Modeling VXX Sebastian A. Gehricke∗ Department of Accountancy and Finance Otago Business School, University of Otago Dunedin 9054, New Zealand Email: [email protected] Jin E. Zhang Department of Accountancy and Finance Otago Business School, University of Otago Dunedin 9054, New Zealand Email: [email protected]

First Version: June 2014 This Version: May 2017

In this paper, we study the VXX Exchange Traded Note (ETN), that has been actively traded in recent years, but has lost 99.84% of its value. Using Zhang, Shu and Brenner’s (2010) formula for VIX futures prices, we develop the first theoretical model for the VXX that links the SPX, VIX and VXX. We show that the roll yield of VIX futures drives the difference between the VXX and VIX returns. The roll yield is a mostly negative process. We then provide a simple yet robust estimation of the market price of variance risk using VXX and VIX futures prices. The model can be used to price VXX options. Keywords: VXX; VIX Futures; Roll Yield; Market Price of Variance Risk; Variance Risk Premium JEL Classification Code: G13 ∗

Corresponding author. We are grateful to Bob Webb (editor) and an anonymous referee whose helpful comments substantially improved the paper. We would also like to acknowledge helpful comments and suggestions from Timothy F. Crack, Jose Da Fonseca, Bart Frijns, Xinfeng Ruan, Anindya Sen, Vladimir Volkov (our AFM discussant), Robert E. Whaley, and seminar participants at the University of Otago, 2014 Auckland Finance Meeting (AFM) in Auckland and 2016 New Zealand Finance Colloquium (NZFC) in Queenstown. Jin E. Zhang has been supported by an establishment grant from the University of Otago.

Modeling VXX

In this paper, we study the VXX Exchange Traded Note (ETN), that has been actively traded in recent years, but has lost 99.84% of its value. Using Zhang, Shu and Brenner’s (2010) formula for VIX futures prices, we develop the first theoretical model for the VXX that links the SPX, VIX and VXX. We show that the roll yield of VIX futures drives the difference between the VXX and VIX returns. The roll yield is a mostly negative process. We then provide a simple yet robust estimation of the market price of variance risk using VXX and VIX futures prices. The model can be used to price VXX options. Keywords: VXX; VIX Futures; Roll Yield; Market Price of Variance Risk; Variance Risk Premium JEL Classification Code: G13

Modeling VXX

1

1

Introduction

The VXX Exchange Traded Note (ETN) was the first VIX futures Exchange Traded Product (ETP), issued in 2009 by Barclays Capital iPath shortly after the inception of the VIX futures indices. The VXX is a non-securitized debt obligation, similar to a zero-coupon bond, but with 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 performance of a position in the nearest and second-nearest maturing VIX futures contracts, which is rebalanced daily to create a nearly one-month maturity. We propose the first stochastic model of the VXX ETN, which accounts for the underlying dynamics of the S&P 500 index (SPX) and the VIX index. This is an important task as the VXX is very heavily traded, but has lost 99.84% of its value since inception. The VIX futures roll yield has been suggested by several authors as the driver of the VXX’s significant under-performance, but none have formalized this argument mathematically, as it is presented in this paper. We use the VIX futures pricing framework of Zhang and Zhu (2006) and Zhang, Shu and Brenner (2010) to derive a stochastic model for the VXX ETN, which leads to the many implications we outline in this paper. Table 1 shows that the VXX has significantly underperformed the VIX index, losing 99.84% of its value since inception, over the same period the VIX index only lost 71.39%. In comparison the SPX had a gain of 162.34%. This underperformance can also be seen graphically in figure 1. Even with the well-documented and easily observed underperformance, the VXX market has made great strides in popularity. In figure 2 we can see that the VXX market has grown significantly since its inception and is still heavily traded. For example, the average daily dollar trading volume for

Modeling VXX

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a randomly chosen month, June 2016, was 1.45 Billion USD. The VXX has been marketed as a diversification tool for equity investors. However, several articles in the literature have shown that equity positions that are combined with VIX Futures and VIX futures ETPs rarely out-perform, and usually under-perform the benchmark portfolio (Alexander and Korovilas, 2012; Deng, McCann and Wang, 2012; Hancock, 2013). The puzzling phenomenon of the highly negative returns of the VXX is also well-documented in the literature (Whaley, 2013). Fernandez-Perez, Frijns, Tourani-Rad and Webb (2015) study the price dynamics of VIX futures. They find that since the introduction of VIX futures ETPs, trades in VIX futures have become less informative and pricing errors have become more persistent. Shu and Zhang (2012) and Frijns, Tourani-Rad and Webb (2015) study the two way Granger causality between VIX and its futures, at the daily and intraday level respectively. At the intraday level Frijns, Tourani-Rad and Webb (2015) find that VIX futures lead the VIX index in price discovery. Bollen, O’Neill and Whaley (2017) show that the VXX leads the VIX futures in price discovery and that VIX futures lead VIX options in price discovery. Bordonado, Moln´ar and Samdal (2016) study the price discovery relationship between 3 pairs of VIX futures ETNs and ETFs, as well as some hedging and trading strategies using these products. In the literature it has been suggested by several articles that the predominantly contango term structure of VIX futures is the cause for the VXX’s underperformance (Alexander and Korovilas, 2013; Liu and Dash, 2012; Whaley, 2013). The roll yield is the return that a futures investor captures when their futures contract converges to the spot price as it matures and is not due to changes in the spot price1 . When the 1

The spot price refers to the price or level of the underlying asset/index of the futures contract, for example the spot price for a VIX futures contract is the level of the VIX index.

Modeling VXX

3

VIX futures market is in backwardation/contango the price rolls up/down to the spot price as maturity approaches; therefore the roll yield will be positive/negative. During normal times the VIX futures market is in contango, but it can be in backwardation during large economic downturns and in this case a long position in the VXX can be profitable, although these profitable periods are short-lived (Whaley, 2013). Although the contango term structure is suggested as the cause of the VXX’s underperformance, until now this argument has not been formalized through a model of VXX returns, nor has the cause of the predominantly negative roll yield been investigated. Simon (2016) designs some VIX options trading strategies to take advantage of the term structure of VIX futures. Eraker and Wu (2017) provide an economic equilibrium model of the variance risk premium and use this to explain the negative returns of VIX futures ETPs, but do not discuss the roll yield. Menc´ıa and Sentana (2013) empirically test the performance of several models for pricing VIX derivatives. In this paper we derive a stochastic model for the VXX return which accounts for the relationship between the SPX, VIX and VXX. The model is based on the VIX futures pricing framework from Zhang and Zhu (2006) and Zhang, Shu and Brenner (2010). Our model shows that the roll yield drives the difference between the constant 30-day to maturity VIX futures contract, which tracks the VIX index closely, and the VXX2 . We establish an explicit relationship between the negative returns of the VXX, the roll yield, and the market price of variance risk. We show that on aggregate when the roll yield is negative, which is the case during normal times, it is because the market price of variance risk is negative. This shows that investors are willing to take these losses in order to be long volatility and benefit from their position during an 2

Figure 1 shows the time series of the VXX, VIX and constant 30-day to maturity VIX futures contract.

Modeling VXX

4

economic downturn. We also provide a new way of estimating the market price of variance risk using the information given by the VXX and VIX futures price data, which can simplify the calibration procedure for many models in the literature. We also explore the effect of the rebalancing frequency of the SPVXSTR index on its returns, as a robustness check for our continuous time model of the daily rebalanced SPVXSTR. The rest of this paper is organized as follows. Section 2 shows the methodology of the SPVXSTR index calculation. In section 3 we review the theory behind pricing VIX futures from Zhang and Zhu (2006) and Zhang, Shu and Brenner (2010) and use this to create a stochastic model for the VXX. We then use this model to show that the roll yield drives the VXX’s underperformance and that the roll yields sign is, on average, driven by the market price of variance risk, in section 4. In section 5, we develop a simple way of estimating the market price of variance risk. We then examine the effect of the rebalancing frequency of the SPVXSTR as a robustness test of our continuous time VXX model in section 6. Lastly, concluding remarks are offered in section 7.

2

The SPVXSTR index

To model the VXX we must first understand the SPVXSTR. In this section, we review the methodology for calculating the SPVXSTR index from S&P Dow Jones Indices (2012). The SPVXSTR index seeks to model the outcome of holding a long position in short-term VIX futures, specifically positions in the nearest and second-nearest maturing VIX futures contracts. The position is rebalanced daily to create a nearly

Modeling VXX

5

one-month maturity VIX futures position (Barclays, 2013). The SPVXSTR index is calculated by: SP V XST Rt = SP V XST Rt−1 (1 + CDRt + T BRt ),

(1)

where SP V XST Rt is the index level at time t, SP V XST Rt−1 is the index level at time t − 1, CDRt is the contract daily return of the VIX futures position and T BRt is the Treasury Bill return earned on the notional value of the position. The T BRt is given by 

1 T BRt = 91 1 − 360 T BARt−1

 ∆91t ,

(2)

where ∆t is the number of calendar days between the current and previous business days. Here, T BARt−1 is the Treasury Bill annual return, which is equal to the most recent weekly high discount rate for 91-day US Treasury Bills effective on the preceding business day. The contract daily return is given by: CDRt =

w1,t−1 FtT1 + w2,t−1 FtT2 − 1, T1 T2 w1,t−1 Ft−1 + w2,t−1 Ft−1

(3)

that is, the CDRt represents the one day discrete return of the underlying VIX futures position3 . Here wi,t−1 is the weight in the ith nearest maturing VIX futures at time t − 1, FtTi is the market price of the ith nearest maturing VIX futures contract at time t. The weights are rebalanced on a daily basis to be: w1,t = 3

dr , dt

(4)

In Equation (3), we put w1,t−1 and w2,t−1 in the numerator. Deng, McCann and Wang (2012) use w1,t and w2,t in their formula for the CDRt , which is inconsistent with the methodology from S&P Dow Jones Indices (2012). When calculating discrete returns of any position, the weights should stay constant over the period for which you are calculating the return and only the prices could change.

Modeling VXX

6

and w2,t = 1 −

dr , dt

(5)

where dr is the number of business days left from the next business day, t + 1, until the business day before the nearest VIX futures settlement date, T1 − 1. Here dt is the total number of business days from the previous VIX futures settlement date, T0 , until the business day before the next VIX futures settlement date, T1 − 1. For the reader’s convenience figure 3 graphically presents the determination of dr and dt. The SPVXSTR methodology mimics a position in the nearest and second-nearest maturing VIX futures contract that is rebalanced every day to maintain a maturity of around one month. Figure 4 shows that the index achieves this goal, as the underlying VIX futures position exhibits a weighted average maturity that fluctuates between 28 and 37 calendar days. The average weighted maturity over the sample period is 31.88 days, with a standard deviation of only 2.4 days. Figure 4 indicates that the underlying futures position time to maturity is nearly one-month, therefore we will use a constant one-month maturity daily rebalanced futures position to model the VXX.

3 3.1

Modeling VXX Review of VIX and VIX futures model

To model the VXX, we need to start with a model for VIX futures. Zhang and Zhu (2006) and Zhang, Shu and Brenner (2010) have developed a model for the VIX and VIX futures using the Heston (1993) framework. We review and combine their results in this section.

Modeling VXX

7

In the physical measure, the SPX St is modeled by: √

P , Vt St dB1,t √ P dVt = κ(θ − Vt )dt + σV Vt dB2,t ,

dSt = µSt dt +

(6) (7)

where Vt is the instantaneous variance of the SPX, µ is the expected return from investing in the SPX, θ is the long-term mean level of the instantaneous variance, κ is the mean-reverting speed of the instantaneous variance and σV measures the P P are two standard Brownian motions that describe and B2,t volatility of variance. B1,t

the random noise in the SPX returns and variance respectively; they are correlated with a constant correlation coefficient ρ. We adopt the standard change of probability measure from the physical one to the risk-neutral one, as follows: µ−r P ∗ dB1,t = dB1,t − √ dt, Vt

P ∗ dB2,t = dB2,t −

λp Vt dt, σV

(8)

where r is the risk-free rate and λ is the market price of variance risk, i.e. λ is the risk premium required by taking the risk of Vt . We thus obtain the risk-neutral dynamics of the SPX as: p ∗ , Vt St dB1,t p ∗ dVt = κ∗ (θ∗ − Vt )dt + σV Vt dB2,t , dSt = rSt dt +

(9) (10)

∗ ∗ where dB1,t and dB2,t are two new standard Brownian motions which are correlated

with the same constant correlation coefficient ρ. The parameter κ∗ is the risk-neutral mean-reverting speed of the instantaneous variance and θ∗ is the risk-neutral longterm mean level of instantaneous variance. The relationship between the physical

Modeling VXX

8

and risk-neutral parameters are given by θ=

κ∗ θ∗ , κ

(11)

and κ∗ = κ + λ.

(12)

The VIX index was designed to measure the markets expectation of the 30-day volatility implied from out-of-the-money SPX options. Under the model described in this section, the squared VIX is equal to the variance swap rate, which is equivalent to the risk-neutral conditional expectation of variance over the next 30 days (Carr and Wu, 2009). Zhang and Zhu (2006) showed that the squared VIX is given by: V

IXt2

=

Et∗



1 τ0

Z

t+τ0

 Vs ds = (1 − B)θ∗ + BVt ,

(13)

t ∗τ

where τ0 = 30/365 years and B = (1 − e−κ

0

)/κ∗ τ0 . Zhang and Zhu (2006) then

obtain the VIX futures price formula as:  p FtT ∗ ∗ ∗ (1 − B)θ + BVT = Et (V IXT ) = Et 100 Z +∞ p = (1 − B)θ∗ + BVT f ∗ (VT |Vt )dVT ,

(14)

0

where the transition probability density of VT conditional on Vt as given by Cox, Ingersoll and Ross (1985) is: f ∗ (VT |Vt ) = ce−u−v

 v q/2 u

√ Iq (2 uv),

(15)

where c=

2κ∗ 2κ∗ θ∗ −κ∗ (T −t) , u = cV e , v = cV , q = − 1, t T σV2 (1 − e−κ∗ (T −t) ) σV2

(16)

Modeling VXX

9

where Iq (.) is the modified Bessel function of the first kind and of order q. Note that T − t is the time-to-maturity, in years, of the VIX futures contract. Equation (14) is an exact formula in an integral form for the VIX futures price based on the Heston(1993) framework, derived by Zhang and Zhu (2006). It appears that the integral can only be computed numerically. Zhang, Shu and Brenner (2010) provide us with a good analytical approximation of the exact formula, equation (14), as follows: FtT = F0 + F1 + F2 , 100

(17)

where ∗ (T −t)

F0 = [θ∗ (1 − Be−κ ∗

∗

] ,

(18)

3

[θ∗ (1 − Be−κ (T −t) ) + Vt Be−κ (T −t) ]− 2 8   −κ∗ (T −t) 2 −κ∗ (T −t) ) ∗ (1 − e 2 −κ∗ (T −t) 1 − e +θ × B Vt e , κ∗ 2κ∗

(19)

5 σV4 ∗ ∗ ∗ [θ (1 − Be−κ (T −t) ) + Vt Be−κ (T −t) ]− 2 16   ∗ −κ∗ (T −t) 2 1 ∗ (1 − e−κ (T −t) )3 ) −κ∗ (T −t) (1 − e 3 3 Vt e + θ ×B , 2 κ∗ 2 2 κ∗ 2

(20)

F1 = −

F2 =

σV2

∗ (T −t) 1 2

) + Vt Be−κ

and F1 + F2 is a convexity adjustment from the Taylor series expansion of equation (14). In order to discuss the different VIX futures approximations we define: A0 :

FtT 100

= F0 ,

(21)

A1 :

FtT 100

= F0 + F1 ,

(22)

A2 :

FtT 100

= F0 + F1 + F2 ,

(23)

and A3 as the exact VIX futures formula, equation (14).

Modeling VXX

3.2

10

Nearly 30-day VIX futures

We now make a case for using an approximation of equation (17) for a nearly 30day-to-maturity VIX futures contract. We show that for such a short maturity the convexity adjustments, F1 +F2 , in A2 make little difference to the result, by calculating the error of the different approximations (i.e. A0 , A1 and A2 ) compared to the exact formula (i.e. A3 ) for a range of reasonable parameter estimates. Table 2 presents the results of our numerical computation of the VIX futures prices using the exact formula, A3 , the closed-form approximation, A2 , and the two simplifications of the closed-form approximation, A0 and A1 . We show that for calculating the 30-day VIX futures price using just A0 creates a very small error when compared to the exact formula A3 . Table 2 shows that this error is always within 3% when θ∗ = 0.1, Vt ranges from 0.04 to 0.2, κ∗ ranges from 4 to 7 and σV ranges from 0.1 to 0.7. There is one outlier when Vt = 0.04, κ∗ = 4 and σV = 0.7, but the error is still small at 3.20%. The root mean squared error (RMSE) from using A0 to approximate the exact formula is 1.29%. It is also interesting to note that the full analytical approximation, A2 , approximates the exact formula very well with a RMSE of 0.23%, for the 30-day-to-maturity VIX futures4 . Given the numerical evidence in Table 2 we can approximate the price of nearly 30-day-to-maturity VIX futures, using A0 , as: 1 FtT ∗ ∗ = [θ∗ (1 − Be−κ (T −t) ) + Vt Be−κ (T −t) ] 2 , 100

(24)

with some small error compared with the exact VIX futures price formula A3 . 4 ∗

θ is set to 0.1 because this is larger than the average value of 0.045 estimated by Luo and Zhang (2012) by using data from 2 Jan 1992 to 31 Aug 2009 and the error of the VIX futures price formula is proportional to θ∗ therefore we are allowing for more error than if we used their θ∗ estimate.

Modeling VXX

11

Figure 5 shows a theoretical term structure of VIX futures prices using A0 , A1 and A2 . We use parameter estimates of θ∗ = 0.1, κ∗ = 5, σV = 0.1425 and Vt = 0.06 to create one example of an upward-sloping VIX futures term structure and so that they fall within the reasonable parameter range we examined in table 2. The difference between the points at t + 30 and t + 29 represents the one-day roll yield of the daily rebalanced one month maturity VIX futures position (i.e. the SPVXSTR index), given these parameter inputs. The spot return is zero when the underlying instantaneous variance Vt is constant, which means that any observed return in this figure is due solely to the roll yield of VIX futures. It can be seen in the diagram that as you step through time from 30 days to maturity to 29 days to maturity, the return will be negative; therefore the one-day roll yield will be negative when the term structure is upward sloping. We then take the natural logarithm in equation (24) to get an expression for the natural log price of nearly 30-day-to-maturity VIX futures given by  ln

3.3

FtT 100

 =

1 ∗ ∗ ln[θ∗ (1 − Be−κ (T −t) ) + Vt Be−κ (T −t) ]. 2

(25)

Model of contract daily return

We can now derive an expression for the daily return of the VIX futures position tracked by the SPVXSTR index, the contract daily return (CDRt ), using the simple log VIX futures price approximation in (25). We can model the change of the nearly 30-day log VIX futures price, using Ito’s Lemma as: d ln FtT =

∂ ln FtT 1 ∂ 2 ln FtT ∂ ln FtT 2 dVt + (dV ) + dt. t ∂Vt 2 ∂Vt2 ∂t

(26)

Modeling VXX

12

where ∂ ln FtT /∂t is defined as the roll yield and the other terms are equal to the return of the constant 30-day-to-maturity VIX futures price5 . The roll yield of the SPVXSTR index is the return of the underlying VIX futures position due to the maturity of the position changing from 30 days to 29 days, that is, it is the return between rebalancing dates which is not due to changes in the spot price. Calculating the partial derivatives and substituting into (26) one gets: d ln FtT

 −1 1 θ∗ ∗ = − θ + Vt dVt 2 Be−κ∗ (T −t)  −2 1 θ∗ ∗ − (dVt )2 ∗ (T −t) − θ + Vt −κ 4 Be   ∗ 1 κ∗ (Vt − θ∗ )Be−κ (T −t) dt. + 2 θ∗ + (Vt − θ∗ )Be−κ∗ (T −t)

(27)

The SPVXSTR index is rebalanced daily to maintain a VIX futures position with one-month maturity; therefore we can model the contract daily return CDRt of the underlying futures position as the log return of a nearly 30-day-to-maturity VIX futures position. Therefore using equation (27) we can get the CDRt process in proposition 1 below. Proposition 1. We can model the contract daily return (CDRt ) of the SPVXSTR as the log return of a nearly 30-day-to-maturity VIX futures position, therefore our model of the CDR is given by: CDRt = d ln FtT

5

T =t+τ0

= d ln Ftt+τ0 + RYt dt,

(28)

The constant 30-day-to-maturity VIX futures price represents a VIX futures contract that has a constant maturity of 30-days. We estimate this price through linear interpolation and show time series of this price in figure 1.

Modeling VXX

13

where d ln Ftt+τ0

 −1 1 θ∗ ∗ = − θ + Vt dVt 2 Be−κ∗ τ0  −2 1 θ∗ ∗ − − θ + Vt (dVt )2 , 4 Be−κ∗ τ0

(29)

and   ∗ 1 κ∗ (Vt − θ∗ )Be−κ τ0 , RYt = 2 θ∗ + (Vt − θ∗ )Be−κ∗ τ0

(30)

where τ0 = 30/365, d ln Ftt+τ0 is the change in the log price of a constant 30-day-tomaturity VIX futures contract, and RYt is the one-day roll yield of the SPVXSTR index going from 30-day maturity to 29-day maturity. Our model of the CDR takes the time step between rebalancing from daily to continuous. If we isolate only the effect of time (the roll yield) of our CDR model and converting it to the discrete time, where ∆t = 1 (i.e. one day), we get the methodology of calculating the CDR from section 2, as shown by RYt dt =

3.4

T lnFtT − lnFt−∆t ∂ ln FtT dt ≈ ∆t ∂t ∆t  T  Ft = ln T Ft−∆t FT ≈ Tt − 1. Ft−∆t

(31)

VXX model

The change in the SPVXSTR index, and therefore the VXX, is composed of the return of the futures position, the CDRt , and a risk-free return on the notional of the futures position, T BRt . Therefore by combining CDRt and a risk-free return we create a model for the SPVXSTR index and consequently for the VXX.

Modeling VXX

14

Proposition 2. We can model the VXX using the CDRt combined with a risk-free return r; d ln V XXt = CDRt + rt dt = =

d ln Ftt+τ0



d ln FtT

T =t+τ0

+ rt dt (32)

+ RYt dt + rt dt

where rt is the risk-free return on the notional value of the futures position on day t. This model of the log VXX price is, to our knowledge, the first attempt in the literature to model the VXX whilst accounting for the intrinsic relationships between the SPX, VIX and VXX. This model can be used to derive the market price of variance risk λ from VXX returns, as described in section 5. We could also use this model to price VXX options, which are essentially Asian options on the underlying instantaneous variance, Vt . In the next section, we use our VXX model to show that the roll yield drives the difference between VIX and VXX returns.

4

VXX roll yield

Whaley (2013), Deng, McCann and Wang (2012) and Husson and McCann (2011) all suggest the roll yield as the reason for the VXX’s underperformance. However these articles never formalize this argument. Figure 1 shows us a comparison between the performance of the VIX, the VXX and a constant 30-day-to-maturity VIX futures contract. The figure shows that the constant 30-day-to-maturity VIX futures contract follows the VIX index very closely but the VXX underperforms these two significantly. From equation (32), we know that the difference between the VXX return and the constant 30-day-to-maturity VIX futures return is equal to the roll yield and riskfree return. Therefore the difference between the VXX and the constant 30-day-tomaturity VIX futures price in figure 1 at any point in time is the roll yield cumulated

Modeling VXX

15

since inception (less the initial difference and the small risk free return). Since the roll yield is negative during normal times the VXX significantly underperforms the VIX index and the constant 30-day-to-maturity VIX futures.

4.1

Roll yield and the market price of variance risk

We reserve t to denote current time and use a new variable s to denote future time, where s > t. The differential form for the stochastic future instantaneous variance Vs is given by equation (7) and in integral form it is given as: Z s p −κ(s−t) P Vs = θ + (Vt − θ)e + σV e−κ(s−u) Vu dB2,u .

(33)

t

Substituting equation (33) into the future instantaneous roll yield formula, equation (30) with t replaced by s, we get the future instantaneous roll yield as: # " Rs √ ∗ P ] − θ∗ )Be−κ τ0 1 κ∗ (θ + (Vt − θ)e−κ(s−t) + σV t e−κ(s−u) Vu dB2,u Rs √ . RYs = P 2 θ + (Vt − θ)e−κ(s−t) + σV t e−κ(s−u) Vu dB2,u ] − θ∗ )Be−κ∗ τ0

(34)

We then take the Taylor series expansion of equation (34) with respect to σV (see Lewis, 2000, chapter 3) and obtain:   ∗ 1 κ∗ (θ + (Vt − θ)e−κ(s−t) − θ∗ )Be−κ τ0 ∂RYs RYs = + 2 θ∗ + (θ + (Vt − θ)e−κ(s−t) − θ∗ )Be−κ∗ τ0 ∂σV

σV + O(σV2 ), (35) σV =0

where ∂RYs = ∂σV

∗

1 κ∗ Be−κ τ0 dt 2 θ∗ +(θ+(Vt −θ)e−κ(s−t) −θ∗ )Be−κ∗ τ0

− 12

κ∗ (θ−θ∗ )Be−κ

×

∗τ 0 dt 2

(θ∗ +(θ+(Vt −θ)e−κ(s−t) −θ∗ )Be−κ∗ τ0 )

√ −κ(s−u) P Vu dB2,u e t Rs √ P × t e−κ(s−u) Vu dB2,u . Rs

We can then compute the expected roll yield conditional on the information available at time t as:   ∗ 1 κ∗ (θ + ξt − θ∗ )Be−κ τ0 + O(σV2 ), Et [RYs ] = 2 θ∗ + (θ + ξt − θ∗ )Be−κ∗ τ0

(36)

Modeling VXX

16

where ξt = (Vt − θ)e−κ(s−t) .

(37)

We can then evaluate the average expected roll yield for the period from t to s as: 1 Et [RYs ] = s−t

Z

s

Et [RYu ]du.

(38)

t

Taking the limit of equation (38) as s → ∞ and using L’Hˆopital’s rule yields the long-term average expected roll yield as: RY

Z s 1 Et [RYu ]du = lim Et [RYs ] = lim s→∞ s→∞ s − t t   ∗ 1 κ∗ (θ − θ∗ )Be−κ τ0 = lim Et [RYs ] = s→∞ 2 θ∗ + (θ − θ∗ )Be−κ∗ τ0 ∗ ∗ 1 λκκ Be−κ τ0 = = RY∞ 2 1 + λκ Be−κ∗ τ0

(39)

where RY∞ denotes the long-term roll yield, which is independent of the information at time t. The long-term roll yield can be regarded as the unconditional expectation of the roll yield. As all of the parameters apart from λ are positive and κ∗ = κ + λ > 0 (Zhang, Shu and Brenner, 2010), from equation (39) we can see that the sign of λ, the market price of variance risk, is the driver of sign for the one-day VXX roll yield, on average. We conclude that the usually negative roll yield of the VXX is driven by the usually negative (as shown in table 3) market price of variance risk λ. The market price of variance risk tends to be negative as investors are willing to accept losses in normal times to insure their positions against high volatility events (market crashes).

Modeling VXX

4.2

17

The market price of variance risk and the variance risk premium

In the recent literature, the existence of a variance risk premium (VRP) is welldocumented. For example, Coval and Shumway (2001) use classical asset pricing theory to study expected option returns. They show that zero beta at-the-money straddles which are long positions in volatility suffer weekly losses on average of about 3%. Bakshi and Kapadia (2003) construct delta hedged portfolios to empirically show that the VRP is negative. While, Carr and Wu (2009) calculate the VRPs for many different stock market indices through replicating variance swaps using options; they find that the VRP on average is negative. Bondarenko (2013) propose a new strategy for replicating discretely sampled realised variance. They empirically study the price of the variance contract using SPX options from January 1990 to December 2009 and find a negative VRP which cannot be explained by known risk factors and options returns. Many models with a stochastic volatility process have a parameter called the market price of variance risk which is used in changing the probability measure from risk neutral to the physical one. Papers which use this concept include, but are not limited to, Johnson and Shanno (1987) Hull and White (1987), Scott (1987) and Heston (1993). The market price of variance risk is estimated by Lin (2007), Duan and Yeh (2010) and Zhang and Huang (2010) and found to be negative as expected. Zhang and Huang (2010) show that the market price of variance risk λ from the Heston (1993) framework, is almost proportional to the VRP, as defined by Carr and

Modeling VXX

18

Wu (2009), as long as λτ0 is small (relative to one). Their result is shown by:  V RP =

    1 ∗ 1 1 ∗ ∗2 ∗ ∗2 2 κ τ0 + O(κ τ0 ) θ + − κ τ0 + O(κ τ0 ) Vt λτ0 + O(λ2 τ02 ), 6 2 3 (40)

We can now interpret the result from section 4.1 in terms of the VRP; the usually negative roll yields of the VXX is driven by the usually negative λ, which is almost proportional to the VRP.

5

Estimating the market price of variance risk

In order to complete the estimation of the Heston (1993) model we have to estimate λ, the market price of variance risk. The market price of variance risk is not observable and there is no clear consensus on the method of its estimation, so far. Table 3 shows some different authors recent estimates for λ and the sample period used. We can see from table 3 that the estimation of λ varies substantially. In this section we develop a new methodology to estimate λ using our model for the VXX. We can rewrite our VXX model in equation (32) as:  d ln

V XXt Ftt+τ0

 = RYt dt + rt dt,

(41)

or, in integral form: 1 T

  Z Z V XXT FTT +τ0 1 T 1 T ln − ln 0+τ0 = RYt dt + rt dt V XX0 T 0 T 0 F0 = RY + rt ,

(42)

where RY is the average roll yield over the sample period from 0 to T , which can be approximated by the long-term average roll yield given by equation (39), and rt is the average risk-free rate over the period.

Modeling VXX

19

¯ as: We define a new variable λ ∗ ∗ ¯ = λκ = λκ , λ κ κ∗ − λ

(43)

and plugging this into equation (39) we get the long-term average roll yield as: ¯ −κ∗ τ0 1 λBe . RY = 2 1 + λ¯∗ Be−κ∗ τ0 κ

(44)

We then solve for λ which leads to proposition 3 below. Proposition 3. We can use VXX prices, constant 30-day-to-maturity VIX futures prices, an estimate of κ∗ and the risk-free rate to estimate λ, as given by: λ=

¯ ∗ λκ ¯, κ∗ + λ

(45)

where ¯= λ

2RY (1 −

2RY κ∗

)Be−κ∗ τ0

,

(46)

with the long-term average roll yield estimated as the average over the sample: 1 RY = T

  FTT +τ0 V XXT − ln 0+τ0 − rt . ln V XX0 F0

(47)

To demonstrate this methodology we collect VXX and VIX futures market prices and the 91 day T-Bill rate for the period from 30 January 2009 to 27 July 2016, from the Bloomberg Professional service. We use the κ∗ estimate of 5.4642 from Luo and Zhang (2012), as this is the most recent estimate in the literature. The market prices are V XXT = 43, V XX0 = 25600, FTT +τ0 = 15.3632 and F00+τ0 = 42.4871 and the average 91 day T-Bill rate over the period is rt = 0.09%. The length of the sample period is T = 7.4959 years and the average roll yield is estimated as RY = −0.7175. Using these inputs in equations (45) and (47) gives us a market price of variance risk

Modeling VXX

20

estimate of λ = −3.7127. This estimate of λ is consistent with prior estimates, refer to table 3. Our method for estimating the market price of variance risk λ makes the calibration of the Heston model simpler, as λ is now a function of VXX prices, VIX futures prices, the risk-free rate and κ∗ .

6

Rebalancing frequency of SPVXSTR

In this section, we explore the effect of the rebalancing frequency of the SPVXSTR index. The effect of rebalancing frequency matters for the robustness of our VXX model. Our model assumes continuous rebalancing, whereas the index is rebalanced at a daily frequency. Therefore, investigating whether the rebalancing frequency affects the level of the index significantly, going from daily rebalancing to more frequent than daily, is important for the robustness of our model. We start by replicating the SPVXSTR index using VIX futures prices from the 20 December 2005 until the 28 March 2014, with the methodology from S&P Dow Jones Indices (2012). This replicated SPVXSTR time series is displayed in figure 6, along with the actual SPVXSTR time series over our sample6 . The actual and replicated indices are almost identical, showing that our replication is adequate. Figure 7 shows four time series of the replicated SPVXSTR index with different rebalancing frequencies of daily, weekly, two-weekly and monthly rebalancing. The figure shows that as the rebalancing frequency is decreased from daily to weekly, twoweekly and monthly, the SPVXSTR’s performance decreases. If this effect exists going from daily to more frequent rebalancing, for example hourly, then this would be a 6

VIX futures prices are available at http://cfe.cboe.com/Data/HistoricalData.aspx#VX; accessed on the 20 April 2014.

Modeling VXX

21

problem for our VXX model which assumes the SPVXSTR is rebalanced continuously. To examine the effect of the rebalancing frequency on the price of the VXX for smaller time steps than daily, we chose to simulate a five-year-long hourly VIX futures price time series. To simulate the hourly time series of VIX futures prices, we first need a time series of instantaneous variance, which we get from the physical measure stochastic process of instantaneous variance, given by dVt = κ(θ − Vt )dt + σV

p Vt dB.

(48)

We then use the full closed-form approximate VIX futures price formula A2 to find a time series of nearest and second-nearest maturing VIX futures prices. As our parameter inputs we use κ∗ = 5.4642, λ = −3.7127, θ = 0.1 and σV = 0.4. For simplicity, we assume that VIX futures mature every 28 days, that there are no nontrading days, trading hours are 24 hours of the day and that the risk-free rate is zero. We then use the methodology from section 2 to calculate the SPVXSTR index for five years with different rebalancing frequencies and a starting value of one. Figure 8 shows the resulting SPVXSTR hourly time series for different rebalancing frequencies ranging from hourly to monthly. We can see in figure 8 that the simulated SPVXSTR time series for hourly and daily rebalancing are almost identical. The rebalancing effect going from daily to hourly rebalancing is very small and therefore not a problem for our continuously rebalanced VXX model. There is, however, a rebalancing effect if the index is rebalanced less often than daily, which is consistent with our findings using market VIX futures prices in figure 7. To show that our conclusion on the rebalancing frequency is robust to the term structure of VIX futures, we repeated the above exercise but holding Vt constant at different levels. This allows us to create a time series of SPVXSTR with an always upward-sloping (contango)

Modeling VXX

22

VIX futures term structure, as shown in figure 9, and an always downward-sloping (backwardation) VIX futures term structure, as shown in figure 10. From figures 9 and 10, we can see that the rebalancing frequency does not significantly impact the SPVXSTR going from daily to hourly rebalancing. However, there is a significant effect when going to less frequent rebalancing. Figures 9 and 10 also show the importance of the roll yield as a driver of the SPVXSTR and subsequently the VXX. The two figures isolate the effect of the term structure and therefore the roll yield, on the returns of the SPVXSTR by holding Vt constant. When the term structure is upward sloping, generating a negative roll yield, the simulated level of the SPVXSTR will tend to zero as in figure 9, and when the term structure is downward sloping, generating a positive roll yield, the simulated level of the SPVXSTR is exponentially increasing as in figure 10.

7

Conclusions and discussions

In this paper we use the VIX futures price approximation from Zhang, Shu and Brenner (2010) and simplify it for the nearly 30-day VIX futures contract. From this simplified formula for VIX futures prices, we develop a simple model for the VXX, which accounts for the fundamental underlying dynamics of the SPX index and the VIX index. All of our results are robust to the choice of process for the instantaneous variance Vt or a time dependent long-term mean level of variance θt as in Zhang, Shu, and Brenner (2010). Our model can be very easily applied to any VIX futures ETPs that use the SPVXSTR as the underlying benchmark. Our VXX model shows that the underperformance of the VXX, when compared to the VIX index and constant 30-day-to-maturity VIX futures price, is due to the

Modeling VXX

23

roll yield as suggested but not formalized in the previous literature. One could use a similar approach to explore the effect of the roll yield on other longer term VIX futures ETNs, but we advise caution in using the simplified closed-form VIX futures price formula A0 and recommend using the full approximation A2 in this case. This is because the convexity adjustments from A2 , which are excluded in A0 , will have more impact as the maturity increases. We also examined the average roll yield of the VXX and showed that the usually negative market price of variance risk λ is the main driver of the VXX roll yield’s sign, on average. From Zhang and Huang (2010) we know that the market price of variance risk is almost proportional to the variance risk premium, so another interpretation of this result is that the usually negative variance risk premium is the main driver of the usually negative VXX roll yield’s sign. Our continuously rebalanced VXX model is adequate for modeling the daily rebalanced VXX, as the effect of the rebalancing frequency is only sizeable at less frequent than daily rebalancing. The VXX model presented in this article could also be used to price options written on the VXX. VXX options can be regarded as Asian options written on the underlying instantaneous variance of the SPX. Bao, Li, and Gong (2012) have created a model for pricing VXX options, but they do not account for the dynamics of the SPX or the VIX, they just model the VXX as a standalone stochastic process. Further research is needed into the calibration technique best used for our model and its accuracy, although it is theoretically sound. Exploring similar approaches to the one in this paper to create models of other VIX futures ETPs could help further develop the literature around these popular investment products.

Modeling VXX

24

References Alexander, C., & Korovilas, D. (2012). Diversification of equity with VIX futures: Personal views and skewness preference. Available at SSRN: https://ssrn.com/abstract=2027580. Alexander, C., & Korovilas, D. (2013). Volatility exchange-traded-notes: Curse or cure? Journal of Alternative Investments, 16(2), 52-70. Bakshi, G., & Kapadia, N. (2003). Volatility risk premiums embedded in individual equity options: Some new insights, Journal of Derivatives, 11(1), 45-54. Bao, Q., Li, S., & Gong, D. (2012). Pricing VXX option with default risk and positive volatility skew. European Journal of Operational Research, 223(1), 246-255. Barclays, (2013). VXX and VXZ Prospectus, Available at: www.ipathetn.com/US /16/en/details.app?instrumentId=259118 . Bollen, N. P., O’Neill, M. J., & Whaley, R. E. (2017). Tail wags dog: Intraday price discovery in VIX markets. Journal of Futures Markets, 37(5), 431-451 Bondarenko, O. (2014). Variance trading and market price of variance risk. Journal of Econometrics, 180(1), 81-97. Bordonado, C., Molnr, P., & Samdal, S. R. (2016). VIX exchange traded products: Price discovery, hedging, and trading strategy. Journal of Futures Markets, 20(10) 1-20. Carr, P., & Wu, L. (2009). Variance risk premiums. Review of Financial Studies, 22(3), 1311-1341. Coval, J. D., & Shumway, T. (2001). Expected option returns. Journal of Finance, 56(3), 983-1009. Cox, J., Ingersoll Jr, J., & Ross, S. (1985). A theory of the term structure of interest

Modeling VXX

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rates. Econometrica, 53(2), 385-407. Deng, G., McCann, C. J., & Wang, O. (2012). Are VIX futures ETPs effective hedges? Journal of Index Investing, 3(3), 35-48. Duan, J., & Yeh, C. (2010). Jump and volatility risk premiums implied by VIX. Journal of Economic Dynamics and Control, 34(11), 2232-2244. Eraker, B., & Wu, Y. (2017). Explaining the negative returns to VIX futures and ETNs: An equilibrium approach. Journal of Financial Economics, Forthcoming. Fernandez-Perez, A., Frijns, B., Tourani-Rad, A., & Webb, R. (2015) Did the introduction of ETPs change the intraday price dynamics of VIX futures? Working paper, Auckland University of Technology. Frijns, B., Tourani-Rad, A., & Webb, R. (2015). On the intraday relation between the VIX and its futures. Journal of Futures Markets, 36(9), 870-886. Hancock, G. D. (2013). VIX futures ETNs: Three dimensional losers. Accounting and Finance Research, 2(3), 53-64. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2), 327-343. Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatilities. Journal of Finance, 42(2), 281-300. Husson, T., & McCann, C. J. (2011). The VXX ETN and volatility exposure. PIABA Bar Journal, 18(2). Johnson, H., & Shanno, D. (1987). Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis, 22(2), 143-151. Lewis, A. (2000). Option valuation under stochastic volatility. Finance Press.

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Lin, Y. N. (2007). Pricing VIX futures: Evidence from integrated physical and riskneutral probability measures. Journal of Futures Markets, 27(12), 1175-1217. Luo, X., & Zhang, J. E. (2012). The term structure of VIX. Journal of Futures Markets, 32(12), 1092-1123. Menc´ıa, J., & Sentana, E. (2013). Valuation of VIX derivatives. Journal of Financial Economics, 108(2), 367-391. Scott, L. O. (1987). Option pricing when the variance changes randomly: Theory, estimation, and an application. Journal of Financial and Quantitative Analysis, 22(4), 419-438. Shu, J., & Zhang, J. E. (2012). Causality in the VIX futures market. Journal of Futures Markets, 32(1), 24-46. Simon, D. P. (2016). Trading the VIX futures roll and volatility premiums with VIX options. Journal of Futures Markets, 37(2), 184-208. S&P Dow Jones indices (2012). S&P 500 VIX futures indices methodology. White paper available at: us.spindices.com/indices/strategy/sp-500-vix-short-term-indexmcap . Whaley, R. E., (2013). Trading volatility: At what cost? Journal of Portfolio Management, 40(1), 95-108. Zhang, J. E., & Huang, Y. (2010), The CBOE S&P 500 three-month variance futures. Journal of Futures Markets, 30(1), 48-70. Zhang, J. E., Shu, J., & Brenner, M. (2010). The new market for volatility trading. Journal of Futures Markets, 30(9), 809-833. Zhang, J. E., & Zhu, Y. (2006). VIX futures. Journal of Futures Markets, 26(6), 521-531.

Modeling VXX

27

Table 1: Summary statistics of the daily returns for the SPX, VIX and VXX. This table shows the summary statistics and correlations of the VXX, SPX and VIX index returns from the 30 January 2009 to the 27 July 2016. Here, rD represents estimates using discrete daily returns and rC represents estimates using continuously compounded daily returns. The annualised standard deviation is calcu√ lated by multiplying the standard deviation by 252. The Holding Period Return (HPR) is the return from the first day to the last day of the sample. The Compound Annual Growth Rate (CAGR) is the constant yearly growth rate that would lead to 1 the corresponding HP R, it is calculated by CAGR = (HP R + 1) T − 1, where T is the length of the sample in years. SPX rD Mean σ Annualised σ Skewness Excess kurtosis HPR CAGR

0.06% 1.10% 17.43% −0.14 4.43 162.34% 13.76%

Correlations SPX VIX VXX

VIX rC

rD

VXX rC

0.05% 0.20% −0.07% 1.10% 7.55% 7.28% 17.44% 119.87% 115.64% −0.24 1.33 0.76 4.42 5.65 3.53 96.45% −71.39% −125.13% − −15.40% − rD

SPX 1 − −

VIX −0.77 1 −

rD

rC

−0.26% −0.34% 4.10% 4.06% 65.05% 64.39% 0.89 0.51 4.99 4.46 −99.84% −643.39% −57.69% − rC

VXX −0.77 0.85 1

SPX 1 − −

VIX −0.78 1 −

VXX −0.78 0.85 1

Modeling VXX

28

Table 2: 30-day VIX futures price estimation. This table shows the VIX futures price estimates using the four different formula and a range of parameter estimates for Vt , σV and κ∗ . For this exercise we keep the time-to-maturity constant at 30 days, τ = τ0 = 30/365 and θ∗ constant at 0.10. The first four columns show the hypothetical θ∗ , Vt , σV and κ∗ parameters used in the futures price estimates. The labels of each of the VIX futures price formula, A0 , A1 , A2 and A3 , are defined in section 3.1. The columns labeled % error, are the percentage differences of the preceding column of futures prices to the prices estimated by the exact formula A3 . The RMSE stands for the root of mean squared error. Parameters θ

∗

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

VIX Futures Price Estimates

Vt

∗

κ

σV

A0

0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

4 4 4 5.5 5.5 5.5 7 7 7 4 4 4 5.5 5.5 5.5 7 7 7 4 4 4 5.5 5.5 5.5 7 7 7

0.1 0.4 0.7 0.1 0.4 0.7 0.1 0.4 0.7 0.1 0.4 0.7 0.1 0.4 0.7 0.1 0.4 0.7 0.1 0.4 0.7 0.1 0.4 0.7 0.1 0.4 0.7

25.14 25.14 25.14 26.32 26.32 26.32 27.26 27.26 27.26 33.51 33.51 33.51 33.20 33.20 33.20 32.95 32.95 32.95 40.17 40.17 40.17 38.88 38.88 38.88 37.79 37.79 37.79

0.07% 1.08% 3.20% 0.05% 0.77% 2.27% 0.04% 0.57% 1.69% 0.05% 0.82% 2.53% 0.04% 0.67% 2.04% 0.03% 0.55% 1.66% 0.04% 0.62% 1.94% 0.03% 0.55% 1.69% 0.03% 0.48% 1.47%

25.12 24.86 24.30 26.31 26.12 25.69 27.25 27.11 26.78 33.49 33.24 32.67 33.19 32.98 32.53 32.94 32.77 32.39 40.15 39.92 39.41 38.87 38.67 38.23 37.77 37.61 37.23

-

1.29%

-

RMSE

% error

A1

% error 0.00% −0.02% −0.25% 0.00% −0.02% −0.17% 0.00% −0.01% −0.12% 0.00% 0.00% −0.02% 0.00% 0.00% −0.04% 0.00% 0.00% −0.04% 0.00% 0.00% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% −0.01% 0.06%

A2

% error

A3

25.12 24.89 24.56 26.31 26.13 25.85 27.25 27.12 26.88 33.49 33.25 32.83 33.19 32.99 32.64 32.94 32.78 32.48 40.15 39.93 39.51 38.87 38.68 38.32 37.77 37.61 37.30

0.00% 0.09% 0.82% 0.00% 0.05% 0.42% 0.00% 0.03% 0.24% 0.00% 0.05% 0.46% 0.00% 0.03% 0.31% 0.00% 0.02% 0.21% 0.00% 0.03% 0.27% 0.00% 0.02% 0.21% 0.00% 0.02% 0.16%

25.12 24.87 24.36 26.31 26.12 25.74 27.25 27.11 26.81 33.49 33.24 32.68 33.19 32.98 32.54 32.94 32.77 32.41 40.15 39.92 39.40 38.87 38.67 38.24 37.77 37.61 37.24

-

0.23%

-

Modeling VXX

29

Table 3: Estimates of λ and κ∗ by various authors. This table shows the estimated value of λ and κ∗ from different authors using different sample periods and estimation methods.

†

Author

Data period

κ∗

λ

Lin (2007) Duan and Yeh (2010) Zhang and Huang (2010) Luo and Zhang (2012) Our estimation

21 Apr 2004 - 18 Apr 2006 2 Jan 2001 - 29 Dec 2006 18 May 2004 - 17 Aug 2007 2 Jan 1992 - 31 Aug 2009 30 Jan 2009 - 27 July 2016

5.3500 -1.7956 1.2989 5.4642 5.4642

-0.3528 -7.5697 -19.1184 †

-3.7127

‡

Luo and Zhang (2012) do not give the estimate for λ. We use the κ∗ = 5.4642 estimate from Luo and Zhang (2012) and assume that it is accurate for our sample period.

‡

Modeling VXX

30

Figure 1: Historical VIX, constant 30-day-to-maturity VIX futures price and VXX price. This figure shows the level of the VIX and the price of the constant 30-day-to-maturity VIX futures on the primary vertical axis and the VXX price on the secondary vertical axis. The 30-day VIX futures contract is the linearly interpolated price of a constant 30-day-to-maturity VIX futures contract, as in Zhang, Shu and Brenner (2010).

Modeling VXX

31

Figure 2: Market capitalization and dollar volume of VXX. This figure shows the daily dollar trading volume and market capitalization of the VXX from the 30 January 2009 to the 27 July 2016, in billions of US dollars.

Modeling VXX

32

Figure 3: Understanding the SPVXSTR roll period. This diagram shows how dr and dt are determined for the calculation of the weights of the VIX futures position underlying the SPVXSTR index. Here, Ti is the settlement date of the ith nearest maturing VIX futures. Here, Ti −1 is the day before ith nearest maturing VIX futures settlement and the last day of the roll period. On the last day of the roll period the nearest settling VIX futures position is eliminated and the second-nearest settling VIX futures contract becomes the nearest. The dr and dt are the factors used in the calculation of the weights of each of the VIX futures contracts in the SPVXSTR, as illustrated in section 2. The roll period represents the time during which the weight in the nearest settling VIX futures contract is gradually replaced by a position in the second-nearest VIX futures contract. At the end of the roll period all the weight will be in the second-nearest VIX futures contract which then becomes the nearest. As the previous nearest contract matures the next roll period starts and the process is repeated.

Modeling VXX

33

Figure 4: Weighted maturity of the SPVXSTR index’s underlying position. This figure shows the weighted maturity of the VIX futures position underlying the SPVXSTR index. The weighted maturity is calculated by taking a weighted average of the nearest and second-nearest VIX futures time-to-maturity (in calendar days). The weights in the weighted average are the same as the weights used to calculate the CDRt , w1,t and w2,t from equation (3).

Modeling VXX

34

Figure 5: VIX futures term structure. This figure shows a hypothetical term structure of VIX futures prices from 1 day to 50 day maturity calculated using the different VIX futures price approximations (i.e. A0 , A1 and A2 ). These estimated VIX futures prices are calculated using the constant parameter estimates of θ∗ = 0.1, κ∗ = 5, σV = 0.1425 and Vt = 0.06. Time-to-maturity varies.

Modeling VXX

35

Figure 6: Replicated vs actual SPVXSTR. This figure shows the actual SPVXSTR time series and our replicated SPVXSTR time series using the methodology from S&P Dow Jones Indices (2012) from the 20 December 2005 until the 28 March 2014.

Modeling VXX

36

Figure 7: Replicated SPVXSTR with different rebalancing frequencies. This figure shows four different time series of our replication of the SPVXSTR index. “SPVXSTR daily” corresponds to daily, “SPVXSTR weekly” to weekly, “SPVXSTR two-weekly” to two weekly and “SPVXSTR monthly” to monthly rebalancing. The final values of the indices are 1178.63 for daily, 1088.38 for weekly, 842.24 for twoweekly and 264.60 for monthly rebalancing.

Modeling VXX

37

Figure 8: Simulated SPVXSTR index using physical process for Vt . This figure shows the simulated SPVXSTR index over our 4 year simulation period using V0 = 0.02, σV = 0.4, λ = −3.7127 the risk-neutral parameter estimates κ∗ = 5.4642 and θ∗ = 0.1, the physical process of dVt as described in equation (48) and the simple VIX futures price formula, A0 . The label of each time series corresponds to the rebalancing frequency used.

Modeling VXX

38

Figure 9: Simulated SPVXSTR index using Vt = θ < θ ∗ . This figure shows the time series of the simulated SPVXSTR index, when the instantaneous variance is set constant at Vt = θ = 0.0476 (< θ∗ = 0.1), forcing an upward sloping VIX futures term structure. To calculate the futures prices we use σV = 0.4, κ∗ = 5.4642 and θ∗ = 0.1 and the simple VIX futures price formula, A0 . The label of each time series corresponds to the rebalancing frequency used.

Modeling VXX

39

Figure 10: Simulated SPVXSTR index using Vt = 0.14 > θ ∗ . This figure shows the time series of the simulated SPVXSTR, when Vt is set constant at Vt = 0.14 (θ∗ = 0.1), forcing a downward sloping VIX futures term structure. To calculate the futures prices we use σV = 0.4, κ∗ = 5.4642 and θ∗ = 0.1 and the simple VIX futures price formula, A0 . The label of each time series corresponds to the rebalancing frequency used.