Real Options: Using Option Theory for Business Decisions
Every chapter so far has been about financial options. Contracts you buy and sell in markets. But the word “option” just means a choice, and choices show up everywhere. Should you shut down a factory that is losing money? Should you invest in a project now or wait? Should you suspend production when prices drop and restart when they recover? Chapter 73 of Wilmott’s book shows that the same math we use for financial derivatives can answer these questions. This is the world of real options.
The Connection
Three things link financial options to business decisions:
- Uncertainty. You do not know the future. Profits, commodity prices, demand. They all fluctuate randomly.
- Timing. You get to choose when to act, and the timing matters. Acting too early or too late costs money.
- Irreversibility. Many decisions cannot be undone. Once you close a mine, reopening it costs real money. Once you invest, you cannot un-invest.
These are exactly the features that make financial options interesting. And the math is essentially the same: diffusion equations, free boundary problems, optimal stopping.
When to Abandon: The Factory Problem
You own a factory that generates profit P continuously. If P is constant and positive, the present value of all future profits is just P/r (for an infinite time horizon). No reason to close. If P is deterministic and goes negative for long enough, you close when the present value hits zero.
But what if P is random? It follows a random walk with some drift a and volatility b. Maybe a is slightly negative because the product is becoming outdated. The profit can go negative (you are losing money), but it might come back.
The value of the factory, V(P, t), satisfies a diffusion equation with P as the space variable. There is a discounting term (present value) and a source term (the ongoing profit P). The final condition is V = 0 at the end of the factory’s natural lifespan.
If you must keep the factory open no matter what, you just solve this equation and accept the result. Sometimes V is negative, meaning you expect to lose money overall.
But you have a choice. You can close the factory when it stops being worth it. This turns the problem into a free boundary problem, exactly like the American option. The factory stays open as long as V > 0. When V hits zero, you shut down. The boundary between “keep open” and “close” depends on P and t, and finding it is part of the solution.
The difference between “must stay open” and “can close optimally” is the value of the option to abandon. Wilmott shows this graphically: the value with the closure option is always at least as high as without it, and the gap is the option value. For a perpetual factory (infinite lifespan), the solution is time-independent and you get clean analytical results.
The punchline: by waiting for the optimal moment to close, you add significant value. If you had to decide right now whether to run forever or close immediately, you would miss the benefit of waiting for new information. The option value comes from the ability to observe P as it evolves and then act at the right moment.
Wilmott adds a cheeky note that this model also applies to marriage, with P representing something other than profitability and T being the expected remaining lifespan. The question becomes when to separate. I will leave the calibration of that model to the reader.
When to Invest: The Entry Problem
Flip the problem around. Instead of deciding when to abandon, decide when to invest. You have the opportunity to pay a fixed amount E to acquire something whose value S fluctuates randomly. If S follows geometric Brownian motion, the value of the investment opportunity V(S, t) satisfies the usual Black-Scholes-like equation.
The boundary condition is: when you invest, you get S minus E. You invest when S is high enough. The free boundary gives you the optimal investment threshold.
If the opportunity lasts forever, the problem is time-independent. The critical question is whether an optimal investment threshold exists, and how it depends on the parameters. With an infinite horizon, you might wait forever because there is always a chance S goes even higher. The answer depends on the drift, volatility, and discount rate.
Suspend and Restart: Costless
What if you could pause the factory whenever it loses money and restart it whenever it becomes profitable? With no cost to stopping or starting, this is simple. You operate when P > 0 and idle when P < 0. The value equation has a source term of max(P, 0) instead of P. You collect all the upside and avoid all the downside.
This is obviously better than permanent closure. The factory value is higher because you never absorb losses. In the perpetual case with constant parameters, the solution involves the Heaviside function applied to the profit.
Suspend and Restart: With Costs
Now the realistic version. It costs C to restart the factory and K to shut it down. You need two value functions: V0 for the idle state and V1 for the active state.
While idle, there is no profit (no source term in the equation). While active, you earn P. The transition conditions are:
- Going from idle to active costs C. So V1 at the switch point equals V0 plus the switching cost.
- Going from active to idle costs K. So V0 at the switch point equals V1 minus the shutdown cost.
On top of these conditions, you want the switches to happen optimally. This creates a free boundary problem with two boundaries: one where you restart and one where you shut down. The profit level at which you restart is higher than the level at which you shut down (because of the costs), creating a hysteresis band. You do not flip back and forth with every small fluctuation in P.
Wilmott shows this numerically for an infinite horizon. The active-value curve and idle-value curve cross at different points depending on C and K. The gap between the restart and shutdown thresholds depends on the size of the switching costs. Higher costs mean a wider gap and more reluctance to change state.
Sequential Investment: Build It in Stages
Some projects are not all-or-nothing. You invest gradually, spending money over time at a rate you control. At any point, you can stop investing and walk away. The question is how fast to invest and when to stop.
The math introduces a variable K for the total remaining investment and a control variable I for the current investment rate. The investment rate is bounded between 0 and some maximum k. The value depends on the project’s worth P, the remaining investment K, and time.
The equation has two extra terms compared to the basic setup. One is a derivative with respect to K (similar to path-dependent options). The other is a sink of money as you spend it on the project. The project only starts generating value when K reaches zero (fully invested).
The optimal investment strategy turns out to be bang-bang control. You either invest at the maximum rate or not at all. No half measures. The decision depends on a single derivative: if the marginal value of investing (reducing K by one unit) exceeds the cost of that unit, you invest at full speed. Otherwise you stop completely.
This is a nice result because it simplifies the problem. Instead of choosing from a continuum of investment rates, you only need to find the boundary between “invest full speed” and “stop.”
The Ashanti Gold Mine
The chapter includes a case study of Ashanti, a gold mining company that hedged gold price risk with options. In the late 1990s, Ashanti may have overhedged so much that an investment in Ashanti was effectively a bet on gold falling, not rising. Not what you expect from a gold mining company.
Wilmott builds a simple model using real options. Gold follows a mean-reverting process (calibrated from 1985 data) with volatility that depends on the price level. Ashanti’s profit is the gold price minus extraction costs (about $235 per ounce). The company value is the present value of expected future profits with the option to close the mine when gold drops too low.
The model says the optimal closure point is $210 per ounce. Below that, it is better to shut down.
Things get interesting when you add the derivatives portfolio. Before September 1999, Ashanti’s share price moved with the gold price, as you would expect. The model fits the actual share price well. But when the derivatives situation became public knowledge, the relationship inverted. Gold went up, Ashanti went down. The derivatives had turned Ashanti from a long-gold bet into a short-gold bet.
The real options framework can incorporate derivatives directly. Buying puts raises the company value for low gold prices (protection against closure). Writing calls funds those puts but caps the upside. The question of what is optimal depends on what the company is trying to achieve: maximize expected value, minimize closure probability, or something else.
The Option Value of Staying Alive
The chapter ends with the most provocative application of real options: the economics of suicide. Wilmott quotes Dixit and Pindyck (1994), who argue that a purely rational model of suicide (end life when expected present value of remaining utility falls below zero) is actually not rational enough. It ignores the option value of staying alive.
Suicide is the ultimate irreversible decision. The future has a lot of uncertainty. Even if the expected direction is downward, there is some probability of improvement. The option value of waiting to see if things get better should be very large. A rational person who properly accounts for this option value would set a much lower threshold for ending life than one who just looks at expected utility.
The argument extends to a general principle of real options: irreversible decisions require a much higher threshold than reversible ones. The more irreversible the action and the more uncertain the future, the more valuable it is to wait. This applies to closing factories, entering markets, making investments, and apparently, to the most fundamental decisions of all.
Key Takeaways
Real options translate the math of financial derivatives into the language of business decisions. The core insight is simple: any situation with uncertainty, choice, and irreversibility has option value. That option value is destroyed if you are forced to decide now instead of waiting for information.
The math is the same as what we have seen. Diffusion equations for the random variable driving the decision. Free boundaries for optimal timing. Constraints linking different states (active, idle, invested, abandoned). The only new ingredient is the source term representing ongoing cashflows rather than a terminal payoff.
The practical lesson for business is: if you can wait, waiting has value. If you can stage an investment, staging it is better than committing all at once. If you can suspend and restart, that flexibility is worth paying for. And if a decision is irreversible, you should demand a much higher hurdle than the breakeven point before pulling the trigger.
Previous post: Energy Derivatives: Oil, Gas, and Why They Are Different
Next post: Life Settlements and Viaticals