Some of the most popular futures contracts traded on the CME Group exchange are the Crude Oil futures contracts. Each contract is for delivery of 1000 barrels of oil, and there are separate contracts for delivery each month. For example: the Dec-20 contract trades until 20-Nov-2020 and the actual physical crude oil is delivered on the rst day of December. The daily prices of four consecutive futures contracts (Oct-20 (CLV0), Nov-20 (CLX0), Dec-20 (CLZ0) and Jan-21 (CLF1)) over the period 2-Jan-20 to 20-Aug-20 are in the Excel le WTIpricesAug20.xlsx provided (the tab Futures Prices Time Series). i) Using the last 3 months of data, estimate the annualized volatilities of each contract. What is the trend of these volatilities? This trend is known as the Samuelson's rule for futures volatilities. ii) Compare the historical volatilities estimated in point (i) of the Dec-20 and Jan-21 futures contracts with the implied volatilities of American options on these futures shown on the tab American Option Prices, also shown below in Fig. 1. Discuss the relationship between historical and implied volatilities observed.

Calls Strike Ticker Bid Ask Last IVM Dec-20 (89d American call 11/17/20); CSize options on Dec2020 1000: CLZO crude oil futures. 43.36 42.5 CLZOC 3.480000019 3.529999971 3.5 36.33633518 43 CLZOC 3.180000067 3.239999771 3.200000048 35.71901043 43.5 CLZOC 2.900000572 2.960000038 2.930000067 35.19056282 44 CLZOC 2.640000105 2.700000048 2.660000086 34.753654 44.5 CLZOC 2.400000095 2.450000048 2.420000076 34.39581761 Jan-21 (118d American call 12/16/20); CSize options on Jan2021 1000; CLF1 crude oil futures 43.63 42.5 CLF1C 4.25 4.320000172 4.289999962 37.53878728 43 CLF1C 3.949999809 4.019999981 3.99000001 37.07461697 43.5 CLF1C 3.670000076 3.74000001 3.700000048 36.65117428 44 CLF1C 3.400000572 3.470000029 3.430000067 36.17005392 44.5 CLF1C 3.140000343 3.200000048 3.170000076 35.73861222 Puts Strike Ticker Bid Ask Last IVM American put options on Dec2020 crude oil futures 42.5 CLZOP 2.609999895 2.660000086 2.609999895 36.22022634 43 CLZOP 2.809999943 2.859999895 2.809999943 35.60360181 43.5 CLZOP 3.029999971 3.089999914 3.039999962 35.19240329 44 CL ZOP 3.269999981 3.329999924 3.269999981 34.75642586 44.5 CLZOP 3.50999999 3.579999924 3.529999971 34.28385997 American put options on Jan2021 crude oil futures 42.5 CLF1P 3.069999695 3.130000114 3.089999914 37.60134235 43 CLF1P 3.279999733 3.339999914 3.289999962 37.08218073 43.5 CLF1P 3.489999771 3.549999952 3.5 36.6366434 44 CLF1P 3.720000267 3.779999971 3.730000019 36.22073597 44.5 CLF1P 3.960000038 4.019999981 0 35.78897826 Figure 1: The prices and implied volatilities (IVM) of American call and put options on the Crude Oil futures Dec-20 (CLZO) and Jan-21 (CLF1) as of 20-Aug-2020. Calls Strike Ticker Bid Ask Last IVM Dec-20 (89d American call 11/17/20); CSize options on Dec2020 1000: CLZO crude oil futures. 43.36 42.5 CLZOC 3.480000019 3.529999971 3.5 36.33633518 43 CLZOC 3.180000067 3.239999771 3.200000048 35.71901043 43.5 CLZOC 2.900000572 2.960000038 2.930000067 35.19056282 44 CLZOC 2.640000105 2.700000048 2.660000086 34.753654 44.5 CLZOC 2.400000095 2.450000048 2.420000076 34.39581761 Jan-21 (118d American call 12/16/20); CSize options on Jan2021 1000; CLF1 crude oil futures 43.63 42.5 CLF1C 4.25 4.320000172 4.289999962 37.53878728 43 CLF1C 3.949999809 4.019999981 3.99000001 37.07461697 43.5 CLF1C 3.670000076 3.74000001 3.700000048 36.65117428 44 CLF1C 3.400000572 3.470000029 3.430000067 36.17005392 44.5 CLF1C 3.140000343 3.200000048 3.170000076 35.73861222 Puts Strike Ticker Bid Ask Last IVM American put options on Dec2020 crude oil futures 42.5 CLZOP 2.609999895 2.660000086 2.609999895 36.22022634 43 CLZOP 2.809999943 2.859999895 2.809999943 35.60360181 43.5 CLZOP 3.029999971 3.089999914 3.039999962 35.19240329 44 CL ZOP 3.269999981 3.329999924 3.269999981 34.75642586 44.5 CLZOP 3.50999999 3.579999924 3.529999971 34.28385997 American put options on Jan2021 crude oil futures 42.5 CLF1P 3.069999695 3.130000114 3.089999914 37.60134235 43 CLF1P 3.279999733 3.339999914 3.289999962 37.08218073 43.5 CLF1P 3.489999771 3.549999952 3.5 36.6366434 44 CLF1P 3.720000267 3.779999971 3.730000019 36.22073597 44.5 CLF1P 3.960000038 4.019999981 0 35.78897826 Figure 1: The prices and implied volatilities (IVM) of American call and put options on the Crude Oil futures Dec-20 (CLZO) and Jan-21 (CLF1) as of 20-Aug-2020