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We investigate the dynamics of the beef-cattle pricing which is affected by several factors such as beef supply, demand for foreign currency, etc. The model incorporates mean-reversion to give insights into the relationship be-tween supply to a developed region (or country) and a third world country’s demand for hard currency. We consider the beef-cattle industry which is seg-mented into two markets: the Farmer-Local Cattle Agency (LCA) market in which the LCA buys cattle from the farmers and the LCA-European Union (EU) market in which the LCA exports beef to the EU. Using the Botswana-EU as an example, we investigate the performance of the Botswana Meat Com-mission (BMC) which buys cattle from the farmers and exports beef to the EU based on the price acceptable to the EU and ask whether the agreed price be-tween the BMC and the EU can ever translate into fair price between the farmer and the BMC. Our study has concluded that the operational problems faced by the BMC are an indication that the BMC is passing on the bulk of what it receives from the EU to the farmers. We have made suggestions on how the BMC can reduce its operational risks.

The Botswana-European Union (EU) beef-cattle trade is an outstanding example of cooperation between a Western block and a third world country (Botswana). Although this trade showed promise, it has of late exhibited cracks as the EU has forged more trade agreements with other beef exporters such as South America, Australia, New Zealand, etc. An example of this occurred when the EU stopped all meat imports from Botswana and demanded a thorough cleanup of the abattoirs [

The export market in Botswana is dominated by a government parastatal organisation called Botswana Meat Commission (BMC). BMC has experienced substantial changes in prices over the past with major surges during the past three decades [

This competition for the EU market has introduced uncertainty in the beef prices. Tothova [

Important contributions have been made by Schwartz [

In this paper we incorporate the aspect of mean reversion and a time dependent stochastic mean reverting process for the price. Unlike stock prices which generally exhibit upward trends in the long run as investors benefit from the long term dividend yields and earnings, beef prices, show a level dependent behavior over a long period since supply and demand dictate the prices. However, one of the fundamental distinguishing characteristics exhibited by beef prices is mean-reverting behavior (for example, Schwartz [

The paper is organized as follows, in Section 2, we present model formulation, section and a preliminary result of the study using a Bivariate analysis for Farmer-BMC price S 1 ( t ) and BMC-EU prices S 2 ( t ) , Section 3, we present the model that we formulated following the influences that affect the pricing process in the Botswana beef-cattle industry. Most of the influences were not explicitly considered, but we assumed they fall under the volatility part. In Section 4, we provide a lemma on the positivity of our model, the proof was built on Lyapunov type function and four cases were considered for our model. Model approximation and calibration are given in Appendix A3, in Section 5 we discuss the findings from our study and perform sensitivity analysis. Subsequently, we concluded in Section 6 and gave some remarks in the Appendix.

In our model we assume that the information available to the BMC is the average price of beef S ¯ ( t ) , whose future trend is not known. In general, Equation (1) below represents the evolution of price S ( t ) at time t based on the EU price.

S ( t ) = S ¯ ( t ) + a i , i = 1 , 2 , (1)

where, S ¯ i ( t ) is the mean price and a i is a random variable.

Prior to our stochastic model we present some preliminary results associated with the price data sets. As noted before that the Botswana beef-cattle industry is divided into two markets segments described earlier.

Let S 1 ( t ) denote the Farmer-BMC price and S 2 ( t ) the BMC-EU price.

Note that while the ratio of the price of Farmer-BMC to BMC-EU was 0.65 in 1992 this ratio declined to 0.30 in 2015. This could be a result of competition in the beef market where the seller had to accept a lower price to secure the export quarter. The mean and standard deviation for each of the two price processes are given in Table1 (see TableA1 in Appendix A1 for an overview of data that was used).

We conclude that the random variable a i , i = 1 , 2 is purely a white noise process. The Pearson’s correlation coefficient for the two price processes is given by,

r = ∑ i = 1 n ( S 1 ( t ) − S ¯ 1 ) ( S 2 , i − S ¯ 2 ) n − 1 ∑ i = 1 n ( S 1 ( t ) − S ¯ 1 ) 2 ∑ i = 1 n ( S 2 , i − S ¯ 2 ) 2 = 0.8999. (2)

S 1 ( t ) | a 1 | S 2 ( t ) | a 2 | |
---|---|---|---|---|

Mean | 12.4133 | 0.0033 | 23.3407 | 0 |

Standard deviation | 10.2059 | 10.2059 | 13.9738 | 13.9738 |

This confirms a strong positive correlation between the Farmer-BMC price S 1 ( t ) and BMC-EU price S 2 ( t ) suggested by

The relationship between S 1 ( t ) and S 2 ( t ) is of interest. We assume a relationship between S 1 ( t ) and S 2 ( t ) of the type:

S 2 ( t ) = ω ( B ) S 1 , t − b . (3)

where, B is the backward shift operator and b is the delay parameter. Our task is to find the transfer function ω ( B ) and b which for this relationship holds.

In the next subsection we discuss the concept of linear prices translation in the Botswana beef-cattle industry using the Box and Jenkins procedure [

Suppose there exists N price observations S i , i = 1 , 2 , where they are collected at equispaced intervals on the time horizon [ 0 , T ] (yearly averages in this case) and pairwise observations as ( S 1 , 1 , S 2 , 1 ) , ( S 1 , 2 , S 2 , 2 ) , ⋯ , ( S 1 , N , S 2 , N ) and finite realization of a discrete bivariate process with the S 1 ( t ) as the independent variable and S 2 ( t ) as the dependent variable (see TableA4 in Appendix A4 for the notations used throughout this paper). We need to find the weights { w k } (response functions), where k = 0 , 1 , ⋯ of the pricing process

S 2 , t = w ( B ) S 1 , t − b . (4)

where w ( B ) = v 0 − v 1 B − v 2 B 2 − ⋯ is called the transfer function. Let, s 1 , t = ∇ d s 1 , t and s 2 , t = ∇ d s 2 , t be incremental changes for the Farmer-BMC prices and BMC-EU prices, respectively. The constant d denotes the degree of differencing, ∇ = ( 1 − B ) , then for any series { M t } , M t − b = B b M t , we can show on differencing Equation (4) that s 2 , t and s 1 , t satisfy the same transfer function model as do S 2 , t and S 1 , t i.e.

s 2 , t = w ( B ) s 1 , t − b . (5)

Writing Equation (5) (the linear filter) in a parsimonious way as in Box and Jenkins [

δ ( B ) s 2 , t = ρ ( B ) s 1 , t − b . (6)

where

δ ( B ) = 1 − δ 1 B − δ 2 B 2 − ⋯ − δ r B r (7)

ρ ( B ) = ρ 0 − ρ 1 B − ρ 2 B 2 − ⋯ − ρ h B h . (8)

We compared Equations (5) and (6) to obtain:

w ( B ) = δ − 1 ( B ) ρ ( B ) (9)

w j = 0 , j < b (10)

w j = δ 1 w j − 1 + δ 2 w j − 2 + ⋯ + δ r w j − r − w 0 , j = 0 , 1 , ⋯ , b (11)

w j = δ 1 w j − 1 + δ 2 w j − 2 + ⋯ + δ r w j − r − ρ j − b , j = ( b + 1 ) , ⋯ , ( b + h ) (12)

w j = δ 1 w j − 1 + δ 2 w j − 2 + ⋯ + δ r w j − r , j > ( b + h ) . (13)

Theoretically, a plot of the weights w k , k = 0 , 1 , ⋯ against lag k provides a pictorial representation of the impulse response function. In reality, however (considering the beef-cattle industry of Botswana in particular) the system involves noise or disturbances whose net effect influences the predicted model by an amount η t , so that the combined translation function-noise model may be written as;

s 2 , t − δ − 1 ( B ) ρ ( B ) s 1 , t − b = η t . (14)

where s 2 , t and s 1 , t are stationary time series for d differencing. Using the Box and Jenkins [

ϕ ( B ) s 1 , t = θ ( B ) α t , (15)

where the variable α t represents a pure white noise process,

α t = θ − 1 ( B ) ϕ ( B ) s 1 , t . (16)

Transforming, to the BMC-EU prices (output series), we obtain

β t = ϕ ( B ) θ − 1 ( B ) s 2 , t . (17)

Calculating, the cross-covariance function of the filtered input and output ( α t and β t and multiplying both sides of Equation (15) by ϕ ( B ) θ − 1 ( B ) gives:

β t = w ( B ) α t + ε t . (18)

where, ε t = ϕ ( B ) θ − 1 n t is the transformed noise series. Multiplying both sides of Equation (18) by α t − k and taking expectation, noting that α t and n t are uncorrelated yields:

w t = E [ α 1 − b , β t ] σ β σ α . (19)

where σ β 2 and σ α 2 are the variances of α t and β t respectively, and E [ α 1 − b , β t ] is the cross-covariance function at lag k. The estimate of the impulse response function ( w k ) determined as outlined above are found to be reliable [

If we consider b = 0 , δ − 1 ( B ) = 1 , ρ ( B ) = ρ = 1 , Equation (14) is transformed into

S 1 ( t ) − S 2 ( t ) = η t . (20)

In

A scatter plot of S 1 ( t ) against S 2 ( t ) (

The bivariate relationship between the Farmer-BMC and BMC-EU prices in 2.2 has shown that the difference between the two prices is a white noise process (

For simplicity of notation, we define the state variables ( x 1 , x 2 ) = ( S 1 ( t ) , S 2 ( t ) ) where x 2 is assumed to be the mean of the price process x 1 ( t ) . The GMR model can be written as:

d x 1 = κ ( x 2 − x 1 ) x 1 d t + σ 1 x 1 d B 1 (21)

d x 2 = σ 2 x 2 d B 2 . (22)

where, σ 1 > 0 , σ 2 > 0 and d B 1 d B 2 = ρ d t and κ is the level dependent mean reverting speed. Note that we assume the mean to be purely stochastic, since the BMC has no knowledge about the price it will be offered by the EU. We can write the system (21)-(22) in matrix form as

d x t = f ( x t ) d t + σ ( x t ) d B , (23)

where x t = ( x 1 ( t ) , x 2 ( t ) ) T

f ( x t ) = ( κ ( x 2 − x 1 ) x 1 0 ) ; σ ( x t ) = ( σ 1 x 1 0 0 σ 2 x 2 ) ; d B = ( d B 1 d B 2 ) .

The solution of (23) is not unique due to nonlinear terms x 1 x 2 and x 1 2 in the drift of the Equation (21). We want to define conditions on f ( x t ) for the system (23) to have a unique solution.

Remark: The solution of the stochastic model (23) does not tend to any nearby steady state (see for example Asfaw et al. [

Let Ω = { ( x 1 , x 2 ) ∈ ℝ + 2 , x i ≥ 0 , i = 1 , 2 } be a positive region under study and consider the system (23) with initial values x 0 = ( x 1 0 , x 2 0 ) ∈ ℝ + 2 . It is easy to show that (23) can be modified (see, Gard [

g ϵ ( s ) = { 0 if s < 0 s if 0 ≤ s ≤ 1 ϵ 1 ϵ if s > 1 ϵ

and ϵ is an arbitrary small number. We denote the modified f ( x t ) in (23) by f ϵ ( x t ϵ ) and consider the following stochastic system

d x t ϵ = f ϵ ( x t ϵ ) d t + σ ( x t ϵ ) d B . (24)

Clearly (see Øksendal 6^{th} edition [

| f ϵ ( x ) − f ϵ ( x ¯ ) | ≤ K ϵ | x − x ¯ |

| σ ( x ) − σ ( x ¯ ) | ≤ K | x − x ¯ |

For all x , x ¯ ∈ ℝ 2 where K ϵ and K are constants.

Denote by τ ϵ the next exit time of x t ϵ from the domain

ℝ 2 ∩ { max 1 ≤ i ≤ 2 x i < 1 ϵ } ,

If this domain contains the initial point x 0 then the system (21)-(22) possesses a nonnegative solution for t > 0 in this domain.

For any finite T > 0 the solution x t ϵ of the system (24) with initial condition x 0 ∈ ℝ t 2 remains in ℝ t 2 for all t < T ∧ τ ϵ (where T ∧ τ ϵ denotes the smaller between T and τ ϵ ) so that the components x t ϵ satisfy x i , t > 0 , if t ∧ T , τ ϵ for i = 1 , 2 . Furthermore, ℙ ( τ ϵ < T ) < C o τ ϵ , where C 0 is a constant independent of x 0 .

Proof

We want to prove first that a non-negative solution does not exit the positive domain Ω. We introduce a Lyapunov type function

V = x 1 − κ 1 ln x 1 + x 2 − κ 2 ln x 2 (25)

d V = ( 1 − κ 1 x 1 ) d x 1 + ( 1 − κ 2 x 2 ) d x 2 + 1 2 κ 1 x 1 2 ( d x 1 ) 2 + 1 2 κ 2 x 2 2 ( d x 2 ) 2 = ( 1 − κ 1 x 1 ) [ κ ( x 1 − x 2 ) x 1 d t + σ 1 x 1 d B 1 ] + ( 1 − κ 2 x 2 ) σ 2 x 2 d B 2 + 1 2 κ 1 σ 1 2 d t + 1 2 κ 2 σ 2 2 d t

= [ κ x 1 x 2 − κ 1 x 1 2 − κ 1 κ x 2 + κ 1 κ x 1 + 1 2 κ 1 σ 1 2 + 1 2 κ 2 σ 2 2 ] d t − κ 1 σ 1 d B 1 − κ 2 σ 2 d B 2 + σ 1 x 1 d B 1 + σ 2 x 2 d B 2 ≤ [ κ x 1 x 2 + κ 1 κ x 1 + 1 2 κ 1 σ 1 2 + 1 2 κ 2 σ 2 2 ] d t + σ 1 x 1 d B 1 + σ 2 x 2 d B 2 = M ( t ) d t + ∑ i = 1 2 σ i x i d B i . (26)

where

M ( t ) = κ x 1 x 2 + κ 1 κ x 1 + 1 2 κ 1 σ 1 2 + 1 2 κ 2 σ 2 2 . (27)

Integrating (26), we obtain

∫ 0 T ∧ τ ϵ d V ≤ ∫ 0 T ∧ τ ϵ M d s + ∑ i = 1 2 ∫ 0 T ∧ τ ϵ σ i x i d B i . (28)

Taking expectation we obtain

E [ V ( T ∧ τ ϵ ) ] ≤ E [ V ( x 0 ) ] + M ( T ∧ τ ϵ ) . (29)

Note that if the path x t ϵ is such that it exits ℝ t 2 at T ∧ τ ϵ then by definition of (25) the function V becomes ∞ at the exit point. In view of (26) the probability of that happening is zero. This completes the proof.

Special CasesWe consider the following four cases of the system (21)-(22):

Lemma 2

For the case σ 1 = σ 2 = 0 , the system (21)-(22) possesses two equilibrium points one of which is stable and the other unstable.

Proof

Take σ 1 = σ 2 = 0 , the system (21)-(22) becomes

d x 1 d t = κ ( x 2 − x 1 ) x 1 (30)

d x 2 = 0 , x 2 = m , m isaconstant .

Equation (30) has two equilibrium points ( 0 , m ) and ( m , m ) The transient solution of Equation (30) for x 1 ( t ) , for x 2 ( t ) = m , is given by

x 1 = m 1 + e − κ m t . (31)

Clearly, as t → ∞ , x 1 → m implying that the movement is away from the unstable equilibrium point ( 0 , m ) towards the stable equilibrium point ( m , m ) .

Remark: When there is no volatility the price x 1 ( t ) would remain constant about the mean price m which economically is not a good situation for either the Farmer-BMC price or the BMC-EU price as neither the farmer nor BMC would generate sufficient capital to expand.

Lemma 3

For the case σ 1 = 0 , σ 2 ≠ 0 , the system (21)-(22) possesses one stable equilibrium point ( 0 , 0 ) .

Proof

For σ 1 = 0 , σ 2 ≠ 0 , the system (21)-(22) becomes,

d x 1 = κ ( x 2 − x 1 ) x 1 d t (32)

d x 2 = σ 2 x 2 d B 2 . (33)

Equation (33) yields the solution

x 2 = exp [ − 1 2 σ 2 2 t + σ 2 B 2 ] (34)

E [ x 2 ] = exp [ − 1 2 σ 2 2 t ] → 0 as t → ∞ . (35)

Solving for x 1 in terms of x 2 , we obtain

x 1 = x 2 exp [ κ x 2 t ] 1 + exp [ κ x 2 t ] = exp [ − 1 2 σ 2 2 t + σ 2 B 2 ] exp [ κ exp [ − 1 2 σ 2 2 t + σ 2 B 2 ] t ] 1 + exp [ κ exp [ − 1 2 σ 2 2 t + σ 2 B 2 ] t ] . (36)

As t → ∞ , x 2 → 0 , x 1 → 0 .

Remark: For σ 1 = 0 , σ 2 ≠ 0 the system (21)-(22) possesses one stable equilibrium point ( 0 , 0 ) which implies the collapse of business. This solution is unrealistic but it cautions against decisions such as price control without taking into account supply and demand. This is the case in the African beef-cattle market where fixing prices is usually a political decision.

Lemma 4

For σ 1 ≠ 0 , σ 2 = 0 the system (21)-(22) is reducible to an Ornstein-Uhlenbeck type process, which yields Ornstein-Uhlenbeck gains or losses.

Proof

Note that for σ 1 ≠ 0 , σ 2 = 0 the system (21)-(22) becomes,

d x 1 = κ ( m − x 1 ) x 1 d t + σ 1 x 1 d B 1 , (37)

which can be written in the form,

d x 1 = f ( x 1 ) d t + g ( x 1 ) d B 1 , (38)

where,

f ( x 1 ) = κ ( m − x 1 ) x 1 and g ( x 1 ) = σ 1 . (39)

Defining,

A ( x ) = f ( x ) g ( x ) − 1 2 g ′ ( x ) , (40)

then based on Gard (1988, chapter 4) [

( ( g A ′ ) ′ A ′ ) = 0. (41)

From (39), it is easy to show that

( ( g A ′ ) ′ A ′ ) = ( g ′ A ′ + g A ″ A ′ ) = ( g ′ A ′ A ′ ) = A ′ ( g ′ A ′ ) ′ − ( g ′ A ′ ) A ″ ( A ′ ) 2 = A ′ ( g ′ A ″ + A ′ g ″ ) − ( g ′ A ′ ) A ″ ( A ′ ) 2 = 0. (42)

Following Gard (1988, chapter 4) [

x 1 = exp { ( m − 1 2 σ 1 2 ) t + σ 1 2 B 1 , t } x 1 , 0 − 1 + m ∫ 0 t exp { ( κ m − 1 2 σ 1 2 ) s + σ 1 B 1 , s } d s . (43)

If we rewrite (37) as,

d x 1 x 1 = κ ( m − x 1 ) d t + σ 1 d B 1 , (44)

then, we can see that the left hand side of (44) represents the gains or losses, while the right hand side represents an Ornstein-Uhkenbeck process. We shall compute the solution (43) and the Ornstein-Uhlenbeck gains or losses (44) numerically.

Proposition 1

For σ 1 ≠ 0 , σ 2 ≠ 0 , the system (21)-(22) yields Ornstein-Uhlenbeck gains or losses that revert to a fluctuating mean, m.

Remark: The system (21)-(22) can be written as,

d x 1 x 1 = κ ( x 2 − x 1 ) d t + σ 1 d B 1 (45)

d x 2 = σ 2 x 2 d B 2 . (46)

Proof

The approach used in lemma (4) can be used to prove proposition (1) and the system (45)-(46) has the following solution,

x 1 = exp { ( x 2 − 1 2 σ 1 2 ) t + σ 1 2 B 1 , t } x 1 , 0 − 1 + x 2 ∫ 0 t exp { ( κ x 2 − 1 2 σ 1 2 ) s + σ 1 B 1 , s } d s (47)

x 2 = exp [ − 1 2 σ 2 2 t + σ 2 B 2 ] . (48)

Clearly, the relative gains or losses in price (right hand side of (45) is of Ornstein-Uhlenbeck type and are level dependent (mean reverting to x 2 given in (34). We present the solution for σ 1 ≠ 0 and σ 2 ≠ 0 in section (5) numerically.

In this section, we present the results of the special cases in Section 4.1. We explore the impact of changes in the level of noise (changes in σ 1 and σ 2 ) on, among other things, the stability of the steady states and its implications.

small values of σ 1 . This represents a stable equilibrium point which is degraded by the noise.

As we increase the volatility σ 1 the fluctuations about a constant mean increase significantly as depicted in Figures 4(b)-(d). Economically, an investor receiving a price x 1 subjected to a constant mean x 2 is subjected to a fair market with non-arbitrage conditions as one is equally subjected to price increase as well as price decrease about the mean.

However, if both noises (σ_{1} and σ_{2}) are large (

_{1}) on the price x_{1} for varying values of σ_{1} and σ_{2}. _{1} and σ_{2} are zero. The mean price x_{2} remains constant (assumed to be zero in this case) but r_{1} increases initially to a peak but declines to zero and remains zero. In the long run if there is no noise, there is no benefit from selling cattle but more importantly the BMC is subjected to no growth in returns.

cattle price and the beef price fluctuate and consequently removing any arbitrage opportunities.

We have shown through a bivariate analysis in Section 2.2 that the price the EU offers to the BMC and the price the BMC offers to the cattle farmers are highly correlated with the difference between them explained by a white noise process. This suggests that the middle agent, in this case the BMC, is left with insignificant amounts to support their operations such as maintenance and salaries. Based on this conclusion, we formulated a stochastic model of the Ornstein-Uhlenbeck type with the BMC-EU price as the stochastic mean of the Farmer-BMC price. We have shown that when both volatilities are zero ( σ 1 = σ 2 = 0 ), the Farmer-BMC price increases to the mean price and remains at that level. This situation is of course uneconomical for BMC as they operate as an intermediary agent which is passing on to the farmer everything received from selling beef to the EU. BMC would obviously not manage to maintain their operations as they would have no capital to service their processing plant and pay salaries. BMC has in recent years run into operational problems, with operations subsidized from government handouts. The situation described for σ 1 = σ 2 = 0 could partly explain some of the difficulties the company is subjected to.

When the volatility for the farmers’ price is kept at σ 1 = 0 , but the volatility σ 2 is nonzero, the farmers’ price fluctuates about the EU mean price. There is then a quasi but variable stability in the sense that the prices x 1 ( t ) and x 2 ( t ) remain close. However, when the volatility σ 2 grows large the difference in the two prices vanishes. This situation is again not conducive to the BMC. When the EU mean price is kept constant (by taking the volatility equal to zero) but the farmer price varies by increasing the volatility of the farmer price σ 1 , there is a big difference between the farmer price and the EU price. We believe that this is the preferred scenario for the farmer. It can be achieved by among other things, increasing and diversifying the number of beef consumers and cattle buyers locally and regionally. The farmers should look for local and regional buyers of their cattle instead of relying on the BMC. The BMC, in turn, should source regional markets for their beef instead of relying on the EU. Sub-Saharan Africa is a large market comprising of over 1 billion consumers. This is the market that BMC should target. We believe that government intervention is one of the reasons, the price that the BMC offers to the farmers is kept artificially high because the government is trying to eradicate poverty by artificially keeping the price of cattle high without due consideration of what price the EU is offering the BMC.

The authors acknowledge with gratitude the support from the Simons Foundation (US) through the Research and Graduate Studies in Mathematics (RGSMA) project at Botswana International University of Science and Technology (BIUST) and the Departments of Mathematics and Statistical Sciences of BIUST.

The authors declare no conflicts of interest regarding the publication of this paper.

Ziwakaya, P.K. and Lungu, E.M. (2021) Modeling Botswana Beef-Cattle Price Dynamics. Journal of Mathematical Finance, 11, 84-106. https://doi.org/10.4236/jmf.2021.111004

Year | S 1 ( t ) | S 1 ( t ) − S ¯ 1 | S 2 ( t ) | S 2 ( t ) − S ¯ 2 |
---|---|---|---|---|

1992 | 4.56 | −7.85 | 7.06 | −16.29 |

1993 | 4.17 | −824 | 7.75 | 15.61 |

1994 | 4.55 | −7.86 | 7.85 | −15.51 |

1995 | 4.62 | −7.79 | 6.94 | −16.42 |

1996 | 4.23 | −8.18 | 11.56 | −11.82 |

1997 | 4.17 | −8.24 | 10.95 | −12.41 |

1998 | 4.78 | −7.63 | 11.19 | −12.16 |

1999 | 4.65 | −7.76 | 12.31 | −11.04 |

2000 | 4.53 | −7.88 | 11.25 | −12.1 |

2001 | 4.45 | −7.96 | 2.45 | −10.91 |

2002 | 4.62 | −7.97 | 14.32 | −9.02 |

2003 | 4 | −8.41 | 15.45 | −7.9 |

2004 | 4.52 | −7.89 | 16.2 | −7.16 |

2005 | 6.71 | −5.7 | 17.19 | −6.17 |

2006 | 8.3 | −4.11 | 20.89 | −2.46 |

2007 | 12.13 | −0.8 | 24.38 | 1.03 |

2008 | 13.51 | 1.1 | 29.34 | 5.98 |

2009 | 19.51 | 7.1 | 32.27 | 8.92 |

2010 | 24.23 | 11.82 | 30.67 | 7.32 |

2011 | 23.64 | 11.23 | 31.31 | 7.96 |

2012 | 14.4 | 1.99 | 37.56 | 14.21 |

2013 | 15.45 | 3.04 | 41.35 | 17.21 |

2014 | 29 | 16.59 | 45.64 | 22.29 |

2015 | 12.5 | 0.09 | 41.78 | 18.43 |

2016 | 33 | 20.59 | 45.9 | 22.54 |

2017 | 32.73 | 20.32 | 42.57 | 19.22 |

2018 | 32.2 | 19.79 | 44.45 | 21.1 |

Mean ( | 12.41 | 23.35 |

The sum of residuals for the two prices are;

1) Farmer-BMC price: ∑ i = 1 n ( S 1 , i − S ¯ 1 ) = 0.00333 and

2) BMC-EU price: ∑ i = 1 n ( S 2 , i − S ¯ 2 ) = 0

Take the residual of b and find its mean and the results are presented in TableA2.

Year | S 1 ( t ) | S 2 ( t ) | b = ( S 2 ( t ) − S 1 ) | a |
---|---|---|---|---|

1992 | 4.56 | 7.062 | 2.5 | −8.44074 |

1993 | 4.17 | 7.75 | 3.58 | −7.36074 |

1994 | 4.55 | 7.85 | 3.3 | −7.64074 |

1995 | 4.62 | 6.94 | 2.32 | −8.62074 |

1996 | 4.23 | 11.56 | 7.31 | −3.63074 |

1997 | 4.17 | 10.95 | 6.78 | −4.16074 |

1998 | 4.78 | 11.19 | 6.41 | −4.53074 |

1999 | 4.65 | 12.31 | 7.66 | −3.28074 |

2000 | 4.53 | 11.25 | 6.72 | −4.22074 |

2001 | 4.45 | 2.45 | 8 | −2.94074 |

2002 | 4.62 | 14.32 | 9.7 | −1.24074 |

2003 | 4 | 15.45 | 11.45 | 0.50926 |

2004 | 4.52 | 16.2 | 11.68 | 0.73926 |

2005 | 6.71 | 17.19 | 10.48 | −0.46074 |

2006 | 8.3 | 20.89 | 12.59 | 1.64926 |

2007 | 12.13 | 24.38 | 12.25 | 1.30926 |

2008 | 13.51 | 29.34 | 15.83 | 4.88926 |

2009 | 19.51 | 32.27 | 12.76 | 1.81926 |

2010 | 24.23 | 30.67 | 6.44 | −4.50074 |

2011 | 23.64 | 31.31 | 7.67 | −3.27074 |

2012 | 14.4 | 37.56 | 23.16 | 12.21926 |

2013 | 15.45 | 41.35 | 25.9 | 14.95926 |

2014 | 29 | 45.64 | 16.64 | 5.69926 |

2015 | 12.5 | 41.78 | 29.28 | 18.33926 |

2016 | 33 | 45.9 | 12.9 | 1.95926 |

2017 | 32.73 | 42.57 | 9.84 | −1.10074 |

2018 | 32.2 | 44.45 | 12.25 | 1.30926 |

Mean ( | 12.4 | 10.94074 | 23.35 |

Where b = S 2 , i − S 1 , i . Note that the sum of the residuals a i is zero.

In line with least squares approximation, the system (21)-(22) can be be written as follows,

x 1 , t + 1 = x 1 , t + κ ( x 2 − x 1 , t ) x 1 , t + σ 1 x 1 , t ϵ 1 , t (50)

x 2 , t + 1 = σ 2 x 2 , t ϵ 2 , t . (51)

where ϵ 1 , t and ϵ 2 , t are standard normal variables. The technique for least squares is tied in estimating parameters by minimizing the squared errors of historical data, from one perspective, and their expected values on the other. The approach we used takes the form of a regression problem, where the variation in one variable, called the dependent variable Y, can be partly explained by the variation in the other variables, called independent variables say X. To estimate the parameters κ, x 2 , σ 1 and σ 2 we used weekly spot prices of beef-cattle from the EU markets for years 2003 to 2018. The choice of 2003 as the initial observation was mainly conditioned by the availability of data. We got weekly beef prices from an official website of the EU [

x 1 , t + 1 x 1 , t x 1 , t = κ x 2 − κ x 1 , t + σ 1 ϵ 1 , t (52)

x 2 , t 1 = x 2 , t σ 2 ϵ 2 , t (53)

Both Equations (52) and (53) bears the characteristics of a linear regression model, with the gain and loss given by x 1 , t + 1 x 1 , t x 1 , t and x 2 , t + 1 as dependent variables, x 1 , t and x 2 , t as explanatory variables. Following the work by [

Parameter | κ | m ( x 2 ) | x 1 ( 0 ) | σ 1 | σ 2 |
---|---|---|---|---|---|

EU Beef-cattle Price | 0.036 | 32.35 | 4.56 | 0.124 | 0.0210 |

Source: Authors’ estimates based on the EU market [

As a good start to the numerical methods for the system (21)-(22) we consider the Euler-Mayurama method to simulate the stochastic differential equations. Given the system (21)-(22), the Euler-Maruyama method generates a discrete sequence x t = { x t j } j ∈ { 1 , ⋯ , d } , which approximates the system on the interval [ 0 , T ] . From the system (21)-(22) we have,

x 1 , t i + 1 j = x 1 , t i j + κ ( x 2 , t i j − x 1 , t i ) x 1 , t i j Δ t , i + x 1 , t i j σ 1 ( W t i + 1 1 − W t i 2 ) = x 1 , t i j + κ ( x 2 , t i j − x 1 , t i ) x 1 , t i j Δ t , i + σ 1 x 1 , t i j t i + 1 − t 1 ϵ j (54)

x 2 , t i + 1 j = x 2 , t i j + x 2 , t i j σ 2 ( W t i + 1 2 − W t i 2 ) = σ 2 x 2 , t i j t i + 1 − t 1 ϵ j . (55)

where X t , 0 = X 0 , X t , 0 = ( x 1 , t 0 , x 2 , t 0 ) , Δ t , i = t i + 1 − t i , for i = 0 , ⋯ , N and ϵ is a Gaussian process with mean 0 and co-variance 𝟙 (identity matrix). It is an iterative technique as the solution of the stochastic differential equation is changed at every step.

In

The data used in this study consisted of n = 835 weekly observations and we obtained x 2 ( 0 ) = 32.35 . Agriculture commodities are transnational products of which the Botswana Beef-cattle is not exceptional and are traded usually having historical data series. In particular, the future volatility rate σ from the system (21)-(22) is estimated from the historic volatility of beef cattle prices, based on the standard deviation of the time series, given by equation:

The drift part of Equation (21) describes the time evolution of beef-cattle prices the continually yield at a rate κ ( x 2 − x 1 ) x 1 , which is represented as a combination of mean-reversion force κ, the average of the EU historical beef prices (assumed to be stochastic) and the cattle prices (Farmer-BMC).

In TableA4 are the notations that were used in this paper.

Notation | Description |
---|---|

BWP | Botswana Pula |

S ( t ) | Beef-cattle price at time t |

S ¯ | Average beef-cattle price |

S 1 ( t ) , S 2 ( t ) | Farmer-BMC price and BMC-EU respectively price at time |

( for simplicity we have denoted by x 1 and x 2 respectively) | |

a 1 , a 2 | Respective residuals for S 1 and S 2 |

r | The Pearson’s correlation coefficient |

r ( t ) | Rate of return |

r 1 , r 2 | Returns for x 1 and x 2 respectively |

x ′ | Prime symbol (for derivative) |

Integral sign | |

d ( . ) | Differential sign |

σ 1 , σ 2 | Rate of volatilities for Farmer-BMC price and BMC-EU respectively price |

ω ( B ) | Transfer function |

∇ S 1 , t d , ∇ S 2 , t d | Incremental changes for Farmer-BMC prices and BMC-EU prices respectively |

E [ . ] | Expectation operator |

T ∧ τ ϵ | Smaller between T and τ ϵ , where ϵ is an arbitrary small number |

β t = ϕ p ( B ) S 1 , t | Autoregressive process AR(p) of order p |

where, ϕ ( B ) = 1 − ϕ 1 B − ⋯ − ϕ p B p | |

α t = θ q ( B ) S 2 , t | Moving average MA(q) of order q |

where, θ ( B ) = 1 − θ 1 B − ⋯ − θ q B q | |

Δ t , i = t i − t i − 1 | Time interval |

ℙ ( . ) | Probability of ( . ) |