Morgan-Stanley-Quant-Finance-笔试

MorganStanleyQuantFinance笔试来源：王汉宁的日志校内上有一篇被疯转的日志“你有去华尔街的资格么”，说的就是我标题里的这个项目。他们的招聘现在有了笔试，今天刚笔完这个，吐血，离“去华尔街的资格”还有好远，权当练习了。。90分钟100道单选，4选1，选错了扣1/4的分。不让用计算器，所以碰到神马ln(2),正态分布各种区间点这种东西之前没记过就无比蛋疼。如果不是知识十分全面且熟练度犹如高考，应该很难做完吧。。。而且据说有A,B,C,D四套题，我没看我的是哪套。几套题之间有交叉，感觉他们应该是有个题库随机出的四套吧。现在把我还记得的题目（居然能回忆出一大半！）贴出来供大家娱乐，有些题目没看懂，有的也没记住，所以难免有叙述不清的情况，凑合看吧。很多不会的题各位大牛指导。Calculus1.(x+2y)在平面上两条曲线间的积分,没记住2.\int(0,pi/12)of(cosx-sinx)/(sinx+cosx)还是神马的3.Extremaoff(x)=x^2+1/x4.Derivativeofx^lnx5.Derivativeofcos(x^2)6.x'(t)=ax+by,y'(t)=cx+dy,(forgotcoefficients)，x(0)=4,y(0)=3,x(1)?(MaybeneedtosolvetwoODEsanddeterminecoefficients?)7.Fastestgrowingdirectionoff(x)=1/(1+ax^2+by^2+cz^2)at(1,1,-2)8.\int(0,inf)ofexp(x^2/2)也可能是我看错题了是exp(-x^2/2)9.Similartothequestion:WhatistheintersectionvolumeoftwounitCylinder?LinearAlgebra1.解二元一次方程组。。。2.Underwhichsituationwillalinearsystemhasnosolutions.3.6variable，3equation，ingeneralhowmanydimsdothesolutionshave?4.(Can'trememberclearly)Positivedefinitematrixdoesn'tnecessarilyhaveproperty:1)allstrictpositivediagonal/eigenvalue2)allstrictpositivediagonalofCholeskydecomposition3)alldatapositive4)invertibleProbability&Stat1.Tworegularcoins,onefakecoinwithbothsidesareheads.Tossonecoinugotheads,what'sthatproboftossingsamecoinandgettingheadsagain?2.Oneregularcoin,onefakecoin.Pickone,tossittwiceandget2heads.Probofpickingthefakecoin?3.52pokercards=4piles，13ineach.2Acesinonepile，theother2Acesintwodifferentpiles,therestpiledoesn'thaveanAce.Prob?4.X,Y,Zhaveequalcorr,thelowerboundofthiscorr?(-0.5)5.RandomvariableX,Y,Z,corr(X,Y)=corr(Y,Z)=c0,whatistheminimumcorrelationbetweenXandZ?(2c^2-1)6.Givendataandsomestatistics,whatistheregressionfunction?7.Uniformdistribution,MLEestimatorofainU[0,a]((n+1)/n*max(xi))8.10000coins,probofmorethan4950heads?(1-N(-1))9.100coins,probofmorethan60heads?(1-N(-2))10.100uniformvariables,...anothercentrallimittheoryquestion11.18socks,12inonecolor,6intheothercolor,probofpick2toformapair？12.Picking2from52pokercards,probofapair?13.Xhave0meanand1var,E(x+2)^2?14.X,Yhave0meanand1var,E(X|X+Y=1)?orundetermined?15.Fillinblank:X1X2areiid,ifforanya,bwecanfindc,dsothataX1+bX2hassamedistributionofcX+d,thenthedistributionis:normal,Poisson,stable,(forgotthelastchoice)16.Twopeople(onefromHarvardandonefromMIT...)arriveuniformlybetween12pm-1pm,theprobofonepersonwaitinganotherforlessthan15min?(1/4,1/8,1/16,...绿皮书)17.Distributionofsamplevolatilitydividedbysamplemean?(chi-square,gamma,F,t)18.Modeof(11,11,29,41,41,41)19.Standarddiviationof(11,11,29,41,41,41)20.25peopleareassignedseatsbuttheysittedrandomly,whatistheexpectednumberofpeoplehavingoriginalassignedseats:1,5,25?StochasticProcess&MathFinance1.ExpectedtimeforWttohitboundary+-1(1)2.GivenzerobondpriceP(0,T)=1/(1+T^2),forwardratebetween(T1,T2)?3.Anasset,(followingGBM?)has$0dollarvaluenow,$1inayear,whatisthepriceatt=0.5:0.5^(0.5),0.5^(0.75),0.5,0.5^(1.25)?4.Radon-Nikodymderivativefromrealprobmeasuretoforwardmeasure,withrespectiveofbondpriceP(0,t)5.TwoindependentBrownianmotion(Bt,Wt),startingatpoint(1,1),whatistheProbabilityofthiscurvehittingpositiveX-axisbeforehittingnegativeX-axis?(3/4)6.(Can'trememberclearly)GiventheprocessGBM(r,sigma?)ofUSD/GBPpriceinUSDriskneutralmeasure,whatistheGBP/USDinGBP?Driftterminchoicesare(-r,-r-sigma^2,-r+sigma^2)7.du/dt-5u=0,backwardEuler-method,whatchoicesofdtcanmakeuunstable:3,5,8,11,(Can'trememberclearly)orallabove?8.X=exp(Wt),E[X]att=2?(e)9.StockpricefollowsGBM,r=0.05,sigma=0.3,spot=100,probofprice50inoneyearfromnow?10.Binarycalloption,Europeanprice=0.1,Americanprice:0.1,0.2,0.3,0.411.Twoequivalentmeasure,PandP',P(A)=0.5,P(A')=?((0,1))12.Thespotis10,ifyouthinkone-monthlater,thepricewillbe8and10withequalprobability,risk-freeinterestrateis5%,whatisthepriceofcalloptionstrikeat10?(between5/1.05and10/1.05becauseofrisk-freeprobability.)13.Gaussiancopula:hazardrateofAandBare1%and2%.Acontractpayyou$1ifAdefaultsearlierthanB.Thepriceofthecontracthaslowestpricewhenthecorrelationis(0,75%,100%,...)?14.MonteCarlo:tofindoutthemeanofA,findavariableB,corr(A,B)=c,andsimulateA+B(E(B)-B)insteadofA.WhatisB?(cov(a,b)/var(a),cov(a,b)/var(b),1,-1)15.MonteCarlo:needtosimulatesomerareevent,souneedtosimulateinadifferentmeasure.Whatisthenameofthistechnique?(sequentialresampling?)16.Varianceof\int(t1,t2)ofWt^2dWt,andsomeotherito’sintegralproblems17.Dividend-payingstock(discretedividend),thepricedifferenceofmodelinginGBMandjump-diffusionFinance(FIandoptions)1.Bond:5%coupon,(10year?)1000face,6%yield,price?2.Bond:3%coupon10yearbondtradedatpar(100),whatisthepriceifrategoesup1bp:(99.91,99.99,100.01,...)3.Whichishigher?FRAvs.Eurodollarfutures4.At2009,atraderbelievesthatdividendin2011islowerthanexpectation,strategy?long/short2010forwardandlong/short2011forward?5.(Can'trememberclearly)AtraderobservedthatimpliedvolofOTMcallsandputsarehigherthanthatofATMoption,strategy:calendarspread,bullspread,bearspread,butterfly?(Iguessedbutterfly..)6.Ifassetpricepositivecorrelatedwithinterestrate,futurepriceishigher/lowerthanforwardprice?(higher)7.Ifinterestrateisdeterministic,futur