1.Aboxspreadisacombinationofabullspreadcomposedoftwocalloptionswithstrikepricesandandabearspreadcomposedoftwoputoptionswiththesametwostrikeprices.a)Describethepayofffromaboxspreadontheexpirationdateoftheoptions.b)Whatwouldbeafairpricefortheboxspreadtoday?Definevariablesasnecessary.c)Underwhatcircumstancesmightaninvestorchoosetoconstructaboxspread?d)Whatsortofinvestordoyouthinkismostlikelytoinvestinsuchanoptioncombination,i.e.ahedger,speculatororarbitrageur?Explainyouranswer.2.Formalongbutterflyspreadusingthethreecalloptionsinthetablebelow.C1X=$90T=180daysC2X=$100T=180daysC3X=$110T=180daysPrice16.330010.30006.0600DELTA0.78600.61510.4365GAMMA0.01380.01810.0187THETA-11.2054-12.2607-11.4208VEGA20.461926.841627.6602RHO30.708525.251518.5394a)Whatdoesitcosttoestablishthebutterflyspread?b)CalculateeachoftheGreekmeasuresforthisbutterflyspreadpositionandexplainhoweachcanbeinterpreted.c)Howwouldyoumakethisoptionportfoliodeltaneutral?Whatwouldbeachievedbydoingso?d)SupposethattomorrowthepriceofC1fallsto$12.18whilethepricesofC2andC3remainthesame.Doesthiscreateanarbitrageopportunity?Explain.3.ConsiderasixmonthAmericanputoptiononindexfutureswherethecurrentfuturespriceis450,theexercisepriceis450,therisk-freerateofinterestis7percentperannum,thecontinuousdividendyieldoftheindexis3percent,andthevolatilityoftheindexis30percentperannum.Thefuturescontractunderlyingtheoptionmaturesinsevenmonths.Usingathree-stepbinomialtree,calculatea)thepriceoftheAmericanputoptionnow,b)thedeltaoftheoptionwithrespecttothefuturesprice,c)thedeltaoftheoptionwithrespecttotheindexlevel,andd)thepriceofthecorrespondingEuropeanputoptiononindexfutures.e)ApplythecontrolvariatetechniquetoimproveyourestimateoftheAmericanoptionpriceandofthedeltaoftheoptionwithrespecttothefuturesprice.NotethattheBlack-ScholespriceoftheEuropeanputoptionis$36.704andthedeltawithrespecttothefuturespricegivenbyBlack-Scholesis–0.442.1064.Afinancialinstitutiontradesswapswhere12monthLIBORisexchangedforafixedrateofinterest.Paymentsaremadeonce...
