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SOLUTIONS MOCK EXAM
Creation date：2024-04-09 14:39:09
Belonging category：经济Economics
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SOLUTIONS MOCK EXAM

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SOLUTIONS MOCK EXAM 

Question1:
(a) See Exercise 2 Tutorial 1.
(b)
> v1 <−runif (6 ,−5 ,5)
> v1
[ 1 ] −3.534955 −2.125690 −3.197416 −2.690970 1.171644 4.615299
> v2 <−runif (14 ,−5 ,5)
> v2
[ 1 ] −1.19118968 0.99891338 1.94085672 −4.69556951
3.59148792 0.02149171
[ 7 ] −2.65920813 4.52136889 −0.60572579 1.77306420 −0.71442411
4.73246855
[ 1 3 ] −1.71294960 −1.27886442
> Marray<−array (c ( v1 , v2 ) ,dim = c ( 3 , 6 , 3 ) )
> Marray
, , 1
[ , 1 ] [ , 2 ] [ , 3 ] [ , 4 ] [ , 5 ]
[ , 6 ]
[ 1 , ] −3.534955 −2.690970 −1.1911897 −4.69556951 −2.6592081
1.7730642
[ 2 , ] −2.125690 1.171644 0.9989134 3.59148792
4.5213689 −0.7144241
[ 3 , ] −3.197416 4.615299 1.9408567 0.02149171 −0.6057258
4.7324685
, , 2
[ , 1 ] [ , 2 ] [ , 3 ] [ , 4 ] [ , 5 ]
[ , 6 ]
[ 1 , ] −1.712950 −2.125690 1.171644 0.9989134 3.59148792
4.5213689
[ 2 , ] −1.278864 −3.197416 4.615299 1.9408567
0.02149171 −0.6057258
1
2[ 3 , ] −3.534955 −2.690970 −1.191190 −4.6955695 −2.65920813
1.7730642
, , 3
[ , 1 ] [ , 2 ] [ , 3 ] [ , 4 ] [ , 5 ]
[ , 6 ]
[ 1 , ] −0.7144241 −1.278864 −3.197416 4.6152986 1.940857
0.02149171
[ 2 , ] 4 .7324685 −3.534955 −2.690970 −1.1911897 −4.695570 −2.65920813
[ 3 , ] −1.7129496 −2.125690 1.171644 0.9989134 3.591488
4.52136889
> Marray [ , 2 , 1 ]
[ 1 ] −2.690970 1.171644 4.615299
> mean( v1 )
[ 1 ] −0.9603482
> print ( Marray [ , Marray [3 , ,3 ] >mean( v1 ) , 3 ] )
[ , 1 ] [ , 2 ] [ , 3 ] [ , 4 ]
[ 1 , ] −3.197416 4.6152986 1.940857 0.02149171
[ 2 , ] −2.690970 −1.1911897 −4.695570 −2.65920813
[ 3 , ] 1 .171644 0.9989134 3.591488 4.52136889
Question2:
(a)
> rA<−0 .11 # expec ted re turn o f s t o c k A
> rB<−0 .14# expec ted re turn o f s t o c k B
> rC<−0 .085# expec ted re turn o f s t o c k C
> sigA<−0 .30 # standard d e v i a t i on o f A
> s igB<−0 .45 # standard d e v i a t i on o f B
> s igC<−0 .30 # standard d e v i a t i on o f C
> sigmaAB<− sigA∗s igB∗0 .3 #
> sigmaAC<− sigA∗s igC∗0 .15 # covar iance between a s s e t s A and C
> sigmaBC<− s igB∗s igC∗0 .45 # covar iance between a s s e t s B and C
> rp<−function (wA,wB){
+ return (wA∗rA+wB∗rB+(1−wA−wB)∗rC)}
> rp (1/3 ,1/3)
[ 1 ] 0 .1116667
> varp<−function (wA,wB){
3+ return (wAˆ2∗sigAˆ2+wBˆ2∗s igBˆ2+(1−wA−wB)ˆ2∗s igCˆ2+
+ 2∗wA∗wB∗sigmaAB+2∗wA∗(1−wA−wB)∗sigmaAC+2∗wB∗(1−wA−wB)∗sigmaBC)
+ }
> sqrt ( varp (1/3 ,1/3) )
[ 1 ] 0 .2607681
> varp (1/3 ,1/3)
[ 1 ] 0 .068
> sigmap<−sqrt ( varp )
> plot ( sigmap , rp , pch=” . ” )
(b)
> r f<−0 .02
> amax<−max( ( rp−r f )/sigmap )# the p o s i t i v e tangency o f sharpe r a t i o
> amax
[ 1 ] 0 .3675796
> x<−seq (0 , 3 ,by=0.01)
> l ines (x , amax∗x+r f )
0.3 0.4 0.5 0.6 0.7 0.8
0.
05
0.
10
0.
15
0.
20
sigmap
rp
(c)
4> varp<−function (wA,wB){
+ return (wAˆ2∗sigAˆ2+wBˆ2∗s igBˆ2+(1−wA−wB)ˆ2∗s igCˆ2+
+ 2∗wA∗wB∗sigmaAB+2∗wA∗(1−wA−wB)∗sigmaAC+2∗wB∗(1−wA−wB)∗sigmaBC)
+ }
> wA<−seq (0 , 1 , by=0.001)
> WBn<−(1/ ( rB−rC ) )∗(0.12−rC−wA∗ ( rA−rC ) )
> Wamin<−match(min( varp (wA,WBn) ) , varp (wA,WBn) )
> wA[Wamin ]
[ 1 ] 0 .56
> WB1<−(1/ ( rB−rC ) )∗(0.12−rC−wA[Wamin ] ∗ ( rA−rC ) )
> WB1
[ 1 ] 0 .3818182
> WC1<−1−wA[Wamin]−WB1
> WC1
[ 1 ] 0 .05818182
Question3:
> bu i ld s tock t r e e<−function (S , u , d ,N){
+ t r e e=matrix (0 ,nrow=N+1, ncol=N+1)
+ for ( i in 1 : (N+1)){
+ for ( j in 1 : i ){
+ t r e e [ i , j ]=S∗uˆ( j −1)∗d ˆ( ( i −1)−( j −1))}}
+ return ( t r e e )}
> QM2<−function ( r , d e l t a t , u , d){
+ return ( (exp( r∗de l t a t)−d)/ (u−d ) )
+ }
> QM1<−function ( r , m, de l t a t , u , d){
+ return (((1+ r/m)ˆ(m∗de l t a t)−d)/ (u−d ) )
+ }
> Bin Op Pri2<−function ( t ree , d e l t a t , r1 , r2 , m, u , d , K){
+ D1<−(1+r1/m)ˆ(−m∗de l t a t )
+ D2<−exp(−r2∗de l t a t )
+ Q1<−QM1( r1 ,m, de l t a t , u , d )
+ Q2<−QM2( r2 , d e l t a t , u , d )
+ Q<−c (Q1,Q2)
+ D<−c (D1 , D2)
+ opt ion1 t r e e=matrix (0 , nrow=nrow( t r e e ) , ncol=ncol ( t r e e ) )
+ opt ion1 t r e e [nrow( opt ion1 t r e e ) , ]=pmax(2∗ (K−sqrt ( t r e e [nrow( t r e e ) , ] ) ) , 0)
5+ for ( i in (nrow( t r e e )−1):1){
+ for ( j in 1 : i ){
+ option1 t r e e [ i , j ]=max(max(2∗ (K−sqrt ( t r e e [ i , j ] ) ) , 0 ) ,
+ (Q[ i ] ∗opt ion1 t r e e [ i +1, j +1]+(1−Q[ i ] ) ∗opt ion1 t r e e [ i +1, j ] ) ∗D[ i ] )
+ }
+ }
+ return ( opt ion1 t r e e )
+ }
> Bin Op Pri va l<−function (S , T, r1 , r2 ,m, K, u , d ,N){
+ Q<−c (QM1( r1 ,m, de l t a t=T/N, u , d ) ,QM2( r2 , d e l t a t=T/N, u , d ) )
+ t r e e<−bu i ld s tock t r e e (S=S , u=u , d=d ,N=N)
+ opt ion<−Bin Op Pri2 ( t ree , d e l t a t=T/N, r1=r1 , r2=r2 , u=u ,m=m, d=d , K=K)
+ return ( l i s t (Q=Q, s tock=tree , opt ion=option , p r i c e=opt ion [ 1 , 1 ] ) )
+ }
> r e s u l t s 1<−Bin Op Pri va l (S=30, T=8/12 , r1 =0.05 , r2 =0.06 ,m=52,
+ u=1.15 ,d=0.8 ,K=6, N=2)
> r e s u l t s 1
$Q
[ 1 ] 0 .6194234 0.6291467
$ s tock
[ , 1 ] [ , 2 ] [ , 3 ]
[ 1 , ] 30 .0 0 .0 0 .000
[ 2 , ] 24 .0 34 .5 0 .000
[ 3 , ] 19 .2 27 .6 39 .675
$opt ion
[ , 1 ] [ , 2 ] [ , 3 ]
[ 1 , ] 1 .154789 0.0000000 0
[ 2 , ] 2 .202041 0.5426693 0
[ 3 , ] 3 .236439 1.4928596 0
$p r i c e
[ 1 ] 1 .154789
Question4:
(a)
> S<−c (50 ,50∗1 .08 ,50∗ 0 . 88 ) #(S(0) , uS (0) , dS (0) )
6> V<−c (max(S [2 ] −48 ,0) ,max(S [3 ] −48 ,0) )
> V
[ 1 ] 6 0
> QM<−function ( r , m, de l t a t , u , d){
+ return (((1+ r/m)ˆ(m∗de l t a t)−d)/ (u−d ) )
+ }
> Q<−QM( 0 . 1 , 1 2 , 3/ 1 2 , 1 . 0 8 , 0 . 8 8 )
> Q
[ 1 ] 0 .7260446
> D1<−function ( r ,m, de l t a t ){(1+ r/m)ˆ(−m∗de l t a t )}
> D<−D1( 0 . 1 , 12 , 3/12)
> D
[ 1 ] 0 .975411
> BinOpP<−(Q∗V[1]+(1−Q)∗V[ 2 ] ) ∗D
> BinOpP
[ 1 ] 4 .249151
> Del<−(V[1]−V[ 2 ] ) / (S [2]−S [ 3 ] ) # compute the d e l t a
> Del
[ 1 ] 0 . 6
(b) For the Hedging strategy, check similar argument as in Homework 5 question 3 (ii).
(c) Assume that C(0) = £4 ¡ £4.2. Then, the value of the option is too cheap
• At time t = 0, short sell 0.6 share and use the proceeds (0.6× 50 = £20) to buy a
call option at £4 and invest the rest 30-4=£26 at the risk free interest rate 10%.
• At time t = T = 3 months, there are two possible outcomes:
: S1(T ) = uS(0) = £48. The call option is exercised. The option buyer collects
26(1+ 0.1
12
)3 ' £26.655 representing the future value of £26 invested at the risk
free interest rate. She additionally short sale 0.4 stock at 54 × 0.4 = £21.6
and use the proceeds 26.655+21.6 = £48.255 to purchase one share at £48 and
close the short position in one share by returning it to the owner. By following
this strategy, there is a risk free profit of pounds 0.255.
: S2(T ) = dS(0) = £44. The option is not exercised. Collect 26(1 +
0.1
12
)3 '
£26.655 representing the future value of £16.5 invested at the risk free interest
rate. Buy 0.6 stock for 0.6 × 44 = £26.4 and close the short position in 0.6
stock by returning it to the owner. By following this strategy, there is a risk
free profit of pounds 26.655− 26.4 ' £0.255.
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 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270M CIS 3207 Programs CSCI 2132 CS270 CSE-381 COMP3230A CIS 212 CSI 333 FINM 326 FINM-36702 Introduction to Software Analysis EEE102 GNG1106 ECS 60 CS 161 CSCI851 CSE1222 EDPS0249 TCHR5003 ANTA02 BBUS1SBY SOSC 1140 MDIA2091 FINS3641 INSY 661 ITEC3040 CPSC 481 computer DEE3519 COMP304 CSE536 Microtheory OFFICE 280 CIVE50003 COSC2673 ECS784U CMPSC/MATH 455 AMA 505 CSE 158 BUS1040 FIT3179 FIT5202 Math 104A Math 111A MBUS107 COMPSCI 761 COMP3710 COMPSCI 316 COMP559 COMP557 CP1 1902 COMP 310 ECED 3403 COMPX523 AGCM 3103 EMET 7001 ENGR3821 CISC 124 IPC comp20005 CS 323 CS105 STAT 326 MATH4007 CS4551 COMP1000 ACIT 4640 ENVX3002 CS2034 FC712 algorithms CSCI 4155 CSI4142 POL 850 MATH20811 2B03 CSCI803 Economies COMP 2016 Architect TRADE INFO1110 FIT1043 networks Robotics ICS 31 ECEC-301 FIT2099 CS 3640 CSE1010 Computational COMP1001 CSE 231 CS10 FIT5166 CSE 400 CSCI561 Math 350 CO-496 ENGG6400 CSCI 1133 COMP3055 MATH 819 CSE 6363 CIS 555 CS4420 COMP0037 ECS 140A Spanning Tree Protocol network 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EBU5376 STAT6118 ECON 20110 COMU7000 average acceleration BUSI4412 BFF3121 BIT233 ECMM462 ETC3430 DESN2348 INTE2584 ENME502 MA3XJ INTE2627 MA2815 MA3AM L0101 Math 3MB3 CPCCBC4004 EEEM061 0502E220 APPLIED 6SSMN310 ENV 3310 PM2PCOL3 CONS6201 MATH502 FIT5003 EEEN313 MATH3001 MECH3460 ECON30009 COSC1252 Assembler COMP 3023 ENGG7601 COMP20005 COMP SCI 1103 ICSI 402 INF2C CS320 COMPSCI7064 ECE 3220 PROGRAMMAMING MIPS Computer Science 2413 AM3611 MSOA TE2101 FFT CMPSC-121 CSE 109 COMP SCI 7064 HCI CMP-5013A CCN2042 COMP3400 EEEN30052 CO4215 EE101 KIT107 CSE 421 CSE521 MCD4720 EECS 280 MIT 403 GRPC VG101 APiE EECE 1080C FIT3143 CSE 325 ISOM 3029 CS3307 State SIT255 Engineering MQF Modelling ACX3150 SIA322 ACCT90004 COMU7201 MGMT90018 WRIT1000 COMP3702 BISM1201 COMP7505 IDEA9106 CSE3SMT BFC5926 BFC5280 BFC5935 QBUS3330 ECON3700 IFN521 ECON2121 COSC2626 MA413 DDES9903 ENGG1300 COMP9337 CO2201 ACCT2002 ICM289 MMM054 MSBA-204 DIGM707 CSC2250 Econ312 EVAT python CAN404 LUBS2840 CSCI-GA.2250 COMSM0093 COMP281 AFM 333 IBUS1001 CP2501 Math 105A VE320 TEC101 SOCY2166 COM240 EFIM20033 ENGI4467 EFIM20034 EC 979 BE432 7EJ502 CSSE6400 COMP90025 Econ 312 ENG2031 ECO0006 EXERSCI 207 CAN209 ECO 137W FT312 ECO 2199 MATH32052 BDW4213 APS 360 SAS 6 CHME0017 PO92Q Economics 01AV MATH39512 MATH 5486 COMP8620 ISE102 ECON235 ECON 206 FI 345 PHAY0020 MATH0099 MATH0085 EFB106 PHY 134 MAE115 circuits Chemistry CSE12 MATH-UA 233 PHAS0038 COMP2140 CONMGNT 7049 COMP151101 ECON 485 ACX5903 ENGN6536 ARCH1162 Law PHAS0042 CCF1C03 MATH 351 Phys 139 AG942 MGT001371 APH106 ECON1051 MA4AM MA3Z7 NUMBER THEORY REST0006 SESS0026 STAT0023 TABL 2741 CMP4008Y EFIM10015 Acct 560 PSYC735 GEOG 5 ACCT3104 CSEC5616 PHIL1012 ITLS6111 ECON8022 5CCM226A compX123 MA4XJ R001 EWB LLP120 COMP643 ENG 103 ECON 23950 ACCT 207 FINC 616 Bio99 Math 137A Psychology BCPM0054 MATH20212 EC3450 CENV6141 PSY 205 CE335 CS365 Calculus ACSD INFT6800 MCIT INFO6030 PHYS 111LV H63EEJ ECON2151 ENGI 2181 MTRX1701 PLIT08009 CS240 Math 270 CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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