The theory of modern finance seeks to allocate and deploy resources across time in an uncertain environment [1]. Traditional finance models such as the Capital Asset Pricing Model [2] show how to optimally deploy assets by determining the optimal path over time given the initial conditions.
These older methods, though, do not adequately capture the value of man- agement having the flexibility to change strategies in response to unfold- ing events. The theory of contingency finance forms the framework of options theory, which addresses this problem by allowing management to revise decisions continuously over time [1]. This new view allows valua- tion when agents (specifically managers) are able to make choices as events unfold. It shows how uncertainty increases the value of a flexible asset deployment strategy.
The theory of options has proven useful for managing financial risk in uncertain environments. To see how options can limit risk, consider the classic call option: It gives the right, but not the obligation, to buy a secu- rity at a fixed date in the future, with the price determined in the past. Buy- ing a call option is the equivalent of betting that the underlying security will rise in value more than the price of acquiring the option. The option limits the downside risk but not the upside gain, thus providing a non- linear payback, unlike owning the security. This implies that options pro- vide increasing value as the uncertainty of the investment grows (that is, as variance in the distribution describing the value of the security increases and the mean is preserved).
Options are a powerful methodology of risk reduction in an uncertain market. They allow limiting loss without capping the potential gain. Figure 5.1 shows graphically what the value of an option is. The non-linear pay- back of the option is the solid line, while the linear payoff of owning the stock is the dashed line. The option holder is able to look at the price of the security when the option is due and decide whether to exercise the option to buy the stock. This protects the option holder by limiting the loss to the cost of acquiring the option, no matter how low the stock price falls. Some risk-averse investors prefer this type of non-linear payback that caps the downside risk, but leaves the upside gain unaltered.
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Figure 5.1 Value of an option.
The following numerical example further illustrates this concept. Sup- pose stock xyzzy is selling at $100.00 on January 1, 2002. The classic call option is to pay a fixed amount per share (say $10) for the option to buy shares of xyzzy on February 1, 2002 at $100.00 per share. Suppose you buy this option on 100 shares of xyzzy for $1,000. If xyzzy rises to $200.00 per share, your profit is the gain in the stock minus the cost of the option:
100*$100 – $1,000 = $9,000.00. If xyzzy does not rise above $110, the option is worthless, and you lose the full cost of the option ($1,000). This is the most you can lose, even if the stock falls to 0. Compare this to owning the security. Buying 100 shares of xyzzy costs $10,000 on January 1, 2002. If its value increases to $200 a share, and you sell it on February 1, $10,000 is the profit. If the price does not change, you lose nothing. If xyzzy files for bankruptcy and the stock drops to zero, you lose the full $10,000. The dif- ference is dramatic between owning the stock and owning an option to buy the security. The options cost $1,000 but protects from a possible loss of
$10,000. If the stock takes off, the upside gain is offset only by the fixed
$1,000 cost of the option. This example shows how options limit the risk without capping the gain of an investment.
The historical variability of the stock price, not the average price of the security, determines the value of the option. This means the variance of the distribution describing the option value is independent of the average value of the asset. This is intuitive if you consider the following example:
0
Stock Price
Strike price
Stock ProfitOption Pay-out
Profit/Loss from Stock sale Excess risk of
stock ownership
Option pay-out
Intuitive View of Options Theory 75
Stock xyzzy is very stable, and very expensive, at $1,000.00 a share. For many years its value has remained constant so its average price is $1,000.
Any option on this stock is worth little because the uncertainty of what the price of the stock will be is so low — it most likely will be very close to
$1,000. On the other hand, consider the stock from zzzz, which has been bouncing around for two years. The low was $10, with a high of $200, and an average price of $20. The option to buy shares of zzzz is valuable because there is a chance of the stock rising again as it did before. This illus- trates that the value of an option is based on the variability of the price, not its average value.
Understanding the complex mathematics underlying the theory of options is unnecessary to gain a good intuitive understanding of how this theory can help convert uncertainty into profit. One key to understanding this options point of view is realizing that the expected value of a correctly structured investment portfolio can increase as uncertainty grows. Options are about the relationship between uncertainty, flexibility, and choice. The idea is that increased uncertainty amplifies the value of flexibility and choice when the correct strategy is crafted.
In technical terms, options help value choice in capital markets.1 The capital market aspect means the players in the market determine the value of each choice, and this value is random. If the values are known, then dynamic programming is the correct valuation technique to use. In simple terms, the complex partial differential equations explaining the theory of options are a mathematical model describing this value of flexibility. It illustrates how more uncertainty about the value of choices in the market increases this value of having choice.
This book is about the value of users having choices, and options are a proven methodology to value this choice. Think about two situations: a user having only one choice for a service or a user having many choices. If there is uncertainty about what the user will like and how much the user will value the service, then the value of the service is random. Options thinking illustrates that this value of giving users choices is the difference between the expected value of each choice, subtracted from the expected value of the best of all the choices. This makes intuitive sense because the expected value of the best choice is the value a user gets if he or she has a choice to pick what they prefer, and the expected value of any particular service is the average value for the service if this user only has one choice. This book and the options framework illustrate how the value of allowing users to pick from among many choices grows as market uncertainty increases.
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1This understanding comes from a conversation I had with Carliss Baldwin at Harvard Business School while working on my thesis.