convergence risk the problem with the new vix futures

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CONVERGENCE RISK: THE PROBLEM WITH THE NEW

VIX FUTURES

Ivelina Pavlova

Doctoral CandidateFlorida International University

Robert T. Daigler*

Knight Ridder Research Professor of FinanceFlorida International UniversityDepartment of Finance RB 208

College of BusinessMiami, Florida 33199

(H) 954-434-2412(O) 305-348-3325

* Corresponding author

Keywords: VIX, convergence, expiration day effects

The authors thank John Hiatt (CBOE) for helpful discussions concerning the VIX futuressettlement procedure. We wish to thank the participants of the 2007 Eastern FinanceMeetings in New Orleans for helpful comments. This paper will also be presented at the2007 Financial Management Meetings in Orlando.

September 29, 2007



CONVERGENCE RISK: THE PROBLEM WITH THE NEW

VIX FUTURES

ABSTRACT

We examine the issue of non-convergence of the VIX futures to thecash VIX, the associated expiration day effects, and their sources.Significant disparities are found between the value of the cash VIXand the VIX futures at settlement of the contract. The reasons for thisdifference include the settlement procedure of the exchange and theunderlying S&P 500 options order imbalances at settlement, thelatter affected by traders unwinding arbitrage positions. We proposean alternative settlement procedure that mitigates these problems.

I. Introduction

VIX futures typically do not converge to the underlying VIX cash index at expiration.

VIX futures settlement prices differ by an average of 26 basis points ($260 per contract)

from the cash VIX index, with a standard deviation of 43 basis points and a maximum

deviation of $1,766. Consequently, the size of the non-convergence problem is large and

economically significant. There is no known method to estimate the size of the non-

convergence ex-ante, with possible forecasting variables possessing little evidence of an

association with the non-convergence bias at settlement. Given the growing importance

of volatility derivatives, the size and ongoing bias is an important issue to this category of

derivatives and to the structure of future derivative instruments. This paper identifies the

issues causing the bias and investigates the historical characteristics associated with the

non-convergence of the futures and cash index.

Previous futures convergence problems fall into three categories. First, squeezes in

commodity markets have caused significant pricing and delivery issues. Examples are

the wheat squeeze of the 1980s (Easterbrook, 1986, and Paul, 1986), the Maine versus

1



Idaho potato controversy (Easterbrook, 1986, and Pirrong, 1993), and the attempted Hunt

silver squeeze (Williams, 1995). Second, more common settlement problems involve

contract development issues, such as futures without a tradable underlying cash market.

Examples of such markets include catastrophy futures, inflation and GNP futures, and

weather derivatives. Such contracts typically fail or only exist as specialized over-the-

counter options instruments. Third, some index-based futures contracts are based on cash

markets that are difficult to match exactly at expiration. The defunct commercial paper

futures were based on the commercial paper of 40 companies, with the specific

composition of the index changing frequently, creating basket-matching problems. In

addition, stock index futures are based on a weighted index of the prices of all stocks in

the index, which is difficult to replicate exactly (for arbitrage) when some of the stocks

do not trade at the futures settlement.

The settlement issues with VIX futures are most like the index-based futures pricing

problems, except the discrepancies for stock index futures are relatively small. The

reason for the large and variable difference between the futures settlement and the

corresponding cash index is the procedure that determines the settlement price.

Specifically, while the cash VIX employs the average of the bid and ask prices of each of

the S&P 500 options that are used to calculate the cash VIX, the futures settlement uses a

special procedure which employs the actual option prices traded at the market open

following the last day of futures trading. Typically, at this open the majority of the option

strikes trade at the bid (or at the ask) for a given expiration settlement, based on the

supply and demand pressures from dealers, arbitrageurs, or hedgers closing their

positions.

2



The uncertainty arising from the futures settlement procedure causes the futures to

trade nearly 2% below the cash index, on average, for several days before futures

expiration.1

The S&P 500 options volume creating the price convergence problem

averages less than 16% of the remaining VIX futures open interest at settlement, showing

that a relatively small position in S&P 500 options can cause a large price divergence

from the cash VIX for the futures settlement price. This raises questions of whether

traders could manipulate the VIX futures settlement price. We propose an alternative

settlement procedure that substantially reduces the price divergence relative to the current

procedure.

II. The Issues and Literature

Futures contracts are settled either via the delivery of the physical asset or through

cash settlement, the latter initiated with stock index futures in the early 1980s. Jones

(1982) explores the economics of futures contracts based on cash settlement, showing

that the major purpose of the delivery procedure is to cause the futures and cash prices to

converge during the delivery period. One of the key differences between physical

delivery and cash settlement is that the settlement price for the physical delivery

procedure can be determined through futures trading,2 while with the cash settlement

procedure the settlement price is determined by the value of the underlying cash market.

Garbade and Silber (1983) discuss the construction of an accurate settlement index as a

1 However, when individual contract behavior is examined for these two days then the direction of thefutures-cash basis is only weakly associated with the direction of the settlement basis. Moreover, there isno association between the size of the ex-ante basis and the settlement basis.2 The settlement price for most commodity futures is based on the futures price during the delivery period.For financial futures the T-bond/T-note settlement price is based on the futures price and the conversionfactor of the bond delivered, with the conversion factor determined from the coupon and maturity of thedelivered bond/note.

3



fundamental requirement for the design of a cash-settled futures contract, with special

consideration being given to the exclusion of intentionally distorted quotations from the

settlement index. For example, Eurodollar futures settle by obtaining twelve bid-ask

quotations on the LIBOR 90-day interest rate on the day of cash settlement; the exchange

then drops the two highest and two lowest quotations to avoid bias, averaging the

remaining eight quotes to determine the futures settlement price.

In the 1990s the focus on cash index-based contracts moved from design issues to

price manipulation. Kumar and Seppi (1992) developed a model in which uninformed

traders can profit by executing trades in the spot market to manipulate the associated

futures price during the delivery period.3

Kumar and Seppi show that profitable

manipulation can take place based on their model, although the profit from manipulation

falls to zero as the number of manipulators grows. However, even in the zero-profit case

the actions of the manipulators influence liquidity, and the existence of manipulation is

robust to a variety of distributional, timing, and preference assumptions.

A number of studies investigate the derivatives expiration day manipulation and

pricing effects in stock index markets. In particular, the so-called “triple watching hour,”

when all index futures, index options and equity options expire at the same time, attracted

considerable attention from academics, regulators and practitioners, especially given the

large price movements that occurred on expiration day.4

The factors found to contribute

to the undesirable expiration day effects include stock index arbitragers liquidating their

3 There are four types of market participants included in their model – a strategic risk-neutral informedtrader, a group of uninformed noise traders, risk-neutral floor traders or specialists, and a risk-neutraluninformed “manipulator.”

4 Price volatility issues associated with the S&P 500 futures expiration caused its settlement to be

moved from its Friday close to the open on the third Friday of the expiration month. See Stoll and Whaley(1991, 1997).

4



positions, stock market procedures related to offsetting sudden imbalances caused by the

liquidation of positions, and deliberate attempts to directly and indirectly manipulate

prices (Stoll and Whaley, 1997). Stoll and Whaley (1987, 1991, 1997), among others,

investigate the impact of these expiration day factors on volume, volatility, and price.

They find that all of the above factors are related to expiration day price behavior.

Specifically, an index arbitrager could attempt to manipulate the cash stock index if he

had a long position in stock and a short position in futures contracts by selling some of

his stocks at non-equilibrium prices in order to reduce the futures settlement price. Stoll

and Whaley observe small (economically insignificant) stock prices movements at

derivative expiration in U.S. and Japanese stock markets, but not in the Australian

market. Illueca and LaFuente (2006) examine the realized volatility Spanish futures

market, finding a significant increase in realized volatility on the expiration day.

Most interesting is the Alkeback and Hagelin (2004) study which examines the

Swedish stock market index futures expiration day effects, where they find no evidence

of price distortions at settlement. The reason they provide for a lack of price distortion is

that the settlement price for the Swedish index is determined by the average volume-

weighted value over the entire last trading day. This longer settlement period is in

contrast with the practice used in the U.S. and other futures markets where settlement

occurs at the opening or closing of a trading day. Hence, the length of time of the

settlement period could have important implications for exchanges and policy makers to

minimize expiration day effects on cash-settled contracts.

5



III. The Cash VIX and VIX Futures Contract

A. The CBOE Volatility Index (VIX)

The CBOE Volatility Index is calculated in real time using the time values of out-of-

the-money options on the S&P 500 cash index and consequently reflects the expectations

of options market participants regarding the future 30-day volatility. The original VIX

(now called the VXO) is still available, and is calculated using the implied volatilities of

only four call and four put strike prices of the S&P 100 index by employing the binomial

option pricing model. The new VIX employs all available strikes of the S&P 500 index

by means of the following non-distributional formula:

5

2

0

2

2 11

)(2∑ ⎥

⎦

⎤⎢⎣

⎡−−

Δ=

i

i

RT

i

i

K

F

T K Qe

K

K

T σ

where

VIX = σ x 100

T = time to expiration (in years, calculated in minutes)

F = the forward index level derived from index option prices (the strike price for the

near-the-money option adjusted for time, the interest rate, and option prices)

K i = strike price of the ith out-of-the-money option

ΔK i = the size of the interval between strike prices

K 0 = the first strike price below the forward index level F; this is the nearest-to-the-

money strike price

R = the risk-free interest rate until option expiration

5 Once the nearest-to-the-money option is determined then all remaining options for the cash VIXcalculation must be out-of-the-money. The nearest-to-the-money option is today’s forward rate of theunderlying index, as shown below. The options must have non-zero bid prices to be used in the calculation.Options are selected until two consecutive strikes have non-zero bids, at which time no more strikes areemployed in the calculation.

6



Q(K i) = the midpoint of the bid-ask spread for the option with strike K i.

Note that the contribution of any given strike price to the calculation of the cash VIX is

proportional to the price of the option and inversely proportional to the strike price. The

put and call option prices used in the calculation of the cash VIX employ the next two

monthly expirations in order to obtain a constant 30-day maturity.6

Jiang and Tian (2007) do show that the procedure to calculate the cash VIX does

cause biases in the estimation of the VIX compared to a model-free implied variance.

However, the cash VIX calculation biases can not be corrected with adjustments to real

world trading procedures using CBOE options.

7

Moreover, Jiang and Tian’s procedure to

correct for these biases would create an untradable VIX index, which would affect

liquidity and pricing for the VIX futures contract (since dealers could not hedge their

positions). In any case, the errors in the cash VIX calculation procedure are not important

here, since it is the difference between the cash VIX and the VIX futures we are

concerned about.

B. VIX Futures

On March 26, 2004, the trading of futures on the VIX index began on the Chicago

Futures Exchange (CFE). The open interest as of May, 2007 was over 44,000 contracts,

which represents 90% of the total open interest for the CFE.8 The Chicago Futures

6 When the nearby option has exactly 30 days until expiration then only one expiration is employed to

calculate the cash VIX. This occurs at the settlement f the VIX futures.7 Jiang and Tian’s simulations show that errors are generally small for typical parameters for actual S&P500 options. In general, they conclude that the CBOE procedures tends to underestimate (overestimate)true volatility when the underlying cash VIX is high (Low). When the true volatility is 20% then the error averages less than three basis points. However, low volatility levels (10%) show simulated Black-Scholes-Merton differences of 19 to 25 basis points, while high volatility (30%) show differences of a negative 35to 51 basis points. Using a stochastic jump model creates even larger differences. However, the non-convergence biases found in our analysis are not strongly associated with the level of volatility.8 Until March 26, 2007 the Volatility Index (VIX) futures were quoted as $100 times the Increased-ValueVIX (VBI), the latter equaling ten times the value of the cash VIX. On that date the VIX futures quotation

7



Exchange (CFE) can list up to three near-term serial months plus eight months on the

February quarterly cycle. The price of a VIX futures contract depends on a number of

factors, such as stochastic volatility, how quickly volatility reverts to the mean, and the

expectation of the market participants regarding future volatility.

C. Settlement Day Procedures

The last trading day of the VIX futures contracts is on the day prior to the settlement

date. The settlement date is the Wednesday that is thirty days before the third Friday of

the next calendar month after the futures contract expires. Typically this is the

Wednesday prior to the third Friday of the futures expiration month.

The final settlement value for the VIX futures contract is determined from a Special

Opening Quotation (SOQ) of the cash VIX, calculated from the relevant opening option

prices used to calculate the VIX index on the settlement date (the average of the bid and

ask are employed for any relevant strike prices not traded; which strikes used for the

settlement value calculation follow the procedure employed to calculate the cash VIX).9

The opening option prices are established by an auction mechanism known as the "Rapid

Opening System" (ROS). ROS is responsible for matching the buy and sell orders in the

Electronic Order Book prior to 8:28 a.m. Central time. If there are no opening traded

prices for the option then the bid-ask quote for each eligible option series is used.10

was re-scaled, with the futures prices divided by 10 to provide direct comparison with the cash VIX.

Consequently, the contract multiplier was changed from 100 to 1000 times the VIX, thus eliminating theuse of the VBI. For simplicity and clarity, the results in this paper adjust previous VIX futures quotes toconform with the number of decimals used for the cash VIX.9 The final settlement value for the VIX futures contract was determined as ten times a Special OpeningQuotation (SOQ) of the VXB until March 26, 2007 (similar to the price quotes discussed above).Multiplying by ten is no longer needed. Here we adjust the VIX futures settlement to be consistent with thecash VIX.10 For the SPX (S&P 500 options) ROS opening on the futures expiration day, all orders that are placed inthe electronic book for this opening can be only on the one SPX option contract month whose prices areused to estimate the VIX, i.e. the options contract month with exactly 30 days until expiration. For

8



IV. Empirical Evidence

A. Cash and Futures Settlement Prices

Analysis of whether VIX futures prices converge to the cash VIX is based on data for

the first 33 expirations of the VIX futures. Table I provides detailed information about

the VIX cash and futures values at the close on the last futures trading day and the open

on the settlement day, as well as their differences. Column 4 provides the official VIX

futures settlement price. As seen in column 5, the difference between the VIX cash and

futures values at the close of the last trading day is greater than 0.30 for ten cases and

greater then 0.50 for five cases. One possible explanation for such large differences is the

expectation in the futures market for supply/demand mismatches of the options at the

next day’s open that will affect the VIX settlement price. However, column 6 shows that

the futures market at the close of trading is a poor forecast of the next day’s settlement

price, as determined by the options opening prices. Also, there is no pattern as to

direction of the forecast error. In fact, column 7 shows that the cash VIX often changed

significantly from the close on the last trading day to the next day’s open. Hence, the

overnight change in the options’ time values does affect the cash VIX. But none of these

factors explains the differences in column 8 between the cash VIX open and the VIX

futures settlement price.

Table I

The focus of this paper is the lack of convergence at expiration between the VIX

futures and cash VIX: column 8 of Table I presents ample evidence of this lack of

example, only June 2007 option prices are needed to calculate the May 2007 VIX settlement price.Moreover, SPX option orders associated with positions in the expiring VIX futures must be received prior to 8 a.m. during the ROS opening procedure.

9



convergence. The difference between the opening value of the cash VIX and the

settlement price of the VIX futures using the special opening procedure was greater than

0.30 ($300 per contract) for 18 out of 33 expirations, and greater than 0.50 ($500 per

contract) on eight occasions. In July 2004, the disparity was a striking 1.77 points or

$1,770 per contract, more than 10% of the total underlying value of the position. These

disparities arise as a result of the settlement procedure of the expiring VIX futures

contract, which is based on the option trade prices during the ROS opening. The

differences are significant and do not decrease with the seasoning of the futures contract;

for example, four of the five expirations in 2007 all had disparities greater than 0.20 and

two of them were greater than 0.30.

Table II reports various descriptive statistics regarding the differences shown in Table

I. The minimum and maximum values show the wide range of differences for this

contract, particularly on the settlement day. The differences between the VIX open and

the futures settlement price is greater than 0.30 for 55% of the expirations and greater

than 0.50 for 24% of the expirations.11

TABLE II

Figure 1 examines the average convergence over time to expiration of the VIX

futures to the cash VIX. For the last three days before settlement the cash VIX is on

average higher than the VIX futures price, and they do not converge. Figures 2 and 3

present the convergence of the two expirations with the largest and smallest differences

11 We examined whether any correlation exists for the differences outlined in tables I and II with the levelsof the cash VIX at the close and next day’s open. Little correlation exists (ρ < 0.08) between higher valuesof the Volatility Index and the differences between the close and next day’s settle of the futures contract.Moreover, larger values of the VIX only possess a moderate association with larger disparities at settlement(ρ < 0.20).

10



between the cash VIX open on settlement day and the futures settle price.12 Comparing

Figures 2 and 3, the difference the day before settlement provides no relation with the

size of the non-convergence bias.

Figures 1 and 2 and 3

B. The Settlement Procedure and Arbitrage

The settlement value of the VIX futures is based on the trade prices of the constituent

options during the special ROS opening.13 Previously we showed that substantial

differences can exist between the VIX futures settlement value based on these option

trade prices and the cash VIX at the open on settlement day. Here we show the cause of

this disparity by examining the option trades during an ROS opening. Figures 4 (puts)

and 5 (calls) depict the bid-ask spread of these options and the difference between the

opening option price and the bid quote for the April 2006 ROS settlement. The figures

show that the opening bid-ask spreads are smaller for options that are more out-of-the-

money. Moreover, the opening price is much closer to the opening bid quote than to the

ask for both put and call options. In fact, for most of the constituent options series the

opening price was at or near the bid. These order imbalances on the bid side of the market

are consistent with arbitrageurs who are selling their options positions to close long

option positions (matching their short futures positions).

Figures 4 and 5

Exhibit I shows the offsetting positions taken by arbitrageurs at initiation and how

they close their positions at futures settlement. Part A shows when futures are priced

12 Traditionally, the major purpose for the delivery or cash settlement procedures of futures contracts is tocause convergence of the price of the futures contract and the cash price during the delivery period, thelatter providing the basis for the futures contract pricing.13 The bid-ask average is included for those options not traded but which still possess a bid-ask spread.

11



high relative to the forward cash VIX at time t, and Part B covers the situation when

futures prices are low compared to the forward cash VIX. The key aspect of the

settlement procedure for dealers and arbitrageurs is the knowledge that the futures and

options positions can be exactly offset at futures expiration, based on the ROS settlement

procedure guaranteeing a profit. However, while this settlement procedure benefits

dealers and arbitrageurs, it causes convergence problems. Specifically, when the

proportion of dealers/arbitrageurs or hedgers unwinding their positions dominates the

options market on the futures settlement date, then the opening options prices will be

biased in one direction. This causes a significant difference between the futures

settlement value and the cash VIX, the latter based on the average of the bid and ask

prices of the same options. Moreover, it may be possible for traders to manipulate

opening options prices in order to magnify profits from holding a position in VIX futures.

Such a strategy would depend on trading a relatively small number of options contracts

relative to the position in futures. Hence, the question becomes whether a small options

position can significantly affect the VIX futures settlement price.

Exhibit I

Table III shows the volume of options traded at the options opening on the futures

settlement days to examine the size of the positions that determine the futures settlement

price. The difference between the cash VIX and the futures settlement price averaged

0.34 over all expirations presented in Table III. More striking is that the average options’

volume at the ROS opening only comprises 15.73% of the VIX futures open interest at

settlement, and only 14.40% for larger differences in the cash VIX vs. the futures

12



settlement price. Hence, the possibility of affecting a much larger futures settlement

open interest position with options trading does exist.

Table III

C. A New Procedure for VIX Cash Settlement

An alternative procedure to the ROS option trade price method is to use the average

of the one-minute cash VIX values on either the last trading day or the settlement day to

determine the settlement value. Table IV presents the results of using such a procedure.

Column (1) shows the difference between the cash VIX and the futures settlement price

using the current ROS method. Columns (2) and (3) show the relevant differences using

the one-minute average VIX value to determine the futures settlement price.14

Columns

(4) and (5) show the one-minute standard deviations for the VIX on these two days.

TABLE IV

For 21 of the 33 expirations in Table IV the average one-minute VIX values over the

last trading day (column 2) provides smaller differences in the cash VIX vs. futures

settlement price than does the current method (column 1). On average these differences

are only -0.10 points, compared to the 0.26 points for the current method. More

importantly, when the current method is superior to the proposed method then the

difference in the two methods was greater than 0.10 for only four expirations.

Alternatively, the differences using the current method are greater than |0.50| for 24% of

the time, while this size difference occurs for the proposed method only 15% of the time.

Consequently, the average VIX procedure on the last trading day provides a viable

alternative to the current settlement method. On the other hand, using the current

14 Column (2) uses the closing cash VIX value for the relevant comparison to that day’s average VIX value.If the settlement price is based on the one-minute average cash VIX value then no opening procedure isneeded to determine the futures settlement price.

13



settlement day (column 3) is only slightly better on average than the current method and

provides substantially greater variability and several very large differences in its cash

VIX versus average VIX values.

V. Conclusions

The empirical investigation of the VIX futures contract expirations conducted in

this study shows a lack of convergence of the futures settlement value to the cash VIX

value, as well as significant disparities between the futures price and cash VIX value. The

size of the non-convergence of futures and cash is unique for futures markets. Previous

non-convergence problems were either isolated supply issues for a given maturity month

or associated with futures that subsequently failed. Thus, identifying and examining this

consistent bias is an important step in eventually modeling a procedure to forecast the

size of the non-convergence.

The reasons for the non-convergence of the futures and cash values include the

ROC settlement procedure (which utilizes actual trade prices of the constituent options)

and the resultant options order imbalances during this procedure. These order imbalances

are caused either by traders unwinding dealer/arbitrage positions, or by speculators

manipulating prices. Traders could close out their positions before settlement. However,

this solution does not solve the underlying problem, since the futures price can deviate

significantly from the cash VIX value before futures expiration. Moreover, traders want

to know the potential fair futures value prior to the settlement, which is not known due to

the effect of the ROS opening procedure.

14



The uncertainty of the settlement price due to the ROS opening procedure creates

risk for futures traders as well as an options bid-ask band around the fair futures price.

On the other hand, such a settlement procedure provides opportunities for dealers and

arbitrageurs, since the ROS method allows them to close their arbitrage positions without

price risk. This procedure allows dealers to stabilize the futures price before futures

expiration and therefore increase the liquidity for the contract, since it reduces risk for the

dealers. Hence, our suggestion to change the settlement procedure would substantially

reduce the price convergence problems but increase the price risk for dealers and

arbitrageurs.

15



References

Alkebäck, Per and Niclas Hagelin. 2004. "Expiration Day Effects of Index Futures and Options:Evidence from a Market with a Long Settlement Period." Applied Financial Economics, 14:6,385-96.

CBOE. 2003. "VIX Primer."

CBOE. 2005. "CFE Information Circular IC05-07." CBOE Futures Exchange.

Easterbrook, Frank H. 1986. “Monopoly, Manuipulaiton, and the Regulatio of Futures Markets.” Journal of Business, 59:2, S103-S127.

Garbade, Kenneth D. and William L. Silber. 1983. "Price Movements and Price Discovery inFutures and Cash Markets." The Review of Economics and Statistics, 65:2, 289-97.

Illueca, Manuel and Juan A. LaFuente. 2006. “New Evidence on Expiration-Day Effects Using

Realized Volatility: An Intraday Analysis for the Spanish Stock Exchange.” The Journal of Futures Markets, 26: 9. 923-938.

Jones, Frank J. 1982. "The Economics of Futures and Options Contracts Based on CashSettlement." Journal of Futures Markets, 2:1, 63-82.

Jiang, George J. and Yisong S. Tian. 2007. “Extracting Model-Free Volatility Prices: AnExamination of the VIX Index.” The Journal of Derivatives, 32:3, 35-60.

Kumar, Praveen S and Duane J Seppi. 1992. "Futures Manipulation with "Cash Settlement." Journal of Finance, 47:4, 1485-1502.

Paul, Allen B. 1986. “Liquidation Bias in Futures Price Spreads.” American Journal of

Agricultural Economics, 68:2, 313-321.

Pirrong, Stephen Craig. 1993. “Manipulation of the Commodity Futures Market DeliveryProcess.” Journal of Business, 66:3, 335-369.

Stoll, Hans R. and Robert E. Whaley. 1987. "Program Trading and Expiration-Day Effects."Financial Analysts Journal, 43:2, 16-28.

Stoll, Hans R. and Robert E. Whaley. 1991. "Expiration-Day Effects: What Has Changed?"Financial Analysts Journal, 47:1, 58-73.

Stoll, Hans R. and Robert E. Whaley. 1997. "Expiration-Day Effects of the All Ordinaries SharePrice Index Futures: Empirical Evidence and Alternative Settlement Procedures." Australian

Journal of Management , 22:2.

Williams, Jeffrey. 1995. Manipulation on Trial: Economic Analysis and the Hunt Silver Case,Cambridge University Press.



Table I

VIX Cash and Futures Values

This table presents the values of the cash VIX and the VIX futures prices on the last trading day and the settlementday of the futures. The last four columns show the differences between the two days and between the cash VIX and

the futures.

(1) (2) (3) (4) (5) (6) (7) (8)

Last Trading Day At the Futures Settlement Difference Between:

Contract

Month

Cash VIX

close

Futures

settle

price

Cash VIX

at the open

Futures

price**

VIX close and

Futures settle

price on last

trading day

Futures price on

last trading day

and Futures settle

price

VIX close on last

trading day and

VIX open on

settle day

VIX open and

Futures settle

price

May-04 19.33 18.36 18.48 18.355 0.08 0.895 0.85** 0.125

Jun-04 15.05 14.00 14.83 13.997 0.05 1.006 0.22 0.833**

Jul-04 14.46 13.13 14.90 13.134 0.15 1.176 -0.44* 1.766**

Aug-04 17.02 17.49 17.55 17.489 0.42* -0.889 -0.53** 0.061

Sep-04 13.56 13.73 13.88 13.725 0.06 -0.225 -0.32* 0.155

Oct-04 15.05 13.16 13.92 13.157 0.03 1.863 1.13** 0.763** Nov-04 13.21 12.84 13.20 12.844 0.63** -0.264 0.01 0

Jan-05 12.47 12.77 12.47 12.772 0.15 -0.452 0.00 -0.302*

Feb-05 11.27 11.29 11.40 11.293 0.02 -0.043 -0.13 0.107

Mar-05 13.15 13.63 13.30 13.626 0.07 -0.546 -0.15 -0.326*

May-05 14.57 13.86 14.11 13.864 -0.08 0.786 0.46* 0.246

Jun-05 11.79 11.01 11.22 11.012 0.63** 0.148 0.57** 0.208

Aug-05 13.52 12.82 13.35 12.819 0.91** -0.209 0.17 0.531**

Oct-05 15.33 15.17 15.63 15.172 0.76** -0.602 -0.30* 0.458*

Nov-05 12.23 12.31 12.22 12.306 0.36* -0.436 0.01 -

Dec-05 11.19 10.18 10.71 10.175 0.04 0.975 0.48* 0.535**

Jan-06 11.91 12.62 12.62 12.615 0.20 -0.905 -0.71** 0.005

Feb-06 12.25 12.04 12.43 12.043 0.30* -0.093 -0.18 0.387*

Mar-06 11.62 11.15 11.71 11.145 0.09 0.385 -0.09 0.565**

Apr-06 11.40 11.94 11.52 11.941 -0.08 -0.461 -0.12 -0.421*

May-06 13.35 14.03 13.83 14.025 0.37* -1.045 -0.48* -0.195

Jun-06 16.69 17.29 16.67 17.285 -0.25 -0.345 0.02 -0.615**

Jul-06 17.74 17.01 17.62 17.005 -0.12 0.855 0.12 0.615**

Aug-06 13.42 12.29 12.69 12.285 0.19 0.945 0.73** 0.405*

Sep-06 11.98 11.29 11.75 11.289 0.16 0.531 0.23 0.461*

Oct-06 11.73 11.43 11.36 11.434 0.50** -0.204 0.37* -0.074

Nov-06 10.50 10.27 10.47 10.268 0.25 -0.018 0.03

Dec-06 10.30 10.05 10.30 10.053 -0.27 0.517 0.00 0.247

Jan-07 10.74 10.71 10.90 10.706 0.15 -0.116 -0.16 0.194Feb-07 10.34 9.95 10.19 9.954 -0.09 0.476 0.15 0.236

Mar-07 13.27 12.98 13.27 12.983 0.21 0.077 0.00 0.287

Apr-07 12.14 12.03 12.48 12.03 0.27 -0.160 -0.34* 0.450*

May-07 14.01 13.63 14.02 13.63 0.05 0.330 -0.01 0.390*

* different by >│0.30│, ** different by >│0.50│ ** The procedure to represent futures prices and settlement prices in March 2007.



Table IISummary Statistics of Differences for the VIX Cash and Futures Contracts

The table presents summary statistics of the differences between the cash VIX and VIX futures prices onthe last trading day and the settlement day.

(1) (2) (3) (4)

Difference Between:

Statistic

VIX close and

Futures close/ on

last trading day

Futures close on last

trading day and

Futures settle price

VIX close on last

trading day and VIX

open on settle day

VIX open and

Futures settle

price

Mean 0.20 0.11 0.05 0.26

Median 0.16 -0.03 0.00 0.25

St. deviation 0.28 0.70 0.40 0.43

Min -0.50 -1.08 -0.71 -0.61

Max 0.87 1.84 1.13 1.77

% of times dif

is above |.3| 30% 39% 55%

% of times dif

is above |.5| 15% 18% 24%

18



Table IIIOptions Trading Volume at the Special Opening on the Futures Settlement Date

The table presents summary statistics regarding the average volume of options per strike traded at theopening of the futures settlement day. Only out-of-the-money options are considered for the settlement.The average of the difference between the VIX and futures prices are absolute averages.

(1) (2) (3) (4) (5)

Average volume

per strikeContract

Calls Puts

Futures open

interest on

settlement day

Difference

between VIX

open and Futures

settle prices

Average Option

volume per strike

as a percentage of

Futures open

interest (%)

Jan-05 202 175 1,534 0.30* 12.29

Feb-05 1,038 1,622 3,214 0.11 41.38

Mar-05 131 279 2,223 0.33* 9.22

May-05 510 768 4,074 0.25 15.68

Jun-05 647 1,160 2,598 0.21 34.78

Aug-05 745 1,173 6,653 0.53** 14.41

Oct-05 114 163 897 0.46* 15.44

Nov-05 312 514 6,0940.09 6.78

Dec-05 263 548 1,548 0.54** 26.20

Jan-06 112 303 831 0.00 24.97

Feb-06 277 455 7,409 0.39* 4.94

Mar-06 96 261 611 0.57** 29.21

Apr-06 8 31 921 0.42* 2.12

May-06 209 670 7,196 0.20 6.11

Jun-06 92 124 2,796 0.61** 3.86

Jul-06 155 283 2,122 0.61** 10.32

Aug -06 1,062 1,454 4,423 0.41* 28.44

Sep-06 403 544 2,753 0.46* 17.20 Nov-06 2,368 2,941 15,458 0.20 17.17

Dec-06 400 335 3,347 0.25 10.98

Jan-07 365 586 3,878 0.19 12.26

Feb-07 868 2,625 8,693 0.24 20.09

Mar-07 824 1,218 5,900 0.29 17.31

Apr-07 1,672 1,280 7,745 0.45* 19.06

Average 536 813 4,288 0.34 15.73 Average if

difference >|.3| 402 521 3,203 0.47 14.40

* different by >│0.30│, ** different by >│0.50│

19



Table IVAlternative Settlement Procedures for VIX Futures

This table shows the results from employing the average of the one-minute cash VIX values as thesettlement value for the futures contract.

(1) (2) (3) (4) (5)

Difference between: Standard Deviation of:

Expiration

Months VIX open and

Futures settle price

Close of the VIX

and Average VIX

on the last trading

day

Open of the VIX

and Average of

the VIX on

settlement day

VIX on the last

trading day

VIX on the

settlement day

May-04 0.13 -0.03 0.48 0.10 0.35

Jun-04 0.83 0.03 0.15 0.17 0.11

Jul-04 1.77 -0.12 1.08 0.13 0.39

Aug-04 0.06 -0.01 1.06 0.15 0.22

Sep-04 0.16 -0.11 -0.53 0.09 0.19

Oct-04 0.76 -0.24 -1.10 0.16 0.37

Nov-04 0.36 -0.15 0.14 0.12 0.13

Jan-05 -0.30 -0.09 -0.21 0.20 0.16

Feb-05 0.11 -0.20 0.11 0.18 0.11

Mar-05 -0.33 0.41 -0.18 0.23 0.15

May-05 0.25 -1.12 0.45 0.52 0.18

Jun-05 0.21 0.05 -0.49 0.13 0.17

Aug-05 0.53 0.48 0.20 0.34 0.16

Oct-05 0.46 0.25 0.47 0.17 0.61

Nov-05 -0.09 0.10 -0.22 0.24 0.10

Dec-05 0.54 -0.02 0.09 0.09 0.18

Jan-06 0.00 -0.24 0.16 0.18 0.11

Feb-06 0.39 -0.24 -0.12 0.25 0.16Mar-06 0.57 0.10 0.33 0.20 0.07

Apr-06 -0.42 -0.53 0.06 0.31 0.11

May-06 -0.20 0.12 -0.99 0.12 0.74

Jun-06 -0.61 -0.22 1.25 0.26 0.39

Jul-06 0.61 -0.88 2.39 0.43 0.52

Aug-06 0.40 0.87 0.29 0.20 0.12

Sep-06 0.46 -0.28 0.25 0.26 0.08

Oct-06 -0.07 -0.04 -0.21 0.14 0.12

Nov-06 0.20 -0.44 0.10 0.26 0.12

Dec-06 0.25 -0.40 0.13 0.25 0.07

Jan-07 0.19 0.01 0.29 0.08 0.14

Feb-07 0.24 -0.29 0.25 0.26 0.14

Mar-07 0.29 -0.70 0.22 0.49 0.68

Apr-07 0.45 0.32 0.25 0.15 0.11

May-07 0.39 0.23 0.26 0.22 0.16

Average 0.24 -0.10 0.19 0.21 0.22

20



Exhibit I: Arbitrage Transactions with VIX Futures

A. When VIX futures price > options time value (forward cash VIX)Futures Transactions Options Transactions

At time t Sell (short) VIX futures Buy a portfolio of out-of-the-money options

At futures expiration Cover short futures position bysettling (“buying”) at expiration

Sell portfolio of options at the bid price atROS settlement

B. When VIX futures price < options time value (forward cash VIX)Futures Transactions Options Transactions

At time t Buy VIX futures Sell (short) a portfolio of out-of-the-moneyoptions

At futures expiration Cover long futures position bysettling (“selling”) at expiration

Buy back the portfolio of options at the ask price at ROS settlement

21



Figure 1Average Convergence of Cash VIX and VIX Futures

-4%

-2%

0%

2%

4%

6%

8%

10%

12%

024681012141618202224262830

( f u t - c a s h ) / c a s h

AverageConvergence

Figure 2Convergence of the Two Contracts with the Largest Difference between the VIX Open on

Settlement Day and the Futures Settlement Price

-20%

-10%

0%

10%

20%

30%

40%

024681012141618202224262830

( f u t - c a s h ) / c a s h

Jun-04

Jul-04

Figure 3Convergence of the Two Contracts with the Smallest Difference between the VIX Open on

Settlement Day and the Futures Settlement Price

-20%

-10%

0%

10%

20%

30%

40%

024681012141618202224262830

( f u

t - c a s h ) / c a s h

Aug-04

Jan-06

Note: In the last two graphs the open VIX is used on the settlement day to compare the VIX open and the futuressettlement price; on the days before expiration the VIX close and futures settlement price are used.

22



Figure 4Out-of-the-Money Put Options

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

1 0 2 5

1 0 7 5

1 1 2 5

1 1 6 0

1 1 7 0

1 1 8 0

1 1 9 0

1 2 0 0

1 2 1 0

1 2 2 0

1 2 3 0

1 2 4 0

1 2 5 0

1 2 6 0

1 2 7 0

1 2 8 0

1 2 9 0

1 3 0 0

Ask - Bid

Open - Bid

Figure 5Out-of-the-Money Call Options

0.00

0.30

0.60

0.90

1.20

1.50

1.80

2.10

1 2 9 0

1 3 1 0

1 3 2 0

1 3 3 0

1 3 4 0

1 3 5 0

1 3 6 0

1 3 7 0

1 3 8 0

1 3 9 0

1 4 0 0

1 4 2 5

Ask - Bid

Open - Bid

Figure 6Average VIX Futures Open Interest

2000

3000

4000

5000

6000

7000

024681012141618202224262830

Average

Open

Interest

convergence risk the problem with the new vix futures

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