Using an option pricing approach to evaluate strategic decisions in a rapidly changing climate: Black–Scholes and climate change

Nature provides critical ecosystem services on which society and businesses rely, but the effort and cost of utilizing those services can change with the climate. Both climatic trend and variance affect these efforts and costs, creating a complex decision space where uncertain future predictions are the rule. Here, we show how these problems mimic option payoffs and demonstrate a modified version of the Black–Scholes option pricing formula (widely used in finance) to analyze these types of business-climate decisions. We demonstrate the method by (1) examining the viability of building ice roads in the Northwest Territories of Canada, where a strong negative warming trend is underway, and (2) applying it to the problem of the ongoing California drought, estimating expected water costs with and without storage. The method is novel and provides a simple and accessible way to make such assessments to at least a first-order approximation. While our focus here is on business situations where decisions are usually based on money, we suggest that a similar approach could be used beyond the business world in examining risk and attributing that risk to climate variance vs. trend.


Introduction
The Earth provides essential ecosystem services on which society and commerce depend (Costanza et al. 2014;Daily 1997). Sunlight and rainfall for lumber and crops are examples where these free services can be utilized with only a limited amount of infrastructure. Generating and transmitting hydroelectric power and large-scale irrigation projects are examples where harvesting nature's bounty requires more extensive infrastructure, such as dams, substations, and canals. In both cases, the natural systems that are the source of the no-cost services respond directly, sometimes dramatically, to changes in climate. Consequently, a changing climate affects not only when, where, and what services will be available but also the extent and cost of infrastructure and human effort that will be required in harvesting and utilizing those services.
Making wise decisions at the intersection of these two systems, one natural and one human, is at the heart of adaptation to climate change (Linnenluecke and Griffiths 2012;Tallis et al. 2008). That intersection space is complex and messy: it includes the coupled worlds of commerce and politics, as well as the physical, chemical, and biological intricacies of the natural world. In addition, climate predictions are usually not accurate or reliable enough for the purpose of informing specific, short-term decisions (Lempert et al. 2004), greatly increasing risk. Consequently, any method or tool that allows for a quantitative valuation of ecosystem services in a volatile and rapidly changing climate is potentially useful.
One way of approaching such decisions has been to create an ensemble of future climate states (Giorgi and Francisco 2000), downscale these (Tebaldi and Knutti 2007; Wilby and Dessai 2010), then apply a range of climate, business, perhaps hydrologic (Blöschl and Montanari 2010;Cervigni et al. 2015;Groves et al. 2013;Harding et al. 2012) and decision models (Herman et al. 2015) to the human-climate system of interest. Are there other, perhaps simpler, tools that can be used to examine this intersection of human and natural systems? Here, we note that many socio-climate problems mimic options and could be evaluated similarly. In addition to a general option valuation approach, we also offer here a useful tool in the form of a modified Black-Scholes (Black and Scholes 1973) option pricing formula (B-S formula), a well-known financial formula that prices stock options. Replacing stock price volatility with climatic variance, and running scenarios to mimic climatic trends, we have found that we can reconfigure the B-S formula in way that allows us to explore the linked behavior of these two climate parameters.
Option pricing uses return distributions to predict future costs in a system (stock options) that has a sharply asymmetric payoff, with zero payoff below a critical point and increasing profit once that threshold is surpassed. Inherent in the approach is the use of a binary payoff function (max[0, S-K] where S is the current stock price and K is the strike price); the same function describes ecosystem services where a change in climate can require a state change in the method of utilizing the service. We modify the inputs in the B-S formula to estimate the expected costs for these socio-climatic systems. For instance, a farmer who incurs no irrigation costs when rainfall exceeds the amount of water needed by his crops incurs increasing costs when rainfall falls below that threshold and he has to pump or purchase water. Our method of exploration, with the resulting analyses (mostly done graphically), clarify in unique ways the combined impact of climate trend and variance on business costs associated with ecosystem service use.
The estimates derived using a modified B-S formula closely match values from the general option valuation approach, indicating that the B-S results, which are easier to derive, can be used with confidence at least as a starting point in evaluating the cost of an anticipated impact or the merits of a particular decision. While the approach is similar in spirit to real options in that the underlying variable is not traded (Myers 1977;Trigeorgis 1993), a key difference is there is no choice on the part of the stakeholders or operators as to whether to exercise the Boption.^1 The B-S formula has been used in climate studies before (Tucker (1997);Fan (2006);and Gersonius et al. (2013)), but our application of the option approach is different.
In developing our method, we have purposely restricted our focus to business and specifically those sectors that rely directly on an ecosystem service. This restriction allows us to couch decision-making on the basis of money, though other factors often play a role. It also allows us to focus on a narrow temporal-spatial realm of climate ( Fig. 1 after Clark 1985). The full realm spans 10 orders of time, while business decisions are made on temporal scales that run from less than a year to typically no more than 10 years, with 4 years being common ( Table 1). The spatial scales of business, while harder to define, range from local (km) to regional. Combined, these limits greatly reduce the overlap between business decisions and climate scales, allowing for considerable simplification. In simplest terms, businesses tend to focus on inter-annual variability (e.g., a parka manufacturer worries more whether next winter will be cold, not whether the climate is changing). What is notable is that the overlap does not include global warming trends, which require about 30 to 50 years to manifest. Given the potential existential threat posed by global warming trends, it is not surprising that these garner a lot of attention, but when it comes to business, climate variance drives the bulk of everyday decisions.
2 Modeling a socio-climatic system using an option-like approach The B-S formula (Black and Scholes 1973;Chriss 1996) 2 , which is a special case of general option theory (Kallianpur and Karandikar 2012), was specifically developed to assist stock traders to decide whether a stock option was priced fairly (Fig. 2). The length of the option contract (e.g., 6 months) and the appropriate interest rate also determine the price. When a put option 3 contract is first initiated, uncertainty in the final stock price is at a maximum, the potential for a larger payoff is greater, and therefore the cost of the option higher. As the option expiration date approaches, the likelihood of large changes in stock price decreases, as does the option price. Upon expiration of the option, the system switches from probabilistic to deterministic: if the stock price is lower than the strike price, the holder of a put option may choose to exercise the option at the current stock price, and the payoff is the difference between the two. If the stock price is higher than the strike price, the option is not likely to be exercised and the payoff is zero. A key point is that upon expiration, predicted values become realized, with the option curve at expiry the realized cost curve. All other (pre-expiry) curves 1 An example makes this distinction clearer: a real option for a mine owner might be to shutter a mine when commodity prices drop. However, if the owner decides to operate the mine, then fuel and supplies must be delivered at the mine regardless of transportation costs, and these may depend on climatic conditions. Our method allows computation of the transportation costs as a function of varying climate conditions. If, after those computations, the owner finds the expected cost too high, he may then exercise the real option of shutting the mine. 2 Development of the B-S formula eventually led to a Nobel Prize for its originators, but not before the paper explaining the formula had been rejected by several journals. 3 Put options pay off when the stock price drops below the strike price (i.e., the buyer of the option bets the stock will be worth less at a later date), while call options pay off when the stock price rises above the strike price. are probabilistic. This family of curves lies en échelon above the realized cost curve, with each higher curve a function of a longer period remaining to expiry and/or increasing stock price volatility (variance). While this is true for most options, the B-S formula provides a closedform solution (for certain distributions) that is easier to compute.

Example I: ice roads and Black-Scholes
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 Fig. 1 Spatial-temporal scales of climate (after Clark 1985). We have superimposed the time and space scales of businesses (see Table 1) denoted by the dashed box. It does not overlap global warming trends 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 Fig. 2 A typical put option graph (black labels) showing the strike price (K) and the payoff (y-axis), which increases linearly as the stock price (S) falls. Option pricing computed using a general option formula are shown for 1 year (red) and 6 months (green) prior to expiry of the option. Longer times to expiry mean greater uncertainty due to volatility, hence higher potential payoffs. The analogous socio-climatic system is suggested by the blue labels  , 1943-2012. In the B-S formula volatility is replaced by climatic variance, in this case AFDD. Winter cold (which freezes and thickens ice) is well represented by AFDD, from which the length of time the ice road stays open can be computed. AFDD has been on a downward trend since 1976, during which the variance of the time series has also increased (Levene Test = 6.2 at a 98% confidence level). The one failure to date (2006: circled in red), which cost the mine operators millions of dollars in airfreight costs, was driven by both increasing variance and downward trend 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 lognormality. 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 Bbest guess^of the potential costs that will be Realized (black) and expected (blue, green, and red) cost curves for the ice road system. Here, the costs are those that would be incurred above and beyond construction and operation expenses. The average operational need is 180,000 tonnes delivered to the mines, which requires the road to be open 33 days, or a supply-to-need ratio of 1 (the OPT). The solid blue line is the expected cost curve for the current climate variance computed using a statistical transformation method; the dashed blue line was computed using the modified B-S formula (Supplements 1 and 2). Between 1943 and 2012, the average ice road-open season dropped by 50 days while the variance increased, raising the expected cost by $74 million (black arrows). With an increase in variance of 1.5× (assumed for purposes of illustration), that figure rises to $109 million (the jump from blue to red cost curves along the 2012 season length vertical dotted line) 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: 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. 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. 4 Example II: California's water supply: more storage?
In this second example, we apply our method to estimate additional water costs in a variable and changing climate. Our particular focus is the system of rivers, dams, and reservoirs that sustain millions of people living in California, as well as a multi-billion dollar agro-business. The majority of the water in this system comes from the Sierra Nevada snow pack (California 2016;Messerli et al. 2004), which accumulates through the winter and then melts slowly in spring and summer, a natural storage facility (Messerli et al. 2004). The system is in a decadelong drought (Diffenbaugh et al. 2015; University of California 2016), primarily due to record low snowfall, that has caused agricultural losses exceeding $2.7 billion (Howitt et al. 2014(Howitt et al. , 2015. The ecosystem service is run-off and it can be measured directly, though the underlying climate metrics are winter precipitation and spring/summer heating, the latter controlling the rate of melt (Hanak et al. 2011).
The Merced River is a good representative of the larger water system. It arises in Yosemite Valley and supplies water to the Merced Irrigation District. 5 The mean annual run-off above all impoundments over the past 98 years has been 557 million cubic meters per year (cmy) and has shown little trend, but year-to-year variability is almost half of the mean (249 cmy) (Fig. 4). We use the detrended time series (mean of 600; standard deviation of 248 cmy) and assume the irrigation district needs a bit more water than the river supplies on average: 650 million cmy (the OPT), and that when there is a shortfall, groundwater will be purchased at $0.73 m −3 (Somer 2014). Trans-district water purchasing has been widespread during the drought, though in reality, if the purchase price of water was to increase too much, farmers have the option of letting their land go fallow.
Mapping this system into an option framework, we examine first the hypothetical case in which there is no storage (i.e., no reservoirs) 6 , after which we examine the effect of partial storage and examine the difference. We again define the strike price (or OPT) as the ratio of the cumulative annual discharge to the amount of water needed by the irrigation district. At ratios less than 1.0, water must be purchased. Expected water costs for an undammed river (Fig. 5, black line with circles) and an OPT of 0.92 ($600 million/$650 million) are over $95 million (Fig. 6). As before, lateral movement (left or right) on the graph corresponds with the impact of trend on the expected cost, while changes in variance move the cost up or down, jumping from one cost curve to another. A 50% reduction in standard deviation (easily achieved by building a dam) reduces the expected water costs to $59 million, so it is easy to see why this river was dammed as early as 1926 (Old Exchequer Dam).
We now model the system with storage. The New Exchequer Dam on the Merced River opened in 1967. The dam holds up to 1.27 billion cubic meters of water, but since it is difficult to pump water when the reservoir level drops too low, we assume the useable volume is about 1 billion cubic meters. Once half full, we assume that only 100 million cubic meters of water can be added per year due to requirements for maintaining adequate capacity for flood mitigation. Including the effect of the reservoir storage and these filling rules, we created an altered time series (Fig. 5, red line and pink filled area), then from this we recalculated the expected costs (Fig. 6). Not surprisingly, the existence of storage substantially changes the discharge time series (Fig. 5), the variance, and produces a dramatic reduction in the expected costs for water purchase. The variance drops from 248 to 146 million cubic meters, which reduces the expected costs by $46 million ($95-$49 million). For the more than 40 other water reservoirs in California (Reservoirs 2016), the same order of valuations would likely be true, conservatively suggesting an overall annual monetary value of water storage to the State of California of at least $2 billion 7 .
For illustration purposes above, we assumed a supply-to-need ratio slightly less than one, and in reality, this is often the case. Need often overtakes supply. For example, within 30 years of the original dam being built on the Merced River (1926), agricultural demand for water had grown to the point where the dam capacity was no longer enough, and the farmers (successfully) advocated for a bigger dam (Hanak et al. 2011).  2014)). The discharge record has a small (15%) increasing trend (green line) that has been removed to produce a detrended time series (black line with circles). The red line with pink fill represents the estimated discharge downstream of the New Exchequer Dam, which in most years can supply the needed 650 cmy. Flood mitigation regulations require that once the reservoir reaches 50% full, water can no longer be impounded, so in exceptionally wet years (e.g., 1983) the discharge exceeds 650 million cmy. The discharge drops below 650 million cmy when the reservoir is so low that water can no longer be withdrawn for irrigation Fig. 6 Realized (black) and expected (blue, green and red) water costs for the Merced River assuming no reservoir. The average operational need (threshold) is assumed to be 650 million cubic meters of water, equivalent to a supply-to-need ratio of 0.92, which in 2014 implies an expected cost of $95 million if there was no dam. With the New Exchequer Dam in place, the expected cost curve shifts downward (indicated by the black arrows) by over $46 million (lower blue line). The results were computed using both a statistical transformation method and the modified B-S formula (Supplement 2), and are nearly identical 5 Harnessing and expanding the power of the option-like approach The basic requirements for harnessing the power of the option-like approach are (1) recognizing that there is an asymmetrical structure in the socio-climatic system, (2) identifying the operational threshold (OPT), (3) understanding and correctly parameterizing the climate metric that affects operations, and (4) finding or developing a climatic time series of sufficient length to obtain the variance.
In the two examples above, we had sufficient data to delineate p(S) and could therefore apply the generalized method, as well as the modified B-S formula, to compute expected costs (Supplement 2), but this is not always the case. Fortunately, the B-S formula is not only fast and easy to use but does not require a long climate time series. For example, in the generalized method, a measured (or simulated) series as long as 50 years in length may be needed, and it may still underestimate the full probability space and variance because it lacks the more extreme excursions. This can lead to an underestimation of expected costs. By using the B-S formula, we are able to evaluate costs as if we had an infinite number of observations, reducing the problem of cost underestimation. Based on the examples we have investigated, means and variances computed from a few decades of data may be sufficient. Even if these values are not well established, they can be varied across a range to explore the set of likely outcomes. In the advent of having only a very limited data record, using the B-S formula is the easier and more efficient method.
Three additional socio-climatic business systems suggest where the method might be applied (see also the table in Supplement 3): Agricultural insurance in Kenya Small-scale farmers in Kenya have long borne the risk of adverse weather, but there is a program (Osgood et al. 2007) that allows them to purchase crop insurance. In this program, payouts are made automatically if rainfall is below a threshold level during the growing season. Similar insurance is available for larger scale farming in the US and other places around the globe. For these programs to remain viable, premiums need to affordable yet still pay for the cost of the program. Climate trends, if present, can bedevil such a program. Over time, if the trend is to drier weather, the program will become more expensive for the insurer, while if the trend is to wetter weather, the farmers may have less incentive to purchase insurance unless the price drops. As noted by Tucker (1997), these Binsurance policies are very much like put options^and the methods described here could be used to assess what the fair price of insurance might be in particular setting or to explore Bwhat ifs^for future rainfall trends and variance.
Ice roads vs. barges in Alaska's north slope oil patch The Prudhoe Bay area of Alaska's north slope is home to many smaller oil fields. Some, like Point Thomson, are beyond the limits of the all-season road network, so they are accessed in winter by an ice road constructed by spraying water on tundra (in contrast to the frozen lakes in NT) and in summer by tug and barge when sea ice is gone. Freezing weather and sea ice conditions affect the viability and duration of both modes of transport. Assuming that the barge costs and ice road costs are not the same, the B-S formula method could be used to explore at what point barge transport should be used vs. tundra ice roads.
Shipping through the northern sea route The Northern Sea Route (NSR) connects the Pacific and the Atlantic by way of the Arctic Ocean to the north of Russia. In recent years, commercial interest in using this route has risen. A major challenge, however, is uncertainty surrounding the opening date for ship transits (i.e., when sea ice concentrations drop below a certain percentage) as well as the length of the shipping season (how long sea ice concentrations remain low). The viability of the NSR could be enhanced by constructing icestrengthened cargo ships. Option pricing could be used to calculate the expected costs of using the NSR vs. alternative routes such as the Suez Canal and also the merits of investing in ice-strengthened vessels as a form of hedging.
In conclusion, we believe there is a large range of problems for which the method we describe can provide actionable information. Researchers analyzing socio-climatic systems with a threshold and an asymmetric payoff structure will find that mimicking these systems using a general option valuation approach or the B-S formula provides useful insights into the impact of trend vs. variance. The use of the B-S formula in particular allows for relatively quick estimates (within minutes) of how expected costs would change for various changes in either the underlying climate or the underlying business or socio-political need. Perhaps most importantly, there is vast literature on applications of option valuation and the B-S formula in finance that could help advance climate analyses related to ecosystem services.

Supplement 1: Diamond Mine Ice Roads: Computing Expected Costs
The Northwest Territories (NT) Transportation Department builds, maintains, and oversees about 2,200 km of ice roads each winter that service remote communities (Fig. S1-1). In addition, commercial ice roads are built to access remote mines, including the diamond mines discussed in the text. The ice road to the diamond mines is built by a joint venture company (http://www.jvtcwinterroad.ca/jvwr/), a partnership between de Beers, BHP, and Rio Tinto mining companies. The commercial ice road is wider and carries heavier loads than its government counterparts, but is still regulated by the NT government. Over 80% of both types of ice roads run on frozen lakes. The commercial ice road achieved a certain amount of notoriety when it appeared in Season 1 of the popular reality TV show, Ice Road Truckers. Only a limited record of season open length is available for the commercial ice road, so we had to develop a longer climate time series based on weather records. We did so by modeling the freezing of lake ice, with and without snow plowing, from which we then computed ice road opening and closing dates for the period 1943-2012. For the ice thickening part of the model we used the parameterized ice freezing equation developed by (Pfirman et al., 2004), in which the daily increase in ice thickness in meters (Dh ice ) is controlled by the air temperature (T air , °C) and the snow and ice thickness: where L freezing is the latent heat of freezing of freshwater (3 X 10 8 J m -3 ), h snow is snow depth, and h ice is the existing ice thickness. Daily increments of ice thickening computed using Equation [S1-1] were accumulated to produce a running total of ice thickness through the winter season. Air temperature and snow depth data came from the Yellowknife airport, available from http://climate.weather.gc.ca/index_e.htm.
To tune and test Equation [S2-1] we used paired ice thickness and snow depth values collected by the Canadian government on the lake ice of Back Bay near Yellowknife (http://www.ec.gc.ca/glaces-ice/). When tuning the model, the thermal conductivity of fresh water ice (k ice ), which does not vary much, was held fixed at 2.3 W m -1 K -1 (Slack, 1980), while the thermal conductivity of the snow (k snow ) was adjusted until modeled ice thickness matched observed thickness. Snow thermal conductivity varies with snow density and texture (Sturm et al., 1997), data not available for the ice roads. Changing snow conditions result in year-to-year variations in k snow . The tuning results (Table S1-1) indicate an average value of 0.216 W m -1 K -1 , which corresponds with a bulk snow density of 0.38 g/cm 3 . That density is typical of a moderately hard wind slab, a realistic type of snow cover for Arctic lakes (Sturm and Liston, 2003). Using this average value and Equation [S1-1] produced satisfactory agreement between observed and modeled ice thicknesses for all 37 cases in which we had both snow depth and ice thickness ( Fig. S1-2). Removal of snow by plowing is the only practical way to create thicker ice over a long route. We simulated plowing by essentially reducing the snow depth to zero. This could not happen until the ice was thick enough through natural growth to support a plow vehicle. Typically, a Bombardier BR-275 Snocat with plow is used for this work. It has a curb weight of approximately 8000 kg and requires a minimum of 0.45 m of ice for support. We applied a safety margin (35%) to account for the fact that the plowing would traverse a long route with uncertain ice conditions in some locations.
Consequently, in our model plowing begins when there is 61 cm of ice. Further, since plowing a 450-km long ice road (even with multiple plow vehicles) takes a finite amount of time, we simulated a delay in the plowing by reducing the snow depth to zero linearly over a 7-day period. Model runs with shorter (2 days) and longer (10 days) delay times had only a minimal impact on the ice road opening date. Mathematically, the delayed reduction in snow depth was achieved by increasing the snow thermal conductivity until it was ten times the normal snow value. In the model the snow was plowed only once and then it was assumed that any additional snow was removed instantaneously. This assumption is realistic because once the road is in place and cleared of initial snow, snow removal by casting plow trucks can be started. These can clear snow much more rapidly than the initial pass done by the Snocats.
We based the road opening criterion on discussions we held with the operators of the Tibbitt to Contwoyto Joint Venture Winter Road (TCWR) and the NT government officials who operate the community ice roads. In point of fact, these ice roads "open" when the ice is thick enough to support a given vehicle load (P[kg]) as predicted by the Gold formula (Gold, 1971): where A is a safety factor set to 4 for government ice roads and 6 for the commercial ice roads, and h is the ice thickness (cm) from Equation [S1-1]. A higher value of A is allowable for the commercial road because the operators can control the speed, load, and frequency of transits, while their government counterparts cannot. Moderate-sized trucks, cars and other light vehicles can begin to transit the ice road immediately after it is plowed, but they can only carry limited cargo, so for purposes of computing load-carrying capacity on the commercial road, the opening is when a truck (semi or a Super-B, which is slightly larger than a semi) can transit the road. In most model years the opening for these two types of trucks is separated by just a few days. For simplicity, we assumed haulage began when the ice reached the thickness required for Super-B (85 cm for A=6: Table S1-2). The difference in accumulated carrying capacity proved negligible because the larger cargo capacity of the Super-B quickly offset any loss in road-open time due to the later Super-B opening date.  Figure S1-3 shows a typical model result with plowing. At the start of plowing the worked ice thickness (red) begins to exceed the natural ice thickness (gold). This results in a Super-B opening date 28 days earlier than would have been the case without plowing and 0.48 m more ice than would have occurred naturally (essential work effort harvesting the ecosystem service sooner). For the 70 years for which we ran the model the ice due to plowing was on average 0.32 m thicker, an increase of 30% over natural ice (Fig. S1-3). Figure S1-

Figure S1-4: Model results showing the percent increase in ice thickness due to removal of snow by plowing. The effectiveness of plowing increases in winters when natural ice thickening (xaxis) is reduced.
Modeling the date the ice road closed was more difficult because "official" closure dates reflect when all material has been hauled up the road, not when the road is physically incapable of being used. The official closure can occur weeks before the latter condition is realized. Sunny, abovefreezing weather produces snow rutting, stuck trucks, and damage to the underlying ground on the portages. This is generally what causes the ice road to close physically. The softening of the snow in a gully just a few tens of meters wide is sufficient to close the whole road. These bad spots arise from vagaries in snow and ice conditions, as well as local weather anomalies, but we had no choice but to model the system using weather data from Yellowknife as a proxy for conditions across the whole road. The match between model results and observed closing dates was limited.
To better simulate closing dates, we examined the closure date of the 7-km long ice road that connects Yellowknife to Dettah, a small aboriginal community across Yellowknife Bay from the capital. Unlike the other ice roads, this road is used by cars and trucks right up until it physically can no longer be used. We computed the date of the start of the thaw based on Yellowknife airport weather data (defined as 3 consecutive days of above-freezing temperatures), then compared the actual road closure date to this thaw metric. As expected, the road closed well before the onset of consistent above-freezing temperatures due to solar effects and local warm pockets. The results (Fig. S1-5) show that the road closed on average 13 days before the weather records indicated a consistent thaw. We then applied this offset to the Yellowknife records for 1943-2012 to produce a time series of closing dates. Combining the opening and closing dates, we computed the Super-B season length, 1943-2012 (Table S1-3). In the simulated season length time series (1943-2012) we found considerable year-to-year variability and a strong secular trend driven by recent higher winter temperatures and more, and earlier, snowfall ( Fig. S1-6). We de-trended the data, then used the residuals (top, Fig. S1-6) to create a trend-free time series. In order to preclude negative season lengths and to match current conditions, we offset the residuals by an amount (+47.3 days) sufficient that the model season length for 2012 equaled the observed season length.
In the simulated time series for road open and closing dates, the mean number of days open dropped from 100 (in 1943) to 50 (in 2012). We removed this trend (-0.76 days per year) by fitting a line to the data then using the residuals from the fit. To these residuals we added a fixed offset of 47.3 days to produce a de-trended but realistic season length record. The offset is the amount needed to match the observed season length of 2012, which allows us to check our calculations against the 10-year observed record. De-trending reduced the standard deviation by 3.3 days (since some of the variance was due to the trend). We then used this offset and detrended series for our calculations. We note that the de-trended series had 9 (out of 70) years (13%) in which there was a "negative" season length. These were truncated to zero, which introduced an insignificant shift in the computed series mean (49.3) and reduced the standard deviation (30.3) by a further 10%, making the resultant series realistic but conservative for variance.
We tested Equation S1-3 using operational data (http://www.jvtcwinterroad.ca/jvwr/) compiled by the operators. The average number of trips/season over the ten years of record was 6,606 and the average amount hauled was 216,643 tonnes/year. Estimates using Equation S2-3 came within 66 trips and 4.7 metric tonnes of these values. We then set the essential amount that had to be hauled to the mines (the need) at 180,000 tonnes/year, a conservative value approximately equal to the amount that was hauled at the height of the recession when worldwide demand for diamonds was at a recent low.
It was assumed that any shortfall below 180,000 tonnes would have to be flown as air cargo. The marginal air cargo costs over driving were computed from the bulk air cargo rate ($2155/tonne), less the cost of truck fuel and driver ($60/tonne), or $2,096/tonne. The total air cargo cost (A t ) was then: If all cargo had to be shipped by air the total bill would be $387,900,000 less the cost of the ice road ($18,600,000), or $369,900,000 net, and absurdly high amount. If all cargo could be shipped by Super-B without resorting to air cargo, then the cost would be the $18,600,000 road construction/maintenance cost plus $10,800,000 (for fuel and driver), or a total of $29,400,000. These end-member transportation costs form a ratio of 13:1 (air cargo to trucking). In Figure S1-7 the red squares are the realized marginal air cargo costs for each year of the 70-years of simulated climate data (air cargo costs less savings in driving costs). They are marginal realized costs because they reflect the amount actually spent on cargo transport (less the savings on not paying drivers or fuel), but they are not very useful for decision-making because they cannot be known until after the haulage season is over.
The expected costs are more useful in this regard. For the same 70-year time series, the expected air cargo marginal costs, which are essentially the probable additional costs incurred, can be computed by assuming each of the 70 realizations has equal probability of occurrence and then averaging the costs for all of them. In this case, the amount is $74 million, shown by the dark blue diamond in Figure S1-7. For our time series, in 20 of the 70 seasons (29%), some air cargo costs were realized, while in 9 of these seasons (13%), the ice road never opened at all for Super-Bs so the full marginal cost was realized.
Figure S1-7: Realized (red) and expected (blue) cost functions for the ice road system where the required tonnage at the mines is 180,000 tonnes, as computed for the time series data using the method described in Supplement 1.
We can vary the mean open season length of the time series in order to examine the impact of a warming or cooling climate change on the expected costs (see Supplement 2). By subtracting or adding a fixed amount (±50, ±100 and so on) from each yearly value, we create a new time series with a different means.
It is important to note that at this point we do not have an option per se, even though the equation above could be, and is used, in general option valuation approaches. As described below, for options, the max [0, (K-S)] term arises due to the choice of whether or not to exercise the option, while in this case, the max operator arises depending on whether or not nature provides sufficient bounty in the form of run-off so that there is no need to supplement water through purchase. Therefore, while the payoff function of the water supply problem mimics a payoff function of an option, in fact there is no strategic choice. The system requires a constant supply of 650 million cubic meters, and by definition there is no choice of whether or not to supply the water, and hence no option. Instead, there is an asymmetric cost structure of supplying water, which is deterministic once the state of the climate is realized. As is noted in more detail below, while this problem looks like it might be a real option problem, real options always contain a strategic choice while in this problem there is no choice at all.
To compute expected costs (mimicking put values) at other supply-to-need ratios, we performed a linear transformation of the original de-trended distribution (X) to create a new distribution (X'): This transformation is similar to the conversion of a temperature series from Celsius (C) to Fahrenheit (F) (F = 32 + 9/5 • C). The process spreads out the data and shifts its mean, but the relative structure of the data remains the same, with all ordering and relative magnitude of spacing in the distribution preserved.
We started by adding (or subtracting) a fixed amount (a) to each value of the series, while holding b = 1. It was necessary to truncate to zero any values that were negative due to the subtraction (since there cannot be negative discharge). While the variance (or standard deviation) of the initial time series is not absolutely preserved during this procedure, the alteration tends to be small, and the procedure is accurate in that it removes the possibility of negative flows. For example, when we increased each annual discharge value by 50 million cubic meters (a=50,000,000; b=1), it reduced the number of years in which there is a shortfall in supply (and therefore an incurred water cost) by 5 years. By adding 50 million cubic meters to each observation in the de-trended series, we effectively created a new distribution, with a new mean (that is 50 million cubic meters higher) but the same variance. Even so, the probability for this new distribution at any given value S was now different from the previous one, and so the values and probabilities at any given S in Equation S2-1 have now changed and so Equation S2-1 needs to be re-run with the new data. Functionally, this means a new average of the value K-S for values below K, and zero otherwise. The expected cost due to this increase in discharge dropped to $74,440,921, and the supply-to-need ratio increased to 1.0 ({600 + 50 million}/650 million=1.0). By continuing to increment (or decrement) the initial de-trended time series (+100 million, +150 million, -50 million, -100 million, etc.) we were able create a complete new expected cost curve. We did this for supply-to-need ratio ratios ranging from 0.5 to 1.5.
In order to compute expected cost curves for higher or lower values of variance we then used the residuals we computed when de-trending the time series, multiplying each of these values by a factor, 0.5 and 1.5 in our case, to decrease or increase the variance. We then added back to each residual a fixed value of 600 million (the average of 98 time series de-trended values) to produce a series that has the original mean but a higher or lower variance.
Mathematically the new distribution (Y) was computed from the original: where the mean was from the original de-trended time series, and b=1. We then altered the standard deviation of this new distribution (Y), which has mean of zero, to produce a second distribution (Z): where c = 0.5 or c = 1.5 for purposes of this paper.
Finally, we added back the mean, to create a new distribution with the same mean as the original but with a different standard deviation: The new distribution, Z, retains the structure of the original distribution, but with an increased or decreased variance. Again we truncated at zero any negative values. Using the new, transformed time series, we then repeated the procedure described in the preceding paragraphs, producing (in our case) two additional expected cost functions, one for a standard deviation that is half the original, and another that is 1.5 times the original (see Figure 6, main text for these curves at difference values of variance). Each of these cost functions is effectively an implementation of Equation S2-1 (the general option valuation approach) for each of these newly created distributions. Figure S2-1: Computing expected costs by transforming the initial distribution incrementally. In order to produce cost curves for alternate values of variance, the same procedure shown here is repeated after applying Equations S2-3 through S2-5.
The previous paragraphs described estimating the costs (which mimic put values) using what is effectively a general option valuation approach. In a seminal paper, Black and Scholes (1973) created a closed form solution for valuing options for log-normal distributions. Since their results provide a closed form solution that is computationally fast and easy to use, we next determined for these socio-climatic distributions the variance values that we would need to map back into the Black-Scholes (1973) option pricing formula (called hereafter the B-S formula) in order to use it to compute expected costs. Again, while these systems are not an option per se, the realized cost functions mimic the asymmetric payoff functions of a put or call option. In the same way as before, we averaged all possible ex post values to get a single number, and the B-S formula ultimately collapses the asymmetric distribution into a single value as wellthe value of the option. By appropriate parameterization and replacement, we can use the B-S formula to summarize the payoffs (or expected costs) of these asymmetric socio-climatic systems as well.
If the underlying distribution of S is log-normal, using the B-S formula will give a solution exactly equivalent to the more general option valuation approach, but since it is unlikely that socio-climate distributions are log-normal, below we demonstrate a heuristic we created that works well even if the underlying distribution is not log-normal.
The "option" at the heart of stock option is represented mathematically in the payoff function by max [0, S-K] where S is the current stock price and K is the strike price. While there is not choice in these socio-climatic systems, a similar max[0,S-K] arises, allowing us to use option valuation approaches such as the B-S formula (appropriately parameterized) to mimic the socio-climatic systems mathematically and therefore provide analogous results. Translated into a socio-climatic system, K becomes the threshold beyond which greater costs are incurred while utilizing the same free ecosystem service (or alternatively, beyond which the costs drop). S becomes the value of the climate metric that governs the way the system functions, and can either be the current value, or some prescribed value for which we need to know the expected costs. With S and K replaced in the B-S formula, it becomes a tool for rapidly summarizing the expected values of the asymmetric payoff function, i.e., we are using the formula by analogy to estimate expected costs.
More formally, the standard B-S formula prices the right (but not the obligation) to call a stock away from someone else at a specified time in the future at a specified price. It assumes returns on a stock are normally distributed with a mean expected return μ and a standard deviation σ. It describes the end result of the movement of stock prices over time and is parsimonious in that it only needs five inputs (S, K, T-t, r, and σ).
For a call option (C) the B-S formula is: where S, K, σ and μ are defined as above, T-t is the time remaining to exercise the option, r is the instantaneous risk-free-rate, and N( ) is the cumulative distribution function operator of the standard normal distribution N(0,1).
The price of a corresponding put option (P) is based on the put-call parity (Stoll, 1969): To use the B-S formula, we need to map the parameters of our socio-climatic system onto the B-S formula parameters. As indicated above, the strike price, K, becomes the operational tipping point (OPT), at a supply-to-need-ratio of 1.0. This parameter reflects the business or monetary cost side of the system. Past the tipping point, costs are incurred at some rate set by commodity, transportation, or other prices (these must be specified by the user). The climate side of the system mimics the stock price and its volatility. For the input for S, we use the mean of the time series itself. is just the standard deviation of the climate time series when the series is lognormal. If it is not, then we must find the standard deviation of an equivalent log-normal distribution with a similar mean value (S) such that it preserves the sum of the products of the probability of the outcome times the expected losses (S-K) of the socio-climate distribution: Note that the right hand side of Equations S2-1 and S2-8 are the same, i.e., both are the integral from zero to K of K-S times the probability of S in the socio-climate series.
The key is finding f, a factor used to adjust the standard deviation estimated from underlying socio-climate distribution so as to find a log-normal distribution where the integral between 0 and K gives the same value. If the socio-climate distribution is lognormal, then f equals 1. If not, finding the correct f can be done via a Monte Carlo simulation, by drawing 10,000 log-normally distributed random numbers using the formulas above and varying f until the average of the 10,000 random numbers and the average of the de-trended record are equal.
An alternate, faster heuristic method for finding the appropriate value of f is as follows: using the formula for μ, σ, and v above, find f such that the Black-Scholes Option Pricing formula gives the same answer for the value of a put option as the average value of the shortfall estimated from the socio-climate series (which was $94,852,747 in the case of the Merced River above). This factor can then be used to find the value of σ to be used as an input into Black-Scholes for other values of S, for example, to derive the Black-Scholes curves quickly for different possible climate means.
With this one-to-one parameter mapping we can then feed our values into any standard B-S formula model (widely available on spreadsheets and on-line) to compute the put price, or in our system, a cost function. This can be done without any reference to the transformation method described above. While theoretically the B-S formula requires a log-normal distribution, our tests ( Fig. S2-2), which were applied to non-log normal distributions using the technique above, resulted in close agreements between the statistically transformed results and the B-S formula computations, though we have not tested this agreement in an exhaustive fashion. Again, while we used the B-S formula with certain modified parameters to get these values, the underlying socio-climate problem did not have any underlying choice, and was therefore not a true option. While the B-S formula was applied to a "real" as opposed to "financial" problem, this was not a strategic choice issue. The application of this technique to socio-climate problems is therefore notably different from standard real option problems, such as the choice to delay drilling in an oil field (see Chapter 22.11 in Ross et al. (2008), or the choice of whether or not to abandon a mine as in Chapter 23.4 of the same text). Without choice, these are not "options" per se, although the payoff structures, and therefore the mathematical valuation techniques, are similar.
The decision whether or not to keep a mine open or whether or not to build a dam is strategic, and therefore a real option. Once that choice is made, however, the cost of supplying the mine or the water to the ultimate user must be undertaken, yet has an asymmetric payoff that mimics an option without actually being an option. The functions are mathematically indistinguishable, and so option pricing methods can be used to value these payoffs. 1 Ultimately, it is the combination climate variability and a specific human need that induces the asymmetry in an expected cost function to mimic an option, and since it mimics an option payoff, it turns out it can be valued as if it were an option, even though it is not an option since there is no 'choice'. While Black-Scholes is only one such option valuation technique and requires certain assumptions, we show that we can relax those assumptions and by appropriate adjustments mimic reasonably closely the estimated value of the option using other methods. By showing that Black-Scholes will give a similar answer to more direct but more complex estimation methods in a system that is non-linear but without choice, we demonstrate that the system that has option-like payoffs does in fact act like different systems that are true options. It also potentially opens a vast area of financial research for use in climate studies.