gives rise to a quadratic correction (seeProposition 2.5), the exponentially affine model still ensures an affine link between the varianceswap rate and the spot variance. Therefore, as in Paragraph 3.2.1, we have already ascertained thatthe exponentially affine framework is a suitable fit for S&P500 data over the period 2006-2018, wededuce that the results in Table 3 confirm the adequacy of the exponentially affine frameworkin capturing the empirical features of S&P500 data. In other words, Table 3 points towards thefact that the extension to the quadratic framework is not necessary to capture the empirical linkbetween the variance swap rate and the spot variance.Furthermore, Table 3 offers additional interesting insight. Recall that the computation of the VIXindex is based on a daily sampling scheme. Thus, it is natural to ask whether the VIX index squaredis adequately approximated by its asymptotic counterpart, namely the future expected quadraticvariation. If this were not the case, one would observe a significant quadratic correction due to thediscrete sampling, see Proposition 2.6. As this is not the case, we infer that the continuous limitrepresents a very good approximation, thus providing empirical support to the numerical result byBroadie and Jain (2008). Table 3. Quadratic framework (24): estimation from S&P500 data over the period 2006-2018(p-values less than 10 − are reported as zero). framework coeff. estimate std. err. t stat. p value R quadratic q , q , q , able 4. Quadratic framework (24): estimation from S&P500 data over yearly subsamples(p-values less than 10 − are reported as zero). Quadratic frameworkyear coeff. estimate std. err. t stat. p value R q , q , q , -0.5633 0.5811 -0.9694 0.33232007 q , q , q , -0.3650 0.0543 -6.7178 02008 q , q , q , q , q , q , -0.3213 0.0665 -4.8310 02010 q , q , q , -0.1861 0.0560 -3.3243 0.00092011 q , q , q , -0.3284 0.1004 -3.2707 0.00112012 q , q , q , q , q , q , -0.3151 0.0872 -3.6135 02014 q , q , q , q , q , q , -0.0919 0.0195 -4.7018 02016 q , q , q , -0.2189 0.0865 -2.5289 0.01142017 q , q , q , -1.8918 0.4822 -3.9233 0.00012018 q , q , q , Polynomial framework. In the last paragraph, the polynomial framework (21) isanalyzed. Before fitting this model, we examine the sample correlation matrix of the regressors,which is shown in Table 5. This table provides empirical evidence of the existence of an almost Table 5. Sample correlation matrix of the regressors of the polynomial form (21) over the period 2006-2018. b V t b V t X t X t X t b V t b V t b V t X t -0.3702 -0.1814 1 X t -0.3615 -0.1765 0.9997 1 X t b V t fullyaffine form in both the log-return and the spot variance.In this regard, recall that it is a well-known stylized fact that asset price series are non-stationary. Indeed, the Augmented Dickey Fuller test, performed at the 90% confidence level,confirms that our log-price series has a unit root. To cope with the non-stationarity of the log-priceseries, we estimate the coefficients in equation (23) after replacing log-prices with their detrendedvalues, i.e., their values minus their sample mean. The estimation results are summarized in Table 6. Table 6. Polynomial (fully affine) framework: estimation from S&P500 data over the period 2006-2018(p-values less than 10 − are reported as zero; p , indicates the coefficient of the detrended price). framework coeff. estimate std. err. t stat. p value R polynomial p , p , p , -0.0015 0.0009 -1.7653 0.0775Based on Table 6, the contribution of the log-price is not statistically significant at the 95%confidence level, but only at 90% level. Overall, this result confirms that the affine framework issufficient to adequately fit for our sample.Finally, it is worth evaluating if the additional contribution of the price in explaining the dynamicsof the VIX index squared is statistically significant on yearly subsamples, i.e., under differenteconomic scenarios. The results of the year-by-year estimation are summarized in Table 7 and arein line with the whole-sample results. 18 able 7. Polynomial (fully affine) framework: estimation from S&P500 data over yearly subsamples(p-values less than 10 − are reported as zero; p , indicates the coefficient of the detrended price). Polynomial (fully affine) frameworkyear coeff. estimate std. err. t stat. p value R p , p , p , -0.0043 0.0019 -2.3374 0.01942007 p , p , p , p , p , p , -0.0322 0.0084 -3.8116 0.00012009 p , p , p , -0.0164 0.0041 -3.9743 0.00012010 p , p , p , -0.0150 0.0043 -3.5235 0.00042011 p , p , p , -0.0402 0.0111 -3.6063 0.00032012 p , p , p , p , p , p , p , p , p , -0.0012 0.0020 -0.5993 0.54902015 p , p , p , -0.0305 0.0057 -5.3606 02016 p , p , p , -0.0088 0.0027 -3.2395 0.00122017 p , p , p , -0.0021 0.0006 -3.5198 0.00042018 p , p , p , -0.0075 0.0036 -2.1090 0.0349 Remark b V mt i . In fact, we have replicated the empirical study using the localizedversion of the two-scale realized variance and bipower variation (see Chapter 8 in A¨ıt-Sahalia andJacod (2014)), in place, respectively, of the Fourier estimator by Malliavin and Mancino (2009) andthe Fourier estimator by Cuchiero and Teichmann (2015). The corresponding results are perfectlyin line with those in Tables 1 − 7, both in terms of the significance of the coefficient estimates and19he R values. 4. Conclusions This paper provides empirical evidence that S&P500 data over the period 2006-2018 are coherentwith the exponentially affine framework, introduced by Kallsen et al. (2011), who analytically provethe existence of an affine relationship between the expected future variance, i.e., the variance swaprate, and the spot variance. This paper collects empirical evidence that this affine relationshipfits the data overwhelmingly well, with statistically significant coefficients and an R larger than0 . 95. Further, this paper provides empirical evidence that the daily sampling used to compute theactual variance swap rates is frequent enough to erase the quadratic correction due to discretesampling. The quadratic correction is expected within the polynomial framework, which includesthe exponentially affine framework as a special case. This empirical non-parametric result confirmsthe result by Broadie and Jain (2008), which was obtained on data simulated from four parametricmodels belonging to the exponentially affine class.The paper focuses also on yearly subsamples, in order to evaluate the sensitivity to events offinancial distress. Empirical results on yearly subsamples are more nuanced. In particular, it emergesthat the exponentially affine framework could be rejected in 2008 and 2011. These two years includethe outbreaks of two global financial crisis sparked, respectively, by the American housing marketand the sovereign debt in the Euro area. 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(of Proposition 2.3)Here we adopt, where no ambiguity arises, the parameter notation introduced in Lemma 4.2of Kallsen et al. (2011) for the exponentially affine model. Using the equation for Ψ Vu and Φ Vu inProposition 2.2 and differentiating the two equations with respect to u , we have ∂ Ψ Vu ∂ u ∂ t ( t ) = γ Ψ Vu ( t ) ∂ Ψ Vu ( t ) ∂ u + β ∂ Ψ Vu ( t ) ∂ u + γ + Z R + × R (cid:18) e x Ψ Vu ( t )+ ux (cid:18) x ∂ Ψ Vu ( t ) ∂ u + x (cid:19) − ∂ Ψ Vu ( t ) ∂ u h ( x ) (cid:19) κ (d x ) , ∂ Φ Vu ( t ) ∂ u = Z t (cid:20) β ∂ Ψ Vu ( s ) ∂ u + γ + Z R + × R (cid:18) e x Ψ Vu ( s )+ ux (cid:18) x ∂ Ψ Vu ( s ) ∂ u + x (cid:19) − ∂ Ψ Vu ( s ) ∂ u h ( x ) (cid:19) κ (d x ) (cid:21) d s . Taking u = 0 and recalling that Ψ V ( t ) = 0 for all t , we obtain the relations satisfied by (Ψ , Φ),that read ∂ Ψ ∂ t ( t ) = β Ψ( t ) + γ + Z R + × R (cid:0) x Ψ( t ) + x − Ψ( t ) h ( x ) (cid:1) κ (d x ) , Φ( t ) = Z t (cid:20) β Ψ( s ) + γ + Z R + × R (cid:0) x Ψ( s ) + x − Ψ( s ) h ( x ) (cid:1) κ (d x ) (cid:21) d s . Note that in our case γ = 0, see (8). Then, splitting the integrals and recalling that h ( x ) is atruncating function, there exists non-negative parameters ( e β , e β ), that is the parameters associatedwith the truncating function h ( x ) = x , such that: ∂ Ψ ∂ t ( t ) = e β Ψ( t ) + γ + Z R + × R x κ (d x ) , Φ( t ) = Z t (cid:20) e β Ψ( s ) + Z R + × R x κ (d x ) (cid:21) d s . Note that Ψ solves a non-homogeneous linear differential equation with non-negative external term γ + R R + × R x κ (d x ). This term is zero if and only if γ = 0 and κ (d x ) = 0, that is in the case ofthe exponential Levy model. We deduce that Ψ( s ) > s , except for the exponentialLevy model, for which volatility is constant and thus the stationary distribution is degenerate.We now turn to Φ and assume Ψ( s ) > 0. Using the integral representation of Φ, we easily obtainthat Φ > e β = 0 or κ (d x ) = 0, see also Filipovic (2001). This is equivalent toassuming that the process V is a continuous-state branching process with immigration. Instead,in the case e β = 0 and κ (d x ) = 0, the process V is a continuous-state branching process withoutimmigration. Without immigration continuous-state branching processes do not have a stationarydistribution, see Theorem 3.20 and Corollary 3.21 in Li (2011). Proof. (of Proposition 2.4)The quadratic variation of X is rewritten as [ X ] t = [ X c ] t + P s ≤ t (∆ X s ) , where X c denotesthe continuous part of the log-price X . According to (12), we have [ X c ] t = R t V s d s . It is easyto show that the variance process V is integrable using Gronwall’s lemma, since the drift of thevariance process is affine and Z is integrable by hypothesis. We now focus on the jump contribution, P s ≤ t (∆ X s ) = P s ≤ t (∆ J s ) . Recalling that the process J is square-integrable, we obtain that theoptional version of the quadratic variation [ X ] t is finite almost surely. Introducing the predictable23ersion h X i t of the quadratic variation and recalling that the optional and the predictable versionof the quadratic variation differ by a martingale, we obtain that E [[ X ] τ ] = E [ h X i τ ] = E (cid:20)Z τ V s d s + Z Z τ ζ ν ( dζ, d s ) (cid:21) . Considering first the jump term, exploiting that ν is affine in the variance process V , we deducethat the jump part is affine in the expectation of the integral of the variance process. Focusing nowon the term E (cid:2)R τ V s d s (cid:3) , we consider the integral version of the SDE (12), i.e. V t − V = κ Z t ( θ − V s )d s + Z t . Taking the expectation, we obtain that E (cid:2)R τ V s d s (cid:3) = κ − ( τ κθ + V − E [ V τ ]) and E [ V t ] satisfiesa linear ODE. This result, combined with the previous result, proves that the expectation of thequadratic variation is an affine function of the initial spot variance. Proof. (of Proposition 2.5)According to the characterization in Proposition 2.12 of Cuchiero et al. (2012), if ( X, V ) is a2-polynomial process then[ X, X ] t = Z t V s d s + Z t Z ζ ν (d ζ, d s ) =: Z t a ( X s , V s )d s, where a ∈ P . Then, the result for E ( x,v ) [[ X, X ] t ] follows from Theorem 3.2 in Cuchiero (2011) andthe application of the stochastic Fubini theorem.In particular, if we consider the quadratic variation of X c , together with the evolution (8), wehave that [ X c ] t = R t V s d s . Taking the expectation and applying the stochastic Fubini theorem, weobtain E [[ X c ] t ] = R t E [ V s ] d s . Now, using the hypothesis that ( X, V ) is 2-polynomial, we see thatthe function f ( x, v ) = v ∈ P , and integrating we obtain the result. Proof. (of Proposition 2.6)Using the definition of realized variance RV m [0 ,τ ] given in (2), we take the expected value of RV m .Due to the finiteness of the sum, the expected value of RV m [0 ,τ ] gives E ( x,v ) h RV m [0 ,τ ] i = m X k =1 E ( x,v ) h(cid:0) X t k − X t k − (cid:1) i . Noting that the function inside the expectation belongs to P , we have that E h RV m [0 ,τ ] i belongs to P2