Visual Option Pricing

By David Harper

As of January 1, 2006, public and private companies will be required to deduct the cost of their employee stock options from reported profits. A vigorous debate over whether companies should be forced to expense options swirled for ten years before the issue was settled. More recently, the focus shifted squarely from “should we?” to the question of “how do we?” price these options (especially because the accounting rules, rather than clamping down on a consistent methodology, put the discretionary burden onto the shoulders of companies). This is a perfect application for Xcelsius because users need help understanding option pricing. The vast majority of people who use the Black-Scholes model disparage it as a “black box” despite its theoretical elegance. Maybe we cannot teach the details of Black-Scholes, but we can use a visual display to help users understand the basic implications.

In countless journals and articles on the subject, you will see an exhibit like the following. It shows the six inputs and their respective directional impacts on the value of an option:

To graph option value as “minimum value plus volatility”

For several years, I have taught an approach to understanding the Black-Scholes that has two steps: first calculate the minimum value of the option, then “plus up” for volatility. The minimum value is what it sounds like; it is the least anybody would pay for an option. In fact, if your volatility is zero, the minimum value is simply the value of the option (private companies used to price with this method, but no more). The “plus up” is a matter of volatility: the higher the volatility, the higher the increase over the minimum value. Graphically, it looks like this -- and so does the Xcelsius model:

Spreadsheet elements

The key inputs are in the upper left-hand corner of the spreadsheet.

The model contains a line chart and a bubble chart. Both use the matrix of calculations that contains one row for each year, from year one to year ten. Each row produces a calculation for both the Black-Scholes and the minimum value. At the bottom, we simply isolate a row for the specific Expected Life that is desired by the user. (The embedded calculations are admittedly complex, but they are drawn directly from a financial text).

Toggle buttons for unobtrusive help

The model also contains three toggle buttons. Toggle buttons are addictively easy to add. You only need three cells in your spreadsheet: two for source data and one to indicate the “display status.” The key is to link the “Insert In:” cell (from the Toggle Button) to the “Display Status” cell in the Dynamic Visibility portion of your desired component. In this case, clicking the toggle button changes the zero to a one, which triggers the display of a simple Label component. The Label component contains some help text.

The model contains a lot of little components that enhance the display. It is a good idea to group related components into folders. In this model, I covered the number display portion of the spinners (to reduce redundancy); the associated rectangles are grouped with their spinners. The other advantage to this is that if we wanted to change to top-to-bottom display order, we can move the folder instead of individual components.

A few enhancements

I did a few things to reduce display redundancy. Because the dividend yield cannot be greater than the riskless rate (and both are less than 10%), we can use a dual slider to capture those inputs:

Also, another thing I like to do is give users both “course control” and “fine control over inputs. Each slider has a corresponding spinner but I covered the numeric portion of the spinner with a white rectangle. The sliders move in large increments but the spinners move in fine increments.

Layered Charts

This model stacks the bubble chart on top of the line chart. The bubble chart simply superimposes a semi-transparent bubble at the location of the actual Black-Scholes value (i.e., where the x-axis is the Expected Life and the y-axis is the option value). I did need to shift the bubble chart around manually to get them to align. This is the next feature I would most like to see in Xcelsius: graphs that snap-to grids so that stacked graphs automatically align.

I also added an advanced feature that displays a few of the so-called greeks. These are sensitivities. Vega, for example, is the sensitivity of the option price to a change in volatility. A vega of seven means this: if the volatility increases by 1%, then the option value will increase by 7%. Greek concepts are normally well beyond the reach of average users, but Xcelsius at least renders a graphical depiction. The idea here is to bury complexity. If the user wants to “peel the onion” for another layer of complexity, he/she can choose to do so.

That’s all! I hope you enjoyed this model.

David Harper is the Principal of Investor Alternatives, LLC, a firm that specializes in investment research, software sector coverage, and derivatives valuation. He is the Editor-in-Chief of Investopedia Advisor, a newsletter devoted to the early recognition of public companies that are likely to be lead future market. He publishes the Bionic Turtle Study Notes, courseware for learning advanced risk management concepts. He is a Charted Financial Analyst (CFA) and Financial Risk Manager (FRM).