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IT IS
ILLEGAL T
O REPRODUCE T
HIS A
RTICLE IN
ANY F
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SPRING 2006 THE JOURNAL OF DERIVATIVES 13
In 1993, the Chicago Board of Options Exchange(CBOE) introduced the CBOE Volatility Index.This index has become the de facto benchmark forstock market volatility. On September 22, 2003,the CBOE revamped the definition and calculationof the volatility index and back-calculated the newindex to 1990 based on historical option prices. OnMarch 26, 2004, the CBOE launched a newexchange, the Chicago Futures Exchange, andstarted trading futures on the new volatility index.Options on the new volatility index are also planned.This article describes the major differences betweenthe old and the new volatility indexes, derives thetheoretical underpinnings for the two indexes, anddiscusses the practical motivations behind the recentswitch. It also looks at the historical behavior of thenew volatility index and discusses the pricing of VIXfutures and options.
In 1993, the Chicago Board of OptionsExchange (CBOE) introduced theCBOE Volatility Index (VIX). This indexhas become the de factor benchmark for
stock market volatility. The original construc-tion of this volatility index uses options dataon the S&P 100 index (OEX) to compute anaverage of the Black and Scholes [1973] optionimplied volatility with strike prices close to thecurrent spot index level and maturities inter-polated at about one month. The market oftenregards this implied volatility measure as a fore-cast of subsequent realized volatility and alsoas an indicator of market stress (Whaley [2000]).
On September 22, 2003, the CBOErevamped the definition and calculation of theVIX and back-calculated the new VIX to 1990
based on historical option prices. The new def-inition uses the S&P 500 index (SPX) to replacethe OEX as the underlying stock index. Fur-thermore, the new index measures a weightedaverage of option prices across all strikes at twonearby maturities. On March 26, 2004, theCBOE launched a new exchange, the ChicagoFutures Exchange (CFE), to start trading futureson the new VIX. At the time of writing,options on the VIX are also planned.
In this article, we describe the major dif-ferences in the definitions and calculations ofthe old and the new volatility indexes. Wederive the theoretical underpinnings of thetwo indexes and discuss the practical motiva-tions for the switch from the old to the newVIX. We also study the historical behavior ofthe new volatility index and analyze how itinteracts with stock index returns and realizedvolatilities. Finally, we discuss how to useoptions on the underlying S&P 500 index todefine valuation bounds on VIX futures andhow to exploit information in the underlyingoptions market and VIX futures to priceoptions on the new VIX.
DEFINITIONS AND CALCULATIONS
The Old VXO
The CBOE renamed the old VIX theVXO, and it continues to provide quotes onthis index. The VXO is based on options onthe OEX. It is an average of the Black-Scholesimplied volatilities on eight near-the-money
A Tale of Two IndicesPETER CARR AND LIUREN WU
PETER CARR
is the director of theQuantitative FinanceResearch group atBloomberg LP and thedirector of the Masters inMathematical Financeprogram at the CourantInstitute of New YorkUniversity, [email protected]
LIUREN WU is anassociate professor ofeconomics and finance atthe Zicklin School ofBusiness, Baruch College,City University of NewYork, [email protected]
Copyright © 2006
14 A TALE OF TWO INDICES SPRING 2006
options at the two nearest maturities. When the time tothe nearest maturity is within eight calendar days, thenext two nearest maturities are used instead.
At each maturity, the CBOE chooses two call andtwo put options at the two strike prices that straddle thespot level and are nearest to it. The CBOE first averagesthe two implied volatilities from the put and call at eachstrike price and then linearly interpolates between the twoaverage implied volatilities at the two strike prices to obtainthe at-the-money spot implied volatility. The interpolatedat-the-money implied volatilities at the two maturities arefurther interpolated along the maturity dimension to createa 22-trading-day volatility, which constitutes the VXO.
The Black-Scholes implied volatility is the annual-ized volatility that equates the Black-Scholes formula valueto the options market quote. The annualization is basedon an actual/365 day-counting convention. Instead ofusing this implied volatility directly, the CBOE intro-duced an artificial “trading-day conversion” into the cal-culation of the VXO. Specifically, let ATMV (t, T ) denotethe time-t Black-Scholes at-the-money implied volatilityas an annualized percentage with expiry date T. TheCBOE converts this percentage to “trading-day” volatilityTV (t, T ) as
(1)
where NC and NT are the number of actual calendar daysand the number of trading days between time t and theoption expiry date T, respectively. The CBOE convertsthe number of calendar days into the number of tradingdays according to the following formula:
NT 5 NC 2 2 3 int(NC/7) (2)
The VXO represents an interpolated trading-day volatilityat 22 trading days based on the two trading-day volatili-ties at the two nearest maturities (TV (t,T1) and TV (t,T2)):
(3)
where NT1 and NT2 denote the number of trading daysbetween time t and the two option expiry dates T1 andT2, respectively.
Since each month has around 22 trading days, theVXO represents a one-month at-the-money impliedvolatility estimate. Nevertheless, the trading-day conver-sion in Equation (1) raises the level of the VXO and makes
VXO TV t TNT
NT NTTV t T
NTNT NTt = ( ) −
−+ ( ) −
−, ,1
2
2 12
1
2 1
22 22
TV t T ATMV t T NC NT, , /( ) = ( )
it no longer comparable to annualized realized volatilitiescomputed from index returns. Thus, the VXO compu-tation methodology has drawn criticism from both acad-emia and industry for its artificially induced upward bias.
The New VIX
In contrast to the old VXO, which is based on near-the-money Black-Scholes implied volatilities of OEXoptions, the CBOE calculates the new volatility indexVIX using market prices instead of implied volatilities. Italso uses SPX options instead of OEX options. The gen-eral formula for the new VIX calculation at time t is
(4)
where T is the common expiry date for all of the optionsinvolved in this calculation, Ft is the time-t forward indexlevel derived from coterminal index option prices, Ki isthe strike price of the i-th out-of-the-money option in thecalculation, Ot(Ki,T ) denotes the time-t midquote priceof the out-of-the-money option at strike Ki, K0 is the firststrike below the forward index level Ft, rt denotes the time-t risk-free rate with maturity T, and DKi denotes the intervalbetween strike prices, defined as DKi 5 (Ki11 2 Ki)/2. Fornotational clarity, we suppress the dependence of rt and Ft
on the maturity date T as no confusion will result.Equation (4) uses only out-of-the-money options
except at K0, where Ot(K0,T ) represents the average of thecall and put option prices at this strike. Since K0 # Ft,the average at K0 implies that the CBOE uses one unit ofthe in-the-money call at K0. The last term in Equation(4) represents the adjustment needed to convert this in-the-money call into an out-of-the-money put using put-call parity.
The calculation involves all available call options atstrikes greater than Ft and all put options at strikes lowerthan Ft. The bids of these options must be strictly posi-tive to be included. At the extreme strikes of the avail-able options, the definition for the interval DK is modifiedas follows: DK for the lowest strike is the differencebetween the lowest strike and the next lowest strike. Like-wise, DK for the highest strike is the difference betweenthe highest strike and the next highest strike.
To determine the forward index level Ft, the CBOEchooses a pair of put and call options with prices that areclosest to each other. Then, the forward price is derivedvia the put-call parity relation:
VS t TT t
KK
e O K TT t
FK
i
i
r T tt i
i
tt, ,( ) =−
∆ ( ) −−
−
−( )∑2 11
20
2
Copyright © 2006
Ft 5 ert(T2t)(Ct(K,T) 2 Pt(K,T)) 1 K (5)
The CBOE uses Equation (4) to calculate VS(t,T )at two of the nearest maturities of the available options,T1 and T2. Then, the CBOE interpolates between VS(t,T1)and VS(t,T2)to obtain a VS(t,T) estimate at 30 days tomaturity. The VIX represents an annualized volatility per-centage of this 30-day VS, using an actual/365 day-counting convention:
(6)
where NC1 and NC2 denote the number of actual days toexpiration for the two maturities. When the nearest timeto maturity is 8 days or fewer, the CBOE switches to thenext-nearest maturity in order to avoid microstructureeffects. The annualization in Equation (6) follows theactual/365 day-counting convention and does not sufferfrom the artificial upward bias incurred in the VXOcalculation.
ECONOMIC AND THEORETICALUNDERPINNINGS
The Old VXO
The VXO is essentially an average estimate of theone-month at-the-money Black-Scholes implied volatility,with an artificial upward bias induced by the trading-dayconversion. Academics and practitioners often regard at-the-money implied volatility as an approximate forecastfor realized volatility. However, since the Black-Scholesmodel assumes constant volatility, there is no direct eco-nomic motivation for regarding the at-the-money impliedvolatility as the realized volatility forecast beyond theBlack-Scholes model context. Nevertheless, a substantialbody of empirical work has found that the at-the-moneyBlack-Scholes implied volatility is an efficient, althoughbiased, forecast of subsequent realized volatility. Exam-ples include Latane and Rendleman [1976], Chiras andManaster [1978], Day and Lewis [1988], Lamoureux andLastrapes [1993], Canina and Figlewski [1993], Fleming[1998], Christensen and Prabhala [1998], and Gwilymand Buckle [1999]. Thus, references to the VXO as aforecast of subsequent realized volatility are based moreon empirical evidence than on any theoretical linkages.
Carr and Lee [2003] identify an economic inter-pretation for at-the-money implied volatility in a theo-retical framework that goes beyond the Black-Scholes
VIX T t VS t TNC
NC NCT t VS t T
NCNC NCt = −( ) ( ) −
−+ −( ) ( ) −
−
100
36530
30 301 1
2
2 12 2
1
2 1
, ,
model. They show under general market settings that thetime-t at-the-money implied volatility with expiry at time Trepresents an accurate approximation of the conditionalrisk-neutral expectation of the return volatility duringthe time period [t, T ]:
ATMV(t,T ) ù EtQ [RV olt,T] (7)
where EtQ [.] denotes the expectation operator under the
risk-neutral measure Q conditional on time-t filtration Ft
, and RVolt,T denotes the realized return volatilityin annualized percentages over the time horizon [t, T ].Appendix A details the underlying assumptions and deriva-tions for this approximation.
The result in Equation (7) assigns new economicmeanings to the VXO, which approximates the volatilityswap rate with a one-month maturity, if we readjust theupward bias induced by the trading-day conversion.Volatility swap contracts are traded actively over thecounter on major currencies and some equity indexes.At maturity, the long side of the volatility swap contractreceives the realized return volatility and pays a fixedvolatility rate, which is the volatility swap rate. A notionaldollar amount is applied to the volatility difference toconvert the payoff from volatility percentage points todollar amounts. Since the contract costs zero to enter, thefixed volatility swap rate equals the risk-neutral expectedvalue of the realized volatility.
It is worth noting that although the at-the-moneyimplied volatility is a good approximation of the volatilityswap rate, the payoff on a volatility swap is notoriously dif-ficult to replicate. Carr and Lee [2003] derive hedgingstrategies for volatility swap contracts that involve dynamictrading of both futures and options.
The New VIX
The new VIX squared approximates the conditionalrisk-neutral expectation of the annualized return vari-ance over the next 30 calendar days:
VIXt2 ù Et
Q [RVt,t130] (8)
with RVt,t130 5 RVol2t,t130 denoting the annualized return
variance from time t to 30 calendar days later. Hence,VIXt
2 approximates the 30-day variance swap rate. Vari-ance swap contracts are actively traded over the counteron major equity indexes. At maturity, the long side of thevariance swap contract receives a realized variance andpays a fixed variance rate, which is the variance swap rate.
SPRING 2006 THE JOURNAL OF DERIVATIVES 15
Copyright © 2006
The difference between the two rates is multiplied by anotional dollar amount to convert the payoff into dollarpayments. At the time of entry, the contract has zero value.Hence, by no-arbitrage, the variance swap rate equals therisk-neutral expected value of the realized variance.
Although volatility swap payoffs are difficult to repli-cate, variance swap payoffs can be readily replicated, upto a higher-order term. The trading strategy combines astatic position in a continuum of options with a dynamicposition in futures. The risk-neutral expected value ofthe gains from dynamic futures trading is zero. The squareof the VIX is a discretized version of the initial cost of thestatic option position required in the replication. The the-oretical relation holds under very general conditions. Wecan think of the VIX as the variance swap rate quoted involatility percentage points.
To understand the replication strategy and appre-ciate the economic underpinnings of the new VIX, wefollow Carr and Wu [2004] in decomposing the realizedreturn variance into three components:
(9)
where St denotes the time-t spot index level, R0 denotesthe real line excluding zero, and m(dx, dt)is a random mea-sure that counts the number of jumps of size (ex 2 1) inthe index price at time t. The decomposition in Equation(9) shows that we can replicate the return variance by thesum of 1) the payoff from a static position in a continuumof European out-of-the-money options on the underlyingspot across all strike prices but at the same expiry T (firstline), 2) the payoff from a dynamic trading strategy holding[2e2rt(T 2 s)/(T 2 t)] [(1/Fs2) 2 (1/Ft)] futures at time s(second line), and 3) a higher-order term induced by thediscontinuity in the index price dynamics (third line).
Taking expectations under the risk-neutral measureQ on both sides, we obtain the risk-neutral expectedvalue of the return variance on the left-hand side. Wealso obtain the forward value of the sum of the start-upcost of the replicating strategy and the replication erroron the right-hand side. By the martingale property, theexpected value of the gains from dynamic futures tradingis zero under the risk-neutral measure. With determin-
RVT t K
K S dKK
S K dK
T t F FdF
T te x
xdx ds
t T T
F
T
F
s ts
t
T
x
t
T
t
t
,
,
=−
−( ) + −( )
+−
−
−−
− − −
( )
+ +∞
−
∫ ∫
∫
∫∫
2 1 1
2 1 1
21
2
20
2
2
0
m
R
istic interest rates, we have
(10)
where « denotes the approximation error, which is zerowhen the index dynamics are purely continuous and oforder O [(dF/F )3] when the index can jump:
(11)
where nt(x)dxdt is the compensator of the jump-countingmeasure m(dx, dt).
The VIX definition in Equation (4) represents a dis-cretization of the integral in the theoretical relation inEquation (10). The extra term (Ft/K0 2 1)2 in Equation(4) is an adjustment for using a portion of the in-the-money call option at K0 < Ft. Appendix B provides aproof for the decomposition in Equation (9) and a justi-fication for the adjustment term in Equation (4). There-fore, the new VIX index squared has a very concreteeconomic interpretation. It can be regarded either as theprice of a portfolio of options or as an approximation ofthe variance swap rate up to the discretization error andthe error induced by jumps.
Practical Motivation for the Switch
The CBOE’s switch from the old VXO to the newVIX was motivated by both theoretical and practical con-siderations. First, until very recently, the exact economicmeaning of the VXO, or the at-the-money impliedvolatility, was not clear in any theoretical framework beyondthe Black-Scholes model. It merely represents a monot-onic but non-linear transformation of at-the-money optionprices. In contrast, the new VIX is the price of a linearportfolio of options. The economic meaning of the newVIX is much more concrete. Second, the trading-day con-version in the VXO definition induced an artificial upwardbias that has drawn criticism from both academia andindustry. Third, although the VXO approximates thevolatility swap rate, it remains true that volatility swaps arevery difficult to replicate. In contrast, Equation (9) showsthat one can readily replicate the variance swap payoffs upto a higher-order error term using a static position in a con-tinuum of European options and a dynamic position infutures trading. Therefore, despite the popularity of theVXO as a general volatility reference index, no deriva-
« E= −−
− − −
∫∫2
12
2
0T t
e xx
v x dxdstx
s
t
TQ
R
( )
EQt t T
r T t tRVT t
eO K T
KdKt
,
,[ ] =−
( )+−( )
∞
∫22
0
«
16 A TALE OF TWO INDICES SPRING 2006
Copyright © 2006
tive products have been launched on the VXO index. Incontrast, just a few months after the CBOE switched tothe new VIX definition, it started planning to launchfutures and options contracts on the new VIX. VIX futuresstarted trading on March 26, 2004, on the CFE.
HISTORICAL BEHAVIORS
Based on historical data on daily closing optionprices on the S&P 500 index and the S&P 100 index, theCBOE has back-calculated the VIX to 1990 and the VXOto 1987. For our empirical work, we choose the commonsample period from January 2, 1990, to October 18, 2005,spanning 5,769 calendar days. We analyze the historicalbehavior of the two indexes during this sample period,with a focus on the new VIX. We also download thetwo stock indexes OEX and SPX and compute the real-ized return volatilities over the same sample period. Foreach day t, we compute the ex post realized volatilityduring the next 30 calendar days according to thefollowing equation:
(12)
where St1j denotes the index level j calendar days afterday t. We follow the industry standard by computing thereturn squared without de-meaning the return and byannualizing the volatility according to the actual/365 day-counting convention. We analyze how the volatility indexescorrelate with the index returns and return volatilities.
RVol S St t t j t jj
, ln /+ + + −=
= × ( )( )∑30 1
2
1
30
10036530
Summary Statistics
Exhibit 1 reports summary statistics on the levels anddaily differences of the two volatility indexes (VXO andVIX), and their corresponding 30-day realized volatilities,RVolSPX and RVolOEX. Since the VXO has an artificialupward bias as a result of the trading-day conversion, wealso compute an adjusted index (VXOA) that scales backthe conversion in the where we approximately regard the 22 trading days ascoming from 30 actual calendar days. All the volatilityseries are represented in percentage volatility points.
Since the VIX squared approximates the 30-dayvariance swap rate on the SPX and the VXOA approxi-mates the 30-day volatility swap rate on the OEX, Jensen’sinequality dictates that the VIX should be higher thanthe VXOA if the risk-neutral expected values of the real-ized volatilities on the two underlying stock indexes (OEXand SPX) are similar in magnitude:
(13)
(14)(15)
Exhibit 1 shows that the sample mean of the realizedvolatility on the OEX is sightly higher than that on theSPX. Nevertheless, the sample average of the VIX ishigher than the sample average of the VXOA owing toJensen’s inequality. The sample average of the originalVXO series is the highest, mainly because of the erroneoustrading-day conversion.
VIX VXOA RVol RVol RVolt t t t tSPX
t t tSPX
t t tOEX2 2
30 30 30− ≅ ( ) [ ] ≅ [ ]+ + +Var if Q Q QE E, , ,
VXOA RVolt t t tOEX≅ [ ]+EQ, 30
VIX RVol RVol RVolt t t tSPX
t t tSPX
t t tSPX2
30
2
30
2
30≅ ( )
= [ ]( ) + ( )+ + +E EQ Q Q, , ,Var
VXO : / ,VXOA VXO= 22 30
SPRING 2006 THE JOURNAL OF DERIVATIVES 17
E X H I B I T 1Summary Statistics of Volatility Indexes and Realized Return Volatilities
Notes: Entries report the sample average (Mean), standard deviation (Stdev), skewness, excess kurtosis, and first-orderautocorrelation (Auto) on the levels and daily differences of the new volatility index VIX, the 30-day realized volatility on SPXreturn (RV olSP X), the old volatility index VXO, its bias-corrected version VXOA, and the 30-day realized volatility on OEXreturn (RV olOEX). Each series has 5,769 daily observations from January 2, 1990, to October 18, 2005. All series are representedin percentage volatility points.
Copyright © 2006
Comparing the volatility index with the corre-sponding realized volatility, we find that on average, theVIX is approximately 5 percentage points higher than therealized volatility on the SPX, and the VXOA is approx-imately 2 percentage points higher than the correspondingrealized volatility on the OEX. To test the statistical sig-nificance of the difference between the volatility indexand the realized volatility, we construct the followingt-statistic:
(16)
where N 5 5,769 denotes the number of observations,X denotes the difference between the volatility index andthe realized volatility, the overline denotes the sampleaverage, and SX denotes the Newey and West [1987] stan-dard deviation of X that accounts for overlapping dataand serial dependence, with the number of lags optimallychosen following Andrews [1991] and an AR(1) specifi-cation. We estimate the t-statistic for (VIX 2 RVolSPX) at14.09 and for (VIX 2 RVolOEX) at 6.72, both of whichare highly significant.
The volatility levels show moderate positive skew-ness and excess kurtosis, but the excess kurtosis for dailydifferences is much larger, showing potential discontin-uous index return volatility movements. Eraker et al.[2003] specify an index dynamics that contains constant-arrival finite-activity jumps in both the index return andthe return variance rate. By estimating the model on SPXreturn data, they identify a strongly significant jump com-ponent in the variance rate process in addition to a sig-nificant jump component in the index return. Wu [2005]directly estimates the variance rate dynamics without spec-ifying the return dynamics by using the VIX and variousrealized variance estimators constructed from tick data onSPX index futures. He also finds that the variance rate
t NXSX
− =stat
contains a significant jump component but that the jumparrival rate is not constant over time; instead, it is pro-portional to the variance rate level. Furthermore, he findsthat jumps in the variance rate are not rare events butarrive frequently and generate sample paths that displayinfinite variation.
Exhibit 2 reports the cross-correlation betweenthe two volatility indexes (VIXt and VXOt) and thesubsequent realized volatilities (RVolt,t130
SPX and RVolt,t130OEX ).
Each volatility index level is positively correlated with itscorresponding subsequent realized volatility, but thecorrelation estimates become close to zero when mea-sured in daily changes. Nevertheless, the two volatilityindexes are highly correlated in both levels (0.98) and dailydifferences (0.86). The two realized volatility series arealso highly correlated in both levels (0.99) and daily changes(0.98). Therefore, just as both stock indexes provide a gen-eral picture of the overall stock market, so both volatilityindexes proxy the overall stock market volatility. Giventhe close correlation between the VIX and the VXO, andthe planned obsolescence of the VXO, we henceforthfocus our analysis on the behavior of the new VIX.
The Leverage Effect
Exhibit 3 plots the cross-correlations between SPXindex returns at different leads and lags and daily changesin the volatility index VIX, with the two dash-dottedlines denoting the 95% confidence band. The instanta-neous correlation estimate is strongly negative at 20.78,but the correlation estimates at other leads and lags aremuch smaller. Careful inspection shows that lagged returns(within a week) show marginally significant positivecorrelations with daily changes in the volatility index,indicating that index returns predict future movementsin the volatility index. However, index returns with
18 A TALE OF TWO INDICES SPRING 2006
E X H I B I T 2Cross-Correlations between Volatility Indexes and Subsequent Realized Return Volatilities
Notes: Entries report the contemporaneous cross-correlation between VIXt, SPX 30-day realized volatility (RVolt,t130SPX ), VXOt,
and OEX 30-day realized volatility (RVolt,t130OEX ) both in levels and in daily differences.
Copyright © 2006
negative lags are not significantly correlated with dailychanges in the volatility index. Therefore, volatility indexmovements do not predict index returns.
The negative correlations between stock returns andstock return volatilities have been well documented.Nevertheless, since return volatility is not observable, thecorrelation can be estimated only under a structural modelfor return dynamics. In Exhibit 3, we use the VIX as anobservable proxy for return volatility and compute thecorrelation across different leads and lags without resortingto a model for return dynamics. The strongly negativecontemporaneous correlation between stock (index)returns and return volatilities captures the “leverage effect”first discussed by Black [1976]: given a fixed debt level, adecline in the equity level increases the leverage of the firm(market) and hence the risk for the stock (index). Var-ious other explanations for the negative correlation havebeen proposed in the literature, for example, Haugen et al.[1991], Campbell and Hentschel [1992], Campbell andKyle [1993], and Bekaert and Wu [2000].
The Federal Open Market CommitteeMeeting Day Effect
Balduzzi et al. [2001] find that trading volume, bid-ask spreads, and volatility on Treasury bonds and bills
increase dramatically around Federal Open Market Com-mittee (FOMC) meeting dates. The Federal Reserve oftenannounces changes in the Fed Funds Target Rate and itsviews on the overall economy during the FOMC meet-ings. The anticipation and ex post reaction to theseannouncements in monetary policy shifts and assessmentscreate dramatic variations in trading and pricing behaviorin the Treasury market. In this section, we use the VIXas a proxy for stock market volatility and investigatewhether stock market volatility also shows any apparentchanges around FOMC meeting days.
We download the FOMC meeting day log fromBloomberg. During our sample period, there were 144scheduled FOMC meetings, or approximately 10 meet-ings per year. Exhibit 4 plots the time series of the FedFunds Target Rates in the left panel and the basis pointtarget changes during the scheduled FOMC meeting daysin the right panel. Among the 144 meetings, 62announced a change in the Fed Funds Target Rate.Among the 62 target moves, the change is 25 bp 45 times,50 bp 16 times, and 75 bp once. On 25 occasions, thechange is positive, representing a tightening of monetarypolicy, and on 37 occasions the change represents a ratecut and hence an easing of monetary policy.
Armed with the list of FOMC meeting days, we sortthe VIX around the FOMC meeting days and compute theaverage VIX level each day from 10 days before to 10 daysafter each FOMC meeting day. The left panel of Exhibit 5plots sample averages of VIX around FOMC meeting days.We observe that the average volatility level builds up beforethe FOMC meeting date and then drops markedly afterward.The volatility index reaches its highest level the day beforethe meeting and drops to the lowest level 4 days after themeeting. To investigate the significance of the drop, wemeasure the difference between the volatility index 1 daybefore and 1 day after the meeting. The mean difference is0.6 percentage volatility points, with a t-statistic of 4.06.
Before the FOMC meeting, market participants dis-agree on whether the Fed will change the Fed FundsTarget Rate, in which direction, and by how much. Thefact that the option-implied stock index volatility increasesprior to the meeting and drops afterward shows that theuncertainty about monetary policy has a definite impacton the volatility of the stock market. This uncertainty isresolved right after the meeting. Hence, the volatilityindex drops rapidly after the FOMC meeting.
Since the VIX squared can be regarded as the vari-ance swap rate on the SPX, we also study whether thetiming of a variance swap investment around FOMC
SPRING 2006 THE JOURNAL OF DERIVATIVES 19
E X H I B I T 3Cross-Correlations between Return and Volatility
Notes: The stem bars represent the cross-correlation estimatesbetween SPX index returns at the relevant number of lags (indays) and the corresponding daily changes in VIX. The twodash-dotted lines denote the 95% confidence band.
Copyright © 2006
meeting days generates different returns. The right panelof Exhibit 5 plots the average ex post payoff from goinglong the swap contract around FOMC meeting days andholding the contract to maturity. The payoff is defined asthe difference between the ex post realized variance andthe VIX squared: (RVt,t1302VIXt
2) We find that the average
payoffs are negative by going long the swap on any day.Therefore, shorting the swap contract generates positivepayoffs on average. Comparing the magnitude differenceson different days, we also find that shorting the swap con-tract 4 days prior to the FOMC meeting day generatesthe highest average payoff and that shorting the variance
20 A TALE OF TWO INDICES SPRING 2006
E X H I B I T 4The Fed Funds Target Rate Changes
Notes: The solid line in the left panel plots the time series of the Fed Funds Target Rate over our sample period. The spikes inthe right panel represents the target rate changes in basis points.
E X H I B I T 5VIX Fluctuation around FOMC Meeting Days
Notes: Lines represent the sample averages of the VIX levels (left panel) and the average payoffs to long variance swap contracts,(RVt,t1302VIXt
2) (right panel), at each day within 10 days before and after the FOMC meeting days.
Copyright © 2006
swap 4 days after the FOMC meeting day generates thelowest average payoff. The difference in average payoffbetween investments on these days is statistically signifi-cant, with a t-statistic of 9.29. Therefore, the evidencesuggests that it is more profitable to short the SPX vari-ance swap contract 4 days before an FOMC meeting than4 days after.
Variance Risk Premia
Up to a discretization error and a jump-inducederror term, the VIX squared is equal to the risk-neutralexpected value of the realized variance on the SPX returnduring the next 30 days:
VIXt2 ù Et
Q [RVt,t130] (17)
We can also rewrite Equation (17) under the statisticalmeasure P as
(18)
where Mt,T denotes a pricing kernel between times t andT. For traded assets, no-arbitrage guarantees the existenceof at least one such pricing kernel (Duffie [1992]).
Equation (18) decomposes the VIX squared intotwo terms. The first term, Et
P [RVt,t130] represents the sta-tistical conditional mean of the realized variance, and thesecond term captures the conditional covariance betweenthe normalized pricing kernel and the realized variance.The negative of this covariance defines the time-t con-ditional variance risk premium (VRPt):
(19)
Taking unconditional expectations on both sides, we have
EP [VRPt] 5 EP [RVt,t130 2 VIXt2] (20)
Thus, we can estimate the average variance risk premiumas the sample average of the differences between the realizedreturn variance and the VIX squared. Over our sampleperiod, the mean variance risk premium is estimated at2158.67 bp, with a Newey and West [1987] serial-depen-dence-adjusted standard error of 17.2. Hence, the meanvariance risk premium is strongly negative.
Risk-averse investors normally ask for a positive riskpremium for return risk. They require stock prices toappreciate by a higher percentage on average if stock
VRPM
MRV RV VIXt t
t t
t t tt t t t t t≡ − [ ]
= [ ] −+
++ +CovP
P
P
EE,
,, ,,30
3030 30
2
VIXM RV
MRV
M
MRVt
t t t t t
t t tt t t t
t t
t t tt t
2 30 30
3030
30
3030≅ [ ]
[ ] = [ ] + [ ]
+ +
++
+
++
E
EE
E
P
P
P P
P
, ,
,,
,
,,,Cov
returns are riskier. In contrast, the negative variance riskpremium indicates that investors require the index returnvariance to stay lower on average to compensate for highervariance risk. Therefore, whereas higher average returnis regarded as compensation for higher return risk, loweraverage variance levels are regarded as compensation forhigher variance risk. Investors are averse not only toincreases in the return variance level but also to increasesin the variance of the return variance.
From the perspective of a variance swap investment,the negative variance risk premium also implies that investorsare willing to pay a high premium or endure an average losswhen they are long variance swaps in order to receive com-pensation when the realized variance is high.
Dividing both sides of Equation (18) by VIXt2, we
can rewrite the decomposition in excess returns:
(21)
If we regard VIXt2 as the forward cost of the investment
in the static option position required to replicate the vari-ance swap payoff, (RVt,t130/VIXt
2 2 1) captures the excessreturn from going long the variance swap. The negativeof the covariance term in Equation (21) represents theconditional variance risk premium in excess return terms:
(22)
We can estimate the mean variance risk premium in excessreturn form through the sample average of the realizedexcess returns, ERt,t130 5 (RVt,t130/VIXt
2 2 1), which is esti-mated at 240.16%, with a Newey and West [1987] stan-dard error of 2.87%. Again, the mean variance risk premiumestimate is strongly negative and highly significant. Investorsare willing to endure a highly negative excess return forbeing long variance swaps in order to hedge away upwardmovements in the return variance of the stock index.
The average negative variance risk premium alsosuggests that shorting the 30-day variance swap and holdingit to maturity generates an average excess return of 40.16%.We compute the annualized information ratio using 30-day-apart non-overlapping data, ,where denotes the time series average of the excessreturn and SER denotes the serial-dependence-adjustedstandard deviation estimate of the excess return. The infor-mation ratio estimates average 3.52, indicating that shorting30-day variance swaps is very profitable on average.
ERIR ER SER= − 12 /
VRPRM
M
RV
VIX
RV
VIXt tt t
t t t
t t
tt
t t
t
≡ − [ ]
=
−+
+
+ +CovP
P
P
EE,
,
, ,,30
30
302
302
1
1 302
30
30
302
=
+ [ ]
+ +
+
+EE
P P
Ptt t
tt
t t
t t t
t t
t
RV
VIX
M
M
RV
VIX, ,
,
,,Cov
SPRING 2006 THE JOURNAL OF DERIVATIVES 21
Copyright © 2006
To further check the historical behavior of excessreturns from this investment, we plot the time series ofthe excess returns in the left panel and the histogram inthe right panel of Exhibit 6. The time series plot showsthat shorting variance swaps provides a positive return89% of the time (5,137 out of the 5,769 daily invest-ments). However, although the historical maximum pos-itive return is 89.53%, the occasionally negative realizationscan be as large as 242.42%. The histogram in the rightpanel shows that the excess return distribution is heavilynegatively skewed. The high average return and highinformation ratio suggest that investors ask for a very highaverage premium to compensate for the heavily nega-tively skewed risk profile. The payoff from shorting vari-ance swaps is similar to that from selling insurance, whichgenerates a regular stream of positive premiums with smallvariation but with occasional exposures to large losses.
To investigate whether the classic Capital AssetPricing Model (CAPM) can explain the risk premiumfrom investing in variance swaps, we regress the excessreturns from being long the variance swap on the excessreturns from being long the market portfolio:
ERt,t130 5 a 1 b (Rmt,t130 2 Rf ) 1 et (23)
where (Rmt,t130 2 Rf) denotes the continuously compounded
excess return to the market portfolio. If the CAPM holds,
we will obtain a highly negative beta estimate for the longvariance swap return. If the CAPM can fully account forthe risk premium, the estimate for the intercept a, whichrepresents the average excess return to a market-neutralinvestment, will not be significantly different from zero.
We proxy the excess return to the market portfoliousing the value-weighted return on all NYSE, AMEX,and NASDAQ stocks (from CRSP) minus the one-monthTreasury bill rate (from Ibbotson Associates). Monthlydata on the excess returns are publicly available at KennethFrench’s online data library from July 1926 to September2005. We match the sample period with our data and runthe regression on monthly returns over non-overlappingdata using the generalized method of moments, with theweighting matrix computed according to Newey andWest [1987].
The regression estimates are as follows, with t-sta-tistics reported in parentheses:
(24)
The beta estimate is highly negative, consistent with thegeneral observation that index returns and volatility arenegatively correlated. However, this negative beta cannotfully explain the negative premium for volatility risk. Theestimate for the intercept, or the mean beta-neutral excess
ER R R e Rt tm
f t= − − − + =
− −
0 3636 3 7999 19 15
65 03 30 10
2. . ) , . %
( . )( . )
(
22 A TALE OF TWO INDICES SPRING 2006
E X H I B I T 6Excess Returns from Shorting 30-Day Variance Swaps
Notes: The left panel plots the time series of excess returns from shorting 30-day variance swaps on SPX and holding the con-tract to maturity. The right panel plots the histogram of excess returns.
Copyright © 2006
return, remains strongly negative. The magnitude of a isnot much smaller than the sample average of the rawexcess return at 238.36%. Thus, the CAPM gets onlythe sign right; it cannot fully account for the large neg-ative premium on index return variance risk. This resultsuggests that variability in variance constitutes a separatesource of risk that the market prices heavily.
To test whether the variance risk premium is timevarying, we run the following expectations-hypothesisregressions, with the t-statistics reported in parentheses:
(25)
Under the null hypothesis of constant variance riskpremium, the first regression should generate a slope ofone, and the second regression should generate a slope ofzero. Zero-variance risk premium would further implyzero intercepts for both regressions. The t-statistics arecomputed against these null hypotheses. Since the dailyseries of the 30-day realized variance constitutes an over-lapping series, we estimate both regressions using the gen-eralized method of moments and construct the weightingmatrix, accounting for the serial dependence accordingto Newey and West [1987] with 30 lags.
When the regression is run on the variance level, theslope estimate is significantly lower than the null value of
RV VIX e
RV VIX VIX e
t t t t t
t t t t t t
, ,
, ,
. .
( . ) ( . )
( / ) . .
( . ) ( . )
+ +
+ +
= − + +− −
− = − + +−
302
30
302 2
30
11 9006 0 6501
0 52 4 79
1 0 4495 0 0001
14 28 1 61
one, providing evidence that the variance risk premiumVRPt is time varying and correlated with the VIX level.When the regression is run on excess returns in the secondequation, the slope estimate is no longer significantlydifferent from zero, suggesting that the variance riskpremium defined in excess return terms (VRPRt) is nothighly correlated with the VIX level.
Predictability of Realized Variance andReturns to Variance Swap Investments
We estimate GARCH(1,1) processes on the S&P500 index return innovation using an AR(1) assumptionon the return process. Then we compare the relative infor-mation content of the GARCH volatility and the VIXindex in predicting subsequent realized return variances:
RVt,t130 5 a 1 b VIX2t 1 cGARCHt 1 et,t130 (26)
where GARCHt denotes the time-t estimate of theGARCH return variance in annualized basis points.Exhibit 7 reports the generalized method of moment esti-mation results on restricted and unrestricted versions ofthis regression.
When we use either VIX2 or GARCH as the onlypredictor in the regression, the volatility index VIX gen-erates an R-squared approximately 10 percentage pointshigher than the GARCH variance does. When we useboth VIX2 and GARCH as predictors, the slope estimateon the GARCH variance is no longer statistically signif-icant, and the R-squared is only marginally higher than
using VIX2 alone as the regressor. Thus, theGARCH variance does not provide muchextra information over the VIX index.
The results in Exhibit 7 show that wecan predict the realized variance using thevolatility index VIX. By using variance swaps,investors can exploit such predictability anddirectly convert them into dollar returns. Weinvestigate whether the predictability of returnvariance has been fully priced into the vari-ance swap rate by analyzing the predictabilityof the excess returns from investing in a 30-day SPX variance swap and holding it tomaturity.
First, we measure the monthly auto-correlation of the excess returns ERt,t 1 30 usingnon-overlapping 30-day-apart data. The esti-mates average 0.12. When we run an AR(1)
SPRING 2006 THE JOURNAL OF DERIVATIVES 23
E X H I B I T 7Information Content in VIX and GARCH volatilities inpredicting future realized return variances
Notes: Entries report the estimation results on restricted and unrestrictedversions of the following relation:
The relation is estimated using the generalized method of moments. Thecovariance matrix is computed according to Newey and West [1987] with30 lags. The data are daily from January 2, 1990, to October 18, 2005,generating 5,769 observations for each series.
RV a bVIX cGARCH et t t t t t, ,+ += + + +302
30
Copyright © 2006
regression on the non-overlapping excess returns, the R-squared estimates average 1.58%. Thus, the predictabilityof excess returns through mean reversion is very low.Although the volatility level is strongly predictable,investors have priced this predictability into variance swapcontracts, so that the excess returns on these swaps arenot strongly predictable.
Exhibit 3 shows that SPX returns predict futuremovements in the VIX. Now we investigate whether wecan predict the excess return on a variance swap invest-ment using index returns. Exhibit 8 plots the cross-correlation between the excess return to the variance swapand the monthly return on SPX based on monthly sam-pled, and hence non-overlapping, data. The stock indexreturn and the return on the variance swap investmentsshow strongly negative contemporaneous correlation, butthe non-overlapping series do not exhibit any significantlead-lag effects. Hence, despite the predictability in returnvolatilities, excess returns on variance swap investmentsare not strongly predictable. This result shows that theSPX options market is relatively efficient.
VIX DERIVATIVES
Given the explicit economic meaning of the newVIX and its direct link to a portfolio of options, the launchof derivatives on this index becomes the natural nextstep. On March 26, 2004, the CBOE launched a newexchange, the Chicago Futures Exchange, and startedtrading futures on the VIX. At the time of writing, optionson the VIX are also being planned. In this section, wederive some interesting results regarding the pricing ofVIX futures and options.
VIX Futures and Valuation Bounds
Under the assumption of no-arbitrage and contin-uous marking to market, the VIX futures price, Ft
vix, is amartingale under the risk-neutral probability measure Q:
Ftvix 5 Et
Q [FT1vix] 5 Et
Q [VIXT1] (27)
We derive valuation bounds on VIX futures that areobservable from the underlying SPX options market,under two simplifying assumptions: 1) the VIX is calcu-lated using a single strip of options maturing at T2 . T1,with T2 2 T1 5 30/365, instead of two strips, and on acontinuum of options prices rather than a discrete numberof options; 2) the SPX index has continuous dynamicsand interest rates are deterministic.
The first assumption implies that the VIX is givenby
(28)
where BT1(T2) denotes the time-T1 price of a zero bond
maturing at T2. The second assumption further impliesthat the equality between the VIX squared and the risk-neutral expected value of the return variance is exact.Alternatively, we can write
(29)
Substituting Equation (29) in Equation (27), we have theVIX futures as
(30)
Then, the concavity of the square root and Jensen’sinequality generates the following lower and upper boundsfor the VIX futures:
F RV t T Ttvix
t T T T= ≤ <E EQ Q
1 1 2 1 2, ,
VIX RVT T T T1 1 1 2= EQ
,
VIXT T B T
O K TK
dKTT
T1
1
2
2 1 2
1 22
0
=−
∞
∫( ) ( )( , )
24 A TALE OF TWO INDICES SPRING 2006
E X H I B I T 8Cross-Correlation between SPX Monthly Returnsand Excess Returns on 30-day Variance Swaps
Notes: The stem bars represent the cross-correlation estimatesbetween SPX returns at different lags and excess returns oninvesting in a 30-day variance swap and holding it to matu-rity. The estimates are based on monthly non-overlappingdata. The two dashed lines denote the 95% confidence band.Positive numbers on the x-axis represent lags in months forindex returns.
Copyright © 2006
(31)
The lower bound is the forward volatility swaprate , which can be approximated by aforward-starting at-the-money option. The proof is sim-ilar to that in Appendix A for the approximation of a spotvolatility swap rate using the spot at-the-money option.The upper bound is the forward-starting variance swap ratequoted in volatility percentage points, which can be determined from the prices on a continuumof options at two maturities T1 and T2:
(32)
The width of the bounds is determined by the risk-neu-tral variance of the forward-starting realized volatility:
(33)
When the market quote on VIX futures (Ftvix) is
available, we can combine it with forward-starting vari-ance swap rates (Ut) to determine the risk-neutral vari-ance of the future VIX:
(34)
Therefore, VIX futures provide economically relevantinformation not only about the future VIX level but alsoabout the risk-neutral variance of the future VIX. Wecan use this information for pricing VIX options.
VIX Options
The VIX futures market, together with the SPXoptions market, provides the information basis forlaunching VIX options. To see this, we consider a calloption on VIX, with the terminal payoff
(VIXT12K )1 (35)
where K is the strike price and T1 denotes the expiry dateof the option. We have shown that we can learn the con-ditional risk-neutral mean (m1t) and variance (m2t) of VIXT1from information in the VIX futures market and theunderlying SPX options market:
Var Vart T t T T T
t T T t T T T t tvix
VIX RV
RV RV U F
Q Q Q
Q Q Q
E
E E E
( )
[ ] ( )
,
, ,
,
,
1 1 1 2
1 2 1 1 2
22 2
= [ ]
= −( ) = −
U L RV RV RVt t t T T t T T t T T2 2
2
1 2 1 2 1 2− = ( ) −( ) = ( )E EQ Q Q
, , ,, , ,Var
U RVT T
T t RV T t V
T T
Ot K T
B T
O K T
B TdKK
t t T T t t T t t T
t
t
t
2
2 12 1
2 1
2
2
1
12
0
1 2 2 1
1
2
= =−
−( ) − −( )[ ]=
−( )
( ) −( )
( )
∞
∫
E E EQ Q Q, , ,,
, ,
U RVt t T T≡ EQ
1 2, ,
L RVt t T T≡EQ
1 2, ,
E EQ Qt T T t
vixt T TRV F RV
1 2 1 2, ,≤ ≤
(36)Thus, under certain distributional assumptions, we canderive the value of the VIX option as a function of thesetwo moments.
As an example, if we assume that VIXT1follows a
log-normal distribution under measure Q, we can usethe Black formula to price VIX options with the twomoments in Equation (36) as inputs:
where
and st is the conditional annualized volatility of ln VIXT1,
which can be represented as a function of the first two con-ditional moments of VIXT1
,
(37)
As another example, if we assume that the risk-neutral distribution of VIXTT1
is normal rather thanlognormal, we can derive the Bachelier option pricingformula as a function of the first two observable momentsof VIXTT1
:
(38)
with . For at-the-money options (K 5Ft
vix), the Bachelier option pricing formula reduces to avery simple form,
(39)
CONCLUSION
The new VIX differs from the old VXO in two keyaspects. First, the two indexes use different underlyings:the SPX for the new VIX versus the OEX for the oldVXO. Second, the two indexes use different formulae inextracting volatility information from the options market.The new VIX is constructed from the price of a portfolio
A B T mt t t= ( ) /1 2 2p
d F K mtvix
t= −( )/ 2
C B T m N d F K N dt t t tvix= + −[ ]( ) ( ) ( ) ( )1 2
9
sT t
m FF
t tvix
tvix
=−
+1
1
22
2ln
( )
( )
dln F K s T t
s T td d s T tt
vixt
t
t1
12
21
1
2 1 1=+ −
−= − −
/ ( ),
C B T F N d KN dt t tvix= ( ) ( ) − ( )[ ]1 1 2
m VIX F
m VIX U FT T
O K T
B T
O K T
B TdKK
F
t t T tvix
t t T t tvix t
t
t
ttvix
1
22 2
2 1
2
2
1
102
2
1
1
2
≡ ( )=
≡ ( ) = −( ) =−
( )−
( )
−( )∞
∫
EQ
QVar,
( )
,
( ).
SPRING 2006 THE JOURNAL OF DERIVATIVES 25
Copyright © 2006
of options and represents a model-free approximation ofthe 30-day return variance swap rate. The old VXO buildson the 1-month Black-Scholes at-the-money impliedvolatility and approximates the volatility swap rate undercertain assumptions. The CBOE decided to switch fromthe VXO to the VIX mainly because the new VIX has abetter known and more robust economic interpretation.In particular, the variance swap underlying the new VIXhas a robust replicating portfolio whose option compo-nent is static. In contrast, robust replication of the volatilityswap underlying the VXO index requires dynamic optiontrading. Furthermore, the VXO includes an upward biasinduced by an erroneous trading-day conversion in itsdefinition.
Analyzing approximately 15 years of daily data onthe two volatility indices, we obtain several interestingfindings on the index behavior. We find that the new VIXaverages about 2 percentage points higher than the bias-corrected version of the old index, although the sampleaverage of the 30-day realized volatility on SPX is 0.66percentage points lower than that of OEX. The differencebetween the new and old volatility indexes is mainlyinduced by Jensen’s inequality and the risk-neutral vari-ance of realized volatility. The historical behaviors of thetwo volatility indexes are otherwise very similar and theymove closely with each other. We also find that dailychanges in the volatility indexes show very large excesskurtosis, suggesting that the volatility indexes contain largediscontinuous movements.
We identify a strongly negative contemporaneouscorrelation between VIX and SPX index returns, con-firming the “leverage effect” first documented by Black[1976]. Furthermore, although lagged index returns showmarginal predictive power on the future movements of theVIX, lagged movements in the volatility index do notpredict future index returns.
When we analyze VIX behavior around FOMCmeeting days, during which monetary policy decisionssuch as Fed Funds Target Rate changes are oftenannounced, we find that the volatility index increasesprior to the FOMC meeting but drops rapidly after themeeting, showing that uncertainty about monetary policyhas a direct impact on volatility in the stock market.
Since the VIX squared represents the variance swaprate on the SPX, the sample average difference betweenthe 30-day realized return variance on the SPX and theVIX squared measures the average variance risk premium,which we estimate to be 2158.67 bp and highly signif-icant. When we represent the variance risk premium in
excess returns form, we obtain a mean estimate of240.16% for being long a 30-day variance swap andholding it to maturity. The highly negative variance riskpremium indicates that investors are averse to variationsin return variance and the compensation for bearing vari-ance risk can come in the form of a lower mean variancelevel under the empirical distribution than under the risk-neutral distribution.
From the perspective of variance swap investors, thenegative variance risk premium indicates that investorsare willing to pay a high average premium to obtain com-pensation (insurance) when the variance level increases.Therefore, shorting variance swaps and hence receivingthe fixed leg generates positive excess returns on average.The annualized information ratio for shorting a varianceswap is approximately 3.52, which is much higher thanfor traditional investments. Nevertheless, the excess returndistribution accessed by being short variance swaps isheavily negatively skewed. Negative return realizationsare few but large. The high information ratio indicates thatinvestors ask for a high average return in order to com-pensate for the heavily negatively skewed risk profile.When we regress the excess returns from being long thevariance swap on the stock market portfolio, we obtain ahighly negative beta. However, the intercept of the regres-sion remains highly negative, indicating that the classicCapital Asset Pricing Model cannot fully account for thenegative variance risk premium. Investors regard vari-ability in variance as a separate source of risk and chargea separate price for bearing this risk. Expectations hypo-thesis regressions further show that the variance risk pre-mium in variance levels is time varying and correlatedwith the VIX level, but the variance risk premium inexcess returns form is much less correlated with the VIXlevel.
We find that the VIX can predict movements infuture realized variance and that GARCH volatilitiesdo not provide extra information once the VIX isincluded as a regressor. Nevertheless, the strong pre-dictability of the realized variance does not transfer tostrong predictability in excess returns for investing in vari-ance swaps.
Finally, we show that the SPX options marketprovides information on valuation bounds for VIX futures.The width of the bounds are determined by the risk-neutral variance for forward-starting return volatility.Furthermore, VIX futures quotes not only provideinformation about the risk-neutral mean of future VIXlevels but also combine with information from the SPX
26 A TALE OF TWO INDICES SPRING 2006
Copyright © 2006
options market to reveal the risk-neutral variance of theVIX. This information can be used to price VIX options.
APPENDIX A
Approximating Volatility Swap Rateswith At-the-Money Implied Volatilities
Let (V, F, Q) be a probability space defined on a risk-neutral measure Q. As in Carr and Lee [2003], we assume con-tinuous dynamics for the index futures Ft under measure Q:
(A-1)
where the diffusion volatility st can be stochastic, but its vari-ation is assumed to be independent of the Brownian motion Wt
in the price. Under these assumptions, Hull and White [1987]show that the value of a call option can be written as the risk-neutral expected value of the Black-Scholes formula, evaluatedat the realized volatility. The time-t value of the at-the-moneyforward (K 5 Ft) option maturing at time T can be written as
(A-2)
where RVolt,T is the annualized realized return volatility over[t, T ]
(A-3)
Brenner and Subrahmanyam [1988] show that a Taylor seriesexpansion of each normal distribution function about zeroimplies
(A-4)
Substituting Equation (A-4) into Equation (A-2) implies that
(A-5)
and hence the volatility swap rate is given by
ATMCF
RVol T tt T tt
t T, ,≈ −
EQ
2p
RVol T tO T tt t T, , (( ) )
−=
−+ −
2 2
3
2
p
NRVol T t
NRVol T t RVol T t
Ot T t T t T, ,−
−
− −
2 2 2
RVolT t
dst T st
T
, ≡− ∫1 2s
NRVol T t
tt T
,,− − −
2
ATMC F NRVol T t
Nt T t tt T t T
,,= −
EQ
2
dF F dWt t t t/ = s
(A-6)
Since an at-the-money call value is concave in volatility,ATMCt,T is a slightly downward-biased
approximation of the volatility swap rate. As a result, the errorterm is positive. However, Brenner and Subrahmanyam showthat the at-the-money implied volatility is also given by
(A-7)
Once again, ATMCt,T is a slightly downward-biased approximation of the at-the-money implied volatility.Subtracting Equation (A-7) from Equation (A-6) implies that thevolatility swap rate is approximated by the at-the-money impliedvolatility:
(A-8)
The leading source of error in Equation (A-6) is partiallycanceled by the leading source of error in Equation (A-7). Asa result, this approximation has been found to be very accurate.
APPENDIX B
Replicating Variance Swaps with Options
The interpretation of the new VIX as an approximationof the 30-day variance swap rate can be derived under a muchmore general setting for the Q-dynamics of SPX index futures:
(A-9)
where Ft2 denotes the futures price at time t just prior to a jump,R0 denotes the real line excluding zero, and the random mea-sure m(dx, dt) counts the number of jumps of size (ex 2 1) in theindex futures at time t. The process {vt(x), x [ R0} compen-sates the jump process Jt ; e0
t eR0 (ex 2 1) m (dx, ds), so that the
last term in Equation (A-9) is the increment of a Q-pure jumpmartingale. To avoid notational complexity, we assume that thejump component in the price process exhibits finite variation:
By adding the time subscripts to st2 and nt(x), we allow bothto be stochastic and predictable with respect to the filtration Ft.To satisfy limited liability, we further assume the two stochastic
(| | ) ( )x v x dxt∧ < ∞∫ 10R
dF F dW e dx dt v x dxdtt t t tx
t/ ( )[ ( , ) ( ) ]− −= + − −∫s m10R
EQt t T t TRVol ATMV O T t, , (( ) )[ ]= + −
32
( / )2p F T tt −
ATMCF T t
ATMV O T tt T
t
t T, , (( ) )=−
+ −2 32
p
( / )2p F T tt −
EQt t T
t
t TRVolF T t
ATMC O T t, , (( ) )[ ]=−
+ −2 32
p
SPRING 2006 THE JOURNAL OF DERIVATIVES 27
Copyright © 2006
processes to be such that the futures price Ft is always non-neg-ative and absorbing at the origin. Finally, with little loss of gen-erality, we assume deterministic interest rates and dividend yields.Under these assumptions, the annualized quadratic variation onthe futures return over horizon [t, T] can be written as
(A-10)
Applying Ito-’s lemma to the function f(F) 5 ln F, wehave
Adding and subtracting 2[(FT/Ft) 2 1] 1 et
T x2 m(dx, dt) and rear-ranging, we obtain a representation for the quadratic variation:
(A-11)
A Taylor expansion with the remainder of ln FT about the pointFt implies
(A-12)
Plugging Equation (A-12) into Equation (A-11), we have
(A-13)
which is the decomposition in Equation (9) that also representsa replicating strategy for the return quadratic variation.
Taking expectations under measure Q, we obtain the risk-neutral expected value of the return variance on the left-handside and the cost of the replication strategy on the right-hand side:
where the first term denotes the initial cost of the static port-folio of out-of-the-money options and the second term is ahigher-order error term induced by jumps.
E EQ Q
R
t t T
r T tt
tx
t
T
sRVeT t
O K T
KdK e x
xv x dxds
t
,
,[ ] =−
( )− − − −
( )
−( ) ∞
∫ ∫∫22 1
220
2
0
T t RVK
K F dKK
F K dK
F FdF
e xx
dx ds
t T T
F
F
T
s tt
T
s
x
t
T
t
t
−( ) = −( ) + −( )
+ −
+ − − −
( )
+∞
+
−
∫ ∫
∫
∫∫
,
,
21 1
21 1
2 12
20
2
2
0R
m
KK F dK
KF K dKT
F
T
F
T
t
− −( ) − −( )∫ ∫+
∞+1 1
20
2
0
ln lnF FF
F FKT t
tT t
F
= + −( ) −1
T t RVFF
FF F F
dF
e xx
dx ds
t TT
t
T
t s tt
T
s
x
t
T
−( ) = − −
+ −
− − − −
( )
−∫
∫∫
, ln
,
2 1 21 1
2 12
2
0R
m
ln ln ,F FF
dF ds x e dx dsT Ts
s
t
T
s
t
Tx
t
T
( ) = ( ) + − + − +[ ] ( )−
−∫ ∫ ∫∫1 12
12
0
s m
R
RVT t
dt x dx dtt T t
t
T T
, ,=−
+ ( )
−∫ ∫∫1 2 2
0 0
s m
R
The VIX definition in Equation (4) represents a dis-cretization of the option portfolio. The extra term (Ft/K0 2 1)2
in the VIX definition adjusts for the in-the-money call optionused at K0 # Ft. To convert the in-the-money call option intothe out-of-the-money put option, we use the put-call parity:
(A-14)
If we plug this equality into Equation (4) to convert all optionprices into out-of-money option prices, we have
(A-15)
where the second term on the right-hand side of Equation (A-15) is due to the substitution of the in-the-money call optionat K0 by the out-of-the-money put option at the same strikeK0. If we further assume that the forward level is in the middleof the two adjacent strike prices and approximate the intervalDK0 by Ft 2 K0, the last two terms in Equation (A-15) cancelout to obtain
(A-16)
Thus, the VIX definition matches the theoretical relation forthe risk-neutral expected value of the return quadratic varia-tion up to a jump-induced error term and errors induced bydiscretization of strikes.
ENDNOTE
The authors thank Stephen Figlewski (the editor), RobertEngle, Harvey Stein, Benjamin Wurzburger, and seminar par-ticipants at New York University and the 20th Annual RiskManagement Conference in Florida for many insightful com-ments. All remaining errors are ours.
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VS t TT t
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e O K Ti
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To order reprints of this article, please contact Dewey Palmieri [email protected] or 212-224-3675.
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