3740. Different axes of revolution Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x2, y = 1, and x = 0 is revolved about the following lines. y = -2
Risk%III Wednesday,*April*6,*2016 13:34 HW 1.Review*in*excel SACF(ret)*from*1*to*10:*correl(:,:)* 44>*graph 44>*no*relation*(today*known,*but*cannot*predict*tomorrow) Linear*Regression: Ret_t(t)=a+b*Ret_(t41) 1) OLS*Estimation: =linest*(selet*5*rows***N…) St.error:*if*n*is*big*enough,*nears*0*(be*cautious*when*n*is*big,*look*at*the*whole*regression) Look*at*R=0 b:*doesn't*explain*(cannot*predict*return* 4Rt;*but*can*predict*Rt 44correlation*graph*is*downward*sloping,*correlated:* cannot*forecast*the*sign,*but*can*forecast*the*range) 2) ML E1=… Same*normal*distribution MAX F(E1..)=F(Ei)= 44>*MAX*log4*4sum* (se2=0.00001,*a,*b*=*0)* 44>*solver: Constraints:*se2>=0.0000000000000001*(e^9) Uncheck*non 4negatives 44>*MAX*log4*4sum* (se2=0.00001,*a,*b*=*0)* 44>*solver: Constraints:*se2>=0.0000000000000001*(e^9) Uncheck*non 4negatives AR(P): T4stat*&*R :*all*are*significant*to*preditt*R 2.Lec*3*22 What*is*volatility t*known44>*to*predict*sigma*in*the*future!*) 10*years*ago:*GARCH*the*best* 4→ •*Realized*Volatility •*HAR*model SP500/Google/apple…:*historical*data*of*returns 10*years*ago:*GARCH*the*best* 4→ •*Realized*Volatility •*HAR*model SP500/Google/apple…:*historical*data*of*returns 1) Fluctuate*around*0* 2) Volatility:*changes*over*time*and*a*lot44 smaller*4 bigger*4 …(!not*constant,*but*stays*some*time/volatility*) Variance*( ! )*can*be*summed*up*(better*than*volatilit!) Rtcan*be*found*easily4*4>*!tcannot*be*seen*directly4*4>*predic!t+1 Rtcan*be*forecasted4*4>**! t Relation:*Var(X)*=*E(X2)* E(X) ,*E(X)=0 Thus,* If*Rt*is*large,*better*estimate Forecast*variance*(Moving*average*model):* 1)*Why*equally*important 2)*How*to*choose*m*m*should*be*big,*not*too*big*(if*m*is*1,*it's*too*noisy;*if*m*is*252,*it's*too*smooth) 3.Lec*3*2 RiskMetrics*model*(JP*Morgan) 0* Optimal*λ*:*0.94*(JP*Morgan) Infinity*4→ Optimal*λ*:*0.94*(JP*Morgan) Infinity*4→ Compared*with*moving*average*model:*better But*t+k Limitations:*the*same/a*constant*variance 4.Lec*3*2 ARCH*and*GARCH*model* One*equation:*1*44>*3*parameters Unconditional*mean: RiskMetrics*α+β=1,*but*: 2 If*k*is*large*enough,*the*best*guess*is*σ : 44>*Use*ML*(not*linear*regression,u*z*σ*like*ghost*we*don't*see Excel:*initial*guesses*should*be*decent* 44 ^w=0.00001 ^α=0.05 ^β=0.3 44 http://vlab.stern.nyu.ed: * projects*for*volatility*data Eg.*S&P Daily*Ret*Rt Daily*std=0.01 Daily*Variance=0.01 =0.0001 Average*var=0.0001*252 Ave*vol=sqr(0.0001*252)=0.158 4.Lec*3*22 beyond*GARCH Independent:* 5*returns*(sub4intervals) 44>*more*intervals 44>*more*intervals If*infinitely,*44>*Realized*variance*(more*frequently,*more*precise*sigma) Cf.*for*the*mean,*only*the*beginning*&*the*end: 0.2*unknown,*but*more*observations,*more*close*to*0.2 RV t* ! t 44>*RV t+! 5.Lec*3*22 HAR*model:*the*best*now*444 Corsi*(2009) =! t+1 Use*daily,*weekly,*monthly*Realized*Variance 44>*Linear*regression Example:*to*get*RV 44>www.google.com/finance/getpricesq=.DJI&x=INDEXDJX&i=60&p=10d&f=d,c,h,l,o,v q=.DJI/APPL… x=INDEXDJX/nasdaq… i=60*seconds,*interval Max:*10days Date,*close,*high,*low,*open,*volume Date:*in*unix*format* 44> Unix*timestamp:*number*of*second*from*01/01/1970. Excel*date4time:*number*of*days*from*01/01/1900.* 44>**=*(UnixDate/86400)*+*date(1970,*01,*01)*+*(TIME*OFFSET*∗ 60)/86400 44>*http://realized.oxford4man.ox.ac.uk Live*data:*from*2000*latest*live*data*(ABD*COLOUMN) <44 HW*(Final):*3*group*members*(1*to*submit);*Excel:*clean*&*precise*44 for*grading;*pdf:*short*is*good
