Three Alternative Methods for Estimating Hedge Ratios

Abstract

This chapter first discusses four different theoretical models, which include minimum variance, mean-variance, expected utility, and value-at-risk method. Then we use S&P 500 data to show how three alternative estimation methods can be used to estimate hedge ratio. These three methods include OLS method, GARCH method, and cointegration and error correction method. We found that OLS method is not sufficient for estimating hedge ratio.

Appendices

Appendix A. Monthly Data of S&P500 Index and Its Futures (January 2005 – August 2020)

Date	SPOT	FUTURES	C_spot	C_futures
1/31/2005	1181.27	1181.7	−30.65	−32
2/28/2005	1203.6	1204.1	22.33	22.4
3/31/2005	1180.59	1183.9	−23.01	−20.2
4/29/2005	1156.85	1158.5	−23.74	−25.4
5/31/2005	1191.5	1192.3	34.65	33.8
6/30/2005	1191.33	1195.5	−0.17	3.2
7/29/2005	1234.18	1236.8	42.85	41.3
8/31/2005	1220.33	1221.4	−13.85	−15.4
9/30/2005	1228.81	1234.3	8.48	12.9
10/31/2005	1207.01	1209.8	−21.8	−24.5
11/30/2005	1249.48	1251.1	42.47	41.3
12/30/2005	1248.29	1254.8	−1.19	3.7
1/31/2006	1280.08	1283.6	31.79	28.8
2/28/2006	1280.66	1282.4	0.58	−1.2
3/31/2006	1294.83	1303.3	14.17	20.9
4/28/2006	1310.61	1315.9	15.78	12.6
5/31/2006	1270.09	1272.1	−40.52	−43.8
6/30/2006	1270.2	1279.4	0.11	7.3
7/31/2006	1276.66	1281.8	6.46	2.4
8/31/2006	1303.82	1305.6	27.16	23.8
9/29/2006	1335.85	1345.4	32.03	39.8
10/31/2006	1377.94	1383.2	42.09	37.8
11/30/2006	1400.63	1402.9	22.69	19.7
12/29/2006	1418.3	1428.4	17.67	25.5
1/31/2007	1438.24	1443	19.94	14.6
2/28/2007	1406.82	1408.9	−31.42	−34.1
3/30/2007	1420.86	1431.2	14.04	22.3
4/30/2007	1482.37	1488.4	61.51	57.2
5/31/2007	1530.62	1532.9	48.25	44.5
6/29/2007	1503.35	1515.4	−27.27	−17.5
7/31/2007	1455.27	1461.9	−48.08	−53.5
8/31/2007	1473.99	1476.7	18.72	14.8
9/28/2007	1526.75	1538.1	52.76	61.4
10/31/2007	1549.38	1554.9	22.63	16.8
11/30/2007	1481.14	1483.7	−68.24	−71.2
12/31/2007	1468.35	1477.2	−12.79	−6.5
1/31/2008	1378.55	1379.6	−89.8	−97.6
2/29/2008	1330.63	1331.3	−47.92	−48.3
3/31/2008	1322.7	1324	−7.93	−7.3
4/30/2008	1385.59	1386	62.89	62
5/30/2008	1400.38	1400.6	14.79	14.6
6/30/2008	1280	1281.1	−120.38	−119.5
7/31/2008	1267.38	1267.1	−12.62	−14
8/29/2008	1282.83	1282.6	15.45	15.5
9/30/2008	1166.36	1169	−116.47	−113.6
10/31/2008	968.75	967.3	−197.61	−201.7
11/28/2008	896.24	895.3	−72.51	−72
12/31/2008	903.25	900.1	7.01	4.8
1/30/2009	825.88	822.5	−77.37	−77.6
2/27/2009	735.09	734.2	−90.79	−88.3
3/31/2009	797.87	794.8	62.78	60.6
4/30/2009	872.81	870	74.94	75.2
5/29/2009	919.14	918.1	46.33	48.1
6/30/2009	919.32	915.5	0.18	−2.6
7/31/2009	987.48	984.4	68.16	68.9
8/31/2009	1020.62	1019.7	33.14	35.3
9/30/2009	1057.08	1052.9	36.46	33.2
10/30/2009	1036.19	1033	−20.89	−19.9
11/30/2009	1095.63	1094.8	59.44	61.8
12/31/2009	1115.1	1110.7	19.47	15.9
1/29/2010	1073.87	1070.4	−41.23	−40.3
2/26/2010	1104.49	1103.4	30.62	33
3/31/2010	1169.43	1165.2	64.94	61.8
4/30/2010	1186.69	1183.4	17.26	18.2
5/31/2010	1089.41	1088.5	−97.28	−94.9
6/30/2010	1030.71	1026.6	−58.7	−61.9
7/30/2010	1101.6	1098.3	70.89	71.7
8/31/2010	1049.33	1048.3	−52.27	−50
9/30/2010	1141.2	1136.7	91.87	88.4
10/29/2010	1183.26	1179.7	42.06	43
11/30/2010	1180.55	1179.6	−2.71	−0.1
12/31/2010	1257.64	1253	77.09	73.4
1/31/2011	1286.12	1282.4	28.48	29.4
2/28/2011	1327.22	1326.1	41.1	43.7
3/31/2011	1325.83	1321	−1.39	−5.1
4/29/2011	1363.61	1359.7	37.78	38.7
5/31/2011	1345.2	1343.9	−18.41	−15.8
6/30/2011	1320.64	1315.5	−24.56	−28.4
7/29/2011	1292.28	1288.4	−28.36	−27.1
8/31/2011	1218.89	1217.7	−73.39	−70.7
9/30/2011	1131.42	1126	−87.47	−91.7
10/31/2011	1253.3	1249.3	121.88	123.3
11/30/2011	1246.96	1246	−6.34	−3.3
12/30/2011	1257.6	1252.6	10.64	6.6
1/31/2012	1312.41	1308.2	54.81	55.6
2/29/2012	1365.68	1364.4	53.27	56.2
3/30/2012	1408.47	1403.2	42.79	38.8
4/30/2012	1397.91	1393.6	−10.56	−9.6
5/31/2012	1310.33	1309.2	−87.58	−84.4
6/29/2012	1362.16	1356.4	51.83	47.2
7/31/2012	1379.32	1374.6	17.16	18.2
8/31/2012	1406.58	1405.1	27.26	30.5
9/28/2012	1440.67	1434.2	34.09	29.1
10/31/2012	1412.16	1406.8	−28.51	−27.4
11/30/2012	1416.18	1414.4	4.02	7.6
12/31/2012	1426.19	1420.1	10.01	5.7
1/31/2013	1498.11	1493.3	71.92	73.2
2/28/2013	1514.68	1513.3	16.57	20
3/29/2013	1569.19	1562.7	54.51	49.4
4/30/2013	1597.57	1592.2	28.38	29.5
5/31/2013	1630.74	1629	33.17	36.8
6/28/2013	1606.28	1599.3	−24.46	−29.7
7/31/2013	1685.73	1680.5	79.45	81.2
8/30/2013	1632.97	1631.3	−52.76	−49.2
9/30/2013	1681.55	1674.3	48.58	43
10/31/2013	1756.54	1751	74.99	76.7
11/29/2013	1805.81	1804.1	49.27	53.1
12/31/2013	1848.36	1841.1	42.55	37
1/31/2014	1782.59	1776.6	−65.77	−64.5
2/28/2014	1859.45	1857.6	76.86	81
3/31/2014	1872.34	1864.6	12.89	7
4/30/2014	1883.95	1877.9	11.61	13.3
5/30/2014	1923.57	1921.5	39.62	43.6
6/30/2014	1960.23	1952.4	36.66	30.9
7/31/2014	1930.67	1924.8	−29.56	−27.6
8/29/2014	2003.37	2001.4	72.7	76.6
9/30/2014	1972.29	1965.5	−31.08	−35.9
10/31/2014	2018.05	2011.4	45.76	45.9
11/28/2014	2067.56	2066.3	49.51	54.9
12/31/2014	2058.9	2052.4	−8.66	−13.9
1/30/2015	1994.99	1988.4	−63.91	−64
2/27/2015	2104.5	2102.8	109.51	114.4
3/31/2015	2067.89	2060.8	−36.61	−42
4/30/2015	2085.51	2078.9	17.62	18.1
5/29/2015	2107.39	2106	21.88	27.1
6/30/2015	2063.11	2054.4	−44.28	−51.6
7/31/2015	2103.84	2098.4	40.73	44
8/31/2015	1972.18	1969.2	−131.66	−129.2
9/30/2015	1920.03	1908.7	−52.15	−60.5
10/30/2015	2079.36	2073.7	159.33	165
11/30/2015	2080.41	2079.8	1.05	6.1
12/31/2015	2043.94	2035.4	−36.47	−44.4
1/29/2016	1940.24	1930.1	−103.7	−105.3
2/29/2016	1932.23	1929.5	−8.01	−0.6
3/31/2016	2059.74	2051.5	127.51	122
4/29/2016	2065.3	2059.1	5.56	7.6
5/31/2016	2096.96	2094.9	31.66	35.8
6/30/2016	2098.86	2090.2	1.9	−4.7
7/29/2016	2173.6	2168.2	74.74	78
8/31/2016	2170.95	2169.5	−2.65	1.3
9/30/2016	2168.27	2160.4	−2.68	−9.1
10/31/2016	2126.15	2120.1	−42.12	−40.3
11/30/2016	2198.81	2198.8	72.66	78.7
12/30/2016	2238.83	2236.2	40.02	37.4
1/31/2017	2278.87	2274.5	40.04	38.3
2/28/2017	2363.64	2362.8	84.77	88.3
3/31/2017	2362.72	2359.2	−0.92	−3.6
4/28/2017	2384.2	2380.5	21.48	21.3
5/31/2017	2411.8	2411.1	27.6	30.6
6/30/2017	2423.41	2420.9	11.61	9.8
7/31/2017	2470.3	2468	46.89	47.1
8/31/2017	2471.65	2470.1	1.35	2.1
9/29/2017	2519.36	2516.1	47.71	46
10/31/2017	2575.26	2572.7	55.9	56.6
11/30/2017	2647.58	2647.9	72.32	75.2
12/29/2017	2673.61	2676	26.03	28.1
1/31/2018	2823.81	2825.8	150.2	149.8
2/28/2018	2713.83	2714.4	−109.98	−111.4
3/30/2018	2640.87	2643	−72.96	−71.4
4/30/2018	2648.05	2647	7.18	4
5/31/2018	2705.27	2705.5	57.22	58.5
6/29/2018	2718.37	2721.6	13.1	16.1
7/31/2018	2816.29	2817.1	97.92	95.5
8/31/2018	2901.52	2902.1	85.23	85
9/28/2018	2913.98	2919	12.46	16.9
10/31/2018	2711.74	2711.1	−202.24	−207.9
11/30/2018	2760.17	2758.3	48.43	47.2
12/31/2018	2506.85	2505.2	−253.32	−253.1
1/31/2019	2704.1	2704.5	197.25	199.3
2/28/2019	2784.49	2784.7	80.39	80.2
3/29/2019	2834.4	2837.8	49.91	53.1
4/30/2019	2945.83	2948.5	111.43	110.7
5/31/2019	2752.06	2752.6	−193.77	−195.9
6/28/2019	2941.76	2944.2	189.7	191.6
7/31/2019	2980.38	2982.3	38.62	38.1
8/30/2019	2926.46	2924.8	−53.92	−57.5
9/30/2019	2976.74	2978.5	50.28	53.7
10/31/2019	3037.56	3035.8	60.82	57.3
11/29/2019	3140.98	3143.7	103.42	107.9
12/31/2019	3230.78	3231.1	89.8	87.4
1/31/2020	3225.52	3224	−5.26	−7.1
2/28/2020	2954.22	2951.1	−271.3	−272.9
3/31/2020	2584.59	2569.7	−369.63	−381.4
4/30/2020	2912.43	2902.4	327.84	332.7
5/29/2020	3044.31	3042	131.88	139.6
6/30/2020	3100.29	3090.2	55.98	48.2
7/31/2020	3271.12	3263.5	170.83	173.3
8/31/2020	3500.31	3498.9	229.19	235.4

Appendix B. Applications of R Language in Estimating Three Different Optimal Hedge Ratios

In this appendix, we show the estimation procedure on how to apply OLS, GARCH, and CECM models to estimate optimal hedge ratios through R language . R language is a high-level computer language that is designed for statistics and graphics. Compared to alternatives, SAS, Matlab or Stata, R is completely free. Another benefit is that it is open source. Users could head to http://cran.r-project.org/ to download and install R language. Based upon monthly S&P 500 index and its futures as presented in Appendix A, the estimation procedures of applying R language to estimate hedge ratio are provided as follows.

First, we use OLS method in term of Eq. (12) to estimate minimum variance hedge ratio. By using linear model (lm) function in R language , we obtain the following program code.

SP500= read.csv(file="SP500.csv") OLS.fit <- lm(C_spot~C_futures, data=SP500) summary(OLS.fit)

Next, we apply a conventional regression model with an AR(2)-GARCH(1,1) error terms to estimate minimum variance hedge ratio . By using rugarch package in R language, we obtain the following program.

library(rugarch) fit.spec <- ugarchspec( variance.model = list(model = "sGARCH", garchOrder = c(1, 1)), mean.model = list(armaOrder = c(2, 0),include.mean = TRUE, external.regressors= cbind(SP500$C_futures)), distribution.model = "norm") GARCH.fit <- ugarchfit(data = cbind(SP500$C_spot), spec = fit.spec) GARCH.fit

Third, we apply the ECM model to estimate minimum variance hedge ratio . We begin by applying an augmented Dickey-Fuller (ADF) test for the presence of unit roots. The Phillips and Ouliaris (1990) residual cointegration test is applied to examine the presence of cointegration. Finally, the minimum variance hedge ratio is estimated by the error correction model. By using tseries package in R language, we obtain the following program.

library(tseries)

# Augmented Dickey-Fuller Test

# Level data

adf.test(SP500$SPOT, k = 1) adf.test(SP500$FUTURES, k = 1)

# First-order differenced data

adf.test(diff(SP500$SPOT), k = 1) adf.test(diff(SP500$FUTURES), k = 1)

# Phillips and Ouliaris (1990) residual cointegration test

po.test(cbind(SP500$FUTURES,SP500$SPOT))

# Engle-Granger two-step procedure

## 1.Estimate cointegrating relationship

reg <- lm(SPOT~FUTURES, data=SP500)

## 2. Compute error term

Resid <- reg$resid

# Estimate optimal hedge ratio using the error correction model

n=length(resid) ECM.fit <-lm(diff(SPOT) ~ -1 + diff(FUTURES) + Resid[1:n-1], data=SP500) summary(ECM.fit)

Finally, we apply a multivariate GARCH(1,1)-BEKK model to estimate dynamic minimum variance hedge ratio.

library(mgarchBEKK) estimated=BEKK(as.matrix(cbind(SP500$C_futures,SP500$C_spot)), order = c(1, 1)) estimated$est.params estimated$asy.se.coef