Advanced Equity Derivatives

Volatility and Correlation

Sebastien Bossu

• Publisher: Wiley
• Year: 2014
• Print ISBN: 9781118750964
• eBook ISBN: 9781118774717
• Edition: 1
• Topic: Business
• Subtopic: Finance
• Index: Business

In Advanced Equity Derivatives: Volatility and Correlation, Sébastien Bossu reviews and explains the advanced concepts used for pricing and hedging equity exotic derivatives. Designed for financial modelers, option traders and sophisticated investors, the content covers the most important theoretical and practical extensions of the Black-Scholes model.

Each chapter includes numerous illustrations and a short selection of problems, covering key topics such as implied volatility surface models, pricing with implied distributions, local volatility models, volatility derivatives, correlation measures, correlation trading, local correlation models and stochastic correlation.

The author has a dual professional and academic background, making Advanced Equity Derivatives: Volatility and Correlation the perfect reference for quantitative researchers and mathematically savvy finance professionals looking to acquire an in-depth understanding of equity exotic derivatives pricing and hedging.

Chapter 1
Exotic Derivatives

Strictly speaking, an exotic derivative is any derivative that is not a plain vanilla call or put. In this chapter we review the payoff and properties of the most widespread equity derivative exotics.

1.1 Single-Asset Exotics

1.1.1 Digital Options

A European digital or binary option pays off $1 if the underlying asset price is above the strike K at maturity T, and 0 otherwise:

In its American version, which is more uncommon, the option pays off $1 as soon as the strike level is hit.

The Black-Scholes price formula for a digital option is simply:

where F is the forward price of S for maturity T, r is the continuous interest rate, and σ is the volatility parameter. When there is an implied volatility smile this formula is inaccurate and a corrective term must be added (see Section 2-1.3).

Digital options are not easy to dynamically hedge because their delta can become very large near maturity. Exotic traders tend to overhedge them with a tight call spread whose range may be determined according to several possible empirical rules, such as:

• Daily volatility rule: Set the range to match a typical stock price move over one day. For example, if the annual volatility of the underlying stock is 32% annually; that is, 32%/√252 ≈ 2% daily, a digital option struck at $100 would be overhedged with $98–$100 call spreads.
• Normalized liquidity rule: Set the range so that the quantity of call spreads is in line with the market liquidity of call spreads with 5% range. The quantity of call spreads is N/R where N is the quantity of digitals and R is the call spread range. If the tradable quantity of call spreads with range 5% is V, the normalized tradable quantity of call spreads with range R would be V × R / 0.05. Solving for R gives
  
  . In practice V is either provided by the option trader...

Table of contents

• Cover
• Series Page
• Title Page
• Copyright
• Dedication
• Foreword
• Preface
• Acknowledgments
• Chapter 1: Exotic Derivatives
• Chapter 2: The Implied Volatility Surface
• Chapter 3: Implied Distributions
• Chapter 4: Local Volatility and Beyond
• Chapter 5: Volatility Derivatives
• Chapter 6: Introducing Correlation
• Chapter 7: Correlation Trading
• Chapter 8: Local Correlation
• Chapter 9: Stochastic Correlation
• Appendix A: Probability Review
• Appendix B: Linear Algebra Review
• Solutions Manual
• Author's Note
• About the Author
• Index
• End User License Agreement