1 Introduction to Derivatives
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Notes 1 INTRODUCTION TO DERIVATIVES 1. As described in Hull (2006, pp. 35–36), in the case of an exchange- traded derivatives contract with physical delivery, a party with a short position in the derivative delivers the physical underlying to a party designated by the exchange who has a long position in the same derivative. In addition, with certain exchange-traded contracts, the party with the short position may have a choice of what underlying to deliver. For example, a seller of a Chicago Board of Trade (CBOT) US Treasury future can deliver one of a set of delivery-grade bonds. For more details, see the Rules and Regulations on the CBOT website, http://www.cbot.com. 2. Hull (2006, pp. 40–41) provides more details on the institutional dif- ferences between futures and forwards. For discrepancies between futures and forward prices, see Hull (2006, pp. 109–110) and see the theoretical discussion in Duffie (1988, pp. 262–263). 3. Another widely traded derivative that pays out linearly in the value of the underlying is a “swap,” which is a generalized version of a forward contract. With a forward contract, there is one exchange of cash on a date in the future, while a swap contract has exchanges of cash on multiple dates in the future. See Chapter 7 of Hull (2006) for more details. 4. In addition to European-style exercise, options are often “American- style” exercise. In this case, the option holder can exercise at many different times up to the expiration of the option. For other exotic- option styles, see Chapter 22 of Hull (2006). 5. We avoid using the term “at-the-money” (which means that the value of the underlying is equal to the strike upon expiration) because using this term with digital options can cause confusion. For example, as we see in moment, a digital call pays out if the underlying is equal to the strike upon expiration while a digital put does not pay out if the underlying is equal to the strike upon expiration. See Hull (2006, 232 NOTES 233 p. 188) for alternative definitions of in-the-money, at-the-money, and out-of-the-money. 6. Digital options are also called “binary” options or “all-or-nothing” options. 7. Section 5.1 will discuss these derivatives in more detail. 8. While derivatives-market participants consider digital options to be exotic,academic researchers typically consider digital options to be the fundamental building blocks of finance.See the upcoming discussions in Sections 2.1 and 5.3. 9. For additional details on customers and market makers, see Harris (2003), particularly Sections 3.1 and 19.3. 10. See Chapter 15 of Hull (2006) for further discussion on delta hedging. 11. See Weber (1999) for details on electronic trading of derivatives on an exchange, and see Chapter 6 of Liebenberg (2002) for details on electronic trading of derivatives OTC. 12. See Harris (2003, pp. 90–91) for a brief introduction to call auctions. Garbade and Silber (1979) show that auctions in general can reduce overall transaction costs. Several papers show that call auctions in particular can be useful for aggregating liquidity. See, among many, Economides and Schwartz (1995), Theissen (2000), and Kalay, Wei, and Wohl (2002). 13. Garbade and Ingber (2005) describe the mechanics of the Treasury auctions. 14. Domowitz and Madhavan (2001) discuss of the mechanics of the opening of NYSE equity markets. 15. See Hull (2006, pp. 36–37) for a brief discussion of these and other order types. For further details, see Chapter 4 of Harris (2003). 16. See Chapter 18 of Bernstein (1996) for an overview of early derivatives trading. 17. See Barbour (1963, pp. 74–84), and De La Vega (1996) for details on trading in Amsterdam during the 1600s. 18. See Paul (1985, pp. 277–278). 19. See Gates (1973) for a history of early options trading in the US, and see Lurie (1979) for a history of the early years of the CBOT. 20. For more details, see Silber (1983), Carlton (1984), Van Horne (1985), Miller (1986, 1992), and Merton (1992b). 21. Paul (1985,pp.276–278) discusses (1) instances of cash settlement out- side of the US prior to 1981 and (2) instances of partial cash settlement in the US before 1981. 22. In fact, the BIS reports daily numbers for April 2004. We convert these to annual numbers by assuming 250 trading days in a year. See Bank for International Settlements (2004, p. 14) for more details. 23. See Commodity Futures Trading Commission (2005, pp. 109–137). 24. See Commodity Futures Trading Commission (2005, p. 13). 234 NOTES 25. Of particular note are the papers by Silber (1983), Van Horne (1985), Miller (1986), and Merton (1992b). Also, see the books by Allen and Gale (1994) and Shiller (1993, 2003). 26. See Anderson (1984) on the development of the CFTC. 27. See Miller (1986) and Telser (1986) for a discussion of the legal aspects involving the CFTC and cash settlement. 28. Of the top ten contracts listed in Burghardt (2005),the following six are cash-settled: KOSPI 200 options, 3-month Eurodollar futures, 28-day Mexican Peso deposit futures, E-Mini S&P 500 index futures, 3-month Euribor futures, and 3-month Eurodollar options. 29. See Cravath, Swaine, and Moore (2001) and White (2001) for more details on the CFMA. 2 INTRODUCTION TO PARIMUTUEL MATCHING 1. Most financial underlyings are likely to require many more than four states. Parimutuel matching can handle such cases, but we chose a small state-space example here to most simply illustrate the properties of parimutuel matching. 2. State claims are also often called “state-contingent” claims or “Arrow– Debreu”securities. 3. In practice, most parimutuelly traded derivatives expire on the same date as the auction so premium and payouts are often exchanged on the same date, satisfying this assumption. See Sections 3.2 and 3.3 for details. 4. In fact, a customer can approximate selling a state claim as follows. For illustration, consider a customer who wants to sell the first state claim.This customer can submit orders to buy every other state claim. In this case, the customer loses money if the first state occurs and may profit if any of the other states occur, as a seller of the first state claim would. However, this customer pays premium up front (as opposed to receiving premium up front as a seller typically would), and this customer receives different payouts depending on what state occurs (since there is no way to guarantee that the customer receives the same payout for every state purchased, since the state prices are unknown at the time the orders are submitted). 5. In fact, it may be the case that a specific state is sufficiently unlikely that no customer invests premium in that state. However, Equation (2.1) is a convenient assumption that simplifies exposition and avoids any dividing-by-zero issues, which would appear for example in Equation (2.8). 6. Equation (2.4) holds because the premium amounts are non-zero, as specified in Equation (2.1). NOTES 235 7. Chapter 6 will add a third no-arbitrage restriction to handle the pricing of derivatives that pay out in multiple states. See Equation (6.4). 8. Equation (2.8) resembles the result from the classical Arrow-Debreu equilibrium, which shows that the ratio of state prices equals the ratio of marginal utilities in those two states. See Arrow (1964). 9. It is not hard to check that the self-hedging conditions of Equation (2.6) and the relative-demand pricing of Equation (2.8) are mathematically equivalent when state prices are positive and sum to one. Appendix 6D will prove this result. 10. It is worth pointing out that parimutuel matching does not always fill all orders when customers can submit limit orders. See the example in Section 5.2. 11. Although this is somewhat non-standard, in this example we fill cus- tomers for a fractional number of contracts to make sure that the self-hedging property of Equation (2.6) is satisfied exactly. 12. Transaction costs may include fees for trading on an exchange,clearing fees, and costs of capital. 13. This example illustrates a very simple no-arbitrage relationship. For a detailed discussion of other no-arbitrage relationships, see Chapter 4 of Cox and Rubinstein (1985). 14. In addition, eliminating arbitrage opportunities helps remove certain pricing anomalies and enforces the “Law of One Price,” an important principle of finance as described in LeRoy and Werner (2001) and Lamont and Thaler (2003). 15. In addition to the research described here, parimutuel matching is closely related to the academic field of “combinatorial auctions.” Section 3.5 will describe combinatorial auctions and compare them to parimutuel matching. 16. The main difference between parimutuel wagering on horses to win and the framework in Section 2.1 is that the horse racing association typically takes 20 cents of every one dollar bet as a fee, whereas the framework in Section 2.1 assumed that fees equal zero. 17. In parimutuel wagering, arbitrage opportunities are generally not possible within one wagering type. 18. See Edelman and O’Brian (2004) for a discussion of arbitrage between the win pool and other parimutuel pools. 19. For empirical support of the favorite-longshot bias, see, among many, Fabricand (1965), Ali (1977), Snyder (1978), and Asch, Malkiel, and Quandt (1982).To explain the favorite-longshot bias,see the work of Ali (1977), Quandt (1986), Thaler and Ziemba (1988), Golec and Tamarkin (1998), Brown and Lin (2003), and Ottaviani and Sorensen (2003).