VXEEM - Emerging Markets Volatility Index Futures Coming To CBOE

CBOE announced that they are launching VXEEM futures on January 9th 2012, and I think there is a good chance the product will take off. At the time when CBOE launched GVZ futures they did not have DPM, however they do have DPM now for VXEEM futures. Also VXEEM has similar dynamics to the VIX, and hopefully that will create some liquidity spillover. Of course I should mention in the timeline of listed volatility products there are many more delisted and inactive volatility products than there are successful ones, but I hope  with coming listings of volatility products (VNKY, VHSI, etc) that we will finally see volatility futures as the first-rate products.

P.S. No date for VXEEM options launch.

VHSI Futures: HSI Volatility Index Futures to Trade on HKEx

Another exciting development in the world of listed volatility products: Hong-Kong Exchange announced listing of futures on VHSI Index at the end of February 2012. This will bring the number of volatility futures products to 5 adding to VIX in the US, VSTOXX in Europe, RTSVX in Russia, VNKY in Japan that are also being launched next year.

News release with product specifications here, Risk magazine article here.

Two months ago I actually mentioned the possibility of VHSI futures on my blog:  "Hang-Seng Indexes Company announced some technical changes to calculation of VHSI - HSI Volatility Index. I am speculating that there is usually no reason to change index methodology unless the exchange is planning to do something, and it may indicate the first step toward Hong-Kong volatility derivatives"

P.S. See my update on VHSI Futures here. 

Expiration Analysis

VIX and VSTOXX finished at their lowest levels in months: VRO - VIX expiration value for December is 21.36, and VSTOXX Index closed Wednesday session at 30.26. Both numbers are siginificatly lower than the forecasts that I made last month, as the markets reacted to a number of good news items from Europe, and possibly end of year seasonality. My direction forecast for VIX was correct, while VSTOXX was not. This unfortunately seems to be the pattern lately - my VIX forecasts are more accurate on the direction than VSTOXX, especially when direction signals disagree.

My forecast for next expiration is for VIX to close at 23.10 vs 24.50 in the futures, and VSTOXX to close at 30.48 vs 32.25 in the futures. Futures prices are Wednesday settlement prices. As before all the forecasts are saved in forecasts tracker page.

Leveraged ETFs: Volatility Skew 2

In the last post I mentioned two models that I developed for translating volatility skew of a main, liquid ETF to volatility skew of a leveraged ETF. Even though these models are not in production (or even production quality, at this stage) I cannot disclose the formulas. However I wanted to show something, so I created these plots to illustrate them on a concrete example: fitting volatility data of SPY etf (january expiration, static snapshot of the data taken few days ago) and creating volatility skews for SSO, SH, SDS, and SPXU.

Market implied volatilities (mid) are in blue. Two models - "red" and "green", with maximum absolute error in vol points between model and market. Neither of the models performs excellent, but red model seems to be particularly bad. Models particularly disagree in the extreme tails, which is not really surprising. If I will have the time, in the following post I will compare results from my models to Zhang's non-parametric approach.

Leveraged ETFs : Volatility Skew

There is very little research available on taking volatility skew from regular (liquid) ETF and producing skew for inverse / leveraged ETF. Another related problem is producing consistent volatility skews between ETFs with the same underlying but different leverage factors.

The only source that I know of is PhD thesis of Jian (Stanley) Zhang, "Path-Dependence Properties of Leveraged Exchange-Traded Funds: Compounding, Volatility and Option Pricing" available for download here. The author presents 3 approaches to the solution.

1. Author calibrates observed liquid options prices to Heston model and computed options on leveraged ETFs using MC. 
2. Second approach is to compute prices of options on leveraged ETF as linear combination of "regular" ETF options, a-la var swap from options prices.
3. Non-parametric skew model that translates "regular" ETF skew into leveraged ETF skew.

While research is certainly groundbreaking, these approaches suffer from some shortcomings. The first two are not directly application to American options, while the third lacks theoretical justification.

In the following posts I would like to elaborate on Dr Zhang's nonparametric approach, as well as two parametric approached that I developed. 

Leveraged ETFs & Decay

There are plenty of things written about leveraged ETFs and their apparent decay based on empirical observation that both bull and bear products often underperform the benchmark. The usual explanation (and the one that makes most sense to me) is that ETFs don't really decay - just appear to be doing so because higher leverage lowers median - the most likely - price, but expected value remains unchanged. Jev Kuznetsov covered the topic few months ago on his blog.

Recently I started working on leveraged ETF models, not for the underlying but for options on leveraged ETFs, and spent more time thinking about the basics. Someone asked me if distribution of "regular" ETF returns is not gaussian, would leveraged ETF price still be a martingale. In retrospect the answer should have been trivial: if E[S*(1+r)]=S, then E[r]=0, and E[L*r]=0, and therefore E[S*(1+L*r)]=S, where S is price, r is return, L is leverage, and E is the expectation operator.  However I did not figure this out right away, and was not sure about the answer. Most of the "basic" illustrations are based on binomial tree that converges to gaussian, and I did not see the same illustration being done for distributions with excess skew or kurtosis.













Here is the usual binomial tree for an ETF. The instrument moves up 10%, or down 10 %. Implicit in the plot are probabilities of up and down move - in this case both equal to 50%. It is easy to verify that the ETF is a martingale, i.e. the expected value does not change. Leveraged version of the ETF is also a martingale.













Now, let's consider a different type of binomial tree, where probabilities are not the same.













Here the ETF moves up x and moves down 1/(1+x). In the first case x = 0.1, and stock moves up to 100 * (1+x) = 100 * 1.1 = 110, and moves down to 100 * 1/(1+x) = 100 * 1/1.1 = 100 / 1.1 = 90.90. Probabilities are of course cannot be the same. To calculate probabilities I solve equation [1 = p*u + (1-p)/u] for p; p = (1/u - 1)/(1/u - u). In our case (1/1.1-1)/(1/1.1-1.1) = 0.47619 , and 100 = 110 * 0.47619 + 90.90909 * (1-0.47619). Similarly for the second step, terminal probabilities are 0.47619^2 = 0.226757, 0.47619 * (1-0.47619) = 0.249433, and (1-0.47619)^2 = 0.274376; expected value is 0.226757 * 121 + 0.249433 * 100 + 0.249433 * 100 + 0.274376 * 82.64463 = 100













Double-leveraged version works as expected. All these examples are trivial - ETF is a martingale, binomial is gaussian in limit. However we can also construct a trinomial tree that can simulate probability distributions other than gaussian.















Up and down moves are chose arbitrarily, while the middle move is set to sqrt(up*down), so the tree recombines. For clarity's sake I drew all nodes. Probabilities are constrained to sum up to 1, and to produce a martingale - p1 * up + p2 * mid + (1-p1-p2)*down = 1. In the illustration my price multipliers and probabilities are:








Leveraged version does not produce a trinomial tree that recombines, but price still remains a martingale.
















In the following posts I will address the topic of constructing consistent volatility skews for regular, inverse, and leveraged ETFs.

Expiration Analysis

Volatility indexes traded in a broad range with politically-driven swings. VIX rose 0.21, and VSTOXX fell 1.81. I managed to be wrong on both directions for the index futures - you can see my complete forecasting track record on Forecasts Tracker page. There is a lot of uncertainly going forward because political factors are driving the market and these things are not something I can forecast with a statistical model. Still - for the next expiration I predict
VIX to close at 32.06 vs 32.35 in the futures, 
VSTOXX to close at 38.31 vs 37.95 in the futures. 
Futures prices are Friday settlement.

Leveraged ETFs

Recently Dr Eric Falkenstein published a post about strategy involving shorting a pair of leveraged ETFs. People observe that since both leveraged and inverse leveraged ETFs tend to underperform, shorting both is a relatively market-neutral way to capitalize on this negative drift. According to the author the strategy generates 14% return with standard deviation of 12%. Image from falkenblog:

 

Unfortunately the strategy does not quite work as well as expected - see readers comments at the end of post. The author notes "I am ignoring the short rebate, which for these may have been highly negative for some of these, but on average these have pretty meager short rates" which is just not true. The main hindrance to profit is borrowing costs - leveraged and inverse ETFs are usually in demand to short, and require a premium (negative rebate), and occasionally are impossible to locate. According to someone who was running such strategy it worked well in 08, but in 09 became flat, and was closed in 10.

While trying to understand leveraged ETFs, and particularly the underlying mathematics I came up with more questions than answers. Two excellent resources I came across is Jian (Stanley) Zhang article with Marco Avellaneda, and Eric Forgy's blog.

Internet Search Queries Predict Stock Market Volatility

Can Internet Search Queries Help to Predict Stock Market Volatility? by Thomas Dimpfl and Stephan Jank, link.

There is a very interesting research on using publicly available Google Trends data to predict realized stock market volatility. The researches use a number of standard models of realized volatility and add search query index results to come up with better forecasts. The most interesting part that I found was that the authors claim significant gain (over standard models) in prediction accuracy during high volatility period in the autumn of 2008. Unfortunately the analysis does not extend to the most recent period of high volatility.

Abstract:

This paper studies the dynamics of stock market volatility and retail investor attention measured by internet search queries. We find a strong co-movement of stock market indices' realized volatility and the search queries for their names. Furthermore, Granger causality is bi-directional: high searches follow high volatility, and high volatility follows high searches. Using the latter feedback effect to predict volatility we find that search queries contain additional information about market volatility. They help to improve volatility forecasts in-sample and out-of-sample as well as for different forecasting horizons. Search queries are particularly useful to predict volatility in high-volatility phases.

I have been working on a similar approach, also using Google Trends and Google Correlate to find drivers for volatility. While intuition was telling me that keywords like "bearish", "negative", "crash", "risk", "drawdown", or just "volatility" would have high predictive power - they did not, it was mostly benign searches like "stock futures" and various references to stock indexes that were most correlated to volatility. I guess people search more when market is crashing, not so much when there is a bull market. I am very interested in advancing research on this topic, and hope to share some of my own findings soon.

A similar topic is analysed in Web search queries can predict stock market volumes, link by a team of researches using proprietary company-specific search query dataset from Yahoo!. They observe one-way causality, that search queries help predict trading volume, but not the reverse.

Abstract:

We live in a computerized and networked society where many of our actions leave a digital trace and affect other people’s actions. This has lead to the emergence of a new data-driven research field: mathematical methods of computer science, statistical physics and sociometry provide insights on a wide range of disciplines ranging from social science to human mobility. A recent important discovery is that query volumes (i.e., the number of requests submitted by users to search engines on the www) can be used to track and, in some cases, to anticipate the dynamics of social phenomena. Successful exemples include unemployment levels, car and home sales, and epidemics spreading. Few recent works applied this approach to stock prices and market sentiment. However, it remains unclear if trends in financial markets can be anticipated by the collective wisdom of on-line users on the web. Here we show that trading volumes of stocks traded in NASDAQ-100 are correlated with the volumes of queries related to the same stocks. In particular, query volumes anticipate in many cases peaks of trading by one day or more. Our analysis is carried out on a unique dataset of queries, submitted to an important web search engine, which enable us to investigate also the user behavior. We show that the query volume dynamics emerges from the collective but seemingly uncoordinated activity of many users. These findings contribute to the debate on the identification of early warnings of financial systemic risk, based on the activity of users of the www.

Expiration Analysis and VIX Forecasts

In the month since last expiration volatility indexes moved in a broad range, but finishing relatively unchanged. VIX index trades as low 28.24 and as high as 45.45, while VSTOXX traded from 34.94 to 50.44. The forecasts that I made last expiration - for VIX to close lower and VSTOXX to finish higher - we half correct: VIX expired at 33.15 (vs 35.70 in futures) but VSTOXX closed on Wednesday at 39.89 (vs 40.70 in the futures). All the forecasts are stored separately on Forecasts Tracker page. I will be updating PL chart in the next day or two.

My forecast for November VIX to close at 36.10 vs 34.05 in the futures market (Thursday close)
My forecast for November VSTOXX to close at 42.76 vs 40.50 in the futures market (Thursday close)

Volatility Derivatives Around The World

The recent document about possible introduction of VXSLV Silver Volatility Futures and Options prompted me to revisit history of listed volatility products. At this time there are only three actively traded volatility products in the world: VIX in the US, VSTOXX in Europe, and RTSVX (usd-denominated!) in Russia.

VSTOXX futures volume has been increasing, however options volume is experiencing some growing pains. Eurex is planning for different ways to boost options volume.

RTSVX volatility futures were listed in the late May 2011, and had a strong start. Unfortunately it seems that trading activity has been falling since the latest economic crisis, probably because some shorts were blown out of their positions when the index rose from 26 to 71 in August (including a heart-attack inducing intraday high of 112 on 8/9, although that was probably just a bad print) No options are planned at this point.

There are few products that are not actively traded, and at this point are set for delisting:
OIV - Nymex Crude Oil volatility derivatives listed on CME. Both futures and options are listed, but there is absolutely no trading activity.
GIV - Comex Gold volatility derivatives, also on CME, also no trading at all.
GVZ - Gold volatility futures are options introduced in April of this year on CBOE, despite a promising début have not traded at all in the last few months.

There are also a few products in the works:
CME has been planning to list two CBOT-based indexes - SIV on soybeans and CIV on corn in the first quarter of 2011, but obviously these have been postponed because of the failure of their other volatility products.
Osaka Securities Exchange announced few weeks ago concrete plans to introduce volatility futures on VNKY Nikkei Volatility Index in the Q1 2012.
Australian Securities Exchange released product specifications and solicited comments from financial community regarding volatility futures and options on ASX 200 VIX Index. According to a private conversation with exchange representative they are looking for committed market makers in the product, in order to insure trading interest and liquidity.
India VIX - NSE started real-time dissemination of their volatility index in July 2010. According to some regulation there was year-long wait period before derivatives could be launched, and since NSE did not announce anything in July 2011 I assume there are no concrete plans for the product.
Hang-Seng Indexes Company announced some technical changes to calculation of VHSI - HSI Volatility Index. I am speculating that there is usually no reason to change index methodology unless the exchange is planning to do something, and it may indicate the first step toward Hong-Kong volatility derivatives.

For a complete list of volatility products and indexes visit Volatility Indexes and ETFs page.

VXSLV / CBOE Is Planning Silver Volatility Derivatives

According to this document filed with SEC CBOE is quietly planning to launch options (and I assume futures) on VXSLV - volatility index based on SLV options. There is no mention of VXSLV derivatives on the CBOE website, or anywhere else that I could find. Hope springs eternal, and CBOE is once again trying to repeat the success of VIX despite failures of other volatility products like RVX, VXN, VXD, GVZ (and OIV and GIV on CME) and others.

How To Manipulate VIX Settlement Price

VIX expiration day often coincides with particularly heavy trading activity in underlying SPX options. VIX settlement value, or VRO rarely matches either the Tuesday close or Wednesday open prices on the "cash" index, prompting pundits to blame VIX settlment for being manipulated. A popular theory is that VIX settlement value is being pushed up or down with huge SPX trades, referred to as "carpet-bombing". Some say that the manipulative trades are concentrated around high-vega strikes, others concerned specifically about puts. In this post I explain why large trades are not likely an explanation for VIX manipulation, and instead how VIX settlement value can be artificially increased for less than one hundred dollars, how VSTOXX futures and options are not subject to such manipulation, and propose a simple modification that makes VIX manipulation too expensive to be profitable.

VIX settlement value is determined by a Special Opening Quotation, based on the opening trades of SPX options instead of quotes. It is true that VIX settlement value can be made higher or lower by placing a big order that would result in a trade. The quantity necessary to move ATM SPX options is significant, and such trade would have to either be maintained and hedged until expiration, or exited immediately possibly with a large slippage. For this reason I don't think that it would make sense for a trader to attempt to manipulate VIX this way - it is very costly, and possibly very risky if the market moves against the trader, and I believe that heavy trading on the open of VIX expiration does not signify VIX manipulation, but rather legitimate SPX trading activity.

However a different form of VIX manipulation is possible. The VIX calculation formula is a weighted sum of option prices, with weight proportional to 1/K^2, where K is the strike. When K is getting smaller, 1/K^2 is getting bigger. While in theory such growth in a weighting term is mitigated with declining put prices (as K is getting smaller puts get cheaper) in practice it seems possible to "blow up" the VIX by placing orders - nickel bids that are most certainly would get executed - at very low strikes. It is a rather small investment - a few cheap options with limited risk since all options are bought - no short positions.

How this would work in practice:
1. On Tuesday before VIX expiration before the close, a trader purchases a significant amount of VIX calls, ATM or slightly OTM, that have the most potential gain from an unanticipated VIX increase.
2. On Wednesday before the open, a trader places 0.05 1-lot bids on low strikes SPX puts for the next month's expiration (the expiration that determines VIX settlement)

To illustrate the idea I downloaded SPX data from the September 2011 VIX expiration available from the CBOE website here. VIX settled at 33.72, with 550 being the lowest strike traded. As I mentioned above, by construction VIX is very sensitive to the low-strike puts, and if a 500 strike had traded at 0.05 VIX would have settled at about 33.73. If 400 and 500 strikes had traded at 0.05 VIX would have settled at about 33.86; adding a 300 strike trade would push VIX to 34.06; adding a 200 strike trade would push VIX to 34.50; adding a 100 strike trade would push VIX to 36.23. To summarize, for a total cost of 5*$5 = $25 a trader can artificially inflate VIX value by 2.51 points.

500	+0.01
400	+0.14
300	+0.34
200	+0.78
100	+2.51

This methodology applies to any VIX expiration. Using settlement data from August 2011 expiration I estimate potential effects of price manipulations as above.

500	+0.15
400	+0.26
300	+0.47
200	+0.92
100	+2.69

Since the CBOE provides what they call "likely VIX series" we can know which SPX strikes were available at the time of expiration, and calculate exactly the effect on VIX index for every expiration for which data is available. The greatest effect on VIX comes from the lowest strike, so for practical implementation the strategy would depend on which SPX strikes are listed.

Economic significance of such manipulation depends on VIX options on the last trading day, and opening price of SPX. Historically the overnight VIX move from Tuesday close to Wednesday morning settlement has been rather volatile (Russell Rhoads did a study of this in his book), but I believe in most cases the trade would have a very large upside with limited downside.

VSTOXX - a pan-European volatility index based on EURO STOXX 50 index is not subject to such manipulation. The contract is settled into an average of index values during a half hour period on the last trading day (here)

The Final Settlement Price is established by Eurex on the Final Settlement Day, based on the average of the index values of the underlying on the Last Trading Day between 11:30 and 12:00 CET.

This certainly makes sense given that VSTOXX expires in the PM. However now that SPXPM options are picking up some volume on C2 I think it would make perfect sense to calculate VIX values based on a similar averaging procedure as VSTOXX that makes manipulation very expensive and practically impossible, or the CBOE to offer a different settlement procedure that is more resistant to manipulation.
 

Nikkei Volatility Futures / VNKY Futures On The Way

Yesterday Nikkei Inc announced that starting January they are going to start real-time calculation of Nikkei Volatility Index and today Osaka Securities Exchange announced plans to launch volatility futures on the main Japanese index. Links: announcement, futures introduction document, proposed contract specifications, index guidebook (calculation methodology). Right now the futures are under month-long public comment period, and are expected to launch sometime in the winter and trade on J-Gate platform. No word on Nikkei VI options yet.

P.S. See my updated post on Nikkei volatility futures here.

Basket of Volatility: Koop Eens Een Mandje Volatiliteit

Readers from Netherlands, or anyone who speaks Dutch can enjoy this concise yet informative introduction from DB analyst Knut Huy to US and European volatility derivatives and how they can act as an insurance for equity portfolios.

Koop Eens Een Mandje Volatiliteit 

Nameer dan een decennium van kommer en kwel op de markten vinden veel beleggers het nog te vroeg om in dit barre beursklimaat volop de kaart van de aandelen te trekken. Alternatieven lijken schaars: staatspapier en cash brengen weinig op en grondstoffen worden steeds meer de speelbal van speculanten. Investeren in volatiliteit kan een uitweg biede.

One things I could add is to be weary of backtested VSTOXX futures performance which (as I have written before) is unrealistic, and is probably based on theoretical marks, and not on actual trades.

As we are talking about VSTOXX I think one things is worth mentioning: Since Monday Sep 26 CBOE extended trading hours for VIX futures. Now they start trading at 8 AM EST. Since VSTOXX futures close at 18:30 Zurich time (12:30 EST), this brings total overlap between contracts to four and a half hours. The contracts are highly correlated, and although exchange rate remains a risk I think there are interesting opportunities for statistical arbitrage between the contracts. I am doing research on such possible strategy, and will blog about it once I get some results.

Volatility Indexes & Exchange Traded Products

I created a new page that I believe has the most complete list of implied volatility indexes including all international VIX-like indexes, commodity volatility indexes, and currency volatility indexes, and listed ETFs and ETNs all over the world including all that are based on the VIX and VSTOXX futures. I will try to keep the list up to date as much as I can and update prices on a weekly basis.

Variance Risk Premium In VIX Options

I just came across a very interesting research paper: Quantifying the Variance Risk Premium in VIX Options by Reed M. Hogan. Even though the work is a college senior thesis, it is a quality research and I believe one of the first papers on the subject!

If you have read Jared Woodard's Options and the Volatility Risk Premium you know about volatility risk premiums for different asset classes like equities and commodities. Now with proliferation of trading products on VIX and VSTOXX, it is important to measure the risk premium embedded in VIX options. The author explains the methodology developed by Peter Carr and Liuren Wu to create synthetic var-swap on VIX options (please note, there is a plus sign missing in the formula [1]), but uses a different formula that is developed by Gatheral. Realized volatility is computed from corresponding futures contract that has at least 12 days until expiration. The author then tests simple call and put writing strategies on the VIX.

Abstract:

This thesis uses synthetically created variance swaps on VIX futures to quantify the variance risk premium in VIX options. The results of this methodology suggest that the average premium is -3.26%, meaning that the realized variance on VIX futures is on average less than the variance implied by the swap rate. This premium does not vary with time or the level of the swap rate as much as premiums in other asset classes.  A negative risk premium should mean that VIX option strategies that are net credit should be profitable.  This thesis tests two simple net credit strategies with puts and calls, and finds that the call strategy is profitable while the put strategy is not.

VIX Expiration

Over the last month we continued to see higher volatility in the market, but since last expiration volatility indexes were pretty much unchanged: VIX expired at 33.72, up 0.99 from 32.73, and VSTOXX closed on 44.14, up 9.10 from 35.04. It is clear that the debt crisis in Europe is adding upward pressure on VSTOXX. While VIX fell from the August highs, VSTOXX rose higher.



While my forecasts were quite off, they were accurate on the signal - both indexes were forecasted to go higher, and both did expire significantly higher than market predicted. For complete history of my forecasts see forecasts tracker page. For the next expiration, October 19 2011 my forecasts are:

        VIX to close at 31.99 vs 35.70 in futures 
        VSTOXX to close at 41.97 vs 40.70 in futures


This is unusual, since the forecast is for VIX futures to finish lower, while for VSTOXX futures to finish higher. While it can make sense from the economic point of view, in the past my forecasts did not perform very well when signs for VIX and VSTOXX were different. Good luck traders, and hedge your deltas!

Week in Volatility

Equity markets rose, and volatility indexes fell this week - $SPX  finished low last Friday, and gained 5% over the week, while Euro Stoxx 50 closed on a multi-month low on Monday, but gained almost 4% for the week. Implied volatility ETFs - VXX and VSXX made new multi-month highs. On Friday VIX futures entered a slight contango, while VSTOXX remains in backwardation, showing lingering concerns about European economic recovery.


Implied volatility of VIX options fell, while VSTOXX remained unchanged. As I wrote before, liquidity in VSTOXX options is lower than the VIX, and combined with options expiration coming on Wednesday I don't think we can read much into the numbers.

Intuitive Understanding Of VIX Formula

Many investors are looking at VIX and VSTOXX indexes as a leading indicators of volatility in equity markets, however many are confused by the formula. According to CBOE VIX white paper or VSTOXX Index guide (page 23) volatility index is calculated using the formula below:
The square root of this number multiplied by 100 gives you the volatility index. This formula is a discrete version of the formula for "fair" strike of variance swap: While this formula may look complicated it is actually really simple if viewed in a chart. Below are 3 charts that will illustrate step by step what the formula means, and I hope will provide intuitive understanding behind the formula. For the charts below I used simulated prices of calls and puts on a hypothetical stock with 36 days until expiration (tenth of a year) and 20% annualized volatility, so we should expect the volatility index  in this theoretical example to be about 20. I plot prices of calls and puts vs strike.

As expected puts increase with strike, and calls decline. The point at which they intersect is the forward price. Now let's consider the area below the two curves. The higher the volatility the higher are options prices, the greater the area under curves.

Now the last step - multiply each value by 2*exp(r*T) and divide by T*K^2 . The adjusted curve will have almost the same shape, but will be different in height. So meaning of the curve does not change, just the scale - the higher the volatility, the greater the area under the curve.

The area under the curve is sigma squared in the first equation. In my calculations the area is 0.040157. The square root of the number multiplied by 100 is 20.0392 which is very close to 20 as we expected.
To summarize - VIX is approximately proportional to the area under call and put curves. Another property of volatility indexes is becomes apparent - ATM, or near ATM options have the greatest contribution to its value.
This post is based on ideas from Simple Variance Swaps by Ian Martin (available here).

Interested in learning how to trade VIX futures and ETFs? Do not miss out the next great opportunity to short vol.
Find out more at VIXMon.com's trader seminars.

Week in Volatility

Stock indexes tumbled across the world on increasingly negative outlook on European debt crisis and prospects of US recovery. VIX futures rose 4.60 points, and VSTOXX rose 9.70. While VIX at 38 is 10 below its recent high, VSTOXX made a new high. Implied volatility ETNs VXX in the US and VSXX in Europe finished week on new highs as well.




Implied volatility of VIX and VSTOXX was also higher than last week, but still lower than at the highs few weeks ago. VSTOXX long-term implied volatility was lower than last week, but I believe it is simply because of limited liquidity of those options, and does reflect true VSTOXX vol levels.

Options Arbitrage In VXX

While doing some analytics last week I realized that there is a pure (not statistical) arbitrage situation in VXX options. By the time I noticed it was too small of an edge to try to trade manually so I did not do anything, and since this is something that is very unlikely to happen again I'll disclose all the details that came up in my analysis. All data obtained from Nanex(nXCore) - a very high quality datafeed. This feed has all messages from exchanges, although to create the charts below I took 1 second snapshots.

When VXX split in Nov 2010 OCC decided to leave the "old" unadjusted strikes, so one can trade both after-split VXX and before-split VXX which has quarter of its value. On August 30th some options were trading out of line with the others, I guess because some market-maker did not properly adjust volatility skews that day. Because options are so far OTM early exercise is not a consideration, and the mispricing is a pure arbitrage.

For example (pre-split) VXX 21 Jan 2012 Call should trade at 1/4 value of VXX 84 Jan 2012 Call, and they do most of the time. The first chart plots bid of VXX 21 Jan 2012 Call times 4, versus ask of VXX 84 Jan 2012 Call. Around lunchtime bid exceeded ask, and traders had an opportunity to capture as much as 12 cents by selling four 21-strike contacts and buying one 84-strike contract. Using conservative estimates of costs for institutional trader this would give 0.12-5*0.01=0.07 profit on 5*0.25 haircut (margin), or 5.6%. Given that at that time there was 143 trading days until expiration, this translates into 5.6 %* (143/365) = 14.3% annualized of riskless profit. The volume on the quotes was sufficient run the spread at least 25 times (0.07 * 100 * 25 = $175) by the most conservative estimate.



 Similar opportunity presented itself between (pre-split) VXX 22 Jan 2012 Call and VXX 87 Jan 2012 Call, which is the highest strike available for regular contracts. Because 22*4 = 88 > 87, 22-strike contract bid multiplied by four should be smaller than ask of 87 strike contract. Here the spread also widened as much as 12 cents. These are of course not common opportunities, but provide an interesting lesson. There were obviously no low-latency arbitrageurs that brought the spread in-line as mispricing persisted for many minutes.
Thanks to Jason S. Williams of Front Run the Delta for his invaluable feedback on this post.