Introducing a New Coffee Futures Pricing Model for the Nairobi Securities Exchange

Transcription

1 Introducing a New Coffee Futures Pricing Model for the Nairobi Securities Exchange Abstract Githinji Rosebell 1 and Muthoni Lucy 2 November 2016 This study intends to aly a different ricing model for ricing coffee futures at the Nairobi Securities Exchange and suggest an imrovement of the existing ricing model currently used at the NSE. This will be done by using Schwartz s model 1 to generate coffee Futures rices, and comare the rices generated by the model with the model currently used to rice coffee futures at the NSE. The difference between the series of rices generated by the two models will be tested for significance, and thus guide on whether introducing a new ricing model for the coffee Exchange will have an imact on the current ricing model. Keywords: Coffee Futures, Futures Pricing Models, Nairobi Securities Exchange 1 BBS Actuarial Science Student, Strathmore Institute of Mathematical Sciences, Strathmore University P. O. Box: , Nairobi rosebell.githinji@strathmore.edu; githinjirose@gmail.com 2 Assistant Lecturer, Strathmore Institute of Mathematical Sciences (SIMS), Strathmore University P. O. Box: , Nairobi LMuthoni@strathmore.edu; lucymuthoni2003@gmail.com

2 INTRODUCTION 1.1. Background Information An agreement, which is standardized, between a seller and a buyer to exchange an underlying item for a articular rice at a re-secified date is called a future contract. The underlying item may be a metal, agricultural commodity, mineral or an energy commodity, financial roduct or a foreign currency. The Futures contract is a derivative security as its value is wholly based on the value of the underlying asset. Unlike forward contracts, futures contracts are exchange traded and regulated, standardized and marked to market. In the last decade, the size of Futures Exchanges have grown significantly in Africa, Asia and Latin America based on the ground that there is need in articular countries for a way of dealing with rice volatility and roviding rice discovery. However, according to Caital Market Authority (2013) reort, 2 out of 3 contracts fail as they have not been designed roerly desite having government and donor agencies suort. Use of futures markets for rice risk management uroses also known as hedging as oosed to seculation, it is a tool to realize better rices, but a way to obtain more certainty about the rices one can exect to realize. Greater redictability, in turn, makes it ossible to make better decisions and to obtain credit at better terms, indirect benefits, rather that better rice realizations, are what allows those who use rice risk management. However according to Lamon and Frida (2007), use of organized futures and otions markets can be cumbersome for the develoing countries roducer considering the stes involved in coming u with an exchange and a clearing house. An exchange is basically like a club whose members are the futures brokers and it is where they execute the trades. The clearing house is resonsible for the settlement and guarantee of all trades on the exchange Coffee in Kenya Coffee is one of Kenya s to foreign exchange earners coming fifth behind tea, tourism, horticulture and remittances from the diasora. According to (22and August 2016), coffee farmers are the last class of economic slaves in Kenya. They have no negotiating ower, knowledge and resonsibility over their cro.

3 The government of Kenya announced that the stes towards develoing institutional and legal frameworks to introduce a Commodities and Futures exchange are underway. A Futures Exchange will offer farmers stable rices and a ready market for their roduce as has been roven by Ethioian and Ugandan Commodities Exchanges, which has resulted in imroved outut in those countries esecially in Ethioia. This study intends to look at the imact the Commodities and Futures Exchange will have on the coffee rices. This will be done by using a futures ricing model to rice the coffee Futures, and comaring the rices generated by the model currently used to generate coffee rices. CHAPTER 2: LITERATURE REVIEW Many models consider the relations between rices of futures contracts and corresonding sot rices for instance Anderson (1983). In Schwartz (1998) article he develoed a one-factor model for the stochastic behavior of commodity rices that retains most of the characteristics of a more comlex two-factor stochastic convenience yield model in terms of its ability to rice the term structures of futures rices and volatilities. The model is based on the ricing and volatility results of the two-factor model. When alied to value long-term commodity rojects, it gives ractically the same results as the more comlex model. The inuts to the model are the current rices of all existing futures contracts (and their maturities) and the estimated arameters of the two-factor model. It only requires, however, the numerical solution corresonding to a simle one-factor model. Many models consider relations between rices of futures contracts and corresonding sot rices, e.g. Anderson (1983), Hirschileifer (1989) and (1990). We also see a textbook by Duffie (1989) trying to exlain the relationshi between the rices of futures contracts and corresonding rices, but alying the concet on ricing of sugar. Schwartz (1997) comared three models of stochastic behavior of commodity rices: a one factor model and three-factor models. Schwartz (1998) develoed a one-factor model that reserves the main characteristics of two-factor models. In this aer, we define the cost as in Black Scholes Merton (1987). For an introduction to the basic concets for the ricing of derivative assets and real otions under the uncertainty and incomlete information, we refer to Bellalah (1995) and (1999b). We use an extension of the analysis in the Schwartz (1997) and (1998) to account for the effects of incomlete information as it aears in

4 the models of Merton (1987) and Bellalah (2001). This aer uses the aforementioned extension to describe the stochastic behavior of commodity rices in the resence of mean reversion and shadow costs of incomlete information. CHAPTER 3: METHODOLOGY 3.1. Data The data that has been used in this study was obtained from Ethioian Coffee Exchange. This is because Ethioia has already already started trading in the coffee futures. Data and information used in the analyses of the roblem was gathered from secondary sources. The reason why we used Ethioian data is mainly because NSE is yet to introduce a futures trading. Other reasons include: The two markets are highly correlated as they are both located in the Eastern art of Africa,they are based on agriculture which face the same challenges in both countries e.g frequent draught and finally, the two countries are third world countries Futures Pricing Model In this model, Schwartz (1997) assumed that the commodity sot rice follows the stochastic rocess: ds = κ(μ ln S)Sdt + σsdz (1) Where dz is an increment to a standard Brownian motion and κ refers to the seed of adjustment. Describing X = ln S, and alying lto s lemma to characterize the log rice by an Ornstein- Uhlenbeck stochastic rocess, equation (1) becomes: dx = κ(α X)dt + σdz (2) With α = μ σ2 2κ (3) Where κ measures the degree of mean reversion to the long run mean log rice α. Under standard assumtions, Schwartz (1997) gives the following dynamics of the Ornstein-Uhlenbeck stochastic

5 rocess under the equivalent martingale measure: dx = κ(α X)dt + σdz (4) Where α = α λ and λ is the market rice of risk. λ Can be interreted as market volatility. We are going to calculate volatility using stochastic methods as illustrated in section 3.4. From equation (4), the conditional distribution of X at time T under the equaivalent martingale measure is normal. The mean and variance of X is: E 0 [X(T)] = e κt X(0) + (1 e κt )α Var[X(T)] = σ2 2κ (1 e 2κT ) (5) When the interest rate is constant, the futures or the forward rice of commodity corresonds to the exected rice of the commodity for the maturity T. Using the roerties of the log-normal distribution, the futures or the forward rice given by: F(S, T) = E[S(T)] = ex (E 0 [X(T)] Var 0[X(T)]) (6) And F(S, T) = ex (e κt ln S + (1 e κt )α + σ2 4κ (1 e 2κT )) (7) This equation can be written in a log form as ln F(S, T) = e κt ln S + (1 e κt ) α + σ2 Equation (7) is solution to the artial differential equation: 4κ (1 e 2κT ) (8). 1 2 σ2 S 2 F SS + (κ(μ λ) ln S (1 e κ T)α + σ2 4k (1 e 2κT ) (9) Under the terminal boundary condition F(S, 0) = S

6 Using this model, we are able to rice the Forward contract or a future contract since we can calculate all the arameters in the equation Volatility Estimation Method It is widely acknowledged that volatility varies with time and is redictable within statistical limits (Hull and White (1987)). Merton (1973) droed the assumtion that volatility in the Black- Scholes model is constant and extended the model to cover the situation in which the volatility is treated as a function of time, that is, volatility follows its own stochastic rocess. Researchers have roosed several stochastic rocesses for volatility. General stochastic volatility models These models assume that the asset rices evolve according to the geometric Brownian motion: ds = μdt + σdz S 1 (10) In which the volatility of the underlying asset evolves according to the Itô rocess given as: dσ = (S, σ, t)dt + q(s, σ, t)dz 2 (11) Where increments dz 1 and dz 2 are unit Wiener rocesses σ 1 = σ 2 = 1, Z 1 ~N(0, t), Z 1 ~N(0, t) The correlation of these rocesses remains an unknown a arameter ρ, imlicit in the theory, to be fitted to data in ractice. Let the value of the otion with stochastic volatility be given as V(S, σ, t), i.e. V is a function of three variables. It should be noted that although volatility is not a traded asset, one can hedge an otion with two other contracts, one being the underlying traded asset, and the other the volatility risk. To illustrate this, we consider a ortfolio which contains one otion of value C(S, σ, t), a quantity- of the underlying asset and a quantity- 1 of another shadow otion whose value is denoted as C 1 (S, σ, t). Here is taken as a coefficient and not as a finite difference oerator. The hedge ortfolio has value π = C(S, σ, t) S 1 C 1 (S, σ, t) (12)

7 The change of value of the ortfolio from time t to t + dt is given as Thus by Itô s lemma we have dπ = dc(s, σ, t) ds 1 dc 1 (S, σ, t) (13) dπ = ( C t σ2 S 2 2 C S 2 + ρσqs 2 C S σ q2 2 C σ 2) dt 1 ( C 1 t σ2 S 2 2 C 1 S 2 + ρσqs 2 C S σ 2 q2 2 C 1 σ 2 ) dt + ( C S 1 C 1 S ) ds + ( C σ 1 In order to eliminate randomness in equation (14), we choose C 1 ) dσ (14) σ ( C C 1 S 1 ) = 0 and ( C C 1 S σ 1 ) = 0 (15) σ The result is then given as: dπ = ( C t σ2 S 2 2 C S 2 + ρσqs 2 C S σ q2 2 C σ 2) dt 1 ( C 1 t σ2 S 2 2 C 1 S 2 + ρσqs 2 C S σ 2 q2 2 C 1 σ 2 ) dt (16) By arbitrage argument, we have the returns of the ortfolio equal to the risk-free rate, i.e. That is dπ = rπdt = r(c S 1 C 1 ) (17) ( C t σ2 S 2 2 C S 2 + ρσqs 2 C S σ q2 2 C σ 2) dt 1 ( C 1 t σ2 S 2 2 C 1 S 2 + ρσqs 2 C 1 S σ q2 2 C 1 σ 2 ) dt = r(c S 1C 1 )dt (18) To searate variables, we collect all the C terms on one side and all the C 1 terms on the other C t +1 2 σ2 S 2 2 C S 2+ρσqS 2 C S σ +1 2 q2 2 C σ 2+rS C S rc C σ = C1 t +1 2 σ2 S 2 2 C1 S 2 +ρσqs 2 C1 S σ +1 2 q2 2 C1 σ 2 rs C 1 S rc 1 C1 σ (19) The left-hand side of equation (19) is in terms of Conly and can ossibly be exressed as a function of indeendent variables S, σ, t (Wilmott (1998), ). Thus we have

8 C t σ2 S 2 2 C S 2 + ρσqs 2 C + 1 S σ 2 q2 2 C σ 2 + rs C S C + ( λq) rc = 0 (20) σ Where the searation constant λ(s, σ, t) is known as the market rice of (volatility) risk. In articular, for an underlying asset, if μ is the growth rate of the tradable asset, then (μ r) σ is the excess rate of return (above the risk-free rate) er unit risk- thus it is known as market rice of risk and is also referred to as Shairo ratio (see Lyuu (2002), 220; Hull (2000), 498; Wilmott (1998), 301). Under the simlifying assumtion that Wiener rocess Z 1 and Z 2 are not correlated then equation (20) becomes: C + 1 t 2 σ2 S 2 2 C + 1 S 2 2 q2 2 C σ 2 + rs C S C + ( λq) rc = 0 (21) σ This is a artial differential equation that is analogous to the Black-Scholes PDE, but accounts through λ for the shadow otion rice C 1. From the ersective of attern recognition for rocesses the PDEs are candidates for fitting a real rice history in which the volatility risk through λ has exerted an influence on market rices, and has to be estimated by aroriate methods Test Statistics The difference between the rices will be tested for significance using two statistical methods which are and rincial comonent analysis and coefficient of determination Princial Comonent Analysis According to (accessed on 23 rd October 2016) rincial Comonent Analysis (PCA) is a statistical rocedure that uses an orthogonal transformation to convert a set of observations of ossibly correlated variables into a set of values of linearly uncorrelated variables called rincial comonents. The number of rincial comonents is less than or equal to the number of original variables.

9 σ 1 2 σ 12 σ 1 Suose we have a random vector X defined as X = (X 1, X 2,, X n ) with a oulation variance- 2 σ covariance matrix defined as var(x) = 21 σ 2 σ 2. Consider the linear 2 σ ( 1 σ 2 σ ) combinations defined below: Y 1 = e 11 X 1 + e 12 X e 1 X (22) Y 2 = e 21 X 1 + e 22 X e 2 X Y = e 1 X 1 + e 2 X e X Where e i = e i1, e i2,, e i are regression coefficients. Y i is random with a oulation variance of var(y i ) = k=1 l=1 e ik e il σ kl = e i e i (23) And Y i and Y j will have a oulation covariance of cov(y i, Y j ) = k=1 l=1 e ik e jl σ kl = e i e j (24) Therefore, the first rincial comonent (PC1) Y 1 is the linear combination of x-variables that has maximum variance among all linear combinations, so it accounts for as much variation in the data as ossible. var(y 1 ) = k=1 l=1 e 1k e 1l σ kl = e 2 1 e 1 = j=1 e ij = 1 (25) The second rincial comonent (PC2) Y 2 is the linear combination of x-variables that accounts for as much of the remaining variation as ossible, with the constraint that the correlation between the first and the second comonent is 0. Thus the variance of PC2 is defined as var(y 2 ) = k=1 l=1 e 2k e 2l σ kl = e 2 e 2 = j=1 e 2j = 1 (26) 2

10 The first and the second rincial comonent will be uncorrelated with one another in the sense that Coefficient of Determination (R 2 ) cov(y 1, Y 2 ) = k=1 l=1 e 1k e 2l σ kl = e 1 e 2 = 0 (27) n 2 /(n k) n i=1(p i P 2 i ) /(n 1) R 2 = 1 i=1 (P i B i ) (28) Where, P is the average rice of all observed futures rices, B is the model rice of a futures i, n is the number of futures traded and k is the number of arameters needed to be estimated. It measures the success of the regression in redicting the values of the deendent variable within the samle. R 2 ranges from 0 to 1 ( accessed on 23 rd October 2016). A very high value of R 2 is therefore associated with a good fit of the observed rices while a small value is associated with a oor fit. Chater 4: Data analysis and finding 4.1. Emirical Results The aim of the study was to imrove on the existing coffee futures ricing model develoed for Kenyan markets Calibration results To calibrate arameter values in the above model, I used one factor Hull-White Model calibration method with a constant mean reversion. Number of observations 36 μ κ α σ σ

11 σ ρ λ ρ These calibrated results had already been estimated. From the estimated arameters I used κ and λ. Since α = μ σ2 2κ and α = α λ, then calculating the future rices is ossible using: F(S, T) = ex {e κt lns + (1 e κt )α + σ2 4κ (1 e 2κT )} (29) Where S is the closing rice, T is calculated at T = t/365 since this is daily data and μ is the rate of return calculated as follows: μ = 1 N (o N i t + c t ) i (30) Where o t + c t = log C t + log O t 4.3. Grahs Fig 4.4.1: Line grah comaring the model s rices and observed rices 1,200 1,100 1, M7 M8 M9 Closing Price Ct Future Prices

12 Fig 4.4.1: Bar grah comaring the model s rices and observed rices 1,200 1,100 1, M7 M8 M9 Closing Price Ct Future Prices As observed in figures and 4.4.2, the futures rices generated by the model are lower than the actual rices. This is because the model assumes that the markets are imerfect and therefore enalizes for the cost of getting information.

13 4.4. Princial comonent analysis Table 4.5.1: Princial Comonent Analysis between the roject s model and observed rices Princial Comonents Analysis Date: 11/01/16 Time: 12:05 Samle (adjusted): 7/22/2010 9/09/2010 Included observations: 36 after adjustments Balanced samle (listwise missing value deletion) Comuted using: Ordinary correlations Extracting 2 of 2 ossible comonents Eigenvalues: (Sum = 2, Average = 1) Cumulative Cumulative Number Value Difference Proortion Value Proortion Eigenvectors (loadings): Variable PC 1 PC 2 CLOSING_PRICE FUTURE_PRICES Ordinary correlations: CLOSING_PRICE_CT FUTURE_P... CLOSING_PRICE FUTURE_PRICES Regression Estimates In this section, we use definitions from (accessed on 23 rd October 2016) to define several arameters found in the generated result-table of regression estimates. These terms are: regression coefficients, standard errors, T-statistic, robability, r-squared, S.E. of regression, sum of squared residuals, Durbin-Watson statistic, Akaike information criterion and Schwartz criterion. After carrying out regression analysis of our data, we were able to generate the following results using eviews, an inbuilt software found in the Microsoft Office ackage, secifically within the Excel ackage. Using the results in the table, we will exlain what each term means and roceed to interret the results deicted.

14 Table 4.6.1: Regression Analysis between the roject s model and observed rices Deendent Variable: CLOSING_PRICE_CT Method: Least Squares Date: 11/01/16 Time: 12:16 Samle (adjusted): 7/22/2010 9/09/2010 Included observations: 36 after adjustments Variable Coefficient Std. Error t-statistic Prob. FUTURE_PRICES R-squared Mean deendent var Adjusted R-squared S.D. deendent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter Durbin-Watson stat Regression Coefficients Each coefficient multilies the corresonding variable in forming the best rediction of the deendent variable. The coefficient measures the contribution of its indeendent variable to the rediction. The coefficient of the series called C is the constant or intercet in the regression. It is the base level of the rediction when all of the other indeendent variables are zero. The other coefficients are interreted as the sloe of the relation between the corresonding indeendent variable and the deendent variable Standard Errors These measure the statistical reliability of the regression coefficients. The larger the standard error, the more statistical noise infects the coefficient. The standard error of the coefficient is which is small enough to make any imact Probability This is the robability of drawing a t-statistic of the magnitude of the one just to the left from a t- distribution. Since the robability that the true value C is zero is is less than our level of significance α = 0.05 we will fail to reject the null hyothesis that the true coefficient is zero.

15 R-squared This measures the success of how a variability of one factor can be caused by the relationshi with another factor. R 2 ranges between 0 and 1, 1 if the regression fits erfectly, and zero if it fits no better than the simle mean of the deendent variable R 2 is the fraction of the variance of the deendent variable exlained by the indeendent variables. It can be negative if the regression does not have an intercet or constant or if two-stage least squares is used. It is calculated using the formulae: Estimated variation Total variation Which can also be written as: 1 SSE TSS From our estimates R 2 is and hence it erfectly fits S.E. of regression This is a summary measure of the size of the rediction errors. It has the same units as the deendent variable. The Standard Error in our estimate is given as Durbin-Watson Statistic This is a test statistic for autocorrelation. The value ranges from 0-4. A value of two imlies there is autocorrelation and values that tend to 4 show negative autocorrelation and values tending to 0 imlies ositive autocorrelation. Since our Durbin-Watson test statistic ( ) less than 2, there is evidence of ositive serial correlation Akaike Information Criterion (AIC) It is a measure of how well a model fits a dataset adjusting the ability of the model to fit any dataset whether related or not. It is based on the sum of squared residuals but laces a enalty on extra coefficients. Under certain conditions, you can choose the length of a lag distribution, e.g. by choosing the secification with the lowest value of AIC. Our value of AIC is

16 Schwarz Criterion This is an alternative to the AIC and also known as Bayesian information criterion with basically the same interretation but a larger enalty for extra coefficients. The smaller the Schwartz Criterion, the better the fit of the model. The value for this model is Discussion and Conclusion The urose of this study was to comare our roject s model with the model develoed for Nairobi Stock Exchange by Muthoni et al (2016), and suggest ossible imrovements in the model Choice of Models Two different models were used. For this study, we used a model that assumes that the commodity sot rice follows the stochastic rocess: ds = κ(μ ln S)Sdt + σsdz (31) And thus the forward rices were given by: F(S, T) = ex (e κt ln S + (1 e κt )α + σ2 (1 4κ e 2κT )) (32) Muthoni et al (2016) used a three factor model; the three factors being the sot rice of the commodity, instantaneous convenience yield and the instantaneous interest rate. To futures rices were calculated using the formulae: F(s, δ, r, T) = S ex[ δ(1 e kt ) k Models Prices vis-à-vis Observed Prices + (r + λs)(1 e αt ) α + C(T)] Below is a comarison of the observed futures rices indicated by closing rice, and the model s futures rices develoed by Muthoni et al (2016). Fig 4.7.1: Muthoni et al (2016) Model vs. Observed Prices

17 2,200 2,000 1,800 1,600 1,400 1,200 1, M7 M8 M9 Closing Price Ct Future Prices Source: Lucy Muthoni, P.O. Box , Nairobi, Kenya one of the authors of Muthoni et al (2016). As the figure suggests, the model s rices are much higher comared to the observed rices. This does not reflect the resence of cost of information, which, when treated as a discount factor, leads to lower model rices comared to observed rices. Comaring the figure above with the figure discussed earlier, we see that this roject s model indeed roduces rices which are lower than the observed market rices, utting into consideration the atmoshere of markets with incomlete information Regression Estimates Results Under Muthoni et al (2016), R 2 calculated was In this study, the calculated R 2 is That means that this thesis model has a higher caability of redicting futures value comared to Muthoni et al (2016) s model Suggested Imrovements to Muthoni et al (2016) s Model Muthoni et al (2016) used a three factor model; the three factors being the sot rice of the commodity, instantaneous convenience yield and the instantaneous interest rate. To futures rices were calculated using the formulae:

18 F(s, δ, r, T) = S ex[ δ(1 e kt ) + (r+λs)(1 e αt ) + C(T)] (33) k α We noticed that under this model, the second and the third terms of the exonents are treated as cumulative function instead of discount functions. From dee analysis of this model, we would like to suggest the following edition: F(s, δ, r, T) = S e ( δ(1 e kt ) + (r+λs)(1 e αt ) +C(T)) k α (34) Equation (34) shows that every term in the exonent is treated as discount factor, therefore reflecting the cost of information which is C(T).

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