Mitigating geographical basis risk of weather derivatives using spatial-temporal regime-switching temperature model

1.   Introduction

Agriculture is the main source of livelihood in Africa ^[1]. However, extreme changes in weather patterns, unpredictable temperature changes, frequent heat wave, and increasing temperature as a result of climate change make agriculture look like a costly gamble in Africa. The report of the Intergovernmental Panel on Climate Change (IPCC) 2007 ^[2] gives a comprehensive evaluation of the impacts of climate change on agriculture in Africa. As stated by the IPCC report, the estimated prediction of agriculture losses by 2100 is between 2% to 7% of gross domestic product (GDP) in parts of the Sahara. The report revealed that Northern and Southern Africa are anticipated to record GDP losses of between 0.4-1.3%, and between 2-4% for the Western and Central Africa. According to ^[3], Africa is extremely affected by climate change from two causes: a limited social, economic, and human abilities needed to adjust to the impact of climate change, and its geographical features of having a predominantly vast land lying across the warming tropics. From planting to harvest, extreme changes in weather can severely affect the quality and the complete production levels of crop yields.

The effects of weather do differ considerably in the agricultural supply and demand chain. Most farmers in Africa have used traditional ways to improve the negative effect of extreme weather conditions on their farmlands. However, most of these farmlands are in the same geographical locations and as a result crop losses are correlated across farmers due to the covariant nature of weather risks. This implies that farmers in the same geographical locations are vulnerable by the same weather event and are probably going to endure extensive losses concurrently. Covariant risk controls the success of traditional risk management techniques of smallholder farmers. Consequently, there is the need for a suitable and efficient risk management tool for farmers to control weather extremes and uncertainties. An emerging shift has been the success of a weather risk management tool-weather derivative, which is use to reduce the financial effects of weather extremes and uncertainties. The uptake of this tool has been lower than expected in the agricultural sector ^[4] due to basis risks. As defined by ^[5], "Basis risk arises when the production pattern of the individual operation is not perfectly correlated with the agammaegated pattern of the area for which the derivative has been designed". Product-design and geographical basis risk are forms of basis risks. Product-design basis risks can easily be mitigated if the appropriate weather variable is used as the underlying index for designing the weather derivative. Geographical basis risk is difficult to mitigate especially in most developing countries due to the unavailability of weather stations at most farming locations under interest. For this reason, the need to develop spatial-temporal mathematical model that considers different farming locations and assigning weights to weather stations relative to the distance from the reference farming locations. This model is also beneficial to farmers with farmlands in the same geographical locations.

Unlike traditional agricultural insurance that is used to hedge against risks from idiosyncratic occurrences, weather derivatives allow agricultural stakeholders (farmers, input suppliers and other stakeholders at the farm level) to hedge against covariant risks. Weather derivative is seen as an efficient tool for mitigating risk that affects most farmers, input suppliers and other stakeholders at the farm level. In a survey report conducted in 2008-2009 and prepared for the weather risk management association, the agricultural sector contributed about $11\%$^* of the total weather derivatives purchase in the weather market. Unlike traditional insurance, the payoff of a weather derivative depends on an index (specially designed measure) that is linked to the risk being hedged against.

^*the second largest percentage after the energy sector

The price of a weather derivative is usually dependent on different weather indices (heating degree days (HDD), cooling degree days (CDD), Pacific Rim (PRIM), cumulative average temperature (CAT), growing degree days (GDD)) that help in pricing weather derivatives. Different authors ^[6,7,8] have used HDD and CDD indices as the major indices for pricing weather derivatives in the energy industry. Similar to the CDD and HDD indices in the energy sector, the GDD and CAT indices are powerful indices that can be use to price weather derivatives for the agricultural sector in Africa. For this reason, CAT and GDD are used as the major indices for the temperature based weather derivative pricing. GDD measures the growth and development of crops, weeds, and insects during a growing season.

The contributions made in this study are: (1) Motivated by ^[9], we develop a temperature dynamics model for spatial-temporal locations in Ghana. This model captures the stylized facts of temperature at different locations. (2) We develop an analytical weather derivative pricing model for basket futures written on CAT and GDD indices. With the proposed basket futures pricing model, it will be cost efficient and pragmatic for farmers to buy weather derivatives contracts for different but correlated farming locations than a single farming location. Geographical basis risk will also be reduced when using this spatial-temporal pricing model.

To the best our knowledge, the proposed analytical weather derivatives basket futures pricing formulas using multi-dimensional regime-switching model is the first of its kind in literature.

2.   Theoretical concepts

Assume $D_B$ represent the random payoff at expiry for the owner of a futures contract. At time $t \le t_1 < t_2$ for a measurement period $[t_1, t_2]$, the holder of the contract enters into the contract. Let $F(t, t_1, t_2)$ represent the price against receiving the random payment $D_B$ at time $t_2$. For a constant continuously compounded interest rate $r > 0$ and a risk-neutral probability measure $\mathbb{Q}$, the arbitrage-free future price for a measurement period $[t_1, t_2]$ on the CAT and GDD can be define as $\mathcal{F}_t$-adapted stochastic process satisfying

Assume that the futures price $F(t, t_1, t_2)$ is $\mathcal{F}_t$ adapted, then the futures price is define as

where $t, t_1, t_2$ are the current, starting and maturity time of the futures contract respectively and $\mathcal{F}_t$ is $\sigma$-algebra up to a specified time $t$. From equation 2.2, it is important to calculate the risk-neutral measure $\mathbb{Q}$ (also referred to as equivalent martingale measure) in order to determine futures prices.

To derive an explicit formula for the future price, we specify $\mathbb{Q}$ to help in calculating the expectation. Following the analysis of ^[11], we use the Girsanov theorem to find a sub-family of probability measures for the normal and shifted regimes. By using the option pricing technique under independent regime-switching model of ^[12], the CAT and GDD futures are priced by splitting the futures price into a normal and shifted price component. The two pricing component are joined up using the idea of weighted mixture of probability in each regime from $t_1$ to $t_2$.

2.1.   CAT futures

Suppose that for a contract period $[t_1, t_2]$, the temperature dynamics follow the temperature model in Lemma 3.1. Then, there is a price dynamic of futures written on a CAT index with $t \le t_1 < t_2$. From equation 2.1, the futures price of CAT is

Since the futures price $F(t, t_1, t_2)$ is $\mathcal{F}_t$ adapted under the measure $\mathbb{Q}$, $F_{CAT}$ is $\mathcal{F}_t$-adapted. We can therefore define the CAT futures price $F_{CAT}(t, t_1, t_2)$ for a weather derivative contract as

2.2.   GDD futures

Similar to the definition of the CAT futures price, the GDD futures price is given as

Using the same idea in deriving the CAT futures price, the price of the GDD futures can be derived as

Where $K$ is the optimal normal temperature at which a crop will develop. The rate of development of most plants depends on the daily air temperature ^[13].

Because the market price of temperature risk remains unchanged for all derivatives that depends on temperature, we use the same risk-neutral measure $\mathbb{Q}$ used in pricing the CAT futures to price the GDD futures.

2.3.   Change of measure

To find a mathematical expression for the futures price of the chosen indices, an arbitrage-free and explicit dynamics for future price of the indices are constructed. The real-world measure $\mathbb{P}$ is changed to a risk-neutral measure $\mathbb{Q}$, in a way that the discounted price process of the underlying is a martingale under $\mathbb{Q}$. To transform the $\mathbb{P}$ to $\mathbb{Q}$, the Girsanov theorem is employed. The Girsanov theorem provides techniques for transforming $\mathbb{P}$ to $\mathbb{Q}$ under the setting of a Brownian motion, where $\mathbb{Q}$ is a second probability measure.

Theorem 2.1 (Girsanov Theorem). Let $W_t$ be a Brownian motion on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and $\lambda = \{\lambda_t : 0 \le t \le T \}$ is an adaptive process satisfying the Novikov condition

then $\mathbb{Q} \sim \mathbb{P}$ can be determined by the Radon-Nikodym derivative

Then we can define the random process

Equivalently,

The process $V_t$ is a Brownian motion under the measure $\mathbb{Q}^{\lambda}$

Proof. See ^[14] for the proof of this theorem.

2.4.   Girsanov's Theorem in $\mathbb{R}^N$

Let $\mathit{\boldsymbol{B}}(t) = \big(B_1(t), B_2(t), B_3(t), \cdots, B_N(t)\big)$ be an $N$-dimensional Brownian motion on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and $\mathit{\boldsymbol{\lambda}} = \big(\lambda_1(t), \lambda_2(t), \lambda_3(t), \cdots, \lambda_N(t)\big)$ be an $N$-dimensional adapted process on $[0, T]$.

Define

where $\mid\mid\mathit{\boldsymbol{\lambda}}(s)\mid\mid^2 = \sum_{i = 1}^{N} \lambda_i(s)^2$.

Let

and suppose that

then $\mathit{\boldsymbol{\tilde{B}}}(t)$ is a N-dimensional standard Brownian motion under the measure $\mathbb{Q}$ defined as

Observe that for each $j$,

The component process of $\mathit{\boldsymbol{\tilde{B}}}(t)$ are independent under the measure $\mathbb{Q}$.

Remark 1.The Novikov condition in the Girsanov theorem makes sure $Z$ is positive martingale and whenever $\mathbb{E}(Z) = 1$. This is referred to as the Radon-Nikodym derivative.

Remark 2. $\lambda$ is referred to as the market price of temperature risk (MPR). Since there is no real weather derivative market in Africa from which the prices can be obtained, $\lambda$ is assumed to be a constant. For a constant $\lambda$, equation 2.11 can be redefined as

3.   Regime-switching temperature model

3.1.   One-dimensional regime-switching model

Different temperature models have been proposed to capture the dynamics of temperature ^{[6,8,9,10,15]}.. The usual assumptions in these models are: volatility of temperature is lower in summer than in winter, temperature is autoregressive, temperature follows a predicted pattern and it goes around a seasonal mean. Early models used autoregressive moving average (ARMA) processes, autoregressive process of order one (AR(1)) processes, mean reverting stochastic differential equations (SDE) ^[9,15,16]. All these models assumed no changes in the state of the dynamics of temperature. That is, they used single-regime model to describe the dynamics of temperature. However, temperature can go through different latent states in a particular period of time and a single SDE can not accurately capture all this states. More complex models have been proposed in recent literatures, an example is the model of ^[7,17,18], which uses a two-state Markov regime-switching model. ^[17] extended the model of ^[7] by replacing the constant volatility in the base regime with heteroscedastic volatility. This was necessary because volatility of temperature changes at different states. ^[18] further extended this model by replacing the Brownian process in the shifted regime with a Lévy process. However, none of these authors proposed a spatial-temporal pricing model for weather derivatives. Motivated by ^[7,17,18], a regime-switching temperature dynamics model is proposed and later extended to a multi-dimensional model. The switching dynamics between the regimes are assumed to be controlled by an unobservable latent variable $R_t$. The model is governed by a two-state regime-switching model $R_t = \{1, 2 \}$ in which a two-state Markov chain controls the characterization of the probability law of switching between $R_t = 1$ and $R_t = 2$ with transition probabilities

The daily average temperature on day $t$, $T_t$ is modelled as the sum of a deseasonalized temperature $\tilde{T}_t$ and a deterministic seasonal component $S_t$,

The deterministic seasonality component at time $t$, $S_t$ is given by

where $a_1, a_2, a_3$ and $\varphi$ are constants.

The proposed model is distinctly appropriate to capture the dynamics of temperature through time. The proposed daily temperature model is given as

where $\sigma_1 {T}_{t, 1}$ is the daily local volatility of the normal regime through time, $\sigma_2$ is the volatility of the shifted regime, and $\beta$ is the mean-reversion rate of the daily temperature in the normal regime which reverses to the long term equilibrium level after the daily temperature has drifted from its equilibrium. The probabilities for the process to be in the normal and shifted regimes are $p_1$ and $p_2$ respectively and $p_1 + p_2 = 1$. $W_t \sim N(0, t)$ is the standard Brownian motion. ${T}_t (T(t))$ is the daily temperature at time $t$. The regimes are assumed to be independent to each other and the futures contract is calculated for each regime model. The final futures price is calculated using the weighted sum of the individual regimes.

Lemma 3.1. If the daily average temperature ${T}_{t, 1}$ follows the proposed model 3.1, then the explicit solution is given by

Proof. For the normal regime,

where $\tilde{T}_{t, 1} = T_{t, 1} - S_{t, 1}$. Using the transformation below, $d\tilde{T}_{t, 1}$ will be evaluated,

By Itô's Lemma and from equation (3.3),

Integrating equation (3.4) over the interval $[0, t]$,

For the shifted regime

3.2.   Spatial-temporal regime switching model

Let $N$ be the spatial locations in the basket, $\left(\omega^{i}\right)_{i = 1}^N$ be the collection of weights for spatial locations $\left(y^{i}\right)_{i = 1}^N$. At time $t$, the basket of the deseasonalized average temperature at the $N$ spatial locations is given as

Where $\sum_{i = 1}^{N} \omega^{i} = 1$.

Assume temperature is spatially correlated across the random noise term. To allow analytical pricing of the basket temperature derivatives contract, assume the risk-neutral distribution of temperature for each location is normally distributed in the temperature model. That is, the basket been a weighted sum of normally distributed temperature is also normally distributed. Consequently, we are able to outwit the principal difficulty associated with pricing basket options of assets when determining the distribution of the sum or average of the underlying assets. From these settings, a spatial-temporal temperature model at each spatial location $y^i$ is proposed,

Model 3.6 can be expressed for locations $i = 1, 2, \cdots, N$ as an $N$-dimensional system,

where $\mathit{\boldsymbol{W}}_t \sim N(0, \mathit{\boldsymbol{\Omega}} t)$. From the property of linear transformation of multivariate normal distribution,

Suppose $Z \sim N(0, \mathit{\boldsymbol{I}}t)$ and $Y = XZ$, then it follows that $Y \sim N(0, XX^Tt)$. By applying Cholesky factorization to $\Sigma$, a lower triangular form for $X$ is derived. Using this theory, $\mathit{\boldsymbol{W}}_t$ can be expressed as an N-dimensional Brownian motion $\mathit{\boldsymbol{B}}_t$,

$\mathit{\boldsymbol{LL^T}} = \mathit{\boldsymbol{\Omega}}$, $\mathit{\boldsymbol{L}}$ is a lower triangular matrix with non-negative diagonal entries, $\mathit{\boldsymbol{L^T}}$ is an upper traingular matrix, and $\mathit{\boldsymbol{B}}_t = (B_t^1, B_t^2, B_t^3, \cdots, B_t^N)^T$ with $dB_t^i dB_t^j = \delta_{ij}dt$. From equation 3.8, equation 3.7 can be reformulated as

Equation 2.8 can be transformed for the normal and shifted regime,

From equation 3.10, and assuming $\lambda$ is a constant for each reference measurement station, it can be inferred that;

Equivalently

Where $\mathit{\boldsymbol{V}}_t^{\lambda}$ is a Brownian motion under the measure $\mathbb{Q}_{\lambda}$

Consequently, the TML model under the equivalent martingale measure $\mathbb{Q}^{\lambda}$ is

Lemma 3.2. For a spatial location $i$, if the dynamics of the daily average temperature follows equation 3.13, then the explicit solution for the $i^{th}$ location $y_i$ is given as

Proof. From the normal regime of model 3.13,

where $\mathit{\boldsymbol{\tilde{T}}}_{t, 1} = \mathit{\boldsymbol{T}}_{t, 1} - \mathit{\boldsymbol{S}}_{t, 1}$. $d\mathit{\boldsymbol{\tilde{T}}}_{t, 1}$ will be evaluated using the transformation in equation 3.16,

By Itô's Lemma and from equation (3.15),

Integrating equation (3.17) over the interval $[0, t]$ gives

At location $y^i$,

From the shifted regime of model 3.13,

At location $y^i$,

hence the lemma.

4.   Pricing futures on temperature basket

The CAT and GDD futures will be priced by splitting the futures price into normal and shifted price components and added up with probabilities in each regime for the contract period.

Where $F^{N}(t, t_1, t_2)$ and $F^{S} (t, t_1, t_2)$ are the futures price of the normal and shifted regimes respectively. $Pr(R_{[t_1, t_2]} = N)$ and $Pr(R_{[t_1, t_2]} = S)$ are the probability of the observed daily average temperature data under the normal and shifted regimes throughout the contract period $[t_1, t_2]$. The splitting is possible since it is assumed that the futures price under the normal regime is independent of the futures price under the shifted regime.

4.1.   CAT and GDD futures on temperature basket

By the same reasoning as in deriving the futures price of a single CAT, the futures price of a basket CAT is given by

Definition 4.1. At a spatial location $y^i$ and a specificied contract period, $t\le t_1 < t_2$, the GDD futures price is define as

Analogously, the futures price of a basket GDD is defined as

Proposition 4.2. At a spatial location $y^i$, the futures contract price on basket of CAT index following the normal regime in equation 3.6 is calculated as

Proof. For $x \ge t$ in Lemma 3.2,

Proposition 4.3. For a specificied contract period, $t \le t_1 < t_2$ at spatial location $y^i$, the futures contract price on basket of CAT index following the shifted regime in equation 3.6 is

Proof. We first integrate the shifted regime in lemma 3.2 at a spatial location $y_i$ over the time interval $[t_1, t_2]$.

For $x \ge t$ and a spatial location $y^i$

Generally, if we assume that the daily average temperature follows model 3.6, then the CAT futures price on temperature basket is computed using Equation 4.1, Proposition 4.2 and 4.3.

Proposition 4.4. For a specificied contract period, $t \le t_1 < t_2$ at spatial location $y^i$, the futures contract price on basket GDD index following the normal regime in equation 3.6 is given by

where $\Phi$ is the cumulative standard normal distribution function, $\phi$ is the standard normal density function,

Proof. Let

For $x \ge t$ in Lemma 3.2,

For convenience, we shall denote the deterministic and random component of equation 4.10 as $G^i(t, x)$ and $H^i(t, x)$ respectively. That is,

Hence

At time $t$, the distribution of the basket $D(x)$ can be computed. However, $G^i(t, x)$ is deterministic. Hence, at time $t$ and by Itô isometry,

But $V_{\lambda}^j(s)$ are independent for each $j$. So the variances can be summed to obtain the variance of $H^i(t, x)$,

Since $\sum_{i = 1}^{N} \omega^{i} H^i(t, x)$ is a sum of normally distributed random variables, it implies that it is normally distributed with the following respective mean and variance:

and

Now, take into account $Cov\big(H^i, H^j\big)$ for $j > 1$. Both $H^1$ and $H^j$ are in the same integral form with respect to the standard Brownian motion $V^1_{\lambda}(u)$. By the independence property of $V^1_{\lambda}(u)$ and $V^j_{\lambda}(u)$, the covariance only exist between these two integrals. Therefore,

Define $\varUpsilon^{ij}(t, x) : = \int_{t}^{x} \sigma^i_1 \sigma_1^j {T}_{s, 1}^i {T}_{s, 1}^j e^{(\beta^i + \beta^j)(x - s)} ds.$

From 4.11,

Let

$D(x)$ can be written in the form of a standard normal random variable $Z \sim N(0, 1)$ as

From equation 4.4, consider

This requires

From Equation 4.14,

From equations 4.4 and 4.14,

Substituting 4.12 and 4.15,

Therefore

where

Proposition 4.5. The price of a futures contract on basket GDD index following the shifted regime in equation 3.6 at time $t \le t_1 < t_2$ is given by

where $\phi$ and $\Phi$ as their usual meaning as in Proposition 4.4.

Proof. The proof of Proposition 4.5 follows in the same way as the proof of Proposition 4.4

In summary, if the daily average temperature follows the regime-switching model in equation 3.6, equation 4.1, Proposition 4.4, and Proposition 4.5 are used to calculate the GDD futures price on the temperature basket.

5.   Conclusion

In this paper, a regime-switching temperature dynamics model for spatial-temporal farming location was developed. To allow analytical tractability of the pricing models, the driving noise of the regimes were captured by a Brownian motion. Based on this model, pricing models for basket futures written on cumulative average temperature (CAT) and growing degree-days (GDD) indices were proposed. Pricing futures on temperature basket provides significant benefit as it mitigates geographical basis risks and changing of contracts relative to risk exposures of hedgers. With the proposed spatial-temporal regime-switching pricing model, investors in the weather derivative market have the opportunity to select the most appropriate composite of weather stations with their desired weight to optimize basis risk.

An extension of this research would be to use the multi-dimensional regime-switching temperature model to price basket options on futures at different locations, also called rainbow options.

Acknowledgment

The first author wishes to thank African Union and Pan African University, Institute for Basic Sciences, Technology and Innovation, Kenya, for their financial support for this research. Authors will like to thank the anonymous reviewers for their insightful comments.

Conflict of interest

The authors declare that there is no conflict of interest.