# Examination of the Cost-of-Carry Formula for Futures Contracts on WIG20. Wavelet and Nonlinear Cointegration Analysis

• Joanna Bruzda
Chapter
Part of the New Economic Windows book series (NEW)

## Abstract

The instantaneous and non-instantaneous dependences between spot and futures index prices has been subject of numerous empirical investigations. The theoretical background of these studies is the cost-ofcarry model introduced by [15]. The cost-of-carry model is an arbitrage relationship based on comparison between two alternative methods of acquiring an asset at some future date. In the first method an asset is purchased now and held until this future date. In the second case a futures contract with maturity on the required date is bought. The present value of the futures contract is invested at the risk free interest rate until delivery of the underlying asset at the maturity date. Arbitrage should ensure that the difference between the current asset price and the futures contract price is the cost of carrying the asset, which involves dividend yields and interest rates. The cost-of-carry formula gives the fair price of the futures contract:
$$F_{t,T} = S_t e^{\left( {r_t - d_t } \right)\left( {T - t} \right)}$$
(1)
where St is the security index price at time t, F t,T is the index futures price at time t with maturity T, r t is the risk free interest rate, d t is the dividend yield on the security index, and (T — t) is the time to maturity of the futures contract. Taking logarithms of both sides of equation (1) we get:
$$f_{t,T} = s_t + \left( {r_t - d_t } \right)\left( {T - t} \right)$$
(2)

## Keywords

Discrete Wavelet Transform Unit Root Test Continuous Wavelet Transform Cointegration Test Future Price
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Abhyankar A. (1998) Linear and Nonlinear Granger Causality: Evidence from the UK Stock Index Futures Market. Journal of Futures Market 18:519–540
2. 2.
Aparicio F.M., Escribano A. (1998) Information-Theoretic Analysis of Serial Dependence and Cointegration. Studies in Nonlinear Dynamics and Econometrics 3:119–140
3. 3.
Arranz M.A., Escribano A. (2004) Outliers-Robust ECM Cointegration Tests Based on the Trend Components. Spanish Economic Review 6:243–266
4. 4.
Beak E., Brock W. (1992) A Nonparametric Test for Independence of a Multivariate Time Series. Statistica Sinica 2:137–156
5. 5.
Biakowski J., Jakubowski J. (2003) The Test of Market Efficiency and Index Arbitrage Profitability on Emerging Polish Stock and Futures Index Markets. Postgraduate Research Programme: Capital Markets and Finance in the Enlarged Europe. Europa-Universität Viadrina 24:1–26Google Scholar
6. 6.
Breiman L., Friedman J.H. (1985) Estimating Optimal Transformations for Multiple Regression and Correlation. Journal of the American Statistical Association 80:580–598
7. 7.
Breitung J. (2001) Rank Tests for Nonlinear Cointegration. Journal of Business & Economic Statistics 19:331–340
8. 8.
Brooks C., Garrett I., Hinich M.J. (1999) An Alternative Approach to Investigating Lead-Lag Relationships between Stock and Stock Index Futures Markets. Applied Financial Economics 9:605–613.
9. 9.
Brooks C., Hinich M.J. (1999), Cross-Correlations and Cross-Bicorrelations in Sterling Exchange Rates. Journal of Empirical Finance 6:385–404
10. 10.
Bruzda J. (2006) Empirical Verification of Money Demand Models: Non-Linear Cointegration Analysis. Dynamic Econometric Models 7:113–123Google Scholar
11. 11.
Bruzda J. (2007) The Cost-of-Carry Model for the FW20 Futures Contracts. Threshold Cointegration Framework in: Milo W, Wrzesiski P (eds) Forecasting Financial Markets and Economic Decision-Making. Acta Universitatis Lodzienskis, Lodz, in pressGoogle Scholar
12. 12.
Bruzda J., Wisniewska E. (2002) Investigating Dependences between Futures and Spot Prices on an Example of Futures Contracts on WIG20 in: Tarczynski W (ed) Capital Market. Effective Investing (in Polish). Wydawnictwo Naukowe Uniwersytetu Szczecinskiego, SzczecinGoogle Scholar
13. 13.
Chang Y., Park J.Y. (2003) A Sieve Bootstrap for the Test of a Unit Root. Journal of Time Series Analysis 24:379–400
14. 14.
Chaouachi S., Dufrenot G., Mignon V. (2004) Modelling the Misalignments of the Dollar-Sterling Real Exchange Rate: A Nonlinear Cointegration Perspective. Economics Bulletin 3:1–11Google Scholar
15. 15.
Cornell B., French K. (1983) The Pricing of Stock Index Futures. Journal of Futures Markets 3:1–14
16. 16.
Diks C., Panchenko V. (2005) A Note on the Hiemstra-Jones Test for Granger Non-Causality. Studies in Nonlinear Dynamics and Econometrics 9: article 4.Google Scholar
17. 17.
Dufrenot G., Mignon V. (2002) Recent Developments in Nonlinear Cointegration with Applications to Macroeconomics and Finance. Kluwer Academic Publishers, Boston
18. 18.
Dwyer G.P., Locke P., Yu W. (1996) Index Arbitrage and Nonlinear Dynamics between the SP500 Futures and Cash. Review of Financial Studies 9:301–332
19. 19.
Gençay R.F., Selcuk F., Whitcher B. (2002) An Introduction to Wavelets and Other Filtering Methods in Finance and Economics. Academic Press, San Diego
20. 20.
Granger C.W.J. (1995) Modelling Nonlinear Relationships Between Extendend-Memory Variables. Econometrica 63:265–279
21. 21.
Granger C.W.J., Hallman J.J. (1991) Long-Memory Processes With Attractors. Oxford Bulletin of Economics and Statistics 53:11–26
22. 22.
Harris D., Inder B. (1994) A Test of the Null of Cointegration in: Hargreaves CP (ed) Nonstationary Time Series Analysis and Cointegration. Oxford University Press, Oxford.Google Scholar
23. 23.
Hiemstra C., Jones J.D. (1995) Testing for Linear and Nonlinear Granger Causality in the Stock Price-volume Relation. Journal of Finance 49:1639–1664
24. 24.
Kawaller I.G., Koch P.D., Koch T.W. (1987) The Temporal Price Relationship between S&P 500 Futures and the S&P 500 Index. Journal of Finance 42:1309–1329
25. 25.
Kuo B-S, Tsong C-C (2005) Bootstrap Inference for Stationarity. Discussion Paper 50, Helsinki Center of Economic ResearchGoogle Scholar
26. 26.
Kwiatkowski D., Phillips P.C.B., Schmidt P., Shin Y. (1992) Testing the Null Hypothesis of Stationarity Against the Alternative of A Unit Root. Journal of Econometrics 54:159–178
27. 27.
Leybourne S.J., McCabe B.P.M. (1993) A Simple Test for Cointegration. Oxford Bulletin of Economics and Statistics 55:97–103
28. 28.
Lin S-J, Stevenson M. (2001) Wavelet Analysis of Index Prices in Futures and Cash Markets: Implication for the Cost-Of-Carry Model. Studies in Nonlinear Dynamics and Econometrics 5:87–102
29. 29.
Martens M., Kofman P., Vorst TCF (1998) A Threshold Error-Correction Model for Intraday Futures and Index Returns. Journal of Applied Econometrics 13:245–263
30. 30.
Misiti M., Misiti Y., Oppenheim G., Poggi J-M (1996) Wavelet Toolbox For Use with MATLAB. The MathWorksGoogle Scholar
31. 31.
Percival D.B., Walden A.T. (2000) Wavelet Methods for Time Series Analysis. Cambridge University Press, Cambridge
32. 32.
Phillips P.C.B., Hansen B.E. (1990) Statistical Inference in Instrumental Variables Regression with I(1) Processes. Review of Economic Studies 57:99–125
33. 33.
Ramsey J.B. (1999) The Contribution of Wavelets to the Analysis of Economic and Financial Data. Philosophical Transactions of the Royal Society of London Series A 357:2593–2606
34. 34.
Ramsey J.B., Lampart C. (1998) The Decomposition of Economic Relationships by Time Scale Using Wavelets: Expenditure and Income. Studies in Nonlinear Dynamics and Econometrics 3:23–42
35. 35.
Robles-Fernandez M.D., Nieto L., Fernandez M.A. (2004), Nonlinear Intraday Dynamics in Eurostoxx50 Index Markets. Studies in Nonlinear Dynamics and Econometrics 8:article 3Google Scholar
36. 36.
Schleicher C. (2002) An Introduction to Wavelets for Economists. Working Paper 2002-3, Bank of CanadaGoogle Scholar
37. 37.
Tsay R.S. (1998) Testing and Modeling Multivariate Threshold Models. Journal of the American Statistical Association 93:1188–1202