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CHAPTER 3. Valuing Vanilla Options

Options and options-embedded instruments trade in abundance and diversity on both the over-the-counter (OTC) and exchange-traded markets globally. Since the underlyings of these options span a multitude of asset classes (e.g., equity, commodity, interest rate, and currency – just to name a few), it is not surprising to expect the option landscape to be a very complex one. Despite the complexity of the landscape, all options can be categorized as either vanilla or exotic:

■ Vanilla options refer to options that allow owners to transact (i.e., buy or sell) the asset underlying the option for a pre-specified price at a prespecified time(s) in the future.

■ Exotic options refer to options that are not vanilla options (e.g., buy the asset for the average price of the asset realized during the life of the option).

Regardless of the nature of the option (vanilla or exotic), all option owners have the ability to exercise at

■ One prespecified time (or option maturity).[1]

■ Any time up to and including the option maturity.[2]

■ Limited times up to and including the option maturity.[3]

The reader is referred to Hull (2012) or any good introductory finance book for further descriptions of these options. For the ease of discussion,

TABLE 3.1 Options Prices (when S&P 500 Value is 1681, T – t = seven days)

Strike

Call

Put

Bid

Offer

Bid

Offer

1660

28.20

30.30

0.45

0.65

1665

23.40

25.40

0.70

0.90

1670

18.90

20.60

1.05

1.25

1675

14.60

16.50

1.65

2.05

1680

11.00

12.30

2.60

3.00

1685

7.50

8.40

3.90

4.40

1690

4.70

5.30

6.00

6.30

1695

2.55

3.10

8.50

9.80

1700

1.30

1.65

12.10

13.40

1705

0.55

0.95

15.60

17.70

1710

0.30

0.50

20.20

22.30

1715

0.10

0.25

25.10

27.20

Source: cboe.com.

unless otherwise mentioned, I will henceforth restrict my discussion to European-style options.

One of the most common and widely traded European-style options is the one linked to the S&P 500 index that trades under the symbol SPXW on the Chicago Board Options Exchange (CBOE).

Table 3.1 shows a snapshot of the S&P 500 (ticker symbol: SPXW) option chain taken during trading hours. As can be seen from the table, the option chain illustrates the impact of varying strike values on the option premiums. Despite this, it is difficult in practice to trade these options if one cannot ascertain how relatively cheap or expensive one option is to another. As a consequence, to be able to trade the options effectively, it is imperative to be able to price these options using a model that professional traders use in practice.

Since the purpose of this chapter is to discuss the application of quantitative methods to price vanilla options, I start with an example to illustrate the derivation and use of the Black-Scholes formulae to price options on nondividend-paying stocks. I then discuss the pricing of similar options when the underlying assets range in diversity from dividend-paying stocks to equity indices to currency rates to commodity prices to interest rates (in particular, swaps, forward rate agreements, and bonds) – and in the process illustrate how practitioners adapt the basic Black-Scholes model to suit their purposes. The chapter then discusses the valuation of options with early- exercise features before concluding with a risk-management application in practice.

  • [1] Such options are called European-style exercise options.
  • [2] Such options are called American-style exercise options.
  • [3] Such options are called Mid-Atlantic options, limited exercise options, discrete style exercise options, and so on.
 
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