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REPRODUCE THE DERIVATIONS OF BLACK- SCHOLES FORMULA TO OBTAN THE FORMULA FOR THE ARBITRAGE FREE PUT OPTION PRICE. . THE FOLLOWNG IS THE UP TO

REPRODUCE THE DERIVATIONS OF BLACK- SCHOLES FORMULA TO OBTAN THE FORMULA FOR THE ARBITRAGE FREE PUT OPTION PRICE. .

THE FOLLOWNG IS THE UP TO DATE (ALMOST) CRUDE OIL PRICES

Date ln(X_t) where X_t= Crude Oil Price

1.12.1995

-2,089761011

1.01.1996

-2,103384597

1.02.1996

-2,095308697

1.03.1996

-1,985125718

1.04.1996

-1,890522111

1.05.1996

-1,996059933

1.06.1996

-2,036350267

1.07.1996

-1,996600206

1.08.1996

-1,968296263

1.09.1996

-1,883057417

1.10.1996

-1,848992252

1.11.1996

-1,901118727

1.12.1996

-1,83517754

1.01.1997

-1,845763957

1.02.1997

-1,972754417

1.03.1997

-2,029876899

1.04.1997

-2,092915271

1.05.1997

-2,038154365

1.06.1997

-2,123076478

1.07.1997

-2,100611687

1.08.1997

-2,087935223

1.09.1997

-2,09746907

1.10.1997

-2,02767777

1.11.1997

-2,081546415

1.12.1997

-2,178367645

1.01.1998

-2,271588993

1.02.1998

-2,311264627

1.03.1998

-2,378213689

1.04.1998

-2,351868597

1.05.1998

-2,392620158

1.06.1998

-2,478050599

1.07.1998

-2,450221092

1.08.1998

-2,503936014

1.09.1998

-2,392101691

1.10.1998

-2,432722965

1.11.1998

-2,547132187

1.12.1998

-2,679271237

1.01.1999

-2,580799877

1.02.1999

-2,618386001

1.03.1999

-2,419609921

1.04.1999

-2,258369226

1.05.1999

-2,235602272

1.06.1999

-2,227745891

1.07.1999

-2,116969627

1.08.1999

-2,061765063

1.09.1999

-1,949731506

1.10.1999

-2,004840787

1.11.1999

-1,908666998

1.12.1999

-1,867545845

1.01.2000

-1,82919088

1.02.2000

-1,756505891

1.03.2000

-1,74413958

1.04.2000

-1,893032388

1.05.2000

-1,783151762

1.06.2000

-1,688247351

1.07.2000

-1,75805481

1.08.2000

-1,71049707

1.09.2000

-1,633928932

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