The following is the up to date (almost) crude oil prices a) Reproduce the derivations of black- scholes formula to obtain the formula for the arbitrage free put option price. b) Use the formula from part (a) to find the put option price with the exercise of 0.2 dollars and with the maturity duration of 6 months after the end date of the following data set, and the nominal interest rate is10%.

(Please check all the assumptions needed to apply the formula, If some of the assumptions do not hold, please just ignore them and continue with applying the formula.) (PLEASE USE R AND SPECFY THE CODES)

Date	Oil_price
1.12.1995	-2,089,761,011
1.01.1996	-2,103,384,597
1.02.1996	-2,095,308,697
1.03.1996	-1,985,125,718
1.04.1996	-1,890,522,111
1.05.1996	-1,996,059,933
1.06.1996	-2,036,350,267
1.07.1996	-1,996,600,206
1.08.1996	-1,968,296,263
1.09.1996	-1,883,057,417
1.10.1996	-1,848,992,252
1.11.1996	-1,901,118,727
1.12.1996	-183,517,754
1.01.1997	-1,845,763,957
1.02.1997	-1,972,754,417
1.03.1997	-2,029,876,899
1.04.1997	-2,092,915,271
1.05.1997	-2,038,154,365
1.06.1997	-2,123,076,478
1.07.1997	-2,100,611,687
1.08.1997	-2,087,935,223
1.09.1997	-209,746,907
1.10.1997	-202,767,777
1.11.1997	-2,081,546,415
1.12.1997	-2,178,367,645
1.01.1998	-2,271,588,993
1.02.1998	-2,311,264,627
1.03.1998	-2,378,213,689
1.04.1998	-2,351,868,597
1.05.1998	-2,392,620,158
1.06.1998	-2,478,050,599
1.07.1998	-2,450,221,092
1.08.1998	-2,503,936,014
1.09.1998	-2,392,101,691