Risk Latte - Vanillas and Exotics - Dimensionality and Order

Vanillas and Exotics - Dimensionality and Order

Rahul Bhattacharya
October 11, 2006

We are all aware of the classification of options and derivatives as "vanilla" and "exotic". Of course, these days it is so difficult to differentiate between a vanilla and an exotic derivative. As one trader recently commented during a training programme 'nothing is vanilla these days and everything is exotic'.

Not long ago, about a decade and a half back, a vanilla derivative was defined as either a European style call option or a put option. Period! Everything else was exotic. A convertible bond, an equity-linked note, a compound option, a barrier option, you name it and they were all exotic products. A European style call or a put was the basic building block of the derivatives universe and their value could easily be expressed as nice closed form formulas. Thus they were vanilla flavour. The exotics on the other hand had path dependency, early exercise features, and what not and needed Monte Carlo simulation and other numerical techniques (it is baffling to us as to how bankers from non-science background still think Monte Carlo simulation is some kind of rocket science) and therefore much more intellectually challenging.

But soon practitioners ran into a problem. Bright Ph.D.s found out closed form solutions - using differential equations mostly - to all the products in the exotics family. Not overnight, but one by one and sooner or later the exotics became vanilla, at least to the small stable of Ph.D. turned quants and some genuinely bright traders. Of course, the bulk of the work was done by some seriously bright academics in either the Universities or in investment banks such as Goldman Sachs. These guys used some complex math to crack the path dependencies and higher order complexities in the exotic options and deduced neat formulas to value them. Bingo!

And there was another problem as well. Soon the exotic traders - yes, in this case it was a handful of bright quantitative traders on Wall Street who got the insight first - realized that many, if not most, of the so called exotic options and derivatives could easily be decomposed into at least one vanilla call or a put option.

If an exotic option or a derivative can be decomposed - or reduced to - into at least one European style vanilla call or a put option then it is not an exotic derivative at all. A knock-out barrier call option can be expressed as a combination of a long vanilla call option and a short knock-in call option. A strict definition of an exotic derivative is one which cannot be further decomposed into vanilla products.

Therefore, a far more robust categorization of options in particular and derivatives in general is based on Dimensionality and Order . This classification is mathematical (and hence sometimes less accepted by the practitioners) but is less erroneous and more stable. The following explanations follow those given by Paul Wilmott, which we find very apt.

What is a "Dimension" of an Option?

Dimensionality, to use Wilmott's definition, refers to the number of underlying independent variables on which the value of an option depends. In a Black-Scholes world where volatility and the interest rates are constant a vanilla call or a put option depends on only two variables, the spot price and the time to maturity. Hence a vanilla call or a put is a two dimensional option (or a derivative).

Strictly speaking, path dependency is an independent variable and a dimension by itself. But many theorists treat weakly path dependent options, such as barrier options, as two dimensional and ignore the path dependency as a separate dimension. This is because path dependency as such cannot be modeled as a tangible "variable" in a math equation unless it expresses itself through some complex form of asset price or time. Strongly path dependent options, such as asian options are three dimensional options whereby the strong path dependency is captured by the average of the asset prices which becomes the third dimension (third variable) of the option.

Another class of three dimensional options (derivatives) are the multi-asset options, i.e. more than one underlying asset. An option on the best of two stock index returns is a three dimensional option.

What is an "Order" of an Option?

This classification is directly related to the modeling issue. Order of an option refers to the dependence of a value of an option on the properties of the underlying asset price path. Broadly speaking, if a derivative's value only depends on an underlying asset's price path then it is of first order . This means that first order options or derivatives are those whose payoffs depend directly on the quantity that we are modeling, i.e. the asset price. Second and ¡§higher order¡¨ options are those contracts whose payoff and value depend on the value of another option or derivative. An example of a second order derivative is a Compound option, i.e. an option on an option. The famous Bankers' Trust Installment Warrants on TSE that had embedded compound options in them and were issued in mid-1990s (search this website for more details on this product) is another example of a second order product. Many structured products are higher order contracts.

Is this any easier on you, dear reader? If not, then keep using the vanilla and the exotic nomenclatures, as long as the product is clear in your mind.

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