Robert L. Kosowski

Chapter 15 – Tools for Volatility Engineering, Volatility Swaps, and Volatility Trading

This chapter introduces tools for volatility engineering. First, we describe positions with volatility exposure based on options such as delta-hedged calls and straddles. We point out the sensitivity of such positions to variables other than volatility. Then, we move to pure variance products. We describe the replication and pricing of a variance swap. We distinguish volatility from variance instruments and highlight the role of convexity adjustments. The GFC affected the market for variance and volatility swaps and we describe how the GFC has affected the market for volatility trading.

Chapter 9 – Mechanics of Options

This chapter introduces the hedging and pricing of options. Delta hedging of options is defined and it is explained how oscillations in the underlying can lead to gamma profits for the market maker. The gains and losses on an option position are shown to be related to a partial differential equation which is shown to be one of several ways of deriving the Black–Scholes option pricing equation. Barrier options are discussed as extensions of simple plain vanilla call and put options. The Greeks (delta, gamma, vega, and theta) are defined and the concept of gamma trading and volatility trading is discussed. Real-life complications related to smile effects and the term structure of volatility are pointed out.

Chapter 12 – Pricing Tools in Financial Engineering

This chapter introduces the fundamental theorem of asset pricing which provides an unified pricing tool for pricing real-world assets. The theorem links asset prices to states of the world and state prices. We define an arbitrage opportunity in this framework and apply it to option pricing. The concepts of real-world and risk-neutral probabilities are introduced. We find that all properly normalized asset prices have a Martingale property under a properly selected synthetic probability measure. We explain how the fundamental theorem helps in specifying explicit stochastic differential equations that can be used in pricing and hedging in practice. Finally we develop the notion of binomial and trinomial trees and present an application of the Martingale property before discussing considerations related to which pricing method to choose in practice.

Dynamic Replication Methods and Synthetics Engineering

This chapter explains the differences between static and dynamic replication. It reviews static replication in the context of bond immunization. We then introduce dynamic hedging in the context of hedging bond positions before moving on to dynamic replication of options using binomial trees. We highlight the distinction between dynamic replication in continuous and discrete time and discuss real-life complications such as bid–ask spreads, jumps, maintenance and operation costs as well as changes in volatility.

Introduction to Interest-Rate Swap Engineering

This chapter expands on the concept of swaps, outlined in the introductory chapter, and their replication. It applies the concept to fixed-income instruments. We discuss the replication of plain vanilla interest-rate swaps. An interest-rate swap can be used to synthetically create its component parts, such as floating rate notes. Analogies to the replication of other swaps, such as equity swaps, are highlighted. The real-world uses of interest-rates swaps are discussed. Real-world complications such as transaction costs are illustrated in an interest-rate swap application involving a new bond issue. Several uses of swaps are discussed in different asset class and the role of taxes and regulation is highlighted.

Chapter 13 – Some Applications of the Fundamental Theorem

This chapter discusses three applications of the fundamental theorem and critically evaluates them in the light of real-life complications. First, we discuss Monte Carlo methods for option pricing including binary FX options. We show how path dependence can be incorporated into Monte Carlo simulations. Second, we explain how stochastic differential equations and tree models can be calibrated to market data using the fundamental theorem. We present the Black–Derman–Toy model and introduce caplets and caps before discussing how to price them. Third, we show how the fundamental theorem can be used for pricing quanto securities such as quanto forwards and options whose price depends on correlations between assets. Finally, building on the fundamental theorem, we derive partial differential equations for quanto instruments.

Chapter 10 – Engineering Convexity Positions

This chapter discusses the concept of convexity, benefits of volatility as well as convexity trading in the context of bonds and options. The notion of a delta-hedged bond portfolio is introduced. A partial differential equation consisting of the gains from convexity of long bonds and costs of maintaining the volatility position is discussed. We describe the intuition behind the Vasicek and Cox–Ingersoll–Ross models before presenting their associated pricing equations. We highlight similarities between a delta-hedged bond portfolio and option pricing and derive the Black–Scholes equation. Different sources of convexity such as mark-to-market, convexity by design, and prepayment options are discussed. Finally, we discuss quantos and quanto swaps and forwards. We explain the role of correlation in their pricing. Quantos are shown to be another class of assets where the non-negligibility of second-order sensitivities leads to dependence of the asset price on variances and covariances.

Chapter 11 – Options Engineering with Applications

This chapter builds on the earlier discussion of option pricing and presents applications of options engineering. First we discuss how options can be used to create synthetic stock positions. We derive the put-call parity theorem and present arbitrage strategies that can result from its violation. The chapter explains yield enhancement strategies such as call overwriting. Moving on to volatility-based strategies, we then discuss options strategies such as straddles and strangles as well as risk reversals. Finally, we introduce exotic options such as barrier options, knock-in and knock-out options and butterfly strategies and explain how they are related and priced relative to plain vanilla call and put options. We discuss trading strategies related to exotic options and risks such as pin risk.

Chapter 22 – Default Correlation Pricing and Trading

The purpose of this chapter is to explain (default) correlation trading to elaborate on the dependence of tranche prices and basket default swaps on default correlation. We first use a simple example with three assets to show how increases in default correlation increase spreads of senior tranches but decrease spreads of equity tranches. We relate this insight to recent movements in tranche spreads. We then introduce the standard market (Gaussian copula) model and apply it. We highlight similarities between compound correlations implied by this model and implied volatility from the Black–Scholes model. Compound correlations are shown to differ from base correlations. Finally, we discuss how views about default correlation can be expressed by means of long and short tranche positions. An investor can trade correlation by delta hedging against movements in the underlyings. Finally we discuss how such positions can be hedged and risk managed and what the different profit sources of such strategies are.

Institutional Aspects of Derivative Markets

The objective of this chapter is to provide the reader with the necessary practical understanding of financial market conventions and institutional aspects to understand financial engineering. Eurocurrency markets and the differences between onshore and offshore markets are explained. Concepts such as trading, clearing, and settlement are introduced. Recent financial regulation regarding the trading and clearing of derivatives and its importance for financial engineering are discussed.