Derivatives: Valuation and Risk Management

By David A. Dubofsky; Thomas W. Miller Jr. | Go to book overview

CHAPTER 15

Option Strategies and Profit Diagrams

A profit diagram depicts the profits and losses from an option investment strategy as a function of the price of the underlying asset at expiration. Profits and losses are on the y axis, and a range of expiration date (time T) prices of the underlying asset are on the x axis. These diagrams are sometimes called payoff patterns or payoff profiles.

The payoff pattern to a long position in an asset is linear. For example, if you own one share of stock and its prices rises by a dollar, your profit is a dollar. In contrast, options offer nonlinear payoff patterns. If you own a call option on one share of stock then, at expiration, your profit rises by one dollar for every one-dollar increase in the stock price only if the option is in the money. If the option is out of the money, your profit is fixed (constant) for any expiration day price of the underlying asset; that is, you have lost the initial premium you paid for the option, and this is independent of how much the option finishes out of the money. Thus, a long call offers two piecewise linear payoff segments. Even more unusual payoff patterns are created when several different options are used, perhaps combined with the underlying asset. Some of these strategies have colorful names: spreads, straddles, strips, straps, butterflies, and condors, for example. This chapter presents many different option strategies and their profit diagrams.

In most of this chapter, we assume there are only two relevant dates for the purchase and/or sale of the assets involved: the initial date (time 0) and the expiration date (time T). Nothing, however, prevents an investor from closing out positions early. We also ignore the possibility that the options will be exercised early. Dividends, commissions, and margin requirements are not explicitly dealt with, but can easily be incorporated into one's preparation of profit diagrams. Remember, too, that because of nonsynchronous trading and bid-asked spreads, prices in the financial press may not reflect prices at which an investor can buy or sell the options. Finally, our discussion focuses on nominal dollar cash flows. This means that the timing of cash flows is ignored—we will simply add dollar cash flows that occur at two dates. Although this is a common practice in preparing profit diagrams, in concept, it is sacrilegious. We all know that a dollar inflow at time 0 is more valuable than a dollar received later.

Profit diagrams can provide additional valuable information if they are prepared on a rate-of-return basis, and also if they are prepared showing profits and losses that result when positions are closed at dates earlier than expiration. These two types of diagram will be illustrated at the end of the chapter.

Table 15.1 contains the notation that will be used in this chapter.


15.1 PROFIT DIAGRAMS FOR LONG STOCK AND SHORT STOCK

The easiest way to illustrate a profit diagram is to begin with the two basic positions in the underlying asset. An investor can buy the asset today for S0 and sell it at a future date at an unknown

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