A general framework for a joint calibration of VIX and VXX options
AA general frameworkfor a joint calibration of VIX and VXX options
Martino Grasselli , Andrea Mazzoran , and Andrea Pallavicini Department of Mathematics, University of Padova, Via Trieste 63 Padova 35121, Italy. Email:[email protected] Devinci Research Center, Léonard de Vinci Pôle Universitaire, 92 916 Paris La Défense, France.Email: [email protected] Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom. Email:[email protected] Financial Engineering, Banca IMI, largo Mattioli 3 Milano 20121, Italy.
December 16, 2020
Abstract
We analyze the VIX futures market with a focus on the exchange-traded noteswritten on such contracts, in particular, we investigate the VXX notes tracking theshort-end part of the futures term structure. Inspired by recent developments incommodity smile modeling, we present a multi-factor stochastic local-volatility modelable to jointly calibrate plain vanilla options both on VIX futures and VXX notes.We discuss numerical results on real market data by highlighting the impact of modelparameters on implied volatilities.
JEL classification codes:
C63, G13.
AMS classification codes:
Keywords:
Local volatility, Stochastic volatility, VIX, VIX futures, VXX.1 a r X i v : . [ q -f i n . M F ] D ec ontents Introduction
In recent years, it has become increasingly common to consider volatility as its own assetclass. The great financial crisis accelerated the demand for volatility instruments andmany different volatility products have been invented and are now available to trade,see Sussman and Morgan (2012). Created by the CBOE in 1993, the VIX index is byfar one of the most popular volatility instruments and it is considered as a landmark bymost market players. It is calculated and disseminated on a real-time basis by the CBOEand represents the market’s expectation of 30-day forward-looking volatility. Derivedfrom the price inputs of the S&P 500 index options, it provides a measure of market riskand investors’ sentiments. Investors, research analysts and portfolio managers look atVIX values as a way to measure market risk, fear and stress before they take investmentdecisions, this being a reason why it is also known as the “Fear Index". Indeed, when theVIX is rising, active investors will typically try to hedge their positions by taking shortposition on the market, while when the VIX is falling, active investors will closely watchthe market to increase their long positions. During its origin, the formula that determinedthe VIX was tailored to S&P 100 index option prices, more precisely it was calculated asa weighted measure of the 30-day implied volatility of eight S&P 100 at-the-money putand call options, when the derivatives market had limited activity and was in growingstages. Ten years later, its calculation has been slightly revised when people started usinga wider set of options based on the broader S&P 500 index, an expansion which allowsfor a more accurate view of investors’ expectations on future market volatility. But apartfrom its role as a risk indicator, nowadays it is possible to directly invest in volatility asan asset class by means of VIX derivatives. Specifically, VIX became tradable via futuresin 2004 and via options in 2006, hence providing market participants with the ability totrade liquid volatility products based on the VIX, see Whaley (2009). Since then, volumeson VIX futures and options drastically increased. For example, in 2015, the VIX becamethe second most traded underlying in the CBOE options market , right after the S&P 500itself.As derivatives on the VIX may be inaccessible to non-institutional players, mainlydue to the large notional sizes of the contracts as currently designed by the CBOE, someexchange-traded notes (ETN) on the VIX were introduced. In 2009, Barclays launchedthe first two ETN’s on the VIX: VXX and VXZ. The VXX ETN was the first exchange-traded product (ETP) on VIX futures, issued shortly after the inception of the VIX futuresindices. The VXX is a non-securitized debt obligation, similar to a zero-coupon bond, butwith a redemption value that depends on the level of the S&P 500 (SPX) VIX Short-Term Futures Total Return index (SPVXSTR). The SPVXSTR tracks the performanceof a position in the nearest and second-nearest maturing VIX futures contracts, whichis rebalanced daily to create a nearly constant 1-month maturity. In other words, VXXmimics the behaviour of the VIX, namely of the 30-day forward-looking volatility, using areplicable strategy based on futures on VIX. Since 2009 then, ETN’s on the VIX flourished:there currently exist more than thirty of them, with several billion dollars in market capsand daily volumes (see Alexander and Korovilas (2013) for a comprehensive empirical See the white paper of CBOE in 2014 titled
CBOE Volatility Index , available at See the white paper of CBOE in 2016 titled
CBOE holdings reports trading volume for 2015 ,available at http://ir.cboe.com/~/media/Files/C/CBOE-IR-V2/press-release/2016/cboe-holdings-volume-report-december-2015.pdf. See the VXX Prospectus of Barclays available at . Even though VXX is very heavily traded, it has lost99.84% of its value since inception and Figure 1 shows the remarkable underperformanceof VXX compared to VIX. Whereas many market players use VXX as a proxy for tradingVIX in a cheap manner, in the VXX Prospectus Barclays warns VXX holders that “YourETN is not Linked to the VIX index and the value of your ETN may be less than it wouldhave been had your ETN been linked to the VIX Index".In literature, attempts to model VIX can be divided in two strands: on the one hand,the consistent-pricing approach models the joint (risk-neutral) dynamics of the S&P 500and the VIX with, most often, the aim of pricing derivatives on the two indices in a consis-tent manner, on the other hand, the stand-alone approach directly models the dynamicsof the VIX.In the former case, we can include the work of Cont and Kokholm (2013) and referencestherein, while more applications can be found in the work of Gehricke and Zhang (2018),where they propose a consistent framework for S&P 500 (modelled with a Heston-likespecification) and the VIX derived from the square root of the variance swap contract.On top of that, they construct the VXX contract from the Prospectus and show thatthe roll yield of VIX futures drives the difference between the VXX and VIX returnson time series. More recently, in the paper of Gatheral et al. (2020), a joint calibrationof SPX and VIX smile has been successfully obtained using the quadratic rough Heston See the white paper of Bloomberg in 2012 titled
SEC said to review Credit Suisse VIX Note available at
One of the main advantages of using local-volatility models (LV) is their natural modellingof plain-vanilla market volatilities. Indeed, a LV model can be calibrated with extremeprecision to any given set of arbitrage-free European vanilla option prices. LV modelwere first introduced by Dupire et al. (1994) and Derman and Kani (1994). Althoughwell-accepted, LV models have certain limitations, for example, they generate flatteningimplied forward volatilities, see Rebonato (1999), that may lead to a mispricing of financialproducts like forward-starting options.On the other hand, stochastic volatility (SV) models, like the well known Hestonmodel, see Heston (1993), are considered to be more accurate choices for pricing forwardvolatility sensitive derivatives, see Gatheral (2011). Although the SV models have desiredfeatures for pricing, they often cannot be very well calibrated to a given set of arbitrage-free European vanilla option prices. In particular, the accuracy of the Heston model forpricing short-maturity options in the equity market is typically unsatisfactory.One possible improvement to the previous issues is considering stochastic local-volatilitymodels (SLV), that take advantages of both LV and SV models properties. SLV models,going back to Lipton (2002), are still an intricate task, both from a theoretical as well as apractical point of view. The main advantage of using SLV models is that one can achieveboth a good fit to market series data and in principle a perfect calibration to the impliedvolatility smiles. In such models the discounted price process ( S t ) t ≥ of an asset followsthe stochastic differential equation (SDE) dS t = S t ‘ ( t, S t ) v t dW t , (2.1)where ( v t ) t ≥ is some stochastic process taking real values, and ‘ ( t, K ) is a sufficientlyregular deterministic function, the so-called leverage function, depending on time and on6he current value of the underlying asset. W t , as usual, is a one-dimensional Brownianmotion, possibly correlated with the noise driving the process v . Obviously, SV andLV models are recovered by setting ‘ ( t, K ) ≡ v ≡
1, respectively. At this stage, thestochastic volatility process ( v t ) t ≥ can be very general and by a slight abuse of terminologywe call ( v t ) t ≥ stochastic volatility as in the SV model.The leverage function ‘ ( t, K ) is the crucial part in this model. It allows in principleto perfectly calibrate the market implied volatility surface. In order to achieve this goal, ‘ needs to satisfy the following relation: ‘ ( t, K ) = σ ( t, K ) E (cid:2) v t (cid:12)(cid:12) S t = K (cid:3) , (2.2)where σ Dup denotes Dupire local-volatility function, see Dupire (1996). The familiar readerwould recognize in Equation (2.2) an application of the Markovian projection procedure,a concept originating from the celebrated Gyöngy Lemma, see Gyöngy (1986), whose ideabasically lies in finding a diffusion which “mimics” the fixed-time marginal distributionsof an Itô process.For the derivation of Equation (2.2), we refer to Guyon and Henry-Labordere (2013).Notice that Equation (2.2) is an implicit equation for ‘ as it is needed for the computationof E (cid:2) v t | S t = K (cid:3) . It is a standard thing in SLV models and this in turn means that theSDE for the price process ( S t ) t ≥ is actually a McKean-Vlasov SDE, since the probabilitydistribution of S t enters the characteristics of the equation. Existence and uniquenessresults for this equation are not obvious in principle, as the coefficients do not typicallysatisfy standard conditions like for instance Lipschitz continuity with respect to the so-called Wasserstein metric. In fact, deriving the set of stochastic volatility parameters forwhich SLV models exist uniquely, for a given market implied volatility surface, is a verychallenging and still open problem. In the sequel, we will limit to investigate numericallythe validity of the assumption just stated in our specific framework. We start this section with a wider look to study the volatility smile of a generic ETPbased on futures strategies. Then, the results shown in this section will be applied to ourspecific VXX case in Section 4.
We analyze an ETP based on a strategy of N futures contracts F t ( T ). If we assume thatthe ETP strategy itself is not collateralized and it requires a proportional fee payment φ t ,we can write the strategy price process V t as given by dV t V t = ( r t − φ t ) dt + N X i =1 ω it dF t ( T i ) F t ( T i ) , (3.3)where r t is the risk-free rate. Moreover, we assume that the futures contracts and theETP are expressed in the same currency as the bank account based on r t . The processes ω it represent the investment percentage (or weights) in the futures contracts, and theyare defined by the ETP term-sheet. Notice that the weights may depend on the wholevector of futures prices. It is straightforward to extend the analysis to more general ETPstrategies based on total-return or dividend-paying indices. Moreover, we can introducesecurities in foreign currencies as in Moreni and Pallavicini (2017).7e can consider as a first example the family of S&P GSCI commodity indices.Exchange-Traded Commodity (ETC’s) tracking these indices are liquid in the market,and they represent an investment in a specific commodity or commodity class. The in-vestment is in the first nearby one-month future, but for the last five days of the contract,where a roll over procedure is implemented to sell the first nearby and to buy the secondone. A second example are the volatility-based ETN’s VXX and VXZ. They representrespectively an investment in short-term or in the long-term structure of VIX futures.The investment is in a portfolio of futures mimicking a rolling one-month contract. InSection 4 we will focus on the VXX case. We wish to jointly model the volatility smile of an ETP and of its underlying futurescontracts. Indeed, market quotes for ETP plain-vanilla options can be found on themarket. The main problem to solve is the fact that the price process of the ETP strategyis non-Markov, since it depends on the futures contract prices, namely the vector process[ V t , F t ( T ) , . . . , F t ( T N )] is Markov.We first start by modelling the marginal distribution of futures prices, then we proceedto describe the full model. The marginal distribution is modelled by following the pro-cedure developed in Nastasi et al. (2020) and originally implemented for the commoditymarket. We consider that the futures prices F st ( T i ) can be modelled in term of a commonprocess s t , which we can identify with the price of a rolling futures contract. Futuresprices are defined in term of s t as given by F st ( T i ) = F ( T i ) (cid:18) − (1 − s t ) e − R Tit a ( u ) du (cid:19) (3.4)where F ( T i ) are the futures prices as observed today in the market, and a is a non-negative function of time. The value of the futures at maturity times t , which are notquoted in the market, can be obtained by interpolation, but they are never used in actualcalculations. The driving risk process s t is modelled by a mean-reverting local-volatilitydynamics, namely ds t = a ( t ) (1 − s t ) dt + η ( t, s t ) s t dW st , s = 1 , (3.5)where η is a positive and bounded function of time and price, to be determined by thecalibration of options of futures, and W st is a standard Brownian motion under the risk-neutral measure. As a consequence, the futures prices follows a local-volatility dynamicsgiven by dF st ( T i ) = η F ( t, T i , F st ( T i )) dW st , (3.6)where we define η F ( t, T i , K ) : = (cid:18) K − F ( T i ) (cid:18) − e − R Tit a ( u ) du (cid:19)(cid:19) η ( t, k F ( t, T i , K )) , (3.7) k F ( t, T i , K ) : = 1 − (cid:18) − KF ( T i ) (cid:19) e R Tit a ( u ) du . (3.8)Thus, plain-vanilla options on futures can be calculated as plain-vanilla options on theprocess s t , which, in turn, can be easily computed by means of the Dupire equation. Moreprecisely, given the dynamics in Equation (3.5), we have that the normalized call price c ( t, k ) := E h ( s t − k ) + i (3.9)8atisfies the parabolic PDE ∂ t c ( t, k ) = (cid:18) − a ( t ) − a ( t )(1 − k ) ∂ k + 12 k η ( t, k ) ∂ k (cid:19) c ( t, k ) , (3.10)with boundary conditions c ( t,
0) = 1 , c ( t, ∞ ) = 0 , c (0 , k ) = (1 − k ) + . (3.11)A more detailed analysis and a proof of this (extended) Dupire equation can be foundin Nastasi et al. (2020, Proposition 4.1). Given the prices of options on futures from themarket, we can plug their corresponding normalized call prices in Equation (3.10), andthen solve it for the function η . Then, we can compute the futures local-volatility function η F ( t, T i , K ), thanks to the mapping in Equation (3.7), and eventually recover the pricesof options on futures.There is a vast literature on how to solve Equation (3.10) with the boundary con-ditions (3.11), here we use an implicit PDE discretization method, as usually done forsolving the Dupire equation, see for example Gatheral (2011). We allow a mean-reversion a ( t ) possibly different from zero when we are calibrating the model to options on futures.On the other hand, options on the ETP spot price can be directly calibrated by a standardlocal-volatility model, that is without a mean-reversion term. Hence, once a ( t ) is chosen,we can calibrate to the market η F ( t, T i , K ) and η V ( t, K ), as previously described.Thanks to the normalized spot dynamics (3.5), we can price all futures options byevaluating the PDE (3.10), and ETP options as well. In this way we implicitly obtain themarginal probability densities of futures prices and ETP prices.We are now looking for the joint probability densities of futures prices, in order to beable to jointly model the dynamics of both the ETP strategy and its underlying futures. Inorder to do that, we need to specify a non trivial correlation structure among the futures.Notice that, for example, a one factor model based on the normalised spot dynamics, as inEquation (3.5), would not be able to produce a non-trivial final correlation among futures,see Nastasi et al. (2020, Section 5.1). Hence, we have to enrich our model by adding newrisk factors.We start by introducing a generic stochastic volatility dynamics for futures prices underthe risk-neutral measure, as given by dF t ( T i ) = ν t ( T i ) · dW t , (3.12)where ν t ( T i ) are vector processes, possibly depending on the futures price itself, and W t is a vector of standard Brownian motions under the risk-neutral measure. The notation a · b refers to the inner product between vectors a and b .In general, we can numerically solve together Equations (3.3) and (3.12) to calculateoptions prices on the ETP or on the futures contracts. Our aim is at calibrating both thefutures and the ETP quoted smile. In the following, we always assume that interest ratesand fees are deterministic functions of time, so that we write: r t := r ( t ) and φ t := φ ( t ). Wecan simplify the problem by noticing that we need only to know the marginal densities ofthe futures and ETP prices, so that we can focus our analysis on the Markovian projectionof the dynamics (3.3) and (3.12). In this way, the model can match the marginal densitiesof futures prices and ETP prices predicted by the local-volatility model. Proposition 3.1.
Given the dynamics (3.12) and (3.3) , the Markov projections of theunderlying processes F t and V t are given by d ˜ F t ( T i ) = η F (cid:16) t, T i , ˜ F t ( T i ) (cid:17) d ˜ W it , η F ( t, T i , K ) := r E h k ν t ( T i ) k (cid:12)(cid:12) F t ( T i ) = K i , (3.13)9 nd d ˜ V t ˜ V t = ( r t − φ t ) dt + η V ( t, ˜ V t ) d ˜ W t , η V ( t, K ) := vuuut E N X i,j =1 ˆ ω it ν t ( T i ) · ν Tt ( T j ) ˆ ω jt (cid:12)(cid:12)(cid:12)(cid:12) V t = K , (3.14) respectively, where ˜ W t and ˜ W it , i = 1 , . . . , N are scalar Brownian motions and ˆ ω it := ω it F t ( T i ) . (3.15)The main ingredient for proving Proposition 3.1 is the Gyöngy Lemma, see Gyöngy(1986), that for the sake of clarity we recall here below. Lemma 3.1 (Gyöngy) . Let X ( t ) be given by X t = X + Z t α ( s ) ds + Z t β ( s ) dW s , t ≥ where W is a Brownian motion under some probability measure P , and α , β are adaptedbounded stochastic processes such that (3.16) admits a unique solution.If we define a ( t, x ) , b ( t, x ) by a ( t, x ) = E [ α ( t ) | X ( t ) = x ] b ( t, x ) = E h β ( t ) | X ( t ) = x i , (3.17) then there exists a filtered probability space ( e Ω , ˜ F , { ˜ F } t ≥ , e P ) where ˜ W is a e P − Brownianmotion, such that the SDE Y t = Y + Z t a ( s, Y ( s )) ds + Z t b ( s, Y ( s )) d ˜ W s , t ≥ admits a weak solution that has the same one-dimensional distributions as X . The process Y is called the Markovian projection of the process X .Proof. (Proposition 3.1) Equation (3.13) follows from equation (3.12) and from Lemma 3.1.Dynamics (3.3) can be rewritten as dV t V t = ( r t − φ t ) dt + N X i =1 ω it ν t ( T i ) · dW t F t ( T i ) , (3.19)and applying Lemma 3.1 we get η V ( t, K ) = E N X i,j =1 ω it ν t ( T i ) F t ( T i ) · ν Tt ( T j ) ω jt F t ( T j ) (cid:12)(cid:12)(cid:12)(cid:12) V t = K , (3.20)while the drift part remains unaffected since both r t and φ t are deterministic functions.The conclusion follows immediately.The functions η F ( t, T, K ) and η V ( t, K ) are respectively the futures and ETP local-volatility functions (we do not show the dependency of η V ( t, K ) on all the futures maturityto lighten the notation). As already discussed previously, we can calibrate the model onplain-vanilla market quotes using the approach of Nastasi et al. (2020), that is η F ( t, T, K )10ill be calibrated using options on VIX futures and η V ( t, K ) using options on the ETPstrategy.So far, we have preserved the marginal densities of futures prices and ETP prices,by applying the Gyöngy lemma directly on the futures and ETP dynamics, as shownin Proposition (3.1). Thus, we can calibrate our model to match the marginal densitiesof futures and ETP prices predicted by the local-volatility model, and in turn to matchthe plain vanilla option-on-futures prices and on ETP prices quoted on the market, byrequiring that the process ν satisfies Equations (3.13) and (3.14). We recall again that thelocal volatility for futures prices is defined in Equation (3.7), while the ETP local volatilitycan be recovered by using a standard local-volatility model with mean-reversion a = 0.Hence, we have to ensure the two constraints given by the Markov projection (3.13)and (3.14) by properly choosing the vector processes ν t ( T i ). More precisely, we need todefine a suitable dynamics for the process ν t ( T i ) satisfying Equation (3.13) and (3.14),which in turn will enable us to preserve the calibration to plain-vanilla options. In thefollowing, we will focus on a specific form, allowing to perform the calibration of the futureslocal volatility independently of the ETP dynamics. A simple way to do that is selectingthe vector process ν t ( T i ) as ν t ( T i ) . = ‘ F ( t, T i , F t ( T i )) √ v t R ( t, T i , V t ) , (3.21)where v t is a scalar process independent of the futures and ETP dynamics. Here, ‘ F ( t, T i , K )is the leverage function referred to the maturity T i , that can be computed as ‘ F ( t, T i , K ) = η F ( t, T i , K ) q E (cid:2) v t (cid:12)(cid:12) F t = K (cid:3) , (3.22)where, again, η F ( t, T i , K ) is the futures local-volatility function. Here, the vector R playsthe role of local correlation and satisfies the condition k R k = 1. It is the main ingredientto model the dependency between different futures and it enables us to correlate them ina natural way. In fact, Equation (3.21) satisfies by construction Equation (3.13), so thatthe model reprices plain-vanilla options on futures correctly. Notice that, as in standardstochastic local-volatility models, the above equation is not a definition since ν t ( T i ) appearsimplicitly also on the right-hand side within the conditional law of the expectation.We will investigate numerically in practical cases the validity of this assumption. InSection 4 we shall introduce a simple specification for R that will be enough for ourpurposes. Then, we shall analyze the constraint on the ETP dynamics by substitutingEquation (3.21) into Equation (3.14). In other words, we are looking for conditions on R under which the model is able to reprice the ETP plain vanilla options correctly. In thisway R becomes a (vector) function of the ETP price. Straightforward computations yield η V ( t, K ) = N X i,j =1 R ( t, T i , K ) · R ( t, T j , K ) E " v t ˆ ω it ˆ ω jt ‘ F ( t, T i , F t ( T i )) ‘ F ( t, T j , F t ( T j )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V t = K . (3.23)In conclusion, to complete the calibration to ETP plain vanilla prices, we have to solvethe above equation for an (admissible) unknown local function R ( t, T i , K ). We consider the specific case of the VIX futures and the VXX ETN strategy.11 .1 Definition of the ETN strategy
The VXX is defined as a strategy on the first and second nearby VIX futures, see Gehrickeand Zhang (2018), with the following weights: ω t . = α ( t ) F t ( T ) α ( t ) F t ( T ) + α ( t ) F t ( T ) , ω t . = α ( t ) F t ( T ) α ( t ) F t ( T ) + α ( t ) F t ( T ) , (4.24)where we define α ( t, T , T ) := ς ( T , − − ς ( t, ς ( T , − − T , α ( t, T , T ) := 1 − α ( t, T , T ) , (4.25)with T being the settlement date of the futures contract just expired (the one precedingthe first nearby), T the settlement date of the first nearby contract, and ς ( t, d ) a calendardate shift of d business days. In this formulae we are considering t < T , otherwise wehave to consider the following pair of future contracts. Remark 4.1. (Definition of the ETP weights.)
In Grasselli and Wagalath (2020) aslightly different definition is adopted, by using the year fraction between the second andthe first nearby futures as reference time interval, instead of the year fraction between thefirst nearby and the previous futures contract (without calendar adjustments), leading to α ( t, T , T ) := T − tT − T . (4.26) We now implement the modelling framework of the previous sections, by exploiting thestructure of the local correlation vector R . A simple way to do that consists in using thefollowing parametrization: R ( t, T , K ) := [1 , , R ( t, T , K ) := [ ρ ( t, K ) , q − ρ ( t, K ) ] , (4.27)which satisfies k R k = 1 by construction. That is, we express the dependence in terms of alocal correlation coefficient ρ ( t, K ) that has to be determined by solving Equation (3.23). Remark 4.2. (Extension to higher dimensions)
The choice in Equation (4.27) relatesto the case N = 2 , namely two futures, but it can be easily extended to an arbitrary number N . In that case, the local correlation vector R ( t, T i , K ) can be found by performing theCholesky decomposition to the corresponding local correlation matrix. Proposition 4.2.
Given the parametrization in Equation (4.27) , the local correlationcoefficient ρ ( t, K ) is given by ρ ( t, K ) = η V ( t, K ) − A ( t, K ) − A ( t, K )2 A ( t, K ) , (4.28) provided that t is not on a futures maturity date, where A ( t, K ) . = E (cid:20) (ˆ ω t ) ‘ F ( t, T , F t ( T )) v t (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) ,A ( t, K ) . = E (cid:20) (ˆ ω t ) ‘ F ( t, T , F t ( T )) v t (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) ,A ( t, K ) . = E (cid:20) ˆ ω t ˆ ω t ‘ F ( t, T , F t ( T )) ‘ F ( t, T , F t ( T )) v t (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) , nd where ˆ ω t = ω t F t ( T ) = α ( t ) α ( t ) F t ( T ) + α ( t ) F t ( T ) , ˆ ω t = ω t F t ( T ) = α ( t ) α ( t ) F t ( T ) + α ( t ) F t ( T ) . (4.29)We recall, once again, that the leverage function ‘ F ( t, T i , K ) , i = 1 , t corresponds to a futures maturity date, Equation (4.28)becomes meaningless, because the weights in (4.29) vanish. Proof.
Exploiting the local-volatility η V ( t, K ) in Equations (3.14) and (3.23), we get η V ( t, K ) = E (cid:20) (ˆ ω t ) k ν t ( T ) k + (ˆ ω t ) k ν t ( T ) k + 2ˆ ω t ˆ ω t ν t ( T ) · ν Tt ( T ) (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) = E (cid:20) (ˆ ω t ) ‘ F ( t, T , F t ( T )) v t + (ˆ ω t ) ‘ F ( t, T , F t ( T )) v t (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) ++ 2 E (cid:20) ˆ ω t ˆ ω t ‘ F ( t, T , F t ( T )) √ v t ‘ F ( t, T , F t ( T )) √ v t ρ ( t, K ) (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) = E (cid:20) (ˆ ω t ) ‘ F ( t, T , F t ( T )) v t (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) + E (cid:20) (ˆ ω t ) ‘ F ( t, T , F t ( T )) v t (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) ++ 2 ρ ( t, K ) E (cid:20) ˆ ω t ˆ ω t ‘ F ( t, T , F t ( T )) ‘ F ( t, T , F t ( T )) v t (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) . (4.30)Therefore, if we define A ( t, K ) . = E (cid:20) (ˆ ω t ) ‘ F ( t, T , F t ( T )) v t (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) ,A ( t, K ) . = E (cid:20) (ˆ ω t ) ‘ F ( t, T , F t ( T )) v t (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) ,A ( t, K ) . = E (cid:20) ˆ ω t ˆ ω t ‘ F ( t, T , F t ( T )) ‘ F ( t, T , F t ( T )) v t (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) , then, when t is not on a futures maturity date, we can obtain ρ ( t, K ) from Equation (4.30)and get the result.We can estimate the conditional expectations by means of the techniques developedin Guyon and Henry-Labordère (2012), namely approximating conditional expectationsusing smooth integration kernels. Thus, we can evaluate the expectation of a process X t given a second process Y t as E [ X t | Y t = K ] ≈ E [ X t δ ε ( Y t − K )] E [ δ ε ( Y t − K )] , (4.31)where δ ε is a suitably defined mollifier of the Dirac delta depending on a smoothingcoefficient ε . We can use such approximation within the Monte Carlo simulation for[ V t , F t ( T ) , . . . , F t ( T N )] to evaluate the diffusion coefficients of the processes. Alternativelywe could use the collocation method developed in Van der Stoep et al. (2014). We stressagain that the existence of a solution for Equation (4.28) with the above coefficients is anopen question, and, moreover, even if the solution exists, we have to check against marketquotes if the chosen parametrization is able to produce a local correlation ρ ( t, K ) ∈ [ − , emark 4.3. (Local volatility case) If we assume that v t = 1 , namely if we remove thestochastic volatility component, then the formulae simplify and we obtain that the optionson futures can be calibrated without any further calculations, since we get ν t ( T i ) . = ‘ F ( t, T i , F t ( T i )) R ( t, T i , V t ) , (4.32) while the local coefficients entering the calculation of the local correlation become A ( t, K ) . = E (cid:20) (ˆ ω t ) η F ( t, T , F t ( T )) (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) ,A ( t, K ) . = E (cid:20) (ˆ ω t ) η F ( t, T , F t ( T )) (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) ,A ( t, K ) . = E (cid:20) ˆ ω t ˆ ω t η F ( t, T , F t ( T )) η F ( t, T , F t ( T )) (cid:12)(cid:12)(cid:12)(cid:12) V t = K (cid:21) . Indeed, in this particular case, the leverage function ‘ F ( t, T i , K ) reduces to the local-volatility function η F ( t, T i , K ) . Remark 4.4. (VXZ strategy)
The above reasoning can be extended to the VXZ ETNstrategy where four future contracts in the long part of the futures term structure areconsidered, in a similar way as the VXX is considering the short term, although this timewe need to model a larger local correlation matrix.
We start from the stochastic local-volatility dynamics for the futures defined in Section 3,namely dF t ( T i ) = ‘ F ( t, T i , F t ( T i )) √ v t · dW t , (4.33)where the stochastic volatility component v t follows a simple one-dimensional CIR process,namely dv t = κ ( θ − v t ) dt + ξ √ v t dZ t , (4.34)with initial condition v (0) = v and where κ is the speed of the mean-reversion of thevariance, θ is the long-run average variance, ξ is the volatility-of-volatility parameter and ρ v is the correlation between W and Z . For the sake of simplicity, we assume the sameBrownian motion Z t for the volatility of each futures F t ( T i ), i = 1 , . . . , N , so we basicallyassume that all futures share the (common stochastic volatility) CIR factor v t .We describe the (static) correlation structure across futures dynamics by means ofa positive semi-definite matrix Σ W ∈ R N × N , where N refers to the number of futures.Therefore, the correlation structure for the tuple of all the Brownian factors ( W t , Z t ) takesthe following form:Σ W,Z = Σ W ρ TW Z ρ W Z ! ∈ R ( N +1) × ( N +1) , Σ W ∈ R N × N , ρ W Z ∈ R × N , (4.35)where ρ W Z = ( ρ v , . . . , ρ v ) is the correlation between the Brownian motions W driving thefutures and the noise Z driving the (common) stochastic volatility component v , so thatthe components of the vector ρ W Z are all equal to the constant ρ v .The parameter ρ v should be chosen in order to ensure the positive semi-definiteness ofthe correlation matrix Σ W,Z . For example, in the case N = 2 and W independent of W ,it turns out that the parameter ρ v needs to belong to the interval h − / √ , / √ i .14IX futures call optionsMaturities Number of options Strikes range (min − max) T : November 20, 2019 40 10 − T : December 18, 2019 40 10 − T : January 22, 2020 35 10 − T : February 19, 2020 35 10 − T : November 7, 2019.VXX call optionsMaturities Number of options Strikes range (min − max) T : November 20, 2019 65 8 − T : December 18, 2019 59 2 − T : January 22, 2020 53 10 − T : November 7, 2019. Remark 4.5. (Extension to a multi-factor stochastic-volatility framework)
Ourone-factor specification for the stochastic volatility turns out to be enough for our purposes,though, of course, it is not difficult to generalise our model to a multi-factor stochasticvolatility framework. What is more, this apparently more general choice does not lead tobetter results in the quality of the fit, up to our numerical experience.
In this section we perform a numerical exercise, based on real data quoted by the marketfor options on the VIX futures and the VXX ETN strategy, where we provide a set ofadmissible parameters, that is leading to an admissible local correlation vector in Equa-tion (4.28) satisfying ρ ( t, K ) ∈ [ − , We test our model against a set of three maturities for VXX call options with a corre-sponding set of four maturities for VIX futures call options. All data were downloadedby Bloomberg on November 7, 2019. Table 1 (resp. Table 2) describes the features of theVIX (resp. VXX) options data set.The different maturities of VIX and VXX options as on November 7, 2019 involvedin the numerical exercise are described in Figure 2. The values of VIX futures and VXXspot price, as observed in the market, are shown in Table 3, while we show in Figure 3(resp. Figure 4) the bid, mid and ask market quotes for call options written on VIX (resp.VXX futures), for the four (resp. three) maturities of the data set.An additional difficulty comes from the peculiar nature of the VIX-VXX options marketdata. In general, one can calibrate on market quotes or choose to implement a smooth-ing technique, like the widely-used SVI parametrization of the implied volatility smile,see Gatheral and Jacquier (2014) and the references therein. With this method, we are15utures and Spot prices Market Value Maturities F ( T ) 14.60 T : November 20, 2019 F ( T ) 16.15 T : December 18, 2019 F ( T ) 17.45 T : January 22, 2020 F ( T ) 18.15 T : February 19, 2020Spot V T : November 7, 2019Table 3: Market values of VIX futures F ( T ) , . . . , F ( T ) and of the VXX spot V as on T (November 7, 2019).Today T V XX T V IX T V IX T V XX T V XX T V IX T V IX Figure 2: Maturities of options on VIX futures and VXX as on November 7, 2019 in ourdata set. T V IXi , i = 1 , . . . , T V XXj , j = 1 , . . . ,
3) refer to VIX futures (resp. VXX)options maturities.guaranteed that the interpolation is arbitrage free. We applied the SVI methodology tosome market quotes of VIX and VXX Call and Put options, as on May 24, 2019. Figure 5shows that VIX market tends to be more irregular for shorter maturities, where a SVIapplication seems to be more needed, while it tends to be more regular as the maturitiesgo further. On the other hand, the VXX option market seems to be more balanced overall,as Figure 6 suggests, so the SVI smoothing is not really necessary.
In this subsection, we perform a sensitivity analysis that will highlight the impact of themodel parameters on the VXX smiles. The aim of this analysis is to find a range ofparameters that guarantees a good fit to the VXX market smile, while maintaining the fiton the VIX market smile. Then we test if these parameters generate a local correlation ρ ( t, K ) ∈ [ − , a , the correlation betweenfutures, ρ W i ,W j , i, j = 1 , . . . ,
4, the volatility-of-volatility ξ and the correlation ρ v betweenfutures and their (common) volatility factor v . Concerning the stochastic volatility com-ponent v , we will provide an admissible choice for the parameters κ , θ and v , such thatthe Feller condition holds and the local correlation ρ ( t, K ) always remains in the interval[ − , a plays a crucial role in order to recover the ATM model implied volatility fromthe VXX options market. The impact of the correlations between futures ρ W ,W is alsosignificant, while the other parameters do not really affect the shape of the VXX smile.We start by the impact of the mean-reversion a . In Figure 7, we present the impact ofthe mean-reversion parameter a on the VXX smile at the first maturity, namely November15, 2019, according to Figure 2. The results are representative also for the other maturities.In order to do that, we fixed the volatility-of-volatility ξ = 1 .
1, the correlation betweenfutures ρ W ,W = 0 .
85 and ρ v = 0 .
75. We then let the mean-reversion speed take thevalues a = 0 , , , , ,
9. Figure 7 shows the market and the model smiles with the differentchoices of the mean-reversion a . We get that the higher the mean-reversion, the lower thesmile. Thus, in order reproduce the market ATM implied volatility level for the VXXsmile, we need a high mean-reversion speed, say around a = 8, according to Figure 7. The(high) value for the mean-reversion speed could seem misconceiving. However, we shouldkeep in mind that we are jointly modelling two markets, the VIX and VXX option books,that display different peculiarities, therefore a somehow non standard level for the meanreversion seems a reasonable price to pay in order to reach our goal.The impact of the correlation between futures ρ W ,W seems to play a pretty importantrole too in the shape of the VXX smile. To see this, we fix ρ v = 0 . ξ = 1 . a = 8 andlet ρ W ,W varying according to − . , , . , . , .
85 and 1. In Figure 8 we see that theimpact of the correlation between futures tends to affect only the right tail of the smile,while leaving unaffected the left one.Next we consider the impact of the volatility-of-volatility ξ and the correlation betweenfutures and their volatilities, namely ρ v . As we are going to see, these two parameters,especially ρ v , seem to play a less pivotal role than the two ones already considered. Westart with the volatility-of-volatility and as shown before, we keep the mean-reversion a = 8 fixed, ρ W ,W = 0 . ρ v = 0 .
75 and let ξ varying within 0 . .
6, with a stepof 0 .
5. Figure 9 shows that the (positive) impact of the volatility-of-volatility is prettysignificant only on the tails of the VXX smile (especially on the left ones).Lastly, we check the impact of the correlation between futures and their volatilities,namely ρ v . We fix ρ W ,W = 0 . ξ = 1 . a = 8 and let ρ F,Z varying with 0 . , . .
75. Figure 10 suggests that the correlations between futures and their volatilities, ρ F i ,Z have a very limited impact on the shape of the VXX smile.In the next subsection, we are going to build up a consistent framework in order to fixan admissible choice for all parameters that allows us to reproduce the market smiles ofboth VIX and VXX. Thanks to the analysis performed in Subsection 5.2, we are now able to pick a cocktail ofparameters that enables us to get a good fit to both the VIX and VXX market smiles andthat leads to an admissible correlation parameter ρ ( t, K ) ∈ [ − , a = 8 (this value revealed to be theoptimal trade-off in order to recover the ATM implied-volatility level for the VXXoptions, while keeping a good fit for the VIX market).• The volatility-of-volatility parameter is fixed as ξ = 1 . a = 0 , , , , , ξ = 1 . ρ W ,W = 0 .
85 and ρ v = 0 . ρ W ,W = − . , , . , . , . , ρ v = 0 . ξ = 1 . a = 8.20igure 9: Impact of the volatility-of-volatility ξ = 0 . , . , . , . ρ W ,W = 0 . ρ v = 0 .
75 and a = 8.Figure 10: Impact of the correlation between futures and their volatilities ρ v =0 . , . , .
75 on the shape of the VXX smile. The other parameters are: ρ W ,W = 0 . ξ = 1 . a = 8. 21aturities Strike ASK BID MID Model Relative error K = 14 1.0911 0.8042 0.9482 1.0739 0.01576November 20, 2019 K = 14 . K = 15 1.1567 0.9297 1.0433 1.1392 0.01513 K = 15 0.8779 0.7519 0.8151 0.8642 0.01560December 18, 2019 K = 16 0.8995 0.7819 0.8406 0.9030 0.00389 K = 17 0.8406 0.8354 0.8702 0.9170 0.09088 K = 16 0.7297 0.6255 0.6778 0.7263 0.00466January 22, 2020 K = 17 0.7619 0.6644 0.7131 0.7540 0.01036 K = 18 0.7861 0.6911 0.7386 0.7839 0.00279 K = 17 0.6992 0.6020 0.6507 0.6896 0.01372February 19, 2020 K = 18 0.7157 0.6361 0.6759 0.7103 0.00754 K = 19 0.7425 0.6513 0.6969 0.7360 0.00875Table 4: Implied volatility comparison between market quotes as on November 7, 2019and the implied volatility obtained via our SLV models for VIX call options, for the fourmaturities and around ATM, according to Table 3.• The (static) correlation structure between the Brownian motions driving the VIXfutures and the parameters describing the leverages, namely the correlation be-tween futures and their common stochastic volatility CIR process, are given bythe following correlation matrix between futures their volatilities: the positions( i, j ) , i = 1 , . . . , , j = 1 , . . . , , i < j represent the correlation between futures, ρ W i ,W j , while the positions ( i, , i = 1 , . . . , ρ v .Σ W,Z = W W W W Z .
85 0 .
85 0 .
85 0 . W ∗ .
85 0 .
85 0 . W ∗ ∗ .
85 0 . W ∗ ∗ ∗ . W ∗ ∗ ∗ ∗ Z • The remaining parameters used in the CIR-like volatility process are chosen by takinginto account the non-violation of the Feller condition. The complete parameter list,where we repeat the entry already set for sake of clarity, is given by: κ = 2 . , θ = 2 . , ξ = 1 . , v = 1 , ρ v = 0 . · paths. In Table 4 (resp. Table 5) we show the model implied volatilities on VIX(resp. VXX) around ATM (according to Table 3), that are generated by the SLV model,and the market ones, with the relative errors.In Figure 11 (resp. Figure 12) we show the whole implied volatilities for all strikesobtained by our SLV model and the market volatilities of VIX futures (resp. VXX) calloptions. Some remarks are in order: first, we calibrated on ask option quotes as they weremore regular and complete, namely we considered real quotes instead of e.g. SVI interpo-lations. In fact, ask quotes are much more informative on the VXX market, according to22aturities Strike BID ASK MID Model Relative error K = 19 0.4254 0.4602 0.4429 0.4602 ∼ K = 19 . K = 20 0.5212 0.5633 0.5394 0.5473 0.02840 K = 19 0.6002 0.6078 0.6040 0.6050 0.00461December 20, 2019 K = 19 . K = 20 0.6267 0.6455 0.6361 0.6329 0.01951 K = 19 0.6378 0.6528 0.6453 0.6450 0.01194January 17, 2020 K = 19 . K = 20 0.6649 0.6796 0.6723 0.6735 0.00838Table 5: Implied volatility comparison between market quotes as on November 7, 2019and the implied volatility obtained via our SLV models for VXX call options, for the threematurities and around ATM, according to Table 3.Figure 4. Despite the irregular shape of the market VXX smiles, the fit is overall good andeven remarkably good for the VIX smiles. Recall that this market is quite irregular, withlarge bid-ask spreads, according to Figure 3. This means that our SLV model reproducesVIX and VXX smiles that fall within the market bid-ask spreads even after a perturbationof (some of) the parameters, according to the sensitivity analysis performed in the previ-ous subsection. What is more, our parameters specification leads to an admissible localcorrelation vector ρ ( t, K ) that falls in the interval [ − , In this paper, we have presented a general framework for a calibration of Exchange-TradedProducts (ETP’s) based on futures strategies. In a numerical simulation based on realdata we have considered as an example of ETP the VXX ETN, which is a strategy onthe nearest and second-nearest maturing VIX futures contracts. The main ingredient toachieve our goal was a full stochastic local-volatility model that can be calibrated on abook of options on a general ETP and its underlying futures contracts in a parsimoniousmanner.A parameters sensitivity analysis allowed us to fix suitable values for the model pa-rameters in order to fit the smile of the VIX and VXX ETN, with a good level of accuracy.Moreover, our specification led to an admissible local correlation function that justifies theconsistency of the whole procedure. The implementation of a full calibration algorithm(in order to find the optimal value of all model parameters) based on neural networkstechniques is currently under investigation. The methodology proposed has been appliedto the case of VIX futures and VXX ETN, but the framework is flexible enough to tacklealso more general ETP’s.Last but not least, the model is based on a simple diffusive dynamics for the VIX andit can be easily extended in order to include jumps, regime switching and other featuresthat could be required for describing the stylized facts of this market.23igure 11: VIX futures model implied volatility against ask VIX call options marketquotes, as on November 7, 2019, for the four maturities November 20, 2019, December 18,2019, January 22, 2020 and February 19, 2020, from the top left corner going clockwise.Figure 12: VXX model implied volatility against ask VXX call options market quotes, ason November 7, 2019, for the three maturities November 15, 2019, December 20, 2019 andJanuary 17, 2020, from the top to the bottom.24 eferences
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