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As noted in previous articles, there has been some talk over the potential skew existing over VIX’s implied volatility curves. The implied volatility over index options is often observed as having a skewed curve with a minima set around ATM strikes. But what would then occur over VIX options, which are themselves a proxy-measure for the implied volatility over S&P500 index options (SPX)?

How does the implied volatility of implied volatility actually measure up in reality?

Reviewing Some Fundamentals

Black and Scholes (1973) does not try and model different changes in volatility. Indeed, one of its core assumptions is that the distribution of an underlying price can be described as following a log-normal distribution with a set volatility estimate. This assumption would therefore imply that volatility is constant across all option strikes.

However, as has been strongly observed over time, equity market security price distributions are far from log-normal. Ongoing discussion regarding the actual distribution of underlying prices is never-ending. Most recently, the debate has pursued some weight towards confronting both kurtosis and skew parameters beyond the standard log-normal price distribution assumptions. Nassem Taleb comes to mind here.

But before we delve further down this route, it is useful to understand the current usage and application of Black & Scholes option pricing assumptions. The continuous and closed-form solution provided by the Black & Scholes model remains both highly used in contemporary finance and valuable as a founding base for derivative pricing and arbitrage theory.

The log-normal pricing assumption, if applied in current markets, therefore push volatility estimates away from a historical parameters and into the realm of core pricing metrics. The financial services industry has responded to this requirement for additional price-discovery with ongoing research and, in effect, a sub-market of supply and demand for volatility.

Illustrating the Skew

Goldman Sachs’ quantitative department presented both an illustration and a response to the issues confronting volatility skews on options market back in the early 90s. The following two diagrams illustrate the effect of such a skew by using binomial pricing trees as a reference point.

Standard Binomial Tree Assuming Normal Distribution

Standard Binomial Tree Assuming Normal Distribution

If one takes the tree branch length of a volatility tree as a constant probability then the following image illustrate how, from the market’s point of view, a skew can occur shifting the actual probability across each branch.

Had the distribution actually been modeled as above, but observed as below, then volatility measures across the higher branches would have recorded back as higher than those read from the lower branches, producing a volatility skew when plotted against underlying prices.

A Binomial Tree Illustrating Distribution Skew

A Binomial Tree Illustrating Distribution Skew

So How Does VIX Fit Into All This

As mentioned before, VIX is a constant maturity estimate over the underlying implied volatility of S&P 500 benchmark options (SPX). The constant maturity adjustment of VIX reflects another facet complicating the constant volatility assumption underpinning Black and Scholes. Term structures also impact variance measure in part due to changes of interest rate expectations, earnings profiles, et al., but also due to changing expectations of volatility over time.

The CBOE Volatility Index (VIX) now is an up-to-the-minute market estimate of expected volatility that is calculated by using real-time S&P 500 Index (SPX) option bid/ask quotes. VIX uses near-term and next-term out-of-the money SPX options with at least 8 days left to expiration, and then weights them to yield a constant, 30-day measure of the expected volatility of the S&P 500 Index.
VIX Methodology, CBOE

As for VIX options, their own implied volatility estimates are then further impacted. As the title of this post would suggest, the options over VIX are in essence measuring the implied volatility of implied volatility. Complications that arise are two folds: first, VIX is far afield from being log-normal – in fact it is mean-reverting, second, VIX options themselves have their own term structures that then compound back over VIX’ original complications.

Mean Reversion Over VIX Options

VIX, by definition, will never reach absolute or near zero unless one is willing to drop the reality of either a random-walk hypothesis completely or a Geometric Brownian Motion assumption that underpin stock market prices. Simply put, as long as the underlying SPX options are variable, which it is predisposed to be, then VIX will always come through above zero. This already pushes it apart from log-normality assumptions that imply the potential for low to near-zero pricing.

This then implies that there is a boundary state above zero which must exist and, furthermore, would therefore be more highly populated than suggested by standard log-normal distributions. In addition, VIX is observed as mean-reverting by exerting back a return motion due to the very nature of volatility.

VIX’s own volatility is, in effect, pulled back by the reality of the SPX’ own volatility: if volatility over the SPX were consistently higher then this would presume the possibility of sequential +/- 10% moves. However, though equity markets may move on an ongoing basis higher, their own tendency to be random implies the unlikely scenario of ongoing sequential large scale moves.

Yet, since 1990 the largest 1-day move in SPX has been -6.9%, and price changes of at least ±5% have occurred only 8 times.
CBOE VIX Options FAQ

Using Black & Scholes As Illustration of Skew

It might seem odd, therefore, that we should then revert back to the Black and Scholes option pricing model to better illustrate the actual skew over VIX options market. However, the Black and Scholes model provides a simplified construct under which to derive implied volatility given its direct impact on theoretical pricing estimates.

The following three graphs illustrate the effective skew in place over VIX options with near expiry and over subsequent expiry periods as of close of trade 11 March 2011. Implied volatility was calculated using the Black and Scholes model to match back a theoretical price with the current market cleared rate. Details of the calculation are available from the following results list.

The three dimensional volatility surface was constructed from the average of both Put and Call implied volatility, when both of these were already priced by the market, or, when market data was insufficient, were either interpolated across their term structure (highlighted in yellow in the table below) or across term structures (highlighted in red). No extrapolation was used to refine the curve beyond known priced strikes.

The Volatility ‘Smirk’ of Near Expiry

Skewed Implied Volatility Smile or Smirk Over VIX options expiring 16th of March 2011

Skewed Implied Volatility Smile or Smirk Over VIX options expiring 16th of March 2011. Note the strong skew, or 'smirk', for At-The-Money Options

The Future Implied Volatility Skew

VIX Implied Volatility Plane as of 11th of March 2011

The Implied Volatility Surface illustrates how variance expectations skew heavily nearer expiry.

The Future Implied Volatility Skew – Changed Perspective

VIX Implied Volatility Plane as of 11th of March 2011

Same data as above but illustrating strike along the horizontal axis with near-expiry at greater depth.

The Future Implied Volatility Skew – Tabular Form

VIX - Impied Volatility Plane - Data Table - 11 Mar 2011

VIX Implied Volatility Surface Data Table: the skew across the data implies a lower volatility at lower strikes further out along the term structure. Yellow cells are interpolated across their own term structure, red cells were interpolated across term structures.

Can a VIX Option Fair-Price be Theoretically Calculated?

Mean-reversion is not a phenomenon unique to VIX. Indeed, a number of other economic and financial variables are considered to have strong mean-reversion factors in their distribution patterns that can not be accounted for by standard log-normal distribution models. GDP growth rates and interest rates are but two examples that come to mind. Continue Reading…

This post was initially going to review some of the supposed skew existing over VIX’s implied volatility curves (see: Tricky Vixy). However, a small roadblock was crossed along the way. Unfortunately Excel 2008 for Mac is without any VBA macro capabilities. This required a little bit of improvisation in order to come through with a suitable financial modeling solution.

The result is a rough applescript dev for a trinomial option pricing tree script, executable under excel, with a fully referenced cell layering and editable trinomial option price lattice allowing additional manual modeling, if so inclined.

The Trinomial Options Pricing Model

The model applied here, proposed by Boyle (1986), is a step-up from the Cox, Ross & Rubinstein (1979) binomial options pricing tree with three distinct possibilities per node: either up, down or stable. The model uses a trinomial price distribution function but still follows the broad Black & Scholes (1973) assumption of a log-normal distribution of the underlying’s asset price. The built-up lattice does allow you to modify some of these assumptions with manual applications of rational bounds and prices boundaries across the pricing tree.

CRR’s binomial and Boyle’s trinomial option pricing models are two models for fair-valuation of American and Bermudan styled options. These are either liberally or discretely exercisable and therefore require a finite pay-offs analysis (the lattice model) for fair-value estimation of the contract.

The Black & Scholes model provides a continuous price distribution model that is formally only applicable to European-styled options (exercised solely at expiration).

Both the binomial, at high-iterations, and trinomial option pricing models, at relatively lower iteration count, will converge back on a Black & Scholes valuation of European options, illustrating the equivalence of the underlying pricing assumptions.

The Equations: Very Quickly

You can find out more on the specifics of these equations over in most finance textbooks. Essentially, Boyle’s trinomial model is based on Cox, Ross and Rubinstein’s own binomial model and follows through on the same assumptions with the previously mentioned stable-branch improvement.

Price distribution is considered to be recombinant across time periods (eg: p at t0 = price up at t1 and price down at t2) as the following three equations demonstrate.

Equations1 Trinomial Options Pricing Model

Underlying prices can move up, down or remain stable across the tree and are recombinant across time periods

Once the pricing distribution tree is constructed, the options are then priced back recursively from the expiry nodes, bringing back to t0 a fair-value estimate over the expected weighted and discounted future possible option prices.

Prices at expiry nodes are the greater of expiry, zero, or intrinsic value: for a call, underlying price minus strike, and for a put, strike minus underlying price.

The expected price point for any point prior is therefore weighted across from the probability equations set out below providing a fair-value estimate of the option’s price at that specific node. This process is conducted until the initial node is reached once more producing a theoretical fair value of the option at time t0.

Equations2 Trinomial

Option pricing is conducted recursively from the end nodes with an expected weighted value applied to all preceding nodes up to the analysis date.

Graphical Demonstration

This process is perhaps best understood when visualising an actual tree. I’ve attached a properly formatted tree below for your consideration (the applescript is so far nude in terms of formatting).

Formatted Trinomial Option Pricing Tree Example

Note the recombinant nature of price moves. The option's theoretical value is then priced back from the expiry nodes through to the analysis date.

Applescript Implementation

This is still early stages, however, initial tests across the lattice have matched up on a one-to-one basis so far compared to manual constructions and valuations from 3rd party solutions. The script is far from optimised just yet and is pretty much just raw code all dumped together. I’ll try and split this out in routines over the course of this week for a more user-friendly read. Also, the output is fairly raw, no fancy formatting just yet.

Once the initial tree is built, all cells should be directly referenced allowing customised analysis to take place. Simply change the core parameters in the top-left of the sheet and the values should dynamically refresh themselves throughout.

Again, technically this script should no longer be necessary for those mac users who have already updated to Excel 2011. The latest version now re-enabled VBA macros and you should therefore be able to either construct your own or adapt other macros out there to your convenience. For those still on Excel 2008, then this is is certainly the only solution to date beyond a manual lattice construction,

Technically, this script does provide a quick alternative to VBA and, all other things considered, could potentially be useful for other applications (eg ports into Xcode et al.). I might further develop the code into something a bit more practical if time permits. NB: remember that Applescript is a bit touchy about new lines versus return carriage, something worth checking if you copy paste from below.

In order to run the code, simply load it in applescript and launch while having a blank spreadsheet running. Ideally, try and save it in the Excel applescript folder for a smoother launch. If all goes well, you should get a result as per the following screen shot.

Trinomial-Option-Tree-Screencap

At the moment, the script is without formatting features. These might be added with basic Greek calcs too.

Continue Reading…

Notwithstanding some concerns over in the Middle-East (see: Vigilant Freedom and Costly Dreams) or some potential for budgetary issues in the US (see: a Gridlock Yield), the US economy appears to be on track. Most readings are now fortunately in line with my previous October article 4 Fundamentals according to 24.

US Real GDP Index Base 100 set on 2007-12-01

'Plucking' Back Up: US Real GDP Index Base 100 set on 2007-12-01. Source St. Louis Fed Economic Research Division.

Returning back to the measures raised in that article is certainly heart warming. As recent ISM readings have shown, sales and inventory turnover are now accelerating. Adding to this momentum are ongoing pressures on capacity slack. The recent GM earnings report provided a quick snapshot: a shift from 48.0% to 89.5% in capacity utilisation over a year.

The return of capital from balance sheet collateral lock-ups is also going well (see: Writing Back the Down Up Redux) and it seems that capital arb is back to play with expected M&A fever rising over the course of this year. NYSE/Deutsche Borse, LSE/TMX and ASX/SGX are all in talks, and that’s only on the stock exchange front.

So the Fundamentals are Sound?

The fundamentals are definitely buoyant at this point in time. However, it is true that investor sentiment out there is perhaps a bit on the sidelines at times. It’s difficult to say always what is the final core driver on that aspect.

Just as an example, I recently had a phone conversation with a client (not related to this website) who was convinced that US money supply had contracted over the past two years. Unfortunately, to this day, I remain baffled as to how he came to form such an opinion.

Though differences of opinion certainly make a market; it was quite perplexing to encounter a person who did not accept the somewhat significant increase of US broad money supply readings. Perhaps this was an illustration of how investor sentiment can indeed skew the perception of even fundamental economic data.

St. Louis Fed Adjusted Broad Based Money Supply Measure

Adjusted Broad Money Supply Index Base 100 set on 2007-12-01 - Source St. Louis Fed Economic Research division.

Drowning in Liquidity Or Buoyant Flotation?

Now, I hear you say, with easy money comes easy problems, but again the data seems reasonably supportive at this stage. The difficulty over this front, is that a slow trickle in price rises arising from a liquid money source can, possibly, pick up pace. And nobody likes mud-slides, let alone floods. One point is starting to come clear, we are no longer in a zero-inflation range. The question then becomes: is this inflation reasonable and acceptable or is the price-rise just too expensive? Continue Reading…

Tricky VIXy

The VIX index is back to its old tricks and pointing its head up again above the 20 mark for two days running.

Again, for those who haven’t been following, the VIX is a measure of the implied volatility over the S&P500 benchmark. Now there is also a VIX curve, which, as alluded to earlier in my previous posts (see: Update on the coming year, Chart of the Month: VIX, VIX Breakout, and the Contango Price Back), already proved quite a page turner over the course of the past few months.

Volatile Vega

In essence, the story on all vol’ traders lips last year was a steepening VIX contango and what it might possibly mean. Externally, it seems that VIX was pricing itself further out on the curve, which is good in a way, but worrying in another.

Volatility has a tendency to both be volatile itself and also to clump its variance together (see: heteroskedasticity). In other words, pushing volatility out further along the curve potentially implied greater implicit worries…

Smirking Smiles

There is some talk, anecdotal at the moment, over the implied volatility curves occurring over VIX options chains themselves with some potential skewing across that smile into a more directional smirk.

This is potentially of greater import for the vega suppliers out there rather than a strong signal to the core underlying benchmark fundamentals just yet. At the moment, broadly speaking, we are still within reasonable parameters (see distribution chart below).

Where Are We Now?

Well, current news flow is shaking markets a little right now: political risk is back on the map and oil, at least over Brent side, is pushing the $120 mark.

This gave a bit of a hit to the good’ol VIX curve with another push-up over the 20 mark. Also, it would seem that the general declining trend might be reviewed a bit on the upside with the MA100 being breached for two consecutive days. This isn’t much to worry about per se but it does indicate a reinforcement of some sentiment out there.

VIX index on 23 Feb. 2011

The VIX index breaks out above the Upper Bollinger and the MA100 for two consecutive trading days.

As the following distribution graph shows, VIX is not yet in the danger zone of the upper 26 and rising range. This could still occur over the course of this quarter; however, as mentioned in previous posts, the economic fundamentals remain broadly positive.

VIX_Distribution_23022011

VIX Distribution Graph 1990 to 2010. VIX Index position for 23 Feb. 2011 is overlaid. All data courtesy of CBOE.

Volatility Hunters

There’s still plenty of vol’ out there to play with for those hungry for that kind of stuff. Indeed, the recent M&A talks in the market have helped in skewering the curves upon good heated economic embers.

It is worth remembering, especially in relation to VIX, that M&A activity is quite often a strong fundamental sign of economic buoyancy. Well, here’s to remaining cautiously positive 🙂

It’s bit been quite disconcerting and disappointing to read, press release after press release, that Egypt’s stock exchange persists on delaying its re-opening.

Expensive Escapism

Unfortunately, though the intention might have been to increase stability and confidence, the reality is: lack of price discovery, especially during volatile periods, is too costly a reality. In Egypt’s new development; the one thing least affordable, at the moment, are added costs.

Markets can, and will, take advantage of pricing volatility when available as the US listed EGPT ETF demonstrated immediately after Mubarak’s fall.

EGX30:IND to EGPT:US

EGX 30 Index relative to the EGPT US based ETF

EGX 30 Index relative to the EGPT US based ETF

This is a core value of liquid and free markets. They will move in while prices are cheap and take advantage while periods are buoyant. However, a closed market provides neither of these values.

You would be mistaken to view this as an overly cold stance given the unfolding reality of current affairs. With the pressing need for reform in the region and the risk of further instability, it is crucial that market liquidity be maintained, both for the sake of the private sector and for those employed therein.

Lead not by Fear but by Example

North Africa’s future requires more than hope. It needs a firm, sound and reasoned mind to go with its newly freed body. When your nearest neighbour, Libya, is besieged by a violent megalomaniac, then it is all the more crucial that you project yourself firmly towards a brighter, better and rational tomorrow.

Markets aim persistently forwards to tomorrow. What greater example can Egypt project than that of a confident, reasoned and liquid tomorrow, an open market?

With ongoing speculation over a possible market float, everyone is currently left wondering: how much is too much or is too much still not enough.

$50bn = 500m x $100 = $0.50 CPC x 100bn = $50 CPM x 1tn?

At 500 million users and counting, would a $100 contribution per member be a valid count towards a valuation estimate? Should it be based on advertising revenue, and from there, by trend growth or historical earnings? What about an FB sales stream: how much could closed gates e-commerce or affiliated marketing sales generate?

Facebook News > Middle-East Streets > TV News?

Facebook’s market cap speculation isn’t really breaking headline news. Still, when 48% of young Americans are following the news first on FB, when North African youths are creating their own news by organising on FB, well, it does make you wonder…

Silver certainly proved quite the heart throb over this recent Valentine’s week. Tired of letting gold hog the spotlight and of its perennial second-place runner-up SPOT in the league tables, it’s decided to put through a sterling performance and go for gold, at least in terms of returns.

Financial puns aside, the past week gave us a startling demonstration of the extensive contango-to-backwardation shift occurring in the Silver precious metals market (see past article: Revlis Spelled Backwards).

A Historic Shift

The LBMA now holds one full month of SIFO backwardation on its books and seems most unwilling to yield up anytime soon. Unsurprisingly, the CME has now caught up in full swing and kept forward rates negative for an equally impressive length of time. It is hard to stress the historical significance of this shift.

Never, and I do mean never, in the history of the CME or the LBMA, has silver held such a lengthened backwardation across its forward curve. Indeed, I would venture as far as to say that never before, in the history of mankind (well, at least as far as my research goes) has silver held such a significant and durable negative forward yield.

Backwardated Preciousness, MK.II

Silver Backwardation 07 February 2011 to 18 February 2011

Silver Backwardation as of 18/02/2011. All prices courtesy of the CME.

Silver Lining Ten Straight Days of Outperformance

On a more pecuniary perspective, the shift since my last article on February 8th brings silver contracts for March expiry up 8.94%. At 5,000 troy ounce/contract, that’s a variation margin of US$ 13,255.00 per March ’11 contract over ten days, with a new 30-year nominal high now down on the books.

Unsurprisingly, the SPAN margin requirements have gone up an impressive 50% as of last Friday’s close to keep up with the heat. Surprisingly, this margin hike did almost nothing to reduce the run-up in prices, which might indicate that the supply/demand imbalance currently pressuring prices may be driven more by core fundamental hedging than speculative momentum positions.

A Lead Lined Gold to Silver Ratio

Gold_To_Silver_Ratio-ThomsonReuters

Unfortunately, this chart is already out of date; Silver closed above $32.29/oz last Friday.

Gold, meanwhile, has pursued a fairly bullish stance as well, though its performance started to lose some shine on a relative basis. Over the course of the past week, gold gained a mere 2.2% compared to silver’s 6.2%. All of this, of course, put further pressure on an extremely downward sloping gold to silver ratio that is now at a record ten year low of 42.99. The next steps down this road involve 1998 lows and then it’s the go-go eighties all over again, which leads to the next question: what is really happening here?

A Mint Sweetener

Demand for silver is fundamentally strong. Notwithstanding that 80% of silver’s use remains for industrial purposes, the marginal impact of value-oriented bullion plays remain significant. Taking into account recent statements by government mints, silver coinage is all the rage at the moment.

The US Mint already produced record sales of American silver eagles over the course of January to meet unexpected demand (potentially explaining some of the retracement witnessed over the CME that month). However, mints are facing a bit of a manufacturing logjam at this stage, with demand seemingly overstepping the minting press’ supply rate.

Are Collector Coins A Significant Indicator?

Well, as previously mentioned, the majority of silver’s demand remains industrial, nonetheless, performance is managed at the margins and it certainly remains somewhat disconcerting that numismatic charms are facing some level of shortage.

Perhaps of greater significance is the resilience of both the backwardation and ongoing SPOT price pressures observed in the London Bullion Market’s OTC vaults. Unlike the CME that does allow cash settlement on contracts in the event of physical duress (which, just in passing, already occurred over the course of December ’10), the LBMA is a bullion only market, where contracts must be met by hard, cold, silver.

Silver’s Loco?

Loco is not just crazy in Spanish, it’s the actual underlying clearing and settlement system that underpins the LBMA market. This settlement system is a physical only market. Basically, each credit and debit is registered on a ledger and, quite litterally, weighed in a vault. The reality is, at the moment, immediate delivery is pushing beyond standard time-elasticity considerations. Demand wants its Silver now, and I do mean, now.

It is difficult to determine just yet how this backwardation will be met by the market. In theory, the incentive is there to push up supply rates. The question then becomes whether the demand, from the numismatic mirror watching consumer to the high-end photovoltaic cell maker, will abate or rise. The next few settlement-months should certainly prove interesting.

Disclaimer: Material posted on 24-something does not contain (and should not be construed as containing) personal financial or investment advice or other recommendations. The information provided does not take into account your particular investment objectives, financial situation or investment needs. You should assess whether the information provided is appropriate to your particular investment objectives, financial situation and investment needs. You should do this before making an investment decision based on the material above. You can either make this assessment yourself or seek the assistance of an independent financial advisor. 24-Something, associated parties and Tariq Scherer accept no responsibility for any use that may be made of these comments and for any consequences that result.