Listed Volatility and Variance Derivatives E-Book

Yves Hilpisch

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Editions
• Sprache: Englisch

Seitenzahl: 181

Veröffentlichungsjahr: 2016

Beschreibung

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.

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Listed Volatility and Variance Derivatives

A Python-based Guide

DR. YVES J. HILPISCH

This edition first published 2017© 2017 Yves Hilpisch

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Contents

Preface

Part One: Introduction to Volatility and Variance

Chapter 1: Derivatives, Volatility and Variance

1.1 Option Pricing and Hedging

1.2 Notions of Volatility and Variance

1.3 Listed Volatility and Variance Derivatives

1.4 Volatility and Variance Trading

1.5 Python as Our Tool of Choice

1.6 Quick Guide Through the Rest of the Book

Chapter 2: Introduction to Python

2.1 Python Basics

2.2 NumPy

2.3 matplotlib

2.4 pandas

2.5 Conclusions

Chapter 3: Model-Free Replication of Variance

3.1 Introduction

3.2 Spanning with Options

3.3 Log Contracts

3.4 Static Replication of Realized Variance and Variance Swaps

3.5 Constant Dollar Gamma Derivatives and Portfolios

3.6 Practical Replication of Realized Variance

3.7 VSTOXX as Volatility Index

3.8 Conclusions

Part Two: Listed Volatility Derivatives

Chapter 4: Data Analysis and Strategies

4.1 Introduction

4.2 Retrieving Base Data

4.3 Basic Data Analysis

4.4 Correlation Analysis

4.5 Constant Proportion Investment Strategies

4.6 Conclusions

Chapter 5: VSTOXX Index

5.1 Introduction

5.2 Collecting Option Data

5.3 Calculating the Sub-Indexes

5.4 Calculating the VSTOXX Index

5.5 Conclusions

5.6 Python Scripts

Chapter 6: Valuing Volatility Derivatives

6.1 Introduction

6.2 The Valuation Framework

6.3 The Futures Pricing Formula

6.4 The Option Pricing Formula

6.5 Monte Carlo Simulation

6.6 Automated Monte Carlo Tests

6.7 Model Calibration

6.8 Conclusions

6.9 Python Scripts

Chapter 7: Advanced Modeling of the VSTOXX Index

7.1 Introduction

7.2 Market Quotes for Call Options

7.3 The SRJD Model

7.4 Term Structure Calibration

7.5 Option Valuation by Monte Carlo Simulation

7.6 Model Calibration

7.7 Conclusions

7.8 Python Scripts

Chapter 8: Terms of the VSTOXX and its Derivatives

8.1 The EURO STOXX 50 Index

8.2 The VSTOXX Index

8.3 VSTOXX Futures Contracts

8.4 VSTOXX Options Contracts

8.5 Conclusions

Part Three: Listed Variance Derivatives

Chapter 9: Realized Variance and Variance Swaps

9.1 Introduction

9.2 Realized Variance

9.3 Variance Swaps

9.4 Variance vs. Volatility

9.5 Conclusions

Chapter 10: Variance Futures at Eurex

10.1 Introduction

10.2 Variance Futures Concepts

10.3 Example Calculation for a Variance Future

10.4 Comparison of Variance Swap and Future

10.5 Conclusions

Chapter 11: Trading and Settlement

11.1 Introduction

11.2 Overview of Variance Futures Terms

11.3 Intraday Trading

11.4 Trade Matching

11.5 Different Traded Volatilities

11.6 After the Trade Matching

11.7 Further Details

11.8 Conclusions

Part Four: DX Analytics

Chapter 12: DX Analytics – An Overview

12.1 Introduction

12.2 Modeling Risk Factors

12.3 Modeling Derivatives

12.4 Derivatives Portfolios

12.5 Conclusions

Chapter 13: DX Analytics – Square-Root Diffusion

13.1 Introduction

13.2 Data Import and Selection

13.3 Modeling the VSTOXX Options

13.4 Calibration of the VSTOXX Model

13.5 Conclusions

13.6 Python Scripts

Chapter 14: DX Analytics – Square-Root Jump Diffusion

14.1 Introduction

14.2 Modeling the VSTOXX Options

14.3 Calibration of the VSTOXX Model

14.4 Calibration Results

14.5 Conclusions

14.6 Python Scripts

Bibliography

Index

EULA

List of Tables

Table 5.1

Table 8.1

Table 8.2

List of Illustrations

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 7.1

Figure 7.2

Figure 7.3

Figure 7.4

Figure 7.5

Figure 7.6

Figure 7.7

Figure 7.8

Figure 7.9

Figure 8.1

Figure 8.2

Figure 8.3

Figure 9.1

Figure 9.2

Figure 9.3

Figure 9.4

Figure 9.5

Figure 9.6

Figure 9.7

Figure 10.1

Figure 10.2

Figure 10.3

Figure 10.4

Figure 10.5

Figure 11.1

Figure 11.2

Figure 12.1

Figure 12.2

Figure 12.3

Figure 13.1

Figure 13.2

Figure 14.1

Figure 14.2

Figure 14.3

Figure 14.4

Figure 14.5

Figure 14.6

Figure 14.7

Figure 14.8

Figure 14.9

Figure 14.10

Figure 14.11

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

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Guide

Cover

Table of Contents

Preface

Volatility and variance trading has evolved from something opaque to a standard tool in today’s 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 OneIntroduction to Volatility and Variance

CHAPTER 1Derivatives, 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, μ the constant drift, σ the instantaneous volatility and Zt 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 [ST − K, 0], K being the fixed strike price. It is

where

The price of a vanilla European put option P with payoff max [K − ST, 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 formula with respect to the value of the risk factor, i.e. . More specifically, we get

When trading takes place continuously, the European call option position hedged by δt index units short is risk-less:

This is due to the fact that the only (instantaneous) risk results from changes in the index level and all such (marginal) changes are compensated for by the delta short index position.

Continuous models and trading are a mathematically convenient description of the real world. However, in practice trading and therefore hedging can only take place at discrete points in time. This does not lead to a complete breakdown of the delta hedging approach, but it introduces hedge errors. If hedging takes place at every discrete time interval of length Δt, the Profit-Loss (PL) for such a time interval is roughly (see Bossu (2014), p. 59)

Γ is the gamma of the option and measures how the delta (marginally) changes with the changing index level. ΔS is the change in the index level over the time interval Δt. It is given by

Θ is the theta of the option and measures how the option value changes with the passage of time. It is given approximately by (see Bossu (2014), p. 60)

With this we get

The quantity is called the dollar gamma of the option and gives the second order change in the option price induced by a (marginal) change in the index level. is the squared realized return over the time interval Δt – it might be interpreted as the (instantaneously) realized variance if the time interval is short enough and the drift is close to zero. Finally, is the fixed, “theoretical” variance in the model for the time interval.

The above reasoning illustrates that the PL of a discretely delta hedged option position is determined by the difference between the realized variance during the discrete hedge interval and the theoretically expected variance given the model parameter for the volatility. The total hedge error over intervals is given by

(1.1)

This little exercise in option hedging leads us to a result which is already quite close to a product intensively discussed in this book: listed variance futures. Variance futures, and their Over-the-Counter (OTC) relatives variance swaps, pay to the holder the difference between realized variance over a certain period of time and a fixed variance strike.

1.2 Notions of Volatility and Variance

The previous section already touches on different notions of volatility and variance. This section provides formal definitions for these and other quantities of importance. For a more detailed exposition refer to Sinclair (2008). In what follows we assume that a time series is given with quotes Sn, n ∈ {0, …, N} (see Hilpisch (2015, ch. 3)). We do not assume any specific model that might generate the time series data. The log return for n > 0 is defined by

realized or historical volatility

: this refers to the standard deviation of the log returns of a financial time series; suppose we observe

N

(past) log returns

R

n

,

n

∈ {1, …,

N

}, with mean return

; the realized or historical volatility

is then given by

instantaneous volatility

: this refers to the volatility factor of a diffusion process; for example, in the Black-Scholes-Merton model the instantaneous volatility σ is found in the respective (risk-neutral) stochastic differential equation (SDE)

implied volatility

: this is the volatility that, if put into the Black-Scholes-Merton option pricing formula, gives the market-observed price of an option; suppose we observe today a price of

C

*

0

for a European call option; the implied volatility σ

imp

is the quantity that solves

ceteris paribus

the implicit equation

These volatilities all have squared counterparts which are then named variance