Do Weather Derivatives Mitigate the Revenue Risk of Farmers?—The Case of Tongliao, Inner Mongolia, China

Abstract

This research probes the potential of weather derivatives as tools for mitigating the variability of crop yields due to climatic uncertainties in China. Centered on Tongliao City in Inner Mongolia, the study exploits a long short-term memory (LSTM) network to dissect and simulate 32 years of local precipitation data, thereby achieving a simulation of high reliability. Further exploration through a multiple linear regression model confirms a marked positive relationship between rainfall amounts and maize yields. By combining precipitation put options and the total revenue function for farmers, mathematical derivations yield specific expressions for optimal trading quantities and risk hedging efficiency. The research findings show that, using an assumption of a maize price that is 3 CNY/kg, when farmers purchase around 6.22 precipitation put options they can achieve 67.9% risk hedging efficiency. This highlights the significant role of precipitation put options under specific conditions in reducing the risk of decreased maize yields due to reduced precipitation. However, in practical markets, variations in maize prices and the price change unit (λ) are inevitable. Through further analysis, this study reveals that as these factors change, the optimal trading quantity and hedging efficiency also undergo varying degrees of adjustment. The investigation lays a theoretical groundwork for the practical application and empirical validation of weather derivatives within China’s agrarian sector. However, the study underscores the necessity of a holistic approach to market dynamics to refine hedging strategies. Future decision making must integrate market fluctuations, and adopting transparent pricing mechanisms is critical for enhanced risk management and the advancement of sustainable agricultural practices.

1. Introduction

Weather factors play a crucial and unavoidable role in the global economy, with direct or indirect influences observed in four-fifths of global economic activities [1]. The economic ramifications of weather span across all sectors, and the sensitivity of different industries to weather factors can be characterized by the extent to which weather impacts sales, production, and costs [2,3]. Categorized by their varying degrees of impact, weather factors can be specifically classified into two groups as follows: disastrous and non-disastrous weather factors. Disastrous weather includes events like droughts, floods, and storms, which, despite having a lower probability of occurrence, can result in substantial losses. In contrast, non-disastrous weather refers to meteorological events with minor deviations from normal weather conditions. While these events may not lead to massive losses, their higher probability of occurrence still has a certain impact on production sectors. Compared with the industrial and service sectors, agriculture is more reliant on weather conditions and is thus more susceptible to weather fluctuations [3,4,5]. In recent years, climate change has become a paramount concern for the international community, particularly due to its severe threat to agriculture. The escalating temperatures, irregular precipitation, and increased frequency of extreme weather events all impact the growing seasons and yields of crops, thus introducing uncertainty to global food supplies. This poses a direct and urgent threat to the dietary security of billions of people worldwide. A report released by the Food and Agriculture Organization of the United Nations highlights that over the past 30 years, crop and livestock production losses due to disaster events have reached an estimated $3.8 trillion, equivalent to 5% of the global annual agricultural output. This figure encompasses losses resulting from extreme weather events such as droughts and floods as well as other non-disaster-related weather risks. The report further notes that the aforementioned weather risks disproportionately affecting developing countries, posing a risk to 15% of their agricultural production. The broad definition of general weather risks in the report may, however, result in the omission of the impact on certain sub-sectors [6]. Given the unique status of agriculture, governments worldwide place significant emphasis on mitigating agricultural weather risks. Agricultural insurance, acknowledged as a potent risk management tool, has found widespread adoption globally. However, the application of agricultural insurance is often tailored to address disastrous weather events, posing challenges in addressing non-disastrous weather. Despite the relatively modest impact of non-disastrous weather on agricultural production, its frequent occurrence requires attention to be paid to its specific effects. Weather derivatives, serving as a valuable complement to agricultural insurance, emerge as an effective solution.

As a potent risk management tool for non-disastrous weather risks, weather derivatives have witnessed substantial growth in the international financial markets in recent years. Their wide-ranging application spans industries including energy, agriculture, and tourism. Weather derivatives stand out for their enhanced flexibility in comparison with other weather risk management tools, providing valuable support to agricultural practitioners in mitigating the risks associated with deviations caused by weather factors. Having been introduced to the financial markets by the Chicago Mercantile Exchange, weather derivatives have achieved noteworthy development in the global financial realm [1]. In comparison with traditional agricultural insurance, weather derivatives offer a means to circumvent the ethical risks and adverse selection challenges encountered by conventional agricultural insurance [7]. Furthermore, they boast noteworthy advantages including lower costs and enhanced transparency [8]. The introduction of weather derivatives effectively addresses the management void in non-disastrous weather risks, contributing to the establishment of a comprehensive and cohesive agricultural risk management framework alongside agricultural insurance [9,10].

In comparison with advanced countries, China presently falls short in utilizing weather derivatives to mitigate risks in agricultural production. Up to now, there has been no practical implementation of weather derivatives in actual risk management. Given China’s status as a crucial market for agricultural production, there is a need for managing non-disastrous weather risks. This underscores the immense untapped market potential for weather derivatives in China. Given that China has not yet introduced weather derivatives, this study simulates a scenario of promoting weather derivatives in the Chinese market. It assumes that farmers purchase weather derivatives to explore whether these instruments can effectively reduce income volatility for farmers. The study also assesses the extent to which derivatives can mitigate income fluctuations. During the research process, it is necessary to fit the meteorological data of the study area and simulate the trends over a future period. To improve simulation effectiveness and draw on previous research, this study compares various weather simulation methods to determine the most suitable one. Ultimately, the study opts for the rapidly advancing neural network model as the simulation tool. The neural network model excels in handling intricate meteorological data and simulating future trends. It autonomously learns patterns and trends within the data, providing more precise simulation results. In contrast to traditional statistical models, neural network models possess greater flexibility and adaptability, enabling better management of nonlinear relationships and multidimensional data. The application of the neural network model enhances the accuracy and predictability in simulating weather data, thereby contributing to an improved understanding and prediction of future meteorological trends. Subsequently, the simulated meteorological data undergo analysis to predict potential fluctuations in agricultural production. The variance of farmers’ agricultural income is computed, and a variance reduction index is determined to evaluate the effectiveness of purchasing weather derivatives in hedging against crop yield volatility.

The subsequent sections of this paper are structured as follows. Section 2 provides a mechanism analysis, elucidating the mechanism through which farmers engaging in the purchase of weather derivatives can effectively hedge against risks. Theoretical evidence is presented to affirm that the application of weather derivatives indeed assists farmers in mitigating income risks. Moving onto Section 3, a comprehensive overview of materials and methods is presented, introducing the research area, data sources, and research methodologies employed in this study. A subsection within this section specifically details the models and methods that will subsequently be utilized in the paper. Section 4 is dedicated to presenting the results obtained from the models and methods discussed in Section 3. These results are then used to draw conclusions, affirming the positive impact of weather derivatives in reducing income risks for farmers. Section 5 delves into a thorough discussion and reflection on the limitations set forth in the preceding sections, validating the study’s feasibility and effectively analyzing the implications of the research conducted in this paper. Finally, in Section 6, the paper is concluded by summarizing its key findings. The insights obtained from the examination of weather derivative implementation in China are delineated, highlighting both the limitations of the current research and suggesting potential directions for future investigations.

2. Mechanism Analysis

Weather derivatives, an innovative branch of financial derivatives, encompass risk management functionalities akin to conventional financial derivatives. In traditional financial derivatives, specific commodities’ futures and spot prices are influenced by the same factors within the exact temporal and spatial dimensions, yielding similar price trends. By capitalizing on these price relationships across both markets, opposing directional trades are executed in future and spot markets. This strategy results in profits in one market while incurring losses in the other one, effectively mitigating risk.

Research has established that weather conditions have a pronounced effect on agricultural output. Jawid A and Wiemer C [11] utilized the Cobb–Douglas model to analyze how seasonal precipitation and annual temperature fluctuations affect agriculture in Afghanistan. Their research uncovered a non-linear relationship between climate factors and agricultural net income. Zhao Xiaofeng et al. [12] analyzed the meteorological data from Inner Mongolia from the past two years. They discovered that the eastern part of Inner Mongolia experienced several large-scale and severe cold weather events in winter. The combination of low temperatures, strong winds, and snowfall increased the cost of heating and insulation for greenhouse facilities, adversely affecting the growth of crops like greenhouse vegetables. The studies mentioned above demonstrate that even without catastrophic weather events, ordinary weather fluctuations can still adversely affect agricultural production, leading to reduced crop yields or increased production costs and thus impacting agricultural income. There is a genuine need for the effective management of non-catastrophic weather risks.

As mentioned, traditional risk management tools often fall short when adequately addressing the crop yield fluctuations resulting from non-catastrophic weather risks. Weather derivatives are designed based on weather indices, which are closely tied to actual weather conditions, and are highly flexible.

As shown in Figure 1, a farmer concerned about the possibility of unusual weather conditions adversely affecting crop yields can acquire corresponding weather derivatives. Suppose that unfavorable weather changes result in reduced crop production. In that case, the farmer can liquidate the weather derivatives in the financial market, thereby generating profits to offset the losses incurred due to diminished crop yields. If the weather behaves contrary to their expectations, the farmer will only suffer the contract fees associated with the derivatives, and these expenses are foreseeable. In other words, by purchasing weather derivatives, the farmer can confine the weather risk to a manageable range, thereby achieving the goal of stabilizing agricultural income. This approach not only aids farmers in reducing income uncertainty but also offers an effective means of addressing the potential impacts of weather fluctuations on agricultural economics. Furthermore, the flexibility and customizability of weather derivatives bestow upon farmers a natural and substantial advantage in addressing the diverse risk management needs faced by various individuals working in agriculture. Notably, the decreased or increased precipitation in the figure refers to non-disastrous weather fluctuations.

In theory, reducing farmers’ income volatility and enhancing their adaptability to unstable weather is feasible by purchasing weather derivatives. Weather derivatives, as a financial instrument, possess a range of advantages and disadvantages in comparison with other tools in agricultural risk management. In terms of their advantages, firstly, weather derivatives can offer precise protection against specific weather events, such as insufficient rainfall or temperature fluctuations, thus enabling agricultural practitioners to address specific meteorological risks more effectively. Secondly, these derivatives are highly customizable, allowing for personalized design based on the specific needs of agricultural production, thus enabling farmers to choose contracts that suit their particular farms and crops. Additionally, the weather derivative market is relatively liquid, allowing participants to trade contracts more flexibly to adjust their risk exposure. However, weather derivatives also come with certain drawbacks. Firstly, their complex financial instruments and contracts may pose a challenge to agricultural practitioners unfamiliar with financial markets as understanding and using these tools may require specialized knowledge. Secondly, compared with other agricultural risk management tools, weather derivatives may involve higher transaction and contract costs, potentially posing an economic burden on small-scale farms or farmers with limited resources. Lastly, the coverage of weather derivatives is limited to weather-related risks and cannot address other agricultural risks, such as market price fluctuations or disease outbreaks. Therefore, agricultural practitioners may need to comprehensively consider a variety of tools to achieve a more comprehensive agricultural risk management strategy. Given the advantages and disadvantages of weather derivatives, whether they can effectively mitigate the risk of crop yield fluctuations in practice requires further empirical research for validation.

Empirical studies, which involve analyzing historical weather data alongside crop yield data, can help uncover the actual effects of weather derivatives under different circumstances. This research can offer a more comprehensive understanding of the potential value of weather derivatives in agricultural risk management and their ability to reduce risk for farmers while increasing agricultural stability. Through empirical research, we can more accurately assess the benefits of weather derivatives as tools for agricultural risk management. This information can offer policymakers and agricultural practitioners more targeted recommendations for better utilizing this tool to hedge against weather-related risks and safeguard agricultural income.

3. Materials and Methods

3.1. Selecting the Study Area and Target Crops

Corn farming exhibits a solid sensitivity to alterations in precipitation and has distinct water requirements during different growth phases. In brief, the months from May to October play a vital role in corn growth concerning rainfall. When rainfall is deficient during these critical growth stages, even subsequent increases in precipitation may struggle to compensate for the decreased crop yields resulting from insufficient water supply during these critical periods.

As shown in Figure 2, the geographical coordinates of Tongliao are between 42°15′ to 45°59′ latitude and 119°14′ to 123°43′ longitude. Tongliao City, situated in the eastern region of the Inner Mongolia Autonomous Region, finds itself within one of the world’s top three golden corn cultivation belts, making it exceptionally suitable for corn farming. As a result, it has consistently been a significant corn production base in China. Corn cultivation plays a pivotal role in the agricultural production of Tongliao City. In 2022 alone, the city’s total planted area for grain crops reached 1.259 million hectares, with corn accounting for 1.1278 million hectares.

Tongliao City, situated in a continental monsoon climate, exhibits a characteristic variability of weather, including uneven precipitation distribution and significant temperature fluctuations. This meteorological profile positions Tongliao City as a typical representative of non-disastrous weather fluctuations. As a key corn production base, Tongliao often faces insufficient rainfall for regular corn cultivation, prompting the implementation of relatively advanced agricultural irrigation measures. However, as discussed in the preceding theoretical analysis, weather fluctuations lead to a reduction in corn yields, resulting in decreased agricultural income for local farmers. Despite the increased resilience of farmers with well-established irrigation facilities to the lack of precipitation, the scarcity of rainfall still increases irrigation costs, indirectly contributing to a decline in agricultural income. Agricultural practitioners routinely navigate the challenges presented by weather fluctuations to agricultural production, highlighting Tongliao City’s suitability as an ideal location for studying non-disastrous weather risk management.

As a result, this paper selects maize as the target crop and designates Tongliao City as the research site. This choice holds greater representativeness concerning weather risk management and agricultural production. The unique characteristics of Tongliao City and its agricultural background make it a crucial case for studying the practical application of weather derivatives in agricultural risk management. Maize cultivation, one of China’s top three staple crops, significantly represents Chinese agricultural production. Investigating maize cultivation in Tongliao City allows for a thorough exploration of how weather derivatives operate within real agricultural production contexts. This research can serve as a valuable reference for effective agricultural risk management in China’s future.

3.2. Research Data

The data used in this study are collected from the Brock Global Agricultural Meteorological Database, encompassing daily precipitation data for Tongliao City from 1 January 1990, to 31 December 2022, with measurements in millimeters. The Chicago Mercantile Exchange offers two primary precipitation weather derivatives as follows: the rainfall volume index (RVD) and the rainfall day count index (RLD). In light of the research aims, this article focuses on the accumulation of rainfall within the primary precipitation season in Tongliao City, which runs from May to October yearly. To more accurately depict the cyclical variations in precipitation in Tongliao City, this paper divides the precipitation data into three distinct time intervals as follows: January to April, May to October, and November to December. This categorization approach offers a more comprehensive insight into Tongliao City’s precipitation characteristics, further enhancing the research’s accuracy. The data needed for the subsequent OLS model, which include corn yield per hectare, rural electricity consumption, and total agricultural machinery power for Tongliao City from 2005 to 2021, have been sourced from the Tongliao Statistical Yearbook.

3.3. Research Methods

3.3.1. Construction of the Weather Derivative Hedging Weather Risk Model

In meteorological data forecasting, numerous methods can be divided into three categories of dynamic, statistical, and hybrid dynamic statistical methods [13]. Schepen [14] utilized hybrid dynamic statistical methods to predict seasonal rainfall in Australia. The study’s outcomes demonstrated that the combined approach of dynamical and statistical techniques yielded superior predictive accuracy compared with using either method in isolation. Mainland China experiences notable periodicity, correlation, uncertainty, and spatiotemporal heterogeneity in its rainfall patterns. Traditional rainfall prediction methods struggle with these complexities. However, Xu Nan Nan’s [15] research in 2021 demonstrates that neural networks can deliver excellent results in simulating these complex characteristics. After years of development, neural network methods have gained increasing recognition within the academic community and have evolved into numerous types of neural network models. Scholars have applied neural networks such as the backpropagation neural network (BP), convolutional neural network (CNN), and deep neural network (DNN) to a wide range of real-world prediction problems [16].

Long short-term memory (LSTM) is a feedback-based neural network model that presents an advanced algorithm, originally introduced to improve the architecture of the conventional recurrent neural network (RNN). Hochreiter [17] pointed out that the original LSTM (long short-term memory) algorithm was initially local in both time and space. The computational complexity at each step was O(1). This characteristic could lead to an unlimited increase in LSTM units, ultimately resulting in network instability. According to Gers’ research [18] in 2000, the pivotal enhancement to the initial neural network architecture was that of incorporating a forget gate. This innovation empowered the neural network to retain information from a considerably earlier timeframe. The outcome of this optimization is the prevalent use of the LSTM model in academic circles today. Greff [19] and numerous other scholars, including Luong [20], have attempted to discover neural network models with better predictive performance than the LSTM model. However, proposed models, like Vanilla-LSTM and GRU, lack the research evidence to demonstrate superior predictive performance compared with the LSTM model. Following comparative studies between LSTM models and other neural network models, researchers like Chung [21] discovered that LSTM models are better suited for processing long time series data. Given the nature of precipitation data as a typical long-time series, LSTM models exhibit distinct advantages when applied to meteorological research. The strength of LSTM models lies in their ability to efficiently capture and manage long-term dependencies within weather indices. They can store and consolidate historical weather patterns, enabling more precise future forecasts. Furthermore, these models exhibit multi-step prediction capabilities, making them highly suitable for forecasting weather trends. Shi [22] found that the LSTM model can achieve good results in precipitation forecasting based on its effective extraction of long-term weather trends. Xiao’s study [23] also underscores the significant applicability of the LSTM model in meteorological forecasting.

Past investigations have distinctly affirmed the remarkable strengths of the LSTM model in handling and forecasting long time series data. It has found widespread application in both theoretical analyses and practical scenarios. Many researchers have successfully utilized the LSTM model to predict weather factors like precipitation, and the results have shown its effectiveness. In this context, this study selects cumulative precipitation as the target variable, which, as a typical example of long time series data, is highly suitable for applying the LSTM model.

The main objective of this research is to simulate the risk hedging potential of precipitation options on the volatility of corn yield in Tongliao City. We require reliable precipitation forecasting to achieve this goal efficiently and accurately, enabling a better understanding and assessment of precipitation’s impact on agricultural output. This, in turn, will aid in offering feasible risk management strategies for farmers. As such, this paper employs the LSTM model to predict cumulative precipitation indices, providing a comprehensive understanding of the potential impact of precipitation variability on corn yield in Tongliao City and the effectiveness of precipitation options in mitigating agricultural risks. The selection of this research approach is grounded in the successful application of the LSTM model in long time series data forecasting. The aim is to provide substantial support for agricultural risk management in Tongliao City and offer valuable experience and methods for research in similar domains.

3.3.2. Precipitation Put Option Payoff Function

Considering that this research focuses on Tongliao City in the Inner Mongolia Autonomous Region, it is evident from historical precipitation data that Tongliao City typically experiences arid and semi-arid conditions, with infrequent instances of excessive rainfall. For agricultural production in Tongliao City, the relatively insufficient precipitation is of greater practical significance as it frequently leads to adverse fluctuations in maize yields. Therefore, through a simulated study of the application of precipitation put options, we can better understand and assess how to utilize this derivative to mitigate agricultural risks stemming from insufficient rainfall. This approach aligns with the actual conditions in Tongliao City, enhances the practical applicability of the research, and, consequently, provides farmers with more feasible risk management strategies. Within this context, the paper has chosen to use the cumulative precipitation put options as the weather derivative to be simulated for purchase.

The cumulative precipitation index is expressed as follows [24]:

$$RVD={\int }_{t_1}^{t_2}RV\left(t\right)dt$$

(1)

In the $RVD$ expression, $RV\left(t\right)$ signifies the cumulative precipitation at time $t$, while $t_1$ and $t_2$ denote the initiation and conclusion times for accumulating rainfall, respectively. Drawing inspiration from the research conducted by Li [25], we establish the profit function for cumulative precipitation put options as follows:

$$F\left(I_t\right)=\left\{\begin{array}{c}-\lambda , Strike-I_t\le 0\\ \lambda Strike-{\lambda I}_t-\lambda , Strike-I_t>0\end{array}\right.$$

(2)

Here, λ functions as a compensation factor (i.e., the unit for price changes). The supply and demand dynamics influence the value of λ in the derivatives market, and it is challenging to make a precise estimation in our simulation. It is important to note that λ only imparts a multiplier effect on the returns of the derivative. Therefore, we set λ to 1, converting the precipitation index into a monetary value. In this framework, $I_t$ represents the precipitation index while $Strike$ is used to denote the predetermined strike price for the put option, which is established as the historical mean of cumulative precipitation in Tongliao City from May to October.

3.3.3. Construction of the OLS Model for Corn Yield per Hectare

This paper establishes an ordinary least squares (OLS) model to analyze the relationship between corn yield per hectare and the RVD index. After consulting the research conducted by Zhang [26], this study section selects the corn yield per hectare in Tongliao City, rural electricity consumption, agricultural diesel usage, fertilizer consumption, pesticide usage, and cumulative rainfall (RVD) as the independent variables for constructing the OLS regression model as follows:

$$Y_t=a+b{RVD}_t+c{Pow}_t+d{Oil}_t+e{Fert}_t+f{Agchem}_t+{\epsilon }_{t }$$

(3)

In this context, $Y_t$ represents the corn yield per hectare (kg) in Tongliao City for the year t; ${RVD}_t$ stands for the cumulative precipitation index (mm) from May to October in Tongliao City for the year t; ${Pow}_t$ denotes the rural average electricity consumption (kWh/ha) in Tongliao City for the year t; ${Oil}_t$ signifies the average agricultural diesel usage (t/ha) in Tongliao City for the year t; ${Fert}_t$ stands for the average fertilizer usage (t/ha) in Tongliao City for the year t; ${Agchem}_{t }$ represents the average pesticide usage (t/ha) in Tongliao City for the year t; and ${\epsilon }_{t }$ denotes the random error term.

3.3.4. Revenue after Farmers Purchase Cumulative Rainfall Put Options

The aim of this study is to investigate the effectiveness of weather derivatives in hedging the risk of weather fluctuations on crop yields. Therefore, it is assumed that other risks faced by farmers, including changes in agricultural product prices, have already been adequately hedged and are thus considered in this study. Consequently, the revenue obtained by farmers after purchasing cumulative rainfall put options is established as follows:

$$R_t=P{Y}_t+NF\left(I_t\right)$$

(4)

Within this framework, $P$ represents the price per kilogram of corn (in CNY); $Y_t$ signifies the corn yield per hectare for year t (in kilograms); $N$ represents the quantity of cumulative rainfall put option contracts traded by farmers; and $F\left(I_t\right)$ denotes the profit function of the cumulative rainfall put options.

3.3.5. Minimum Variance Strategy

In this paper, an evaluation of the effectiveness of weather derivatives in hedging crop yield fluctuations is conducted using the minimum variance strategy. In this strategy, when the variance of the total revenue function obtained by farmers through the purchase of cumulative rainfall put options reaches its minimum, it signifies the lowest level of revenue volatility for farmers. Therefore, the study establishes the following relationship:

$$Var\left(R_t\right)=P^{2}Var\left[Y_t\right]+N^{2}Var\left[F\left(I_t\right)\right]+2PNCov\left(Y_t,F\left(I_t\right)\right)$$

(5)

Taking the derivative of $N$ in the above equation, the total revenue variance is minimized when the derivative equals zero, and the optimal trade quantity of $N$ is achieved. Therefore, the optimal trade quantity is as follows:

$$N=-\frac{PCov\left(Y_t,F\left(I_t\right)\right)}{Var\left[F\left(I_t\right)\right]}$$

(6)

Furthermore, the adoption of variance reduction indices is employed to evaluate the efficiency of derivatives in hedging weather risk:

$$VR=1-\frac{Var\left(R_t\right)}{Var\left(P{Y}_t\right)}$$

(7)

4. Results

4.1. Prediction of the RVD Using the LSTM Model

This paper utilized the daily precipitation data for Tongliao City from 1 January 1990, to 31 December 2022, in a time sequential order. As previously explained, these data are categorized into three temporal segments of January to April, May to October, and November to December for each year spanning from 1990 to 2022. During the preprocessing phase, 99 valid data points are obtained. To facilitate the adequate training and validation of the LSTM model, the conventions established in the other LSTM-related literature are follows, and 90% of the valid data is assigned to the training set. In comparison, the remaining 10% is designated for the testing set. This translates to 9 data points for testing, with the remaining 90 data points used for training the model.

In model construction, to extract the latent data features from the historical precipitation data to the greatest extent, this paper employs a two-layer long short-term memory (LSTM) network. Research on the number of neurons suggests a typical range from 50 to 200, and this paper selects 150 neurons through multiple simulations to find the optimal choice. Regarding the determination of the number of training iterations, training iterations typically span from 100 to 2000 given the data volume in this study and insights from related research. Following multiple simulations to identify the optimal value, this paper ultimately decides on 500 training iterations. After 500 training iterations, the LSTM model established in this paper converges, thereby indicating that it can make predictions based on the specified configurations and the available data.

To validate the predictive performance of the LSTM model, the paper plots the nine test values alongside the nine predicted values generated through the LSTM model in Figure 3 for comparison.

The comparison of test values and predicted values as shown in figure below.

Figure 3 provides a lucid representation clearly showing that the predictions generated through the LSTM model closely resemble the nine test values that were reserved. Whether it is due to the overall patterns or the specific numerical values, the concordance is striking. This underscores the exceptional fitting performance of the LSTM model and strong predictive accuracy.

4.2. OLS Model for Corn Yield per Hectare

4.2.1. The Results of the Model

The estimated results for the OLS model configuration of corn yield per hectare are shown in

Table 1. Results of the OLS Model for corn yield per hectare.

Variables	Model One	Model Two	Model Three	Model Four
Cumulative Precipitation	0.127 *** (3.31)	0.116 * (2.22)	0.108 * (1.99)	0.067 * (2.04)
Rural average electricity consumption	0.512 *** (3.93)	0.562 ** (2.65)	0.624 ** (2.75)	0.863 *** (6.06)
Average fertilizer usage		−0.060 (−0.31)	0.064 (0.26)	0.398 ** (2.45)
Average pesticide usage			−0.099 (−0.86)	−0.093 (−1.42)
Average agricultural diesel usage				−0.373 *** (−3.96)
Constant	5.623 *** (7.47)	5.85 *** (5.51)	4.988 ** (3.40)	3.782 *** (4.27)
Adj-R2	0.6126	0.5694	0.5547	0.856

Note: ***, **, * indicate significance levels at 1%, 5%, and 10%, respectively, with t-values in parentheses.

.

As indicated through the table above, the RVD (cumulative precipitation) significantly impacts corn yield per hectare. The adjusted R-squared values for all the models are relatively high, meaning that these models can effectively explain the variation in corn yield per hectare.

4.2.2. The Stationarity of Time Series Data

It is noteworthy that the core explanatory variable in the aforementioned time series models is the cumulative precipitation. To avoid spurious regression, it is imperative to verify the stationarity of the time series. Unit root tests were conducted on each variable of the model, revealing issues of non-stationarity in the constructed time series model. Consequently, differencing was applied, and all variables passed the unit root tests after first-order differencing. However, considering that the focal variable in this study is the cumulative precipitation, employing differencing methods for stationarity might result in the loss of information regarding the impact of cumulative precipitation on maize yield, potentially compromising the model’s performance. To mitigate this, residual predictions were conducted on the initially constructed model, and the stationarity of model residuals was verified. Combining the fact that each variable belongs to a first-order integrated process, it can be inferred that there exists a long-term cointegration relationship among the variables, mitigating the risk of spurious regression and effectively ameliorating the errors in parameter estimation caused by stationarity. Consequently, this study adheres to the initial model specification, constructing a model for maize yield and cumulative precipitation.

4.2.3. Robustness Testing

In the model examining the relationship between cumulative precipitation and maize yield, measures were taken to address heteroscedasticity through BP and White tests, thus yielding p-values of 0.864 and 0.4637, respectively. These outcomes suggest the robustness of the model. Following the consideration of other potential factors affecting maize yield, this study introduced evaporation as an alternative variable for a robustness check. The model results are shown in

Table 2. Results of robustness test.

Variables	Model One	Model Two	Model Three	Model Four
Evaporation	0.1284 *** (3.28)	0.1176 * (2.20)	0.109 * (1.98)	0.0689 * (2.07)
Rural average electricity consumption	0.5119 *** (3.91)	0.5630 ** (2.65)	0.6254 ** (2.75)	0.8632 *** (6.11)
Average fertilizer usage		−0.0616 (−0.32)	0.0660 (0.27)	0.4011 ** (2.48)
Average pesticide usage			−0.1009 (−0.87)	−0.0941 (−1.45)
Average agricultural diesel usage				−0.374 *** (−4.01)
Constant	5.2920 *** (6.42)	5.5429 *** (54.71)	4.6919 ** (3.04)	3.585 *** (3.93)
Adj-R2	0.6095	0.5661	0.5528	0.8581

Note: ***, **, * indicate significance levels at 1%, 5%, and 10%, respectively, with t-values in parentheses.

, in comparison with Table 1, there is no significant deviation in the coefficients and significance levels of the core explanatory variable and other control variables. The results from the two robustness tests indicate the resilience of the cumulative precipitation and maize yield model, thereby affirming their high reliability.

4.3. Analysis of Optimal Trading Quantity and Hedge Efficiency of Cumulative Rainfall Put Options on Corn Yield Volatility

This section of the study assumes that the cumulative rainfall put option contracts have a contract period from 1 May to 31 October. Looking at historical data, the average annual cumulative rain from May to October in Tongliao City from 1990 to 2022 is 526.38 mm, corresponding to the $\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{k}\mathrm{e}$ value in Equation (2).

A pattern of variations is identified both using the LSTM model and based on the historical precipitation data for Tongliao over the past 32 years. This pattern is then utilized to make 1000 backward predictions, resulting in 1000 different precipitation values, $I_t$. These values are subsequently input into the profit function (Equation (2)) of the rainfall put options, yielding derivative profits, $F\left(I_t\right)$. The variance of the profit function, $Var\left[F\left(I_t\right)\right]$, is calculated.

Following this, the predicted precipitation data are incorporated into the previously mentioned OLS model (Equation (3)) assuming that other variables remain at their 2021 levels. This process results in predictions for different corn yields, $Y_t$.

Based on the historical corn price in China, let us assume the corn price is p = 3 CNY/kg. We are substituting $P$, $F\left(I_t\right)$, $Var\left[F\left(I_t\right)\right]$, and $Y_t$ into Equation (6), and we find that the optimal quantity of weather derivatives for farmers to hedge against yield fluctuations is $N$ = 6.22 under the condition of a corn price of 3 CNY per kilogram in Tongliao. Therefore, Farmers achieve the most favorable risk mitigation and income stability by acquiring 6.22 units of weather derivatives per hectare.

By substituting $P$, $Y_t$, $F\left(I_t\right)$, and $N$ into Equations (4) and (7), the calculations are made to determine farmers’ total income, $R_t$, the variance of total income, $Var\left(R_t\right)$, and the variance of farmers’ agricultural income, $Var\left(P{Y}_t\right)$. The variance reduction index is determined to be $Vr$ = 0.679. This indicates that under a 3 CNY/kg corn price and the purchase of 6.22 units of cumulative rainfall put options per hectare, farmers achieve an optimal hedging efficiency of 67.9%. It demonstrates that purchasing cumulative rainfall put options can effectively hedge weather risk to a certain extent.

The findings of this study align with those of Yao [24], underscoring that farmers can effectively hedge income risks arising from general weather uncertainties under optimal hedging conditions. This reinforces the theoretical analysis presented in Section 2 of this paper regarding the efficacy of weather derivatives in mitigating income risks for farmers. Even in the presence of the potential for negative returns from derivatives, under the meteorological conditions in Tongliao, farmers’ adoption of weather derivatives proves instrumental in mitigating the impact of weather-related factors on income. This resonance with international financial market practices suggests that the implementation of weather derivatives in China can yield effective risk management outcomes. The validation of the feasibility of weather derivatives in China offers academic support for the prospective introduction of this financial instrument in the country.

5. Discussion

However, corn prices and price change units (λ) are unlikely to remain constant in the real market. Although this study adopts fixed numerical values to provide an intuitive illustration of the hedging efficacy of derivatives, this approach may not mirror the complex realities of the market. Therefore, the subsequent discussion in this paper explores the potential changes in research outcomes when both variables are allowed to vary.

5.1. The Impact of Corn Price on the Hedge Efficiency and Optimal Trading Quantity of Cumulative Rainfall Put Options

As previously discussed, when the corn price is 3 CNY per kilogram, each hectare of corn farmers buying 6.22 units of derivative contracts for cumulative rainfall put options can effectively hedge 67.9% of the yield fluctuation risk. Nevertheless, given the price volatility of corn, it is essential to investigate whether cumulative rainfall put options can sustain their risk mitigation effectiveness at various price levels. The chart below displays the price trend of corn in the Inner Mongolia Autonomous Region from 2007 to 2023.

Figure 4 shows that the corn price has fluctuated and risen from 1.47 CNY/kg to 2.72 CNY/kg. According to the average prediction line, the price still exhibits a continuing upward trend.

A more in-depth analysis of Equation (6) reveals that the variable P directly multiplies with other components, while the other features remain unaffected by $P$. This implies that $P$ acts as a multiplier, leading to an increase in $N$ as $P$ increases. However, changes in hedging efficiency are challenging to analyze through Equation (7) mathematically. To further confirm the derived results, this study assumes that corn prices fluctuate from 1 to 10 CNY/kg while keeping other parameters constant. The objective is to investigate how weather derivatives’ hedging efficiency and the optimal trading quantity change at different corn prices. The results are depicted in following Figure 5 and Figure 6. When corn prices fluctuate between 1 and 10 CNY/kg, the optimal quantity of weather derivatives does, as previously analyzed, increase with rising prices. At the optimal trading quantity, the risk mitigation efficiency of derivatives remains constant. Therefore, derivatives can still effectively fulfill their role in weather risk management under different price scenarios.

5.2. The Impact of Price Change Unit λ on Hedge Efficiency and Optimal Trading Quantity

In the preceding research, the price change unit $\mathsf{\lambda }$ was established as 1. Nonetheless, the price of derivatives is subject to change. Consequently, fluctuations in $\lambda$ will impact hedging efficiency and the optimal quantity for hedging cumulative rainfall put options. Hence, it is imperative to explore the different situations involving $\lambda$. By isolating $\lambda$ from Equation (2), Equation (6) is transformed as follows:

$$N=-\frac{PCov\left(Y_t,F\left(I_t^{\prime }\right)\right)}{\lambda Var\left[F\left(I_t^{\prime }\right)\right]}$$

(8)

Since $\lambda$ is in the denominator of Equation (8), Equation (8) will decrease as $\lambda$ increases. In other words, the number of derivative contracts that farmers must purchase to achieve optimal hedging efficiency will decrease as $\lambda$ increases.

Likewise, after isolating $\lambda$, Equation (5) undergoes the following transformation:

$$Var\left(R_t\right)=P^{2}Var\left[Y_t\right]+N^2{\lambda }^{2}Var\left[F\left(I_t^{\prime }\right)\right]+2\lambda PNCov\left(Y_t,F\left(I_t^{\prime }\right)\right)$$

(9)

In Equation (9), the first part is independent of $\lambda$, the second part increases with ${\lambda }^2$, and the third part, with $Cov\left(Y_t,F\left(I_t^{\prime }\right)\right)$ being negative, changes with $\lambda$. As a result, the second part of Equation (9) is more sensitive to changes in $\lambda$ compared with the third part. Its magnitude is greater than that of the third part, leading to an overall increase in Equation (9) as $\lambda$ increases. In summary, it can be concluded that, with all the other factors being held as constant, Equation (7) will decrease as $\lambda$ increases. In other words, risk hedging efficiency will reduce with increasing $\lambda$.

6. Conclusions and Implications

6.1. Conclusions

This study explores weather derivatives’ effectiveness in hedging crop yield volatility in the potential Chinese market. Inner Mongolia’s Tongliao City was chosen as the research area, and a reliable simulation model was constructed by fitting 32 years of precipitation data using an LSTM model. Through validation with a test dataset, the model demonstrates a high predictive performance.

Subsequently, this research establishes an OLS model between cumulative precipitation and maize yield, revealing a significant positive correlation. Further mathematical derivations are conducted to build upon this model and incorporate elements such as the profit function of precipitation put options and the farmers’ total revenue function. The study arrives at specific expressions for the optimal trading quantity and risk hedging efficiency through these derivations. Finally, for a more intuitive presentation of the research results, the study assumes a maize price of 3 CNY per kilogram and a price change unit ($\lambda$) of 1. The research demonstrates that for every hectare of maize, farmers can achieve a risk hedging efficiency of 67.9% when they purchase approximately 6.22 units of precipitation put options. This result underscores the significant value of precipitation put options under specific conditions, proving their ability to effectively hedge the risk of reduced maize yield due to decreased precipitation. Farmers who purchase precipitation put options can better manage income fluctuations resulting from reduced precipitation. This conclusion has important implications for decision making of agricultural operators.

However, in real market operations, the maize price ($P$) and the price change unit ($\lambda$) cannot remain constant. While this study employed fixed numerical values to provide an intuitive illustration of the hedging effects of derivatives, this approach cannot account for the various situations in the market. Therefore, the study discusses the potential variations in research results when both $P$ and $\lambda$ change. The results indicate that, with other factors being held as constant, as the maize price ($P$) fluctuates between 1 and 10 CNY, the optimal trading quantity of the derivatives increases with the rising costs, but the risk hedging efficiency remains constant. Similarly, when other factors are unchanged, an increase in the price change unit ($\lambda$) results in a reduction in the optimal trading quantity and a decrease in hedging efficiency.

6.2. Implications

Based on the research presented, this paper provides significant insights into the hedging effectiveness of weather derivatives in mitigating crop yield volatility risks in the potential Chinese market. Firstly, the study’s simulation model and the OLS model that relates precipitation to maize yield validate the potential application value of weather derivatives in crop risk management. Through accurate data analysis and model establishment, this paper unveils the effectiveness of precipitation put options under specific market conditions, offering the prospect of partially hedging crop yield volatility and reducing income fluctuations for farmers. Secondly, considering the fluctuations in maize and derivative market prices, adjusting the optimal trading quantity can help maintain hedging efficiency to some extent. However, as the price change unit ($\lambda$) increases, hedging efficiency gradually decreases, thus underscoring the significant impact of derivative prices on their risk management role. Finally, this study provides a new perspective on agricultural risk management and offers theoretical support for introducing weather derivatives into the Chinese market. It demonstrates the feasibility of weather derivatives in the Chinese market and provides theoretical backing for constructing a diversified agricultural risk management system. This approach addresses both catastrophic and non-catastrophic weather risks, which is paramount for China in achieving more effective risk hedging and sustainable agricultural development.

In addition, this study has its limitations and provides avenues for further in-depth exploration in the future. The limitations of this study primarily encompass three aspects. Firstly, the predictive accuracy of the LSTM model can significantly improve with an increase in the volume of data. Due to data availability constraints, the meteorological data used for model training in this study only span 30 years. The predictive precision of the LSTM model could be further enhanced through a larger dataset. Secondly, the study area is confined to Tongliao. While Tongliao is representative of maize cultivation in Chinese cities, there is still a deviation in validating the feasibility of introducing weather derivatives to China as a whole. Future research will delve deeper into this issue. Additionally, this study solely focuses on weather derivatives related to cumulative precipitation. While precipitation is a crucial weather factor influencing maize cultivation, other factors such as sunshine duration also play a significant role. Subsequent research should expand to include other weather factors and explore derivatives that combine multiple weather elements.