In this post, we’ll discuss a topic that’s strongly emphasized in all MBA programs – real options.
Real options receive heavy emphasis in business school for at least two reasons. First and most obviously, it’s strategically important that managers be able to assign a value to flexibility. In any business context, an option gives the option holder the right, but not the obligation, to take a certain action (termed exercise).
Naturally, choice and flexibility are most significant when the stakes are high. If the price of poker is $10 million, for example, the freedom to invest some or all of that amount over time is vastly preferable to a single Yes/No, now-or-never window of opportunity.
And therein lies the importance of real options.
Of course, real options are fundamental to quintessential high stakes enterprises such as petroleum exploration and big pharma. But opportunities to invest a significant portion of the company’s net worth are apt to come up in almost any business model. When these investments lend themselves to staged commitments, the discrete opportunities present a series of decision points (called levers). The process of strategic decision-making includes assigning the appropriate value to each lever.
Another reason why business schools emphasize real options is related to the option value equation. Naturally, business schools teach students how to value options (also known as derivatives) on publicly traded stock. As it turns out, options are options to a great extent, and the same basic equation that’s used to calculate the value of a vanilla call option on AAPL can be used to quantify the value of an option to purchase a competitor’s factory.
The Black-Scholes-Merton Equation
Throughout this discussion, we’ll focus on one particular method of valuing options: the Black-Scholes equation, as modified by Robert Merton (BSM). The BSM equation has enjoyed wide popularity ever since its creation for at least three reasons. First, it’s simple to code in any computer language. Second, it’s a closed-form representation. And third, provided the user understands and remains cognizant of BSM’s inherent limitations (especially in the financial options context), it mostly works.
Since this is an article written primarily for non-MBA entrepreneurs and business owners, I’ll forgo a discussion of the math that underlies BSM. Interested readers can find a good exposition in a paper that’s freely downloadable here.
In the world of financial options there are a number of other algorithms and methods that quantitative analysts use to calculate option value. It should come as no surprise that one can purchase an option to buy or sell publicly traded stock on almost any terms imaginable. For example, Mathematica‘s FinancialDerivative function includes several dozen option types that are built-in:
Those financial options that include some combination of unorthodox payoff structures, valuation criteria, term length and/or exercise provisions are called exotic. Generally speaking, the more exotic the option, the less useful BSM is in valuing it. Most exotic options are valued using complex numerical calculation methods.
Part of BSM’s appealing simplicity is its use of only six variables. Within the VBA module below, I’ve added a seventh variable to capture whether you’re pricing a call or put option. For our purposes in this article, an option to acquire would be considered a call option, while an option to divest would be deemed a put option.
In the table that follows, I’ve set out each of the six along with the value that each parameter represents in the context of real options, and then financial options.
|Parameter||Real Option||Financial Option|
|S||Present value of future cash flows||Current stock price|
|X||Present value of future fixed costs||Option strike (exercise) price|
|sigma||Uncertainty of future cash flows||Volatility of underlying stock|
|t||Option life from today to expiry (years)||Same as real option|
|delta||Diminution of option value over time||Dividend rate of underlying stock|
|r||Risk-free interest rate||Same as real option|
These parameters are largely self-descriptive, but I’ll expound briefly upon each from the standpoint of real options. I’ll also explain how each parameter affects overall option value. In the interest of brevity, I’ll assume in each instance that we’re considering a call option. Just be aware that an increase in the value of any parameter would have the opposite effect on a put option.
S represents the expected net present value (NPV) of future cash flows from the investment opportunity. We’ve discussed NPV at various times in previous articles, but the easiest way to understand NPV is to actually calculate it. Here’s a spreadsheet I’ve written that calculates expected NPV (note that the workbook uses macros). An increase in S increases real option value.
X is similar to S, except that you’re aggregating the fixed costs associated with the investment instead of expected cash flow. As long as you start from scratch and substitute fixed costs for revenue, you can calculate X with the same spreadsheet you used to calculate S. An increase in X decreases real option value.
In many ways the parameter sigma is the key to BSM valuation. We use sigma to specify the volatility (or uncertainty) that you attach to your model. Somewhat counter-intuitively, the greater the uncertainty, the more an option is worth. Yet it’s uncertainty that makes flexibility so valuable. In the course of valuing your real options, I encourage you to play with different values for sigma and take note of its significance and influence. As noted, an increase in sigma increases real option value.
For purposes of real option valuation, t is ordinarily the time interval between today, and the final drop-dead date for exercising a real option. An increase in t increases real option value. You always want to express t in terms of years, so you would divide n number of days by 365 and use that value. You are of course free to specify some value for t other than the day on which the whole thing turns into a pumpkin. If you’d like to value only a certain lever in terms of real options, you can specify instead the deadline for that discrete investment (or other action). Just make sure that your other values are consistent with the same interim period.
The parameter delta is Merton’s contribution to the basic Black-Scholes equation, and it was originally designed to account for options to buy or sell stocks that pay a shareholder dividend. Not all stocks pay dividends, of course, so in many cases the value of delta is zero. The same is true in a real options context. With respect to real options, delta stands for all of those factors that chip away at the value of a real option, and which come about as a function of time. For example, perhaps you have an option to buy a depreciating asset. Or perhaps it will require interim investment on your part to either keep the option open or have it remain a viable strategy. Any tangential detractor of option value should be included within your assigned value for delta. An increase in delta decreases real option value.
The last parameter is r, or the risk-free interest rate. An increase in r increases real option value. Unlike our use of r in previous articles to cover a wider and more eclectic series of factors, r with respect to real options is intended to represent only the (risk-free) time value of money. As we’ve noted in earlier posts, the usual benchmark for r is the interest rate on 3 month U.S. T-Bonds, which has remained below one percent for many consecutive months. You’re free to prognosticate with respect to r; just remember that you’ll skew the calculation if you allow factors other than the expected risk-free rate to affect your estimate.
Black-Scholes-Merton VBA Code
As noted above, the BSM equation is remarkably easy to code. To use the VBA function below in Excel for Mac 2011, simply click the Developer tab, then the Editor button, then select Insert > Module and then copy-and-paste. Don’t forget to save your workbook in .xlsm format so that the VBA module is saved.
Public Function realOption(S As Double, X As Double, sigma As Double, _ t As Double, delta As Double, r As Double, cp As Long) As Double ' Uses Black-Scholes-Merton to calculate ' real option value ' Enter cp = 1 for Call option or cp = 2 for Put option Dim d1 As Double, d2 As Double With Application.WorksheetFunction d1 = (Log(S/X) + (r-delta+((sigma^2)/2) * t)) / (sigma * Sqr(t)) d2 = d1 - (sigma * Sqr(t)) If cp = 2 Then realOption = X * Exp(-r*t) * .NORMSDIST(-d2) - S * Exp(-delta*t) * _ .NORMSDIST(-d1) Else realOption = S * Exp(-delta*t) * .NORMSDIST(d1) - X * Exp(-r*t) * _ .NORMSDIST(d2) End If End With End Function
Here’s a graphic showing the setup in Excel: