Model-driven statistical arbitrage on LETF option markets
aa r X i v : . [ q -f i n . S T ] S e p September 22, 2020 Quantitative Finance manuscript
Published in
Quantitative Finance , Vol. 19, No. 11, 2019, 1817–183710.1080/14697688.2019.1605186
Model-driven statistical arbitrage on LETF optionmarkets
S. NASEKIN ∗ † and W. K. HÄRDLE ‡ † Lancaster University Management School, Lancaster University, Lancaster, United Kingdom,LA14YX ‡ C.A.S.E.- Center for Applied Statistics & Economics, Humboldt-Universität zu Berlin,Spandauer Str. 1, 10178 Berlin, Germany ( This is a post-peer-review, pre-copyedit version of an article published in Quantitative Finance. The finalauthenticated version is available online at: http://dx.doi.org/10.1080/14697688.2019.1605186 )In this paper, we study the statistical properties of the moneyness scaling transformation byLeung and Sircar (2015). This transformation adjusts the moneyness coordinate of the implied volatilitysmile in an attempt to remove the discrepancy between the IV smiles for levered and unlevered ETF op-tions. We construct bootstrap uniform confidence bands which indicate that the implied volatility smilesare statistically different after moneyness scaling has been performed. An empirical application showsthat there are trading opportunities possible on the LETF market. A statistical arbitrage type strategybased on a dynamic semiparametric factor model is presented. This strategy presents a statistical decisionalgorithm which generates trade recommendations based on comparison of model and observed LETFimplied volatility surface. It is shown to generate positive returns with a high probability. Extensiveeconometric analysis of LETF implied volatility process is performed including out-of-sample forecastingbased on a semiparametric factor model and uniform confidence bands’ study. It provides new insightsinto the latent dynamics of the implied volatility surface. We also incorporate Heston stochastic volatilityinto the moneyness scaling method for better tractability of the model.
Keywords : exchange-traded funds; options; implied volatilities; moneyness scaling; bootstrap; dynamicfactor models; trading strategies
JEL Classification : C00, C14, C50
1. Introduction
Exchange-traded funds (ETFs) are financial products that track indices, commodities, bonds, bas-kets of assets. They have become increasingly popular due to diversification benefits as well asthe investor’s ability to perform short-selling, buying on margin and lower expense ratios than, for ∗ Corresponding author. Email: [email protected] eptember 22, 2020 Quantitative Finance manuscript instance, those of mutual funds.Leveraged ETFs (LETFs) are used to generate multiples or inverse multiples of returns on theunderlying asset. For instance, the LETF ProShares Ultra S&P500 (SSO) with a leverage ratio β = +2 is supposed to grow 2% for every 1% daily gain in the price of the S&P500 index, minusan expense fee. An inverse leveraged ETF would invert the gain/loss of the underlying index andamplify it proportionally to the ratio: the ProShares UltraShort S&P500 (SDS) with leverage ratio β = − eptember 22, 2020 Quantitative Finance manuscript
2. Confidence analysis of moneyness scaling2.1.
Moneyness scaling
We begin by introducing basic results on (L)ETF options and moneyness scaling. The dynamics ofthe underlying asset is assumed to follow a stochastic process under a risk-neutral measure Q : dS t S t = ( r − δ ) dt + σ t dW Q t , (1)where r is the risk-free interest rate, δ the dividend yield, ( σ t ) t ≥ is some stochastic volatility process.The moneyness scaling technique proposed by Leung and Sircar (2015) proposes a coordinatetransformation for the LETF option implied volatility and potentially reflects the increase of risk inthe underlying index.Figure 1 compares empirical implied volatilities for SSO, UPRO, SDS, SPXU before moneynessscaling has been applied and afterwards. In this example, the log-moneyness LM def = log( K/L t ) isused, where K is the strike of the LETF option and L t the LETF price at time t . After re-scaling,there are still visible discrepancies between the implied volatilities for the SPY ETF and its leveragedcounterparts. The moneyness scaling procedure yields a more coherent picture when the LETF andETF implied volatilities overlap visually better.Based on the assumption that the distribution of the terminal price of the β -LETF dependson the leverage ratio β , the moneyness scaling formula includes an expectation of the β -LETFlog-moneyness conditional on the terminal value of the unleveraged counterpart. For the LETFlog-moneyness LM ( β ) (consider ETFs as LETFs with β = 1) the result linking the log-moneynesscoordinates LM ( β ) and LM (1) of the leveraged and unleveraged ETF is written as follows: LM ( β ) = βLM (1) − { r ( β −
1) + c ∗ } T − β ( β − EQ (Z T σ t d t (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) S T S (cid:19) = LM (1) ) , (2)where T is the time to maturity/expiration (TTM), c ∗ = c + δ is the LETF expense ratio c correctedfor dividend yield δ . The expense ratio c is expressed in percent and approximates an annual feecharged by the ETF from the shareholders to cover the fund’s operating expenses.More generally, for two LETFs with different leverage ratios β , β the expression (2) takes theform: LM ( β ) = β β LM ( β ) + (cid:20)(cid:26) β β ( β − − ( β − (cid:27) r + β β c ∗ − c ∗ (cid:21) T + β ( β − − β ( β − EQ (Z T σ t d t (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) S T S (cid:19) = LM (1) ) , (3)with c ∗ k = c k + δ k , k = 1 , eptember 22, 2020 Quantitative Finance manuscript increase. This can cause, e.g. for both long and short ETFs to have negative cumulative returns overa longer horizon. In this study we use declared leverage ratios. This is motivated by the use of shorttime horizons in the empirical study of LETF option portfolios’ returns in Section 5. Confidence bands
Cont and da Fonseca (2002), Fengler et al. (2007), Park et al. (2009) studied the implied volatilityas a random process in time, so that the data generating process includes some non-parametricfunction m : Y t = m ( X t ) + ε t , t = 1 , . . . , T, (4)or can be driven by a latent factor process Z t : Y t = Z ⊤ t m ( X t ) + ε t , t = 1 , . . . , T, (5)where Y t stands for an implied volatility process, the covariates X t can be one- or multi-dimensional,including, for instance, moneyness and time-to-maturity.The statistical properties of the estimators b m ( X t ) and b Z ⊤ t b m ( X t ) for the models (4) and (5) havebeen outlined, respectively, in, e.g., Härdle (1990), Ruppert and Wand (1994) and Park et al. (2009).To study the consistency of the implied volatility difference between the ETF and the moneyness-scaled LETF case, one needs to consider statistical differences of the corresponding estimators.Confidence band analysis may provide an insight into the matter. An important issue for smoothconfidence bands for functions is the correct probability of covering the "true" curve.The approach of Härdle et al. (2015) proposes a uniform bootstrap bands construction for a wideclass of non-parametric M and L -estimates. It is logical to use a robust M -type smoother for theestimation of (4) for implied volatility, as IV data often suffer from outliers. The procedure runsas follows: considering the sample { X t , Y t } Tt =1 , where Y t denotes the IV process, X t is taken to beone-dimensional and includes the log-moneyness covariate LM ( β ) , do the following:(i) compute the estimate b m h ( X t ) by a local linear M -smoothing procedure (see Appendix 7.1)with some kernel function and bandwidth h chosen by, e.g., cross-validation, and obtainresiduals b ε t def = Y t − b m h ( X t ),(ii) do bootstrap resampling from b ε t : for each t = 1 , . . . , T , generate random variables ε ∗ t,b ∼ b F ε | X t ( z ) for b = 1 , . . . , B according to the conditional edf b F ε | x ( z ) def = P Tt =1 K h ( x − X t ) { b ε t ≤ z } P Tt =1 K h ( x − X t ) , (6)which is further centered as shown in Härdle et al. (2015). Then construct the bootstrapsample Y ∗ t,b as follows: Y ∗ t,b = b m g ( X t ) + ε ∗ t,b , (7)with an "oversmoothing" bandwidth g ≫ h such as g = O ( T − / ) to allow for bias correction,(iii) for each bootstrap sample { X t , Y ∗ t,b } Tt =1 compute b m ∗ h,g using the bandwidth h and construct4 eptember 22, 2020 Quantitative Finance manuscript the random variable d b def = sup x ∈ J | b m ∗ h,g ( x ) − b m g ( x ) | q b f X ( x ) b f ε | X t ( ε ∗ t ) qb E Y | x { ψ ( ε ∗ t ) } , (8)where J is a finite compact support set of b f X and ψ ( u ) = ρ ′ ( · ) as described in Appendix 7.1;the conditional expectation b E Y | x ( · ) is defined with respect to the edf b F Y | x ( z ) def = P Tt =1 K h ( x − X t ) { Y t ≤ z } P Tt =1 K h ( x − X t ) , (9)where b f ε | X t ( · ) and b f X ( x ) are consistent estimators of conditional density corresponding to(6) and the density f X ( x ), respectively; for more details, see Härdle et al. (2015),(iv) calculate the 1 − α quantile d ∗ α of d , . . . , d B ,(v) construct the bootstrap uniform confidence band centered around b m h ( x ): b m h ( x ) ± qb E Y | x { ψ ( ε ∗ t ) } d ∗ α q b f X ( x ) b f ε | X t ( ε ∗ t ) . (10)Such an approach utilizes bootstrap confidence bands while the distribution of the original datais "mimicked" via a pre-specified random mechanism achieving both uniformity and better cover-age. Additionally, it performs better than asymptotic confidence bands which generally tend tounderestimate the true coverage probability, see Hall and Horowitz (2013). Compared to a Bonfer-roni approach, bootstrap uniform confidence bands would be less conservative and make use of thesubstantial positive correlation of the curve estimates at nearby points, see Härdle (1990).
3. Moneyness scaling under Heston stochastic volatility3.1.
An analytical approach
In the study of moneyness scaling, one needs to estimate the following conditional expectation: EQ (Z T σ t dt (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) S T S (cid:19) = LM (1) ) . (11)Taking σ t = σ constant, one obtains σ T . As empirical evidence shows, constant volatility is nota plausible assumption, therefore one needs to determine the measure Q for the case of randomvolatility under a model which allows random dynamics of σ t . Second, one needs to estimate theintegrated variance Z T σ t dt, (12)Stochastic volatility presents a viable alternative to the constant case. One could choose among5 eptember 22, 2020 Quantitative Finance manuscript different specifications of stochastic volatility models. Popular special cases include specificationsof Heston (1993), Hull and White (1987), Schöbel and Zhu (1999). An example of a more generalstochastic volatility system is given in Leung and Sircar (2015). Simpler models tend to generatesemi-closed-form solutions for return distributions. For instance, a solution for the Heston model byHeston (1993) was proposed by Dragulescu and Yakovenko (2002).We use the Heston model to compute the quantity in (11). As noticed in Leung and Santoli (2016),this approach allows for tractability and efficient numerical pricing of options on LETFs. Stochasticvolatility framework also allows to better assess volatility decay, i.e. value erosion due to the increaseof the realized variance with the holding horizon.The Heston model with risk-neutral dynamics under a risk-neutral measure Q and zero volatilityrisk premium is described by a two-dimensional system of stochastic differential equations dS t = ( r − c − . S t dt + p V t S t dW Q S,t , (13) dV t = κ ( θ − V t ) dt + σdW Q V,t , (14)where we have put V t = σ t ; r − c are costs of carry on S t , θ is the long-run variance level, κ is therate of reversion to θ , σ is the "volatility of the volatility" parameter which determines the varianceof V t ; W S,t , W V,t are correlated with parameter ρ . The tails of the Heston-implied densities for log-returns x t = log( S t /S t − ) are exponential and heavier than those of the normal distribution withthe dispersion parameter equal to the long-term variance θ , see, i.e. Cizek et al. (2011).An analytical solution to (11) requires knowledge of the conditional distribution of the integratedvariance (12) given the logarithm of terminal stock price log( S T ). If we define: e V def = Z T V t dt,X T def = log( S T ) , then we can write EQ (Z T V t dt (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) S T S (cid:19) = LM (1) ) = EQ (cid:26) e V (cid:12)(cid:12)(cid:12)(cid:12) X T = g LM (1) (cid:27) = Z ∞ f e V | X T (cid:18)e v (cid:12)(cid:12)(cid:12)(cid:12) x T = g LM (1) (cid:19) d e v, (15)where f e V | X T ( e v | x T ) is the conditional density of e V given X T under the measure Q and g LM (1) =log( S ) + LM (1) .Unfortunately, f e V | X T ( e v | x T ) does not assume a simple form and is ultimately expressed in termsof Fourier transforms of characteristic functions of these quantities. Technical details are given inAppendix 7.2. As follows from the details, four improper integrals have to be estimated. Numericintegration methods can be used to approximate (11).Additional complexity arises from the necessity to evaluate a modified Bessel function of the firstkind which takes a complex argument. Numerical approximation methods for such evaluations suchas the trapezoidal rule are outlined in Broadie and Kaya (2006).6 eptember 22, 2020 Quantitative Finance manuscript Considering the complexity of the density estimation, we consider a Monte-Carlo approach toevaluate (11). This method is feasible and straightforward from a practical point of view.
A Monte-Carlo approach
Alternatively, the conditional expectation in (11) can be computed using Monte-Carlo simulations.The simulations are performed using the Heston model and the calibrated parameters obtainedminimizing the squared difference between theoretical Heston prices C Θ ( K, τ ) obtained from themodel and observed market prices C M ( K, τ ),min Θ ∈ R N X i =1 (cid:16) C Θ i ( K i , τ i ) − C Mi ( K i , τ i ) (cid:17) (16)where Θ def = ( κ, θ, σ, v , ρ ) Heston parameters, N number of options used for calibration, K strikesand τ times-to-maturity. Theoretical prices C Θ ( K, τ ) are obtained via numeric integration of theHeston characteristic function.The Monte-Carlo algorithm is motivated by van der Stoep et al. (2014) and can be formulated asfollows:(i) Generate N pairs of observations ( s i , v i ), i = 1 , . . . , N .(ii) Order the realizations s i : s ≤ x ≤ . . . ≤ s N .(iii) Determine the boundaries of M bins ( l k , l k +1 ], k = 1 , . . . , M on an equidistant grid of values S ∗ def = S e LM (1) (iv) For the k th bin approximate the conditional expectation (11) by EQ Z T σ t d t | S T ∈ ( l k , l k +1 ] ! ≈ hN Q ( k ) H X i =1 X j ∈J k V ij , (17)where h is the discretization step for V t , J k the set of numbers j , for which the observations S T are in the k th bin and Q ( k ) is the probability of S T being in the k th bin.The results of the simulation are presented in Figure 5. Polynomial smoothing is applied to producethe smoothed version of SCO LETF realized variance. The generated expected realized variance hasthe form of a "smile" which confirms the intuition behind using average square implied volatility inthe case of constant-volatility moneyness scaling approach.We use the Monte-Carlo approach given Euler discretization scheme for the empirical applicationin later sections given its tractability and theoretical justification. Both methods, analytical andMonte-Carlo, introduce errors into the calculation of (11). For the analytical method, discretizationand truncation errors appear when the integral is estimated at discrete points and is truncated tobe approximated as a finite sum. If the trapezoidal rule is used to approximate the integrals in theanalytical method, then the discretization error is of order O ( M − ) where M − d increases, the discretization error orderincreases to O ( M − /d ) ("curse of dimensionality").For the current example, the discretization error order for (11) becomes O ( M − / ), which matchesthe convergence order of Monte-Carlo discretization bias. Additionally, analytical approximation of7 eptember 22, 2020 Quantitative Finance manuscript (11) effects a truncation error which is potentially significant due to the oscillatory nature of theintegrand. Monte-Carlo approach inherently induces a statistical error of order O ( N − / ) which canbe made sufficiently small by taking a large number N of samples. As noted in Higham and Mao(2005), Lord et al. (2010), a simple discretization scheme such as the Euler scheme converges to thetrue process under certain conditions on the discretization size. This was shown to be true for theHeston in particular by Higham and Mao (2005).
4. Dynamic semiparametric factor model4.1.
Model description
A generalized version of the model in (4) represented by (5) assumes the implied volatility Y t to bea stochastic process driven by a latent stochastic factor process Z t contaminated by noise ε t . To bemore specific, define J def = [ κ min , κ max ] × [ τ min , τ max ], Y t,j implied volatility, t = 1 , . . . , T time index, j = 1 , . . . , J t option intraday numbering on day t , X t,j def = ( κ t,j , τ t,j ) ⊤ , κ t,j , τ t,j are, respectively, amoneyness measure (log-, forward, etc.) and time-to-maturity at time point t for option j . Then the dynamic semiparametric factor model (DSFM) is defined as follows: assume Y t,j = Z ⊤ t m ( X t,j ) + ε t,j , (18)where Z t = (1 , Z ⊤ t ), Z t = ( Z t, , . . . , Z t,L ) ⊤ unobservable L -dimensional stochastic process, m =( m , . . . , m L ) ⊤ , real-valued functions; m l , l = 1 , . . . , L + 1 are defined on a subset of R d . One canestimate: b Y t = b Z ⊤ t b m ( X t ) (19)= b Z ⊤ t b A ψ ( X t ) , (20)with ψ ( X t ) def = { ψ ( X t ) , . . . , ψ K ( X t ) } ⊤ being a space basis such as a tensor B-spline basis, A isthe ( L + 1) × K coefficient matrix. In this case K denotes the number of tensor B-spline sites: let( s u ) Uu =1 , ( s v ) Vv =1 be the B-spline sites for moneyness and time-to-maturity coordinates, respectively,then K = U · V . Given some spline orders n κ and n τ for both coordinates and sets of knots ( t κi ) Mi =1 ,( t τj ) Nj =1 , one of the Schoenberg-Whitney conditions requires that U = M − n κ , V = N − n τ , seede Boor (2001). The usage of the parameter K is roughly analogous to the bandwidth choice inFengler et al. (2003) and Fengler et al. (2007); however the results of Park et al. (2009) demonstrateinsensitivity of DSFM estimation results to the choice of K , n .The estimates for the IV surfaces b m l are re-calculated on a fine 2-dimensional grid of tensor B-spline sites: the estimated coefficient matrix b A is reshaped into a U × V × L + 1 array of L + 1matrices b A of dimension U × V . Factor functions m l can then be estimated as follows: b m l ; i,j = U X i V X j b A l ; i,j ψ i,k κ ( κ i ) ψ j,k τ ( τ j ) , (21)where k κ , k τ are knot sequences for the moneyness and time-to-maturity coordinates, respectively.8 eptember 22, 2020 Quantitative Finance manuscript The estimated factor functions b m l together with stochastic factor loadings b Z t are combined intothe dynamic estimator of the implied volatility surface: c IV t ; i,j = b m i,j + L X l =1 b Z l,t b m l ; i,j , (22)where b Z l,t can be modeled as a vector autoregressive process. It should be noted that b m l and b Z l,t are not uniquely defined, so an orthonormalization procedure must be applied.An indication of possible mispricing of LETF options allows to test a trading strategy based onthe comparison of the theoretical price obtained from the moneyness scaling correction as well asthe application of the DSFM model and the market price. Such a strategy would mainly exploitthe two essential elements of information from these two approaches. The first element is obtainingevidence of statistical discrepancies resulting from the mismatch between ETF and LETF IVs. Themoneyness scaling approach allows to estimate LETF IV using richer unleveraged ETF data whichalso would make the DSFM IV estimator more consistent. The second element is implied volatilityforecasting. The DSFM model allows to forecast a whole IV surface via the dynamics of stochasticfactor loadings Z t . Model estimation
The DSFM model is estimated numerically. The number of factors has to be chosen in advance. Oneshould also notice that for m l to be chosen as eigenfunctions of the covariance operator K ( u, v ) def = Cov { Y ( u ) , Y ( v ) } in an L -dimensional approximating linear space, where Y is understood to be therandom IV surface, they should be properly normalized, such that k m l ( · ) k = 1 and h m l , m k i = 0 for l = k .The choice of L can be based on the explained variance by factors: EV ( L ) def = 1 − P Tt =1 P J t j =1 n Y t,j − P Ll =0 b Z t,l b m l ( X t,j ) o P Tt =1 P J t j =1 ( Y t,j − Y ) . (23)The model’s goodness-of-fit is evaluated by the root mean squared error (RMSE) criterion: RM SE def = vuuut P t J t T X t =1 J t X j =1 ( Y t,j − L X l =0 b Z t,l b m l ( X t,j ) ) . (24)The prediction quality at time point t + 1 is measured by the root mean squared prediction error(RMSPE) given by RM SP E def = vuuut J t +1 J t +1 X j =1 ( Y t +1 ,j − L X l =0 b Z t +1 ,l b m l ( X t +1 ,j ) ) . (25)9 eptember 22, 2020 Quantitative Finance manuscript
5. Empirical application5.1.
Data description
For the purpose of an empirical application, we use data on SPY, SSO, UPRO and SDS (L)ETFcall options in the period Nov 2014 - June 2015. The data summary statistics are outlined in Table2 below. The data were taken from the Datastream database by Thomson Reuters.To give an impression of leveraged ETF option tradability, we give an illustration of the existingbid-ask spreads and actual trades of the SSO LETF, as these data will be used for the tradingstrategy example below. Figures 13 and 14 show variation of existing trade prices and volumes forvarious option contacts based on exercise price and time to expiration. We can see that shorter-termcontracts are traded more broadly. It has been also found that trades predominantly occur at ornear mid-quotes. In Figure 15 we show bid-ask spreads for the same range of option contracts, whichtend to be quite high, but somewhat lower for longer-term contracts.The option data we use for the empirical application are trade-based data, i.e. each observationcorresponds to an actual trade, not price quotes or settlement data. Implied volatility and optionprices are taken from the database and computed in accordance with standard conventions used bymarket participants using the midpoint of the best closing bid price and best closing offer price forthe options, taking account of liquidity and dividends.Additionally, we remove data which may contain noise, potential misprints and other errors. Suchdata include anomalous and outlier data resulting, e.g., from artificial extrapolation of implied volatil-ities for non-traded options or feature lower liquidity for the out-of-the-money or options which aredeeply in-the-money.
Confidence bands
We use the data described above to construct bootstrap confidence bands for the M -smoother ofimplied volatility Y given log-moneyness X , according to methodology described in Section 2.2.Accordingly, X is transformed using (3). The results are shown in Figures 2 and 3 for time-to-maturity 0.5 and 0.6 years, respectively. In Figure 4 combined bands are provided.We can observe clear discrepancy between the implied volatilities of leveraged ETFs and theirunleveraged counterpart SPY. For all LETFs, non-overlapping confidence bands imply that thereis a statistically significant difference between IV functions at the significance level α = 0 .
05. It ismore pronounced for in- and out-of-the-money options. This phenomenon may occur due to lowerliquidity of in- and out-of-the-money options compared to at-the-money options. On the other hand,as shown, e.g. in Etling and Miller (2000), the relationship between option moneyness and liquidityis more complex than quadratic, maximized for at-the-money options. Therefore, liquidity need notbe the only reason for this fact.We can see from Figure 4 that the bands for SSO demonstrate particularly strong deviation fromthose of SPY. This implies that discrepancies not removed by the moneyness scaling procedure arethe largest for this LETF. Therefore we conclude this section with a trading strategy which is meantto exploit such statistical discrepancies on the market of SPY and SSO options.10 eptember 22, 2020 Quantitative Finance manuscript
DSFM estimation and forecasting
The EV , RM SE and
RM SP E criteria are displayed in Table 3. The model order L = 3 is chosenfor estimation. The data for the SPY ETF option are used with parameters n κ , n τ = 3; M = 9, N = 7, so that U = 6, V = 4, K = 6 × b Z t in time. Two largest "spikes" in the value of the third stochasticloading in the beginning of the period correspond to the period of relatively large values of theCBOE volatility index (VIX). The second of the "spikes" precedes in time an increase in the VIXvalue implying that the model has predictive value with respect to market instability dynamics. Thisshows that DSFM captures leading dynamic effects as well as can explain effects like skew or termstructure changes.Theoretical and simulation results in Park et al. (2009) justify using vector autoregression (VAR)analysis to model b Z t . To select a VAR model, we computed the Schwarz (SC), the Hannan-Quinn(HQ) and the Akaike (AIC) criteria, as shown in Table 4. All three criteria select the VAR(1) model.Furthermore, the roots of the characteristic polynomial all lie inside the unit circle, which shows thatthe specified model is stationary. Portmanteau and Breusch-Godfrey LM test results with 12 lagsfor the autocorrelations of the error term fail to reject residual autocorrelation at 10% significancelevel.The degenerate nature of implied volatility data is reflected by the fact that empirical observationsdo not cover estimation grids at given time points. This is due to the fact that contracts at certainmaturities or strikes are not always traded. The DSFM fitting procedure introduces basis functionswhich approximate a high-dimensional space and depend on time. This allows to account for allinformation in the dataset simultaneously in one minimization procedure which runs over all b m l and b Z t and avoid bias problems which would inevitably occur if some kernel smoothing procedure suchas Nadaraya-Watson were applied for this type of degenerate data. Option trading strategy5.4.1.
Description.
Ability to forecast the whole surface of implied volatility can be used incombination with the moneyness scaling technique to exploit potential discrepancies in ETF andLETF option prices or implied volatilities to build a trading strategy. A suitable strategy wouldbe the so-called "trade-with-the-smile/skew" strategy adapted for the special case of ETF-LETFoption IV discrepancy. It would use the ETF option data to estimate the model (theoretical) smileof the leveraged counterpart and the information from the IV surface forecast to recognize the future(one-period-ahead) possible IV discrepancy.Going back to the results in Section 5.2, we see that the largest statistical discrepancy betweenleveraged and unleveraged ETF implied volatilities is the one between SPY and SSO, so we considerthese two options in the strategy setup. The strategy can be outlined as follows: choose a movingwindow width w ; then for each t = w, . . . , T ( T is the final time point in the sample) do the following:(i) given two leverage ratios β SP Y = 1, β SSO = 2, re-scale the log-moneyness coordinate LM ( β SPY ) according to the moneyness scaling formula (3) to obtain d LM ( β SSO ) . This willbe the "model" moneyness coordinate for DSFM estimation,(ii) map the space [ d LM ( β SSO ) min , d LM ( β SSO ) max ] × [ τ SP Ymin , τ
SP Ymax ] to [0 , × [0 ,
1] using marginal transfor-11 eptember 22, 2020 Quantitative Finance manuscript mation,(iii) estimate the DSFM model (18) on [0 , × [0 , c IV SSO , . . . , c IV SSOt ,(iv) forecast the IV surface estimate c IV SSOt +1 using the VAR structure of the estimated stochasticloadings b Z t and the factor functions b m l ,(v) choose a time-to-maturity τ ∗ at time point t , take the corresponding real-world values of SSOlog-moneyness LM ( β SSO ) and map them to [0 ,
1] using the marginal distribution of d LM ( β SSO ) ;denote the output as LM ( β SSO ) τ ∗ ; M ,(vi) using the marginally re-scaled grid [ d LM ( β SSO ) min , d LM ( β SSO ) max ] × [ τ ∗ , τ ∗ ] and c IV SSOt +1 , obtain inter-polated values c IV SSOt +1; LM ( βSSO ) τ ∗ ; M ,τ ∗ corresponding to LM ( β SSO ) τ ∗ ; M , τ ∗ ,(vii) compare the "theoretical" values c IV SSOt +1; LM ( βSSO ) τ ∗ ; M ,τ ∗ with known real-world implied volatilities IV SSOt ; LM ( βSSO ) τ ∗ ; M ,τ ∗ corresponding to LM ( β SSO ) τ ∗ ; M , and construct a delta-hedged option portfolio: • if c IV SSOt +1; LM ( βSSO ) τ ∗ ; M ,τ ∗ > IV SSOt ; LM ( βSSO ) τ ∗ ; M ,τ ∗ for all LM ( β SSO ) τ ∗ ; M , then buy (long) options corre-sponding to the largest difference D long def = c IV SSOt +1; LM ( βSSO ) τ ∗ ; M ,τ ∗ − IV SSOt ; LM ( βSSO ) τ ∗ ; M ,τ ∗ , • if c IV SSOt +1; LM ( βSSO ) τ ∗ ; M ,τ ∗ < IV SSOt ; LM ( βSSO ) τ ∗ ; M ,τ ∗ for all LM ( β SSO ) τ ∗ ; M , then sell (short) options corre-sponding to the largest difference D short def = IV SSOt ; LM ( βSSO ) τ ∗ ; M ,τ ∗ − c IV SSOt +1; LM ( βSSO ) τ ∗ ; M ,τ ∗ , • if it holds that both c IV SSOt +1; LM ( βSSO ) τ ∗ ; M ,τ ∗ > IV SSOt ; LM ( βSSO ) τ ∗ ; M ,τ ∗ and c IV SSOt +1; LM ( βSSO ) τ ∗ ; M ,τ ∗ For the purpose of the estimation of the strategy from the previoussection, the DSFM model parameters are taken to be the same as in Section 5.3. The rolling windowwidth is assumed to be w = 100 and the forecasting horizon is 1 day ahead.The dynamic strategy performance in the period April 2015 - June 2015 is displayed in Figure7. Out of 55 investment periods, in 30 cases long-only portfolios were constructed, the remaining25 cases short and long positions were taken; net portfolios were short portfolios in 42 cases, longportfolios in the remaining 13 cases.For the sake of illustration, let us go through one step from the outlined strategy in a numericalexample. Assume that we are at the step t = 147 of the sample, which corresponds to June 18, 2015.At this point, we have a training sample of 100 days for DSFM estimation, encompassing 14,859observations of the option data for contracts with various strike prices and time-to-maturity. Thehistogram and density estimates for SPY log-moneyness LM ( β SPY ) , "theoretical" SSO log-moneyness d LM ( β SSO ) (that is, rescaled LM ( β SPY ) ) and its marginally transformed version are given in Figure 8.Further we proceed as proposed in the strategy above:(i) estimate (18) and perform a forecast to obtain c IV SSO on June 19, 2015 (day 148 in thesample),(ii) choose τ ∗ = 0 . 6; we have 37 values of LM ( β SSO ) for τ ∗ = 0 . 6. We use the marginal distributionof d LM ( β SSO ) shown in Figure 8 to calculate the corresponding "theoretical" values LM ( β SSO ) τ ∗ ; M implied by SPY data and the moneyness scaling procedure, both shown in Figure 9,(iii) using the forecast IVS c IV SSO , we can determine "theoretical" IV values corresponding to LM ( β SSO ) τ ∗ ; M ∈ [0 , c IV SSO LM ( βSSO ) τ ∗ ; M ,τ ∗ and the real-world IV values IV SSO LM ( βSSO ) τ ∗ ; M ,τ ∗ corre-sponding to the same LM ( β SSO ) τ ∗ ; M through the mapping of LM ( β SSO ) , described above,(iv) the resulting c IV SSO LM ( βSSO ) τ ∗ ; M ,τ ∗ and IV SSO LM ( βSSO ) τ ∗ ; M ,τ ∗ are demonstrated in Figure 10. We buyan option corresponding to the largest D long = 0 . 052 at the closing price C long ;147 = $27 . D short = 0 . 163 at the closing price C short ;147 =$2 . P ortf ;147 =∆ long ;147 − ∆ short ;147 = 0 . − . 461 = 0 . 398 and the underlying SSO which has a closingprice of L = $66 . C long ;147 − C short ;147 − ∆ P ortf ;147 × L = $27 . − $2 . − . × $66 . 960 = − . C long ;148 and C short ;148 as well as the price of the underlying L . We find that on June 19, 2015, C long ;148 = $28 . 450 and C short ;148 = $0 . L = $68 . L stillequal to the previous-day portfolio delta ∆ P ortf ;147 , the portfolio is now worth C long ;148 − C short ;148 − ∆ P ortf ;147 × L = $28 . − $0 . − . × $68 . 300 = 0 . eptember 22, 2020 Quantitative Finance manuscript portfolio in an offsetting trade and have secured a gain of $2 . C long gainedin value while C short went down in value. The coupled gain was larger than an offsetting lossof $0 . 533 from the delta hedge which resulted in the total gain.In the end, the cumulative gain of the strategy applied daily as shown in the illustrating exampleabove, is 19.043 after 55 investment periods. It occurs that 39 out of 55 investment decisions correctlydetermined the direction of one-step-ahead implied volatility smile change. In the strategy, this smilechange is anticipated according the relation between the "model" IVS computed using the moneynessscaling approach and the real-world IVS of a LETF option. There is a high positive chance ofgenerating positive cumulative returns exploiting statistical deviations of leveraged and unleveredimplied volatility smiles in the ETF option market.It should be mentioned that the sample period includes the day of an underlying SSO stock 2-for-1 split which took place on May 20, 2015. The split was implemented to attract a wider rangeof buyers at the resulting lower price per share. It has been shown by many researchers, such asOhlson and Penman (1985), Sheikh (1989), Desai et al. (1998) that stock splits result in post-splitincreases of implied stock volatilities. For instance, Ohlson and Penman (1985) show that stock splitscause short-term increases in volatility upon announcement and long-term increases in volatility afterthe date the split is effective.In Figure 11, we show real-world and DSFM-forecast IVS on two different dates: before (19 June,2015) and after the split (21 June, 2015). It can be seen that the model anticipates a significantincrease in implied volatilities after the split which indeed takes place. Robustness check. The performance of the option trading strategy obtained in Section5.4.2 above may seem to have occurred purely by chance. Therefore some sort of a robustness checkis necessary.We perform a bootstrap resampling exercise on the time series of the underlying prices of SPYand SSO (L)ETFs and re-run the strategy on the resampled data. Overlapping block bootstrapapproach proposed by Künsch (1989) is applied. It works as follows: given the data observations { X i : i = 1 , . . . , T } , a block size b is specified. With overlapping blocks of length b , block 1 is thenobservations { X j : j = 1 , . . . , b } , block 2 is { X j +1 : j = 1 , . . . , b } and so on. Random sampling isthen performed on the level of blocks.Overlapping block bootstrap assumes the data to be stationary. However, in the current casewe do not do inference on the resampled data, so this stringent assumption is less relevant here.Nevertheless, we run standard stationarity tests on SPY and SSO price series in the period fromNovember 2014 to June 30, 2015 such as Phillips-Perron, augmented Dickey-Fuller and KPSS tests.The first two tests have presence of unit root in the series as a null hypothesis, while the KPSS testtests trend stationarity as the null.We perform a series of tests for each approach using the number of lags from 1 to 10 in the Newey-West estimator of the long-run variance. In Table 5 the results of stationarity tests are demonstrated.Most of this evidence does not reject the hypothesis of possible trend stationarity of price series ofthe (L)ETFs. Therefore we proceed with the bootstrap.We run 500 bootstrap iterations on 2-dimensional series of SPY and SSO prices in the periodfrom November 2014 to June 30, 2015 and take the block size equal to 5. In Figure 12 cumulativeperformance of the strategy on bootstrapped time series is shown. At each of the 155 time steps14 eptember 22, 2020 Quantitative Finance manuscript the values of 2.5% and 97.5% empirical percentiles are found. We can see that at the end of theperiod, positive performance occurs with more than a 95% probability. Positive performance withthis probability occurs from period 23 onwards until the end of of the test sample, which yields 32periods out of 55 in total. 6. Conclusion In this paper, we provide statistical and econometric analysis of the moneyness scaling transfor-mation for leveraged and unlevered exchange-traded funds’ options’ implied volatility smiles. Thistransformation adjusts the moneyness coordinate of the smile in an attempt to remove the discrep-ancy between the levered and unlevered counterparts.We incorporate stochastic volatility into the moneyness scaling method by explicit estimation ofthe conditional expectation of the realized variance. We present two approaches to implement thisestimate: via an analytical approach and using a Monte-Carlo method.We construct bootstrap uniform confidence bands which reveal a statistically significant discrep-ancy between the implied volatility smiles, even after moneyness scaling has been performed. We findthat this discrepancy is stronger for in- and out-of-the-money options which, however, is unlikely tobe explained by liquidity issues alone.This discrepancy allows to define a theoretical statistical equilibrium value of LETF moneyness.Based on deviations from this equilibrium, possible trading gain opportunities on the (L)ETF marketwhich can be exploited. We construct a trading strategy based on a dynamic semiparametric factormodel. This model-based statistical arbitrage strategy utilizes the dynamic structure of impliedvolatility surface allowing out-of-sample forecasting and information on unleveraged ETF options toconstruct theoretical one-step-ahead implied volatility surfaces.The proposed strategy has the potential to generate trading gains due to simultaneous use of theinformation from the discrepancies between the forecast "theoretical" (model) SSO LETF impliedvolatilities and the historical ("true") ones. It protects the portfolio against unfavorable moves inthe underlying asset through delta-hedging and aims to gain from forecast moves in volatility. Thestrategy is shown via bootstrap technique to generate positive returns with a high probability. Funding This research was supported by the Deutsche Forschungsgemeinschaft under Grant "IRTG 1792".15 eptember 22, 2020 Quantitative Finance manuscript References A. Ahn, M. Haugh, and A. Jain. Consistent pricing of options on leveraged ETFs. SIAM Journal onFinancial Mathematics , 6(1):559–593, 2015.Y. Aït-Sahalia, P. J. Bickel, and T. M. Stoker. Goodness-of-fit tests for kernel regression with an applicationto option implied volatilities. Journal of Econometrics , 105(2):363–412, 2001.M. Avellaneda and J.-H. Lee. Statistical arbitrage in the US equities market. Quantitative Finance , 10(7):761–782, 2010.M. Avellaneda and S. Zhang. Path-dependence of leveraged ETF returns. SIAM Journal on FinancialMathematics , 1(1):586–603, 2010.M. 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Giventhe model Y i = m ( X i ) + ε i , (26)where Y i ∈ R , X i ∈ R d , ε i def = σ ( X i ) u i , u i ∼ (0 , X def = { ( X i , Y i ); 1 ≤ i ≤ n } , the local linear M -smoothing estimator is obtained from:min α ∈ R ,β ∈ R p n X i =1 ρ n Y i − α − β ⊤ ( X i − x ) o W ih ( x ) , (27)where W hi ( x ) def = h − K ′ { ( x − X i ) /h } b f h ( x ) − K h ( x − X i ) b f ′ h ( x ) b f h ( x ) (28)is a kernel weight sequence with b f ′ h ( x ) def = n − P ni =1 K ′ h ( x − X i ), h is the bandwidth, K is a kernelfunction; R K ( u ) du = 1, K h ( · ) def = h − K ( · /h ). The function ρ ( · ) is designed to provide more robust-ness than the quadratic loss. An example of such a function is given by Huber (1964), see also Härdle(1989): ρ ( u ) = (cid:26) . u , if | u | ≤ c ; c | u | − . c if | u | > c. , (29)with the constant c regulating the degree of resistance. Derivation of the conditional density of Heston integrated variance giventerminal log-price As pointed out in (15), we require the conditional density f ˜ V | X T (˜ v | x T ), under the measure Q , to analytically determine the expression (11). As noted by Broadie and Kaya (2006) andGlasserman and Kim (2011), the conditional distribution of X T is conditionally normal given e V and R T √ V t dW Q V,t : X T ∼ N ( r − c − . T − . e V + ρ Z T p V t dW Q V,t , q − ρ e V ! , ∼ N (cid:18) ( r − c − . T − . e V + ρσ ( V T − V − κθT + κ e V ) , q − ρ e V (cid:19) , (30)18 eptember 22, 2020 Quantitative Finance manuscript where (30) follows from Z T p V t dW Q V,t = σ − ( V T − V − κθT + κ e V ) , which, in turn, follows from (14).Expression (30) yields the conditional density f X T | e V ,V T given Heston parameters κ , θ , V , σ and ρ .The joint density f e V ,X T ,V T can then be obtained by simply multiplying f X T | e V ,V T by f e V ,V T , the jointdensity of e V and V T given some starting Heston variance level V .Broadie and Kaya (2006) have derived the characteristic function of the distribution of e V givenvariance endpoint values V T and V . This function is quite complex and involves modified Besselfunctions of a complex variable of the first kind: ϕ e V | V T ( ω ) = γ ( ω ) e − . γ ( ω ) − κ ) T (1 − e − κT ) κ (1 − e − γ ( ω ) T ) exp ( V T + V σ κ (1 + e − κT )1 − e − κT − γ ( ω )(1 + e − γ ( ω ) T )1 − e − γ ( ω ) T ) ×× B κθσ − − (cid:16) √ V T V γ ( ω ) e − . γ ( ω ) T σ (1 − e − γ ( ω ) T ) (cid:17) B κθσ − − (cid:16) √ V T V κe − . κT σ (1 − e − κT ) (cid:17) , (31)where γ ( ω ) = √ κ − σ iω , i = √− B ν ( z ) is the modified Bessel function of the first kindgiven by B ν ( z ) def = ( z/ ν ∞ X j =0 ( z / j j !Γ( ν + j + 1) , where Γ( x ) def = R ∞ t x − e − t dt is the gamma function.Using the inversion formula for characteristic functions, we can compute the density f e V | V T ,V asfollows: f e V | V T ( e v | v T ) = 12 π Z + ∞−∞ e − i e vω ϕ e V | V T ( ω ) dω. To find the joint density f e V ,X T ,V T , the transitional density f V T | V is required. As noted by Cox et al.(1985) in the context of short interest rate process, V T given V follows a scaled non-central chi-squared distribution: V T = σ (1 − e − κT )4 κ e χ d κe − κT V σ (1 − e − κT ) ! , (32)where e χ d ( λ ) stands for the non-central chi-squared random variable with d degrees of freedom andnon-centrality parameter λ . The probability density function of e χ d ( λ ) is defined using B ν ( z ): f e χ d ( λ ) ( x ) = 0 . e − . x + λ ) ( xλ − ) . d − . B . d − ( √ λx )19 eptember 22, 2020 Quantitative Finance manuscript Using a change-of-variables technique, it is straightforward to show that the density f V T | V takesthe form: f V T | V ( v T | V ) = 2 κσ (1 − e − κT ) exp ( κ θTσ − . κT − κ ( v T + e − κT V ) σ (1 − e − κT ) ) ×× (cid:18) v T V (cid:19) κθσ − − . B κθσ − − κe − . κT σ (1 − e − κT ) p V v T ! . Using known rules for computing joint densities via conditional and marginal densities, it followsthat f e V ,X T ,V T ( e v, x T , v T ) = f X T | e V ,V T ( x T | e v, v T ) f e V | V T ( e v | v T ) f V T | V ( v T | V ) f e V ,X T ( e v, x T ) = Z ∞ f e V ,X T ,V T ( e v, x T , v T ) dv T Therefore we have for f e V ,X T estimated at e v , x T , given the Heston parameters: f e V ,X T ( e v, x T ) = 2 κ (2 π ) / p (1 − ρ ) e vσ (1 − e − κT ) Z ∞ exp ( κ θTσ − . κT − κ ( v T + e − κT V ) σ (1 − e − κT )+ ( v T + V ) κ (1 + e − κT ) σ (1 − e − κT ) − (cid:0) x T − log( S ) − ( r − c − . T + 0 . e v − ρσ ( v T − V − κθT + κ e v ) (cid:1) − ρ ) e v ) × (cid:18) v T V (cid:19) κθσ − − . Z + ∞−∞ e − i e vω " γ ( ω ) e − . γ ( ω ) − κ ) T (1 − e − κT ) κ (1 − e − γ ( ω ) T ) exp ( − γ ( ω )(1 + e − γ ( ω ) T )( v T + V ) σ (1 − e − γ ( ω ) T ) ) × B κθσ − − p v T V γ ( ω ) e − . γ ( ω ) T σ (1 − e − γ ( ω ) T ) ! dωdv T Finally, the density f e V | X T ( e v | x T ) is found as f e V | X T ( e v | x T ) = f e V ,X T ( e v, x T ) f X T ( x T ) , where f X T ( x T ) is the probability density of X T estimated at x T . This marginal density under therisk-neutral measure Q is again found via inversion of the characteristic function, see Rouah (2013): ϕ ( ω ) X T = exp { C ( ω ) + D ( ω ) V + iω log( S ) } , eptember 22, 2020 Quantitative Finance manuscript where C ( ω ) = riωT + κθσ ( ( κ − ρσiω + d ( ω )) T − − g ( ω ) e − d ( ω ) T − g ( ω ) !) ,D ( ω ) = ( κ − ρσiω + d ( ω )) σ − e − d ( ω ) T − g ( ω ) e − d ( ω ) T ! ,g ( ω ) = κ − ρσiω + d ( ω ) κ − ρσiω − d ( ω ) ,d ( ω ) = q ( ρσiω − κ ) + σ ( iω − ω ) . eptember 22, 2020 Quantitative Finance manuscript 8. Tables (L)ETF Ticker Lev. ratio Exp. ratio (%) Div. yield (%)SPDR S&P 500 SPY +1 0.090 1.867ProShares Ultra S&P500 SSO +2 0.900 0.440ProShares UltraPro S&P500 UPRO +3 0.950 0.263ProShares UltraShort S&P500 SDS − − Table 1. Summary financial information on (leveraged) ETFs on S&P 500 underlying index eptember 22, 2020 Quantitative Finance manuscript Min. Max. Mean Stdd. Skewn. Kurt. SPY τ . 258 2 . 364 1 . 202 0 . 515 0 . 421 2 . LM − . 061 0 . − . 381 0 . − . 513 5 . σ I . 086 2 . 677 0 . 271 0 . 195 3 . 228 18 . SSO τ . 208 2 . 236 1 . 239 0 . − . 044 1 . LM − . 704 0 . − . 484 0 . − . 089 2 . σ I . 154 1 . 340 0 . 363 0 . 091 1 . 774 12 . UPRO τ . 208 2 . 236 1 . 205 0 . 585 0 . 043 1 . LM − . 182 0 . − . 168 0 . − . 360 2 . σ I . 250 1 . 669 0 . 503 0 . 099 1 . 335 9 . SDS τ . 208 2 . 236 1 . 146 0 . 581 0 . 196 1 . LM − . 738 0 . 858 0 . 187 0 . − . 276 2 . σ I . 107 1 . 262 0 . 424 0 . 129 0 . 792 4 . Table 2. Summary statistics on (L)ETF options data ( τ is time to maturity, LM log-moneyness, σ I impliedvolatility) eptember 22, 2020 Quantitative Finance manuscript Criterion L = 2 L = 3 L = 4 L = 5 EV . 915 0 . 921 0 . 925 0 . RM SE . 090 0 . 088 0 . 087 0 . RM SP E . 095 0 . 096 0 . 099 0 . Table 3. EV , RMSE and RMSP E criteria for different model order sizes eptember 22, 2020 Quantitative Finance manuscript Model order n AIC( n ) HQ( n ) SC( n )1 − . ∗ − . ∗ − . ∗ − . − . − . − . − . − . − . − . − . − . − . − . Table 4. The VAR model selection criteria. The smallest value is marked by an asterisk eptember 22, 2020 Quantitative Finance manuscript Table 5. Stationarity tests’ statistics for SPY, SSO price series Lags SPY SSOPP ADF KPSS PP ADF KPSS1 − . ∗∗ − . ∗∗ . ∗∗ − . ∗∗ − . ∗∗ . ∗∗ − . ∗∗ − . ∗∗ . ∗∗ − . ∗∗ − . ∗∗ . ∗ − . ∗∗ − . ∗∗ . ∗ − . ∗∗ − . ∗∗ . − . ∗∗ − . ∗ . − . ∗∗ − . ∗∗ . − . ∗∗ − . ∗ . − . ∗∗ − . ∗∗ . − . ∗∗ − . 133 0 . − . ∗∗ − . ∗ . − . ∗∗ − . 970 0 . − . ∗∗ − . ∗ . − . ∗∗ − . 908 0 . − . ∗∗ − . 094 0 . − . ∗∗ − . 897 0 . − . ∗∗ − . 103 0 . − . ∗∗ − . 003 0 . − . ∗∗ − . 125 0 . ∗ ∗ ∗ , ∗∗ , ∗ : significant on 1%, 5%, 10% level, respectively26 eptember 22, 2020 Quantitative Finance manuscript 9. Figures LM -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 I m p li ed V o l SSO, 207 days to maturity: before scaling LM -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 I m p li ed V o l SSO: 207 days to maturity: after scaling LM -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 I m p li ed V o l UPRO, 207 days to maturity: before scaling LM -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 I m p li ed V o l UPRO: 207 days to maturity: after scaling LM -0.5 0 0.5 1 I m p li ed V o l SDS, 207 days to maturity: before scaling LM -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 I m p li ed V o l SDS: 207 days to maturity: after scaling LM -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 I m p li ed V o l SPXU, 207 days to maturity: before scaling LM -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 I m p li ed V o l SPXU: 207 days to maturity: after scaling Figure 1. SPY (blue) and LETFs (red) implied volatilities before (left column) and after scaling (right column) onJune 23, 2015 with 207 days to maturity, plotted against log-moneyness e p t e m b e r , Q u a n t i t a t i v e F i n a n ce m a nu s c r i p t − − − moneyness i m p v o l SPY − − − moneyness i m p v o l SSO − moneyness i m p v o l UPRO − moneyness i m p v o l SDS Figure 2. Fitted implied volatility and bootstrap uniform confidence bands for 4 (L)ETFs on S&P500; τ : 0.5 years e p t e m b e r , Q u a n t i t a t i v e F i n a n ce m a nu s c r i p t − − − moneyness i m p v o l SPY − − moneyness i m p v o l SSO − − − moneyness i m p v o l UPRO − − moneyness i m p v o l SDS Figure 3. Fitted implied volatility and bootstrap uniform confidence bands for 4 (L)ETFs on S&P500; τ : 0.6 years eptember 22, 2020 Quantitative Finance manuscript − − − moneyness i m p v o l − − − moneyness i m p v o l Figure 4. Combined uniform bootstrap confidence bands for SPY, SSO, UPRO and SDS after moneyness scaling( τ = 0 . τ = 0 . eptember 22, 2020 Quantitative Finance manuscript -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10.0820.0840.0860.0880.090.0920.0940.0960.0980.1 E x pe c t ed c ond i t i ona l I V -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10.0820.0840.0860.0880.090.0920.0940.0960.0980.1 E x pe c t ed c ond i t i ona l I V Original IVSmoothed IV Figure 5. Upper panel: estimated value of EQ ( R T σ t d t | log( S T /S ) = LM (1) ); lower panel: smoothed estimate e p t e m b e r , Q u a n t i t a t i v e F i n a n ce m a nu s c r i p t Sep14 Nov14 Dec14 Feb15 Apr15 May15 Jul15-0.25-0.2-0.15-0.1-0.0500.050.10.150.20.25 101214161820222426 Figure 6. Time dynamics of b Z t, , b Z t, , b Z t, , VIX index eptember 22, 2020 Quantitative Finance manuscript P o r tf o li o W ea l t h Figure 7. Cumulative performance of the trading strategy eptember 22, 2020 Quantitative Finance manuscript Figure 8. LM ( β SPY ) (top panel), d LM ( β SSO ) (middle panel), d LM ( β SSO ) after marginal transformation (bottom panel) e p t e m b e r , Q u a n t i t a t i v e F i n a n ce m a nu s c r i p t L M ( β SS O ) , L M ( β SS O ) τ ∗ ; M Values of LM ( β SSO ) , LM ( β SSO ) τ ∗ ; M LM ( β SSO ) LM ( β SSO ) τ ∗ ; M Figure 9. Values of LM ( β SSO ) and LM ( β SSO ) τ ∗ ; M e p t e m b e r , Q u a n t i t a t i v e F i n a n ce m a nu s c r i p t Moneyness I V Values of IV SSO LM ( β SSO ) τ ∗ ; M , τ ∗ , c IV SSO LM ( β SSO ) τ ∗ ; M , τ ∗ on 20150618 c IV SSO LM ( β SSO ) τ ∗ ; M , τ ∗ IV SSO LM ( β SSO ) τ ∗ ; M , τ ∗ SELLBUY