Jeffrey Lange

Introduction to Derivatives

This chapter introduces the various aspects of derivatives trading, focusing on topics that we will draw on later in the book. Section 1.1 reviews the basics of derivatives contracts, and Section 1.2 discusses derivatives trading. Section 1.3 describes the recent growth in deriva`tives markets, and Section 1.4 discusses some of the causes of that recent growth. In these sections and throughout the remainder of the book, we assume that the reader has some familiarity with derivatives as is found, for example, in Chapters 1, 2, and 8 of Hull (2006). Where relevant, we provide additional references for readers interested in exploring specific topics in more detail.

Introduction to Parimutuel Matching

This chapter introduces the main features of parimutuel matching. Parimutuel matching has been used successfully for over 100 years for wagering on horse races, and it has been in use in the derivatives markets since October 2002. This chapter describes parimutuel matching in its simplest form, because doing so allows us to introduce most of the main features of parimutuel matching without too much detailed mathematics and notation. We defer the more detailed mathematics of parimutuel matching to the later chapters in this book.

Mathematical Properties of Parimutuel Equilibrium Prices

This chapter describes the mathematical properties of the prices in the Parimutuel Equilibrium Problem (PEP). Specifically, for a given set of opening orders and customer orders (the “auction inputs”), this chapter shows that the PEP has a unique set of state prices and derivative strategy prices. The fact that prices are unique is a very strong result mathematically. In addition, it is a very desirable result pragmatically, as auction participants are likely to find comfort that a particular set of orders can only generate one set of prices.1

A Case Study Using Nonfarm Payrolls

This chapter analyzes parimutuel derivatives auctions in more detail, presenting a case study on the monthly change in nonfarm payrolls (NFP), a particularly influential US economic statistic. We proceed as follows. Section 4.1 provides background on NFP. Section 4.2 discusses why hedgers and speculators trade short-dated derivatives on NFP. Section 4.3 argues that parimutuel derivatives auctions have a comparative advantage over bilateral matching for trading NFP derivatives. Section 4.4 describes the design, mathematics, and dynamics of parimutuel derivatives auctions on NFP.

The Parimutuel Equilibrium

The Parimutuel Equilibrium Problem (PEP) represents a flexible approach for trading derivatives using parimutuel methods. Chapter 5 introduced the first two mathematical principles of the PEP, and this chapter presents the three remaining mathematical principles of the PEP. We proceed as follows. Section 6.1 describes the opening orders, which have several uses in a parimutuel derivatives auction. Section 6.2 presents the third, fourth, and fifth mathematical principles of the PEP. The third principle is that the PEP prices are arbitrage-free. The fourth principle is that the prices and fills are “self-hedging,” which means that the net premiums collected fund the net payouts, regardless of the value of the underlying at expiration.1 The fifth principle is that the PEP maximizes a measure of auction volume called “market exposure.” Based on the five PEP principles, we can present the complete mathematical specification of the PEP, which is done in Table 6.7 and in shorthand form in Equation (6.28). Section 6.3 describes in more detail the features of the PEP. Throughout this chapter, we once again illustrate the material using an example based on the US Consumer Price Index (CPI) as an underlying.2

The Solution Algorithm for the Parimutuel Equilibrium Problem

In the parimutuel wagering framework described in Chapter 2, the equilibrium prices can be calculated easily using arithmetic. As discussed in the more recent chapters, a parimutuel derivatives auction has considerably more flexibility than the parimutuel wagering framework. Computing the equilibrium prices and fills is significantly harder in this framework, primarily due to the presence of limit orders. This chapter is devoted to numerically solving this more difficult problem, which we call the Parimutuel Equilibrium Problem (PEP). The numerical algorithm presented in this chapter has two parts. Part one solves for the equilibrium state prices and the equilibrium derivative strategy prices using an iterative and nonlinear algorithm. Part two uses the equilibrium prices from part one and a linear program (LP) to solve for the customer fills that maximize the market exposure, a measure of auction volume. After the LP determines the customer fills, the remaining unknown variables can be solved for in a straightforward fashion. Table 7.1 summarizes the properties of the two parts of the solution algorithm.1

Mathematical Properties of Customer Fills in the Parimutuel Equilibrium

For a given set of opening orders and customer orders (the “auction inputs”), the Parimutuel Equilibrium Problem (PEP) has a unique set of state prices and derivative strategy prices, as shown in Chapter 8. For a given set of auction inputs, Chapter 8 also shows that there always exists at least one vector of customer fills that satisfy the PEP.1 There can, in certain cases, be multiple vectors of customer fills that satisfy the PEP for a given set of auction inputs. In this chapter, we describe the conditions under which such multiple vectors of customer fills exist.

Derivative Strategies and Customer Orders

Chapter 2 introduced parimutuel pricing in its most basic form, a framework in which customers can only trade single state claims (no trading of vanilla options) and customers can only submit market orders (no submitting of limit orders). In this framework, pricing was based on two mathematical principles — the principle of “no-arbitrage” and the principle of “self-hedging.” Although this framework is widely used for wagering, it is not flexible enough to be useful for trading derivatives.