Abstract
Nowadays one cannot understand modern finance and financial markets without a solid understanding of derivatives. This chapter introduces the basic building blocks of derivatives: forwards, futures, swaps (a brief introduction only) and options with a balance of theoretical and practical perspectives. The approach focuses on understanding the contracts and strategies, with an emphasis on options. The pricing aspect will be discussed in the next chapter.
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Notes
- 1.
These assumptions are often either unreasonable or unrealistic from a practical point of view (e.g., a time t is assumed to be continuous, but in reality it is discrete).
- 2.
Later we will show that the definition of options coincides with the definition of contingent claim in the financial dictionary: “a claim that can be made only if one or more specified outcomes occur.”
- 3.
A commodity is a raw material used in commerce or primary agricultural product that can be bought and sold such as copper, silver, crude oil, natural gas, wheat, beef cattle, and coffee.
- 4.
A callable bond is an example of a bond with an embedded call option.
- 5.
Section 2.10
- 6.
- 7.
One can think of a call option (see Definition 7.8). It is defined in a later section, but this example should be readily understandable.
- 8.
It is understood that by a portfolio we mean a self-financing portfolio (i.e., no money is added to or withdrawn from the portfolio after time 0).
- 9.
One of those versions is given as follows:
A portfolio Π is said to be a statistical arbitrage portfolio, or simply a statistical arbitrage if it satisfies either of the two sets of conditions:
$$\displaystyle{(a)\ \varPi (0) = 0\quad \text{and}\quad \mathbb{E}(\varPi (T) > 0) > 0,\quad (b)\ \varPi (0) < 0\quad \text{and}\quad \mathbb{E}(\varPi (T) \geq 0) > 0.}$$ - 10.
In old times, people would most often shake hands when agreeing on deals.
- 11.
We leave out of the theoretical arguments such as whether a no-arbitrage condition can be verified when the underlier is the S&P 500 Index.
- 12.
The cost of carry is the cost incurred by holding the underlying asset such as storage costs or insurance as well as other incidental costs.
- 13.
For example, being able to take advantage of shortages of the underlying asset.
- 14.
Note that in the equality \(F_{T}(0)\mathrm{e}^{-\mathsf{r}T} = S(0)\,\mathrm{e}^{-qT}\), the LHS is the present value of \(F_{T}(0)\), and the RHS is the present value of \(S(T)\) because \(\mathrm{e}^{-qT}\) shares at time 0 grow to 1 share at time \(T\) and, during the same time period, the stock value changes from S(0) to \(S(T)\).
- 15.
The bond here is a zero-coupon bond. For our purpose, the position of long 1 par zero maturing at time \(T\) can be interpreted as “lend \(\$S(0)\mathrm{e}^{-qT}\) at interest rate \(\mathsf{r}\).” Similarly, a position of short zero here would be interpreted as “borrow \(\$S(0)\mathrm{e}^{-qT}\) at interest rate \(\mathsf{r}\).” Using the bond terminologies will provide us convenience (e.g., for letter expression of long or short position) later.
- 16.
To compare to a familiar environment, simply consider, in a housing market, the difference between the market exposure of “for sale by owner” and that of “for sale by real-estate agency.” Generally speaking, the bigger the market exposure, the higher the level of liquidity.
- 17.
Futures contracts allow fewer delivery options than forward contracts.
- 18.
Precisely speaking, the futures price in the second column should be the daily settlement price or simply settlement price, which is defined by the exchange. There are different types of settlement procedures. Each derivative exchange has a set of procedures used to calculate the settlement price. Margin requirements are based on the daily settlement price, not the daily closing price. For our purpose, we consider the settlement price to be essentially the closing price on that day.
- 19.
The SEC requires that every short sale transaction be entered at a price that is higher than the price of the last trade.
- 20.
The name plain vanilla swap reflects that those swaps do not possess any special or unusual features.
- 21.
See Section 1.1.2 on page 4.
- 22.
Such a loan is called a floating rate loan or a variable or adjustable rate loan, which is a debt, such as a bond, mortgage, or credit, that does not have a fixed interest rate over the life of the debt. The interest rate on a floating rate loan is referred to as a floating interest rate, or variable or adjustable rate.
- 23.
Company Y enjoys a lower borrowing cost in both markets because we assume that company Y has a better credit rating than company X.
- 24.
A swap bank is a generic term for a financial institution that facilitates swaps between counterparties and serves as a broker or a dealer for the trading.
- 25.
We mean the European-style option (to be defined shortly), which is the basic option style from which other styles of options derive.
- 26.
The Chicago Board Options Exchange (CBOE), a spin off from the Chicago Board of Trades, first traded standardized options in 1973 and NYSE in 1982.
- 27.
There are other option styles such as Asian or Bermuda. We only consider American and European options because they are the most actively traded options.
- 28.
Indeed, American-style stock options tend to cost more than equivalent European-style options for the same stock in practice. Almost all exchange-traded stock options are American-style options, whereas stock index options can be issued as either American or European options (e.g., S&P 100 index options are American options, and Nasdaq 100 index options are European options).
- 29.
Option pricing done by the Black-Scholes-Merton model applies to European options, not to American options, and reflects the risk associated with having to wait to exercise the option, which is not appropriate for American options because of the possibility of early exercise.
- 30.
The name vanilla reflects that calls and puts do not possess any special or unusual features. In contrast, exotic options have more complex features.
- 31.
For standard S & P 500 index futures, the multiplier is 250 (index level × 250 = price); for E-mini SPX futures (smaller contract), the multiplier is 50. Multiplier varies for indices.
- 32.
If the third Friday is a market holiday, then those options expire on the third Thursday.
- 33.
A neutral option strategy is a strategy that is designed to profit from either a rise or fall (non-directional) in the underlier price.
- 34.
This explains the name “price spread” (more precisely, “strike price spread”). Since the strike prices are listed vertically by the news media, a price spread is also referred to as a vertical spread.
- 35.
This explains the name “calendar spread” or “time spread.” Since the expiration months are listed across the top of the newspaper page horizontally, a calendar spread is also referred to as a horizontal spread.
- 36.
A perfect hedged portfolio is a portfolio with complete risk elimination.
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Petters, A.O., Dong, X. (2016). Derivatives: Forwards, Futures, Swaps, and Options. In: An Introduction to Mathematical Finance with Applications. Springer Undergraduate Texts in Mathematics and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3783-7_7
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