It was while studying ice roads in the Northwest Territories (NT) of Canada that we first noticed the strong similarity between put option curves (Fig. 2) and the cost curves we were constructing to describe the operation of ice roads that supply diamond mines north of Yellowknife (Figs. 3 and 4). Nature provides a free ecosystem service (in the form of winter cold) for this road network in a region that is otherwise roadless. Once the roads are constructed, truck transport using the roads is a fraction of the cost of air cargo, the only other way of moving material to these highly profitable but remote mines. If the winter is cold enough, sufficient ice thickness can be produced so that the ice roads will last until all required cargo has been delivered to the mines. In a mild winter, the ice roads have failed early (Fig. 3), forcing the remaining cargo to be delivered by air at a steep premium. There has been a significant reduction in winter cold during the past 40 years (Fig. 3), suggesting (Stephenson et al. 2011) that ice roads might soon cease to be viable. Is this true?
To answer the question, we mapped the ice road system to a general option formula as well as to the B-S formula (Fig. 4). First, we replaced the strike price (K) with the minimum season length sufficient to bring up the all the material needed by the mines by truck. This open season length is controlled by winter cold (Fig. 3), but that is only half the story (Supplement 1). Human effort, in the form of plowing snow, can enhance ice growth and increase the length of time the road stays open, while the number of trucks and the efficiency with which they are used can reduce the length of time needed to transport all cargo. Consequently, the strike price is actually a function of an ecosystem service (cold) and human effort (plows and trucks). But unlike the supply of cold, the human component is not free. It comes at a cost of C$18.6 million each year in construction and operation costs. Composite (human plus nature) strike prices seem to be the norm for socio-climatic systems.
Next, we replaced payoff (P
T
, the y-axis in Fig. 2) with an equivalent fiscal metric, in this case the marginal cost of flying cargo to the mines. In the case of the diamond mines, this is the marginal cost of air over truck transport, or $2096/tonne, which when multiplied by the number of tonnes that need to be flown, produces the values on the y-axis in Fig. 4, which shows values from the general option approach in blue, green, and red.
To employ the B-S formula, we replace stock price volatility with the variance of the relevant socio-climatic metric (e.g., Fig. 3). When the distribution of this variance (or alternately, the standard deviation) is log-normal, it can be used directly in the B-S formula, which assumes an underlying log-normal distribution. But for the diamond mines it was not, so we had to employ an adjusted value (Supplement 2). We have found that this approach works with reasonable accuracy even for distributions that differ considerably from log-normality. The relevant metric was an adjusted value of the variance of the open season length. This metric does not appear explicitly in the graphical results but rather produces a family of curves in which higher variance results in a curves delineating increasingly higher expected costs. These expected costs represent a “best guess” of the potential costs that will be incurred before the actual climate conditions can be known and they form the basis of most business decisions, which invariably need to be made before the actual conditions (climate or business) have materialized. In mathematical terms, this mapping corresponds directly to a general option valuation formula:
$$ {P}_T = {\displaystyle {\int}_0^{\infty } \max \left[0,\left(K-S\right)\right]p(S)ds={\displaystyle {\int}_0^K\left(K-S\right)p(S)ds}} $$
(1)
where K is the current value of the socio-economic metric and p(S) is the distribution of that metric (typically represented by the variance or standard deviation).
In socio-climatic systems, the strike price corresponds to the operational tipping point (OPT), a point where an abrupt increase (or decrease) in the cost and/or work effort of utilizing the ecosystem service occurs. The OPT is largely determined by human needs, and in many cases (based on past operations) is well known for short periods, though over a period of a few decades the value is likely to change. More problematic is developing a quantitative climate time series that captures past, or future, climate conditions, and this is difficult whether using our method or GCMs. Nevertheless, we think the OPT is an extremely useful concept when considering responses to climate change across a wide range of businesses. Defining it forces one to think clearly about the ecosystem service that is being utilized, how climate moderates that service, and what labor and costs we incur in harvesting the service. Mathematically, we express OPT as the ratio between what nature is able to supply to what is needed, so the tipping point occurs at a value of 1. In the case of the NT ice roads, this corresponds to being open for 33 days, which requires (in general) that nature deliver an ice thickness of 85 cm or more (Supplement 1).
Two fundamental climate-affected decisions face the owner/operators of the mine: (1) What is the financial risk of an ice road failure in the current climate, and (2) what would the risk be in a climate that is warmer (colder) and/or more (less) variable? The owner/operators must decide in late summer whether to commit to building ice roads the following winter, long before even a seasonal weather forecast is available. For many reasons (letting of construction and trucking contracts; hiring of road workers and drivers, obtaining permits, etc.), they cannot reduce their time to expiry: in other words, they are effectively always a half year out in the sense that they must assume maximum variance. What the operators do know (based on weather records) is the historical trend and variability of the winter weather and cold (Fig. 3), which is exactly analogous to the stock trader who knows the historic trend and volatility of a stock.Footnote 4 Implicit is the assumption of some type of climate stationarity: that the past climate record is related in some functional way to future climate behavior. Mapped into the option formulation, perhaps across a range of climate scenarios, the resulting calculations and graphs allow the owners to address their questions and decide if an ice road is a viable solution.
The results are surprising and non-intuitive. First, even in the pre-climate warming world of 1971 (before the diamond mines were discovered in the 1990s), the expected flight costs of an imaginary ice road system would have been non-zero ($4.4 million). By 2012, after years of warming, with the diamond mines in place for 14 years, the expected costs had risen to $74 million, yet the road-open season length had not dropped below the OPT: the decision to build the road had merely gotten riskier. This risk is real: in 2006, the ice road system failed prematurely, resulting in what we estimate to be millions of dollars in air cargo costs. Should the warming trend continue as before, the mean road-open season length will drop below the OPT in 2031, but it is a mistake to assume ice roads would no longer be built after that point is passed. As long as the ice road system manages to allow enough transport by truck that the differential in the cost of the amount of cargo delivered over the road vs. by air exceeds the construction and operation costs of the road itself, ice road use will remain financially advantageous, just risky.
Figure 4 also clarifies the impact climatic variance has on the financial risk, and it is as large, if not larger, than the impact of climate trend. For example, were the variance to drop to half of current value, the change would reduce the expected cost to $22 million, a reduction by a factor of 3. Conversely, increasing the variance by 50% would push the expected costs well past values that would otherwise not be realized for another 20 years of warming, to over $100 million.
As cited above, alternative ways of achieving the same insights gave been developed. For instance, conducting a series of ensemble or multi-model runs of global circulation or regional climate models, then down-scaling these and integrating them with run-off or business models (or both) also allows for examination of the impact of both climate trend and variance on costs and outcomes and can be applied in more sophisticated ways (e.g. (Cervigni et al. 2015)) but these alternatives are typically more expensive and require greater computing tools and resources to implement. In contrast, option pricing, and particularly the use of the modified B-S formulation, can be performed using a simple spreadsheet. Once parameterized, the effect of changes in the variance or mean of the climate metric can be calculated in seconds.
One additional concept needs introducing and some explanation before we present additional examples. The concept is that of hedging through storage, a concept that has been known for thousands of years (see the Bible, Genesis 41). Unfortunately, in the case of ice roads, hedging through storage is not practical. The main cargo transported up the roads is fuel, and storage capacity at the mines is sufficient for only 1 year’s operation. Expanding storage is too expensive without greater certainty in the lifespan of the mines. This is one reason why the system has an asymmetrical cost structure: excess costs in warm years cannot be offset by stockpiling in cold years. While systems that can be fully hedged through storage or financial markets would not typically have an asymmetric payoff profile, it turns out that many business systems that rely on nature’s bounty in some way (see Supplement 2) are asymmetrical.