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.

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.

#### 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).

#### 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.

**Attachments**- Trinomial Option Pricing Tree Applescript
- 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.
- Clifford, P., Zaboronski, O., “Pricing Options Using Trinomial Trees”, 17/11/2008, Warwick University.
- Cox, J.C., Ross, S. and Rubinstein, M. (1979), ‘‘Option pricing: a simplified approach’’, Journal of Financial Economics, Vol. 7, pp. 229-64.
- Direct Line Source Code Available Below

## 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. Continue Reading…

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 | Comments Off