We introduce a new financial weather derivative—a drought option contract—designed to protect agricultural producers from potential income loss due to agricultural drought. The contract is based on an index that reflects the severity of drought over a long period. By modeling temperature and precipitation, we price a hypothetical drought contract based on data from the Jinan climate station located in a dry region of China.
Since the dawn of agriculture, farmers have been vulnerable to the effects of drought. These effects include not only any immediate financial losses that result from reduced crop production, but also the adverse effects that those losses may have on their operational viability going forward.
Historically, the only financial instruments available to mitigate the risks associated with drought were crop insurance contracts. Those contracts have the obvious advantage of minimizing the financial losses incurred by farmers, thus stabilizing their operations. But because the marginal benefits of farmers’ crop damage-mitigating strategies are offset by reductions in the expected value of the insurance claim, the private benefits they realize from those efforts will fall short of the social benefits. From a social perspective, the resulting moral hazard then leads to an inefficiently low level of damage-mitigating effort and an inefficiently high level of realized crop loss. Moreover, because of drought’s high positive spatial correlation, regional insurers may themselves have difficulty meeting their obligations in the event of drought, thus undermining the ability of insurance contracts to reduce the drought risks farmers face.
By the mid-1990s, however, new financial contracts known as weather derivatives had emerged as a way of reducing particular weather-related risks, including those resulting from uncertain temperature, precipitation, and frost. Because the payoffs of these derivatives depend only on weather variables over which the farmer has no control, these instruments have the advantage of sustaining the damage-mitigation incentives that crop insurance contracts may undermine, thus mitigating real crop losses [
Weather derivatives are very similar to traditional derivatives on financial assets. But because the weather indices they depend on have no underlying measurable value [
In this paper, we propose, model, and price a weather derivative, which we call a “drought option,” that is based not only on multiple weather factors (i.e., precipitation and temperature), but also on an agricultural drought index that depends on the particular crop growth being hedged as a way of reducing this basis risk.
The balance of the paper is organized as follows. In Section
We model our derivative contract as a European put option on an agricultural drought index,
Because no-arbitrage pricing methods are unsuitable, we use the equilibrium approach and price the option as the present value of the expectation of its payoff [
To simplify the problem, we will assume that
Because our interest lies in developing a tool to help farmers hedge against the risks of reduced agricultural yield, our approach will utilize the Reconnaissance Drought Index (
Potential evapotranspiration is difficult to measure because it depends on the type of plants, the type of soil, and the climatic conditions. For practical purposes, we will therefore use the
This gives us a new
To estimate the
We now wish to calculate possible values of our
For Historical Burn Analysis, the basic assumption is that the historical data reasonably approximate future scenarios, allowing the price of the option to be calculated from data we already have. From (
Our Index Value Simulation approach involves finding the best fit distribution for index values calculated using the Historical Burn Analysis approach and then sampling this distribution to produce possible future values of the index. The option price will then be calculated using (
Mean reversion stochastic processes are used to simulate the daily mean temperature and the speed of monthly rainfall over the entire year. The period’s rainfall and evapotranspiration are then obtained by summing the related simulated values. Finally, the option payoff is computed based on the simulation where the average discounted payoff is used as the option price.
Although based on (
Average daily temperature is, of course, subject to seasonal changes. For one specific day in each year, however, we will assume that the temperature fluctuates around some mean value, which means we can choose a mean reversion stochastic process to simulate the behavior of daily temperature [
We use the following Stochastic Differential Equation (SDE) to model daily temperature:
To account for local warming trends observed in the data,
We then estimate the parameters in (
Estimates of
The term
As we see,
If we discretize (
To estimate the mean reversion parameter
To then solve (
Because it is difficult to use the integral solution in (
While precipitation does not behave continuously, the speed of precipitation can be assumed to change continuously in time. When rain starts, the speed of precipitation increases continuously from zero to its peak and decrease continuously to zero where it remains at zero until it rains again. In this formulation, the total amount of precipitation can be obtained by integrating the speed of precipitation over the period of interest, as given by the formula:
The monthly rainfall, then, is essentially the monthly mean speed of precipitation with units (mm/m2·month). In order to simulate monthly rainfall, it will be sufficient to simulate the speed of rainfall using a mean reversion process.
In the previous model of monthly temperature, diffusion was modeled as a piecewise function
The data we will utilize are for the Jinan climate station in eastern China, obtained from the China Meteorological Data Sharing Service System (
Figure
The parameters in
We now wish to find an unbiased estimator of the mean reversion parameter
The diffusion parameters can be estimated by first squaring equation (
Mean speed of monthly rain over 56 years.
After the parameter estimation, based on the implicit Milstein method, the monthly precipitation can be simulated by the following:
For both temperature and precipitation models, the distance between the average of historical records and the simulated mean curve can also be simulated with another sinusoidal function like (
Figures
Simulated daily temperature comparison.
Simulated mean speed of rain comparison.
Since
If
If we can find a numerical method to solve this model which has an eternal life, as long as the initial value of historical precipitation is positive, we can make sure that all the following simulated points are positive. This is another reason to choose the implicit Milstein method to do the simulation [
For the mean reversion process given by
Compared to (
We now use the daily data from Jinan station with
Simulated speed of monthly rain in Jinan.
Therefore, in order to ensure positivity,
Continuing with our use of the Jinan data as an example, the results are shown in Figure
Adjusted simulated speed of monthly Rain in Jinan.
To use the new Change ( After calculating all the simulated points, we need to change the whole list back down to the original level:
Using temperature and precipitation data from the Jinan climate station, and the curve of consumptive crop coefficient based on McGuinness and Bordne [
Option price comparison for months 1–12.
1951–2006 | 1967–2006 | 1977–2006 | 1987–2006 | |
---|---|---|---|---|
Historical Burn Analysis | 0.1764 | 0.1813 | 0.1799 | 0.1561 |
Index Value—Weib. Distribution | 0.1769 | 0.1816 | 0.1805 | 0.1558 |
Stochastic Simulation | 0.1666 | 0.1782 | 0.1739 | 0.1502 |
Option price comparison for months 4–8.
1951–2006 | 1967–2006 | 1977–2006 | 1987–2006 | |
---|---|---|---|---|
Historical Burn Analysis | 0.1872 | 0.1877 | 0.1797 | 0.1453 |
Index Value—Ext. Val. Distribution | 0.1897 | 0.1931 | 0.1843 | 0.1535 |
Stochastic Simulation | 0.1846 | 0.1988 | 0.1873 | 0.1584 |
Option price comparison for months 4–6.
1951–2006 | 1967–2006 | 1977–2006 | 1987–2006 | |
---|---|---|---|---|
Historical Burn Analysis | 0.1963 | 0.1927 | 0.1686 | 0.1505 |
Index Value—Ext. Val. Distribution | 0.1928 | 0.1897 | 0.1647 | 0.1522 |
Stochastic Simulation | 0.1616 | 0.1676 | 0.1520 | 0.1447 |
Comparing the option prices for the full 56-year span, we see that the results from the Historical Burn Analysis and the Index Value approaches are very similar to one another, while the Stochastic Simulation method yields different results. This difference is attributable to the inaccuracy of the simulation of the speed of monthly rain. The Stochastic Simulation method does, however, yield results that have the lowest variance across the four columns.
Looking at the 1987–2006 results in the last column in each of Tables
To simplify our analysis, we have ignored a number of financial factors. Future work could incorporate the relationship between a drought index and farmers’ profits, the possibility of changing interest rates, and the potential market price of risk [
In addition, a number of improvements could be made to the weather models we have used to price drought contracts. In particular, the price of the drought option clearly depends on the joint distribution of the temperatures and the precipitation at maturity. To simplify our analysis, we have simulated these two variables independently. In future work, a joint model could be developed. We have also not considered the location limitations of the climate models. In particular, the climate stations which collect precipitation data are typically located in large urban areas, far from the rural areas where farming is concentrated. Because the levels of precipitation might be quite different between even neighboring urban and rural areas, future work should incorporate this spatial basis risk.
M. Pollanen and K. Abdella are partially supported by NSERC Canada.