Chapter 19. Volatility Options

We are facing extreme volatility.

— Carlos Ghosn

Volatility derivatives have become an important risk management and trading tool. While

first-generation financial models for option pricing take volatility as just one of a number

of input parameters, second-generation models and products consider volatility as an asset

class of its own. For example, the VIX volatility index (cf.

http://en.wikipedia.org/wiki/CBOE_Volatility_Index), introduced in 1993, has since 2003

been calculated as a weighted implied volatility measure of certain out-of-the-money put

and call options with a constant maturity of 30 days on the S&P 500 index. Generally, the

fixed 30-day maturity main index values can only be calculated by interpolating between a

shorter and a longer maturity value for the index — i.e., between two subindices with

varying maturity.

The VSTOXX volatility index — introduced in 2005 by Eurex, the derivatives exchange

operated by Deutsche Börse AG in Germany (cf. http://www.eurexchange.com/advanced-

services/) — is calculated similarly; however, it is based on implied volatilities from

options on the EURO STOXX 50 index.

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This chapter is about the use of the DX derivatives analytics library developed in Chapters

15 to 18 to value a portfolio of American put options on the VSTOXX volatility index. As

of today, Eurex only offers futures contracts and European call and put options on the

VSTOXX. There are no American options on the VSTOXX available on public markets.

This is quite a typical situation for a bank marketing and writing options on indices that

are not offered by the respective exchanges themselves. For simplicity, we assume that the

maturity of the American put options coincides with the maturity of one of the traded

options series.

As a model for the VSTOXX volatility index, we take the square_root_diffusion class

from the DX library. This model satisfies the major requirements when it comes to the

modeling of a quantity like volatility — i.e., mean reversion and positivity (see also

Chapters 10 , 14 , and 16 ).

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In particular, this chapter implements the following major tasks:

Data collection

We need three types of data, namely for the VSTOXX index itself, the futures on the

index, and options data.

Model calibration

To value the nontraded options in a market-consistent fashion, one generally first

calibrates the chosen model to quoted option prices in such a way that the model

based on the optimal parameters replicates the market prices as well as possible.

Portfolio valuation

Equipped with all the data and a market-calibrated model for the VSTOXX volatility