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

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

#### 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. Note the strong skew, or 'smirk', for At-The-Money Options

#### The Future Implied Volatility Skew

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

#### The Future Implied Volatility Skew – Changed Perspective

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 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.

#### Where Would Three Roads Lead To?

As noted in my previous post, finite lattice based models provide one possible avenue over which to explore and model a more skewed distribution pattern.

Ideally, the purpose of such a model would be to achieve a more reliable fair-value estimate of the derivative at the analysis date. Two aspects that could be confronted are set price boundaries, perhaps based on observed distributions, and modified distribution probabilities to price back recursively across the tree.

The Hull-White (1990) model, still popularly used for term-structure related fixed income securities and derivatives valuation, tries to take this into account by offering a lattice based framework to model both *kurtosis* and *mean-reversion* factors into the probabilistic branching across the trinomial tree allowing the same recursive logic and limited initial parametric inputs. There are also a number of *relaxed* and *modified* trinomial lattice based models available that either build-upon or adjust the assumptions set out by CRR (1979), Boyle (1986) and Hull & White (1990) such as Ji & Wade Brorsen (2009) “A relaxed lattice option pricing model: implied skewness and kurtosis” effort covering agricultural related commodity options.

#### Or Do All Roads Lead To Monte-Carlo?

Derivative pricing models have been developed to confront these issues, in part by providing a framework for *assumptions* modeling instead of becoming a setting for built-in assumptions (eg: Rombouts & Stentoft (2010) “Multivariate Option Pricing With Time Varying Volatility and Correlation”).

These models offer strong performance but do suffer both in their complexity and ongoing requirements of specialized assumptions testing, potentially leading to a limitless number of pricing factors beyond the simplified parametric nature of Black & Scholes. Multivariate Monte-Carlo modeling methods can face the risk of a limitless number of external factors impacting the derivative’s theoretical value.

#### A Quick Aside On Systems And Modeling

My previous article noted some of the limitations with Mac Excel 2008 encountered primarily due to my laziness of not installing a dual-OS on my computer. Personally, apart for my free time and this website spent on a mac, most of my professional financial modeling experience is spent on a PC environment.

I am therefore taking this opportunity of giving the nod at FinCAD a supplier to a previous employer of mine, SimCorp (who themselves have a fairly advanced pricing library and financial modeling module). FinCAD offers a professional/institutional level excel modeling library for in-depth analytical pricing and SimCorp offers a broad system-wide integrated pricing universe for large first-tier buy-side institutional requirements (such as those of another past employer: Macquarie Funds Group).

Also, more geared towards the personal and smaller investor range is Hoadley’s set of investment analytic Excel add-ins, provided via a simple PC Excel library. His analytical tool-kit provides the basic building blocks sufficient for excel-level portfolio management and should be extensible by anyone with rudimentary VBA skills.

As for the Apple platform, there is a current gap in the market, it would seem, for basic modeling libraries. Though Excel 2011 might re-integrate VBA support, I am curious whether a native built library or more sturdy stand-alone platform might not be preferable. Of course there is always Matlab and Mathematica that are aimed for the more *pointy-end* quant work.

**Background Links:**
- Calculation Results: Implied Volatility Calculations – 11 March 2011
- Black, F. and Scholes, M. (1973), ‘‘The pricing of options and corporate liabilities’’, Journal of Political Economy, Vol. 81, pp. 637-59.
- Boyle, P.P. (1986), ‘‘Option valuation using a three-jump process’’, International Options Journal, Vol. 3, pp. 7-12.
- Cox, J.C., Ross, S. and Rubinstein, M. (1979), ‘‘Option pricing: a simplified approach’’, Journal of Financial Economics, Vol. 7, pp. 229-64.
- Hull, J. & White, A. (1990), “Valuing Derivatives Using the Explicit Finite Difference Method”, Journal of Financial and Quantitative Analysis, Vol. 25, No. 1, pp 87-100.
- Ji, D. & Wade Brorsen, B. (2009), “A relaxed lattice option pricing model: implied skewness and kurtosis”, Agricultural Finance Review, Vol. 69, No. 3, pp 268-283.
- Rombouts, J.V.K. & Stentoft, L. (2010) “Multivariate Option Pricing With Time Varying Volatility and Correlation”,
*Série Scientifique*, Cirano: Montreal, Mai 2010.
- Derman, E. & Kani, I. (1994) “The Volatility Smile and Its Implied Tree”, Goldman Sachs Quantitative Strategies Research Notes
- VIX White Paper, CBOE 2003
- VIX Options FAQ, CBOE 2011

## Implied Volatility Squared

Mar 14th, 2011 by Tariq Scherer

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

kurtosisandskewparameters 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

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

## 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.

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

randomimplies the unlikely scenario of ongoing sequential large scale moves.## 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. Note the strong skew, or 'smirk', for At-The-Money Options

## The Future Implied Volatility Skew

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

## The Future Implied Volatility Skew – Changed Perspective

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 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.

## Where Would Three Roads Lead To?

As noted in my previous post, finite lattice based models provide one possible avenue over which to explore and model a more skewed distribution pattern.

Ideally, the purpose of such a model would be to achieve a more reliable fair-value estimate of the derivative at the analysis date. Two aspects that could be confronted are set price boundaries, perhaps based on observed distributions, and modified distribution probabilities to price back recursively across the tree.

The Hull-White (1990) model, still popularly used for term-structure related fixed income securities and derivatives valuation, tries to take this into account by offering a lattice based framework to model both

kurtosisandmean-reversionfactors into the probabilistic branching across the trinomial tree allowing the same recursive logic and limited initial parametric inputs. There are also a number ofrelaxedandmodifiedtrinomial lattice based models available that either build-upon or adjust the assumptions set out by CRR (1979), Boyle (1986) and Hull & White (1990) such as Ji & Wade Brorsen (2009) “A relaxed lattice option pricing model: implied skewness and kurtosis” effort covering agricultural related commodity options.## Or Do All Roads Lead To Monte-Carlo?

Derivative pricing models have been developed to confront these issues, in part by providing a framework for

assumptionsmodeling instead of becoming a setting for built-in assumptions (eg: Rombouts & Stentoft (2010) “Multivariate Option Pricing With Time Varying Volatility and Correlation”).These models offer strong performance but do suffer both in their complexity and ongoing requirements of specialized assumptions testing, potentially leading to a limitless number of pricing factors beyond the simplified parametric nature of Black & Scholes. Multivariate Monte-Carlo modeling methods can face the risk of a limitless number of external factors impacting the derivative’s theoretical value.

## A Quick Aside On Systems And Modeling

My previous article noted some of the limitations with Mac Excel 2008 encountered primarily due to my laziness of not installing a dual-OS on my computer. Personally, apart for my free time and this website spent on a mac, most of my professional financial modeling experience is spent on a PC environment.

I am therefore taking this opportunity of giving the nod at FinCAD a supplier to a previous employer of mine, SimCorp (who themselves have a fairly advanced pricing library and financial modeling module). FinCAD offers a professional/institutional level excel modeling library for in-depth analytical pricing and SimCorp offers a broad system-wide integrated pricing universe for large first-tier buy-side institutional requirements (such as those of another past employer: Macquarie Funds Group).

Also, more geared towards the personal and smaller investor range is Hoadley’s set of investment analytic Excel add-ins, provided via a simple PC Excel library. His analytical tool-kit provides the basic building blocks sufficient for excel-level portfolio management and should be extensible by anyone with rudimentary VBA skills.

As for the Apple platform, there is a current gap in the market, it would seem, for basic modeling libraries. Though Excel 2011 might re-integrate VBA support, I am curious whether a native built library or more sturdy stand-alone platform might not be preferable. Of course there is always Matlab and Mathematica that are aimed for the more

pointy-endquant work.Background Links:Série Scientifique, Cirano: Montreal, Mai 2010.Posted in Analysis, Article Review, Market Comments | Tagged in , At-The-Money, ATM, Binomial Options Pricing Models, Black & Scholes, Boyle, CRR, derivatives, European Options, Hull & White, Implied Volatility, Index Options, Kurtosis, Log-Normal Distribution, Mean Reversion, Monte-Carlo Modeling, Multivariate Modeling, Options Pricing, OTM, Out-of-The-Money, Skew, SPX, Term Structures, Trinomial Options Pricing Models, VIX, Volatility, Volatility Curves, Volatility Skew, Volatility Surfaces