Hilpisch - Listed volatility and variance derivatives : a Python-based guide

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This edition first published 2017
2017 Yves Hilpisch

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Preface

Volatility and variance trading has evolved from something opaque to a standard tool in todays financial markets. The motives for trading volatility and variance as an asset class of its own are numerous. Among others, it allows for effective option and equity portfolio hedging and risk management as well as straight out speculation on future volatility (index) movements. The potential benefits of volatility- and variance-based strategies are widely accepted by researchers and practitioners alike.

With regard to products it mainly started out around 1993 with over-the-counter (OTC) variance swaps. At about the same time, the Chicago Board Options Exchange introduced the VIX volatility index. This index still serves today after a significant change in its methodology as the underlying risk factor for some of the most liquidly traded listed derivatives in this area. The listing of such derivatives allows for a more standardized, cost efficient and transparent approach to volatility and variance trading.

This book covers some of the most important listed volatility and variance derivatives with a focus on products provided by Eurex. Larger parts of the content are based on the Eurex Advanced Services tutorial series which use Python to illustrate the main concepts of volatility and variance products. I am grateful that Eurex allowed me to use the contents of the tutorial series freely for this book.

Python has become not only one of the most widely used programming languages but also one of the major technology platforms in the financial industry. It is more like a platform since the Python ecosystem provides a wealth of powerful libraries and packages useful for financial analytics and application building. It also integrates well with many other technologies, like the statistical programming language R, used in the financial industry. You can find links to all Python resources under http://lvvd.tpq.io.

I thank Michael Schwed for providing parts of the Python code. I also thank my family for all their love and support over the years, especially my wife Sandra and our children Lilli and Henry. I dedicate this book to my beloved dog Jil. I miss you.

YVES

Voelklingen, Saarland, April 2016

Part One
Introduction to Volatility and Variance

CHAPTER 1
Derivatives, Volatility and Variance

The first chapter provides some background information for the rest of the book. It mainly covers concepts and notions of importance for later chapters. In particular, it shows how the delta hedging of options is connected with variance swaps and futures. It also discusses different notions of volatility and variance, the history of traded volatility and variance derivatives as well as why Python is a good choice for the analysis of such instruments.

1.1 Option Pricing and Hedging

In the Black-Scholes-Merton (1973) benchmark model for option pricing, uncertainty with regard to the single underlying risk factor S (stock price, index level, etc.) is driven by a geometric Brownian motion with stochastic differential equation (SDE)

Throughout we may think of the risk factor as being a stock index paying no dividends. St is then the level of the index at time t, the constant drift, the instantaneous volatility and Zt is a standard Brownian motion. In a risk-neutral setting, the drift is replaced by the (constant) risk-less short rate r

In addition to the index which is assumed to be directly tradable, there is also a risk-less bond B available for trading. It satisfies the differential equation

In this model, it is possible to derive a closed pricing formula for a vanilla European call option