Listed Volatility and Variance Derivatives

A Python-based Guide

Yves Hilpisch

• Publisher: Wiley
• Year: 2016
• Print ISBN: 9781119167914
• eBook ISBN: 9781119167938
• Edition: 1
• Topic: Business
• Subtopic: Finance
• Index: Business

Leverage Python for expert-level volatility and variance derivative trading

Listed Volatility and Variance Derivatives is a comprehensive treatment of all aspects of these increasingly popular derivatives products, and has the distinction of being both the first to cover European volatility and variance products provided by Eurex and the first to offer Python code for implementing comprehensive quantitative analyses of these financial products. For those who want to get started right away, the book is accompanied by a dedicated Web page and a Github repository that includes all the code from the book for easy replication and use, as well as a hosted version of all the code for immediate execution.

Python is fast making inroads into financial modelling and derivatives analytics, and recent developments allow Python to be as fast as pure C++ or C while consisting generally of only 10% of the code lines associated with the compiled languages. This complete guide offers rare insight into the use of Python to undertake complex quantitative analyses of listed volatility and variance derivatives.

• Learn how to use Python for data and financial analysis, and reproduce stylised facts on volatility and variance markets
• Gain an understanding of the fundamental techniques of modelling volatility and variance and the model-free replication of variance
• Familiarise yourself with micro structure elements of the markets for listed volatility and variance derivatives
• Reproduce all results and graphics with IPython/Jupyter Notebooks and Python codes that accompany the book

Listed Volatility and Variance Derivatives is the complete guide to Python-based quantitative analysis of these Eurex derivatives products.

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. S_t is then the level of the index at time t, μ the constant drift, σ the instantaneous volatility and Z_t is a standard Brownian motion. In a risk-neutral setting, the drift μ 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 C maturing at some future date T with payoff max [S_T − K, 0], K being the fixed strike price. It is

where

The price of a vanilla European put option P with payoff max [K − S_T, 0] is determined by put-call parity as

There are multiple ways to derive this famous Black-Scholes-Merton formula. One way relies on the construction of a portfolio comprised of the index and the risk-less bond that perfectly replicates the option payoff at maturity. To avoid risk-less arbitrage, the value of the option must equal the payoff of the replicating portfolio. Another method relies on calculating the risk-neutral expectation of the option payoff at maturity and discounting it back to the present by the risk-neutral short rate. For detailed explanations of these approaches refer, for example, to Björk (2009).

Yet another way, which we want to look at in a bit more detail, is to perfectly hedge the risk resulting from an option (e.g. from the point of view of a seller of the option) by dynamically trading the index and the risk-less bond. This approach is usually called delta hedging (see Sinclair (2008), ch. 1). The delta of a European call option is given by the first partial derivative of the pricing...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• Preface
• Part One: Introduction to Volatility and Variance
• Part Two: Listed Volatility Derivatives
• Part Three: Listed Variance Derivatives
• Part Four: DX Analytics
• Bibliography
• Index
• EULA