Exam MFE/3F Spring Answer Key

Transcription

1 Exam MFE/3F Sping 9 Answe Key Quesion # Answe E B 3 B 4 D 5 E 6 D 7 D 8 B 9 E C C E 3 A 4 E 5 C 6 A 7 A 8 A 9 D C

2 Answe: E We have S, δ 5, σ 3, 5, an h By, u exp[ δ h σ exp[ δ h σ h] exp[5 5 3 h] exp[5 5 3 ] e ] e 3 3 By 5, δ h e e e p* u e e The sock pices an call pices ae lise a each noe below: S C 46 S u Ealy execise: Coninuaion: S 7488 C S uu 888 C uu 888 S u C u S C Fo he calculaion of C u, we have C u e 5 [p*c uu p*c u ] 33796, bu ealy execise woul be opimal a a value of The ime- pice of he call is C e 5 [p*c u p*c ] 464 Remak: Fo a given volailiy σ, if u an ae compue using he meho of fowa ee, hen δ h δ h δ h σ h σ h σ h e e e e e p*, δ h σ h δ h σ h σ h σ h σ h σ h u e e e e e e an hence p* As a esul, e p * < ; his povies a check fo p* σ h

3 Answe: B i The aveage of he sock pice is: The payoff is hus ii The call is knocke-ou on Oc 3, 8, when he sock pice is 5 The payoff is hus iii The call is knocke-in on Feb 9, 8, when he sock pice is The payoff is hus max5, 5 The maximum iffeence is Remak: While i is incoec o say ha an opion ha goes ou of exisence has an unefine payoff, some saemens in he ex can be confusing Fo he May 9 exam, A was also accepe as a coec answe 3

4 3 Answe: B Since u 55/5 an 4/5 8, we have, by 5, δ h 5 e e 8 p* 54 u 8 The no-abiage pice of he call is h 5 C e [ p * C p* C ] e u > 9 As a esul, an abiageu woul buy he unepice call an hen hege he isk of he sock in oe o obain iskless abiage pofi This ules ou A, D an E Since he ela of he call is posiive see Figue on page 3, he abiageu mus sho sell shaes o eliminae he sock pice isk This ules ou C Alenaive meho: We eemine he eplicaing pofolio of he call opion Suppose ha a, he eplicaing pofolio has Δ shaes an B ollas in a bank accoun eaning a isk-fee ae of inees Since he sock pays iviens, by invesing all iviens in he sock, he numbe of shaes woul gow o Δe δh afe h yeas 55e 4e 5e 4e δ Δe Su Be Cu δ Δe S Be C 5 e Δ 5 5 e B 5 Δ 5 B Δ 36 B 683 The no-abiage pice of he call is C ΔS B 4 > 9 Theefoe, Michael can make an abiage pofi by puchasing he call opion a $9 an sho selling he eplicaing pofolio Since Δ > an B <, shoing he eplicaing pofolio, in his case, means shoing 36 shaes of he sock an lening $683 a he isk-fee ae δh Cu C Remak: In a binomial moel, Δ e Su S In he Black-Scholes moel, Δ e δh N 4

5 4 Answe: D We can consuc an oinay K-sike Euopean pu by buying K unis of a K-sike Euopean cash-o-nohing pu an selling a K-sike Euopean asse-o-nohing pu: Oinay u K Cash-o-nohing u Asse-o-nohing u The wo ems on he igh-han sie above coespon o he wo ems in he Black- Scholes fomula Ke T N Se δt N The pice of he asse-o-nohing pu is Se δt N, which is a fomula ha can also be foun in he mile of page 76 We ae given S, K 6 no 4, δ, 5, σ %, an T Thus, ln / Fom he nomal able, N 68 N ice of us,,, e 37 3,66, million 68 5 Answe: E ah isk-neual pobabiliy ime- pice of he bon ime- payoff e e e The pu pice is e e e 9 5

6 6 6 Answe: D Le f x, /x so ha Y f, Then f xx, /x, f xxx, /x 3, an f x, By fomulas 7a, b, c, [] [8 8 ] Z 8 [Z] 64 By Iô s lemma, 8 ] 8 [ ] 8 [8 ] [ Z Y Y Y Y Z Z Y which means ha α y 64y 3 8y y an β y 8y Thus, α½

7 7 Answe: D Le Q u Q be he pice of a secuiy ha pays $ when he up own sae occus, an be he coninuously compoune isk-fee inees ae Then Q Qu Qu u Q 8Q e 3 By using he 3 equaion, we ge Q u 565 uing back ino he n equaion, we ge Q , 8 an hence i follows fom he s equaion ha e Now if S is 6 insea of 8, hen because an S ae unchange, he sysem of simulaneous equaions becomes Qu Q Qu 6Q Qu 9675 C Eliminaing Q fom he n equaion by using he s equaion, we ge Q u Thus, C Q u 398 Remak: The esul is inepenen of he ue pobabiliy of an up-move Analogously, he Black-Scholes equaion an fomulas o no epen on α Alenaive meho: The ime- pice of he call opion is h e 8 8e C e [ p * Cu p* C ] e p* 8 8e Seing 3, we ge e 9675 o 33% If S 6, hen 6, an e 6 6e C e

8 8 Answe: B The picing fomula fo he eivaive secuiy, V, mus saisfy he Black-Scholes paial iffeenial equaion V V V δ S σ S V s s Fo Vs, e ln s, we have V e ln s V, V s e, V ss e s s Thus, he DE becomes V S e σ δ S e V S S δ e σ e δ σ, yieling δ σ Alenaive meho: As he eivaive secuiy oes no pay iviens, we have, fo T, V[S, ] FT, V[ S T, T ] In paicula, V[S, ] F, T V[ S T, T ], which, in his poblem, means Une he isk-neual pobabiliy measue, ln[s] E*[e T e T ln ST] E*[ln ST] ln ST ~ Nln S δ ½σ T, σ T, which is a esul given a he op of page 65, wih α eplace by δ Thus, he coniion ln[s] E*[ln ST] means ha δ ½σ T, yieling he same soluion as befoe Thi meho no in he syllabus: By he funamenal heoem of asse picing, he sochasic pocess {e V[S, ]} is a maingale wih espec o he isk-neual pobabiliy measue, yieling he coniion E*[ln S] ln S 8

9 9 Answe: E By pu-call paiy equaion 94 on page 86, bu eplacing S by he foeign exchange ae x an he ivien yiel δ by he foeign isk-fee inees ae f, he pice of a 4- yea olla-enominae Euopean call opion on yens wih a sike pice of $8 is x exp 5 e 3764 f T K exp T 5 4 8e 3 4 Noe ha 5 /8 By cuency opion pu-call ualiy equaion 97 on page 9, he pice of a 4-yea yen-enominae Euopean pu opion on ollas wih a sike pice of /8 is Alenaive meho: Noe ha 5 /8 By cuency opion pu-call ualiy equaion 97 on page 9, he pice of a 4-yea yen-enominae Euopean call opion on ollas wih a sike pice of /8 is By pu-call paiy, he pice of a 4-yea yen-enominae Euopean pu opion on ollas wih a sike pice of 5 is C F, T K exp f T x e 5e Answe: C Obseve ha S S If we long uni of S, hen we mus long 8S S Z 95S 5S Z S ΔS has no Z em 9 Δ S 8S shaes of S so ha 5S S

10 In ems of olla amoun, shae of S o 8S /S shaes of S S ollas invese in sock o 8S ollas invese in sock : 8 Thus, he pecenage in sock is % Remaks: i The wo expece aes of eun, 8 an 95, ae no use in eemining he popoion They ae neee fo eemining ii The popoion is inepenen of Answe: C To fin he pice, we nee o fis eemine he negaive consan a Fom ii, we know ha ue sock pice pocess is a geomeic Bownian moion wih α 5 an σ By 35 bu eplace by α o by he momen-geneaing funcion fomula fo a nomal anom vaiable, he expece value of he coningen claim a ime T is a a E[ S T ] S exp{[ a α δ a a σ ] T} Subsiuing T, S 5, δ, α 5 an σ ino he equaion above, a 5 exp[5a a a ] 4 a ln 5 5a a a ln4 a 3 ln 5 a ln4 a o 3366 ejece By he fis pa of oposiion 3 on page 667, he ime- pice of he coningen claim is, [ a F S ] a e S exp[ a a a σ ] e e 37 E[ S 3 a 4 e ]exp[ a α] Alenaively, one can calculae he ime- pice using he fomula E*[e S a ], whee he aseisk signifies ha he expecaion is aken wih espec o he isk-neual pobabiliy measue

11 Answe: E By 93, call pice is a eceasing funcion of K Thus, C5, T C55, T By he foonoe on page 3, C5, T C55, T 55 5e T Thus, I is coec Fo II an III, we sa wih hei mile expession: 45, T C5, T S While hee is no a iec elaion beween 45, T an C5, T, we can use pu-call paiy o expess 45, T in ems of C45, T, 45, T C5, T S [C45, T S 45e T ] C5, T S C45, T C5, T 45e T Simila o I, we have C45, T C5, T 5 45e T, which is equivalen o 45e T C45, T C5, T 45e T 5e T Thus, III is coec Since III is coec, II mus be incoec 3 Answe: A 8 monhs afe puchasing he opion, he emaining ime o expiaion 4 monhs ln85 / / 9888, N 846, 6 4 / σ T / , N 878 A ime of puchase, C SN Ke T N e 5 4/ Hence, 8-monh holing pofi is e 5 8/

12 4 Answe: E By 43, F, [, 3],3, In a Black-Deman-Toy moel,, u an uu ae in a geomeic pogession Thus, uu u u 8 u u 8 4% u Because he isk-neual pobabiliy of an up an a own move ae boh 5, an Thus,, 3 5, 5 5 u 6 u uu u u 4963 F, [, 3] u

13 5 Answe: C The sho-ae pocess is a Vasicek moel wih a, b 8, an σ 5 We fis eemine he Shape aio φ, By 4 an 49, if he ue sho-ae pocess is a σ Z, hen he isk-neual sho-ae pocess is [a σφ, ] σ Z, whee Z Z φ, The sochasic pocess { Z } is a sana Bownian moion une he isk-neual pobabiliy measue By compaing he if of he ue pocess wih ha of he isk-neual pocess, we ge σφ, 5 Since σ 5, we have φ, fo all an Now i follows fom 47 ha α4, q4,, 5 4, 5 So we nee o fin q4,, 5 By 4,,,,, T q T σ,, T When he bon pice has an affine sucue as in he case of Vasicek an CIR moels, we have,, T B, T,,, T o q,, T B, T σ Fo he Vasicek moel, Hence, B, T a exp[ a T ] e a T foce of inees a α4,,

14 6 Answe: A By line 3 on page 74, he isk-neual pobabiliy ha ST > K is N, whee ln S / K δ ½ σ T σ T As a esul, he ue pobabiliy ha ST > K is N ˆ, whee ˆ ln S / K α δ ½ σ T, σ T which is 84 Now, S, α, σ 3, δ, T 75, an K 5, giving ˆ ln /5 ½ The answe is N 7 N Alenaive meho: Une he ue pobabiliy measue, ln ST ~ Nln S α δ ½σ T, σ T, which is a esul given a he op of page 65 ST > K ln ST > ln K ln K [ln S α δ ½ σ T] Z > whee Z ~ N, σ T ln5 ln ½ 3 75 Z > 3 75 ln Z > 3 75 Z >

15 7 Answe: A T δt i By he Black-Scholes fomula, S, T Ke N S e N Since he -yea pu is a-he-money an he sock is nonivien-paying, we have S K an δ This yiels S, T T e N N e N N, S ln[ S / K] δ ½ σ T ½σ whee an σ σ T σ ii Dela of a pu opion is e δt N [ N ] As a esul, we have an e N N < 5 N 4364 Equaion implies ha N 5636, o 6, which means Equaion implies ha o N > 577, o > ½σ 6 σ σ 3σ 4 σ o σ N < e , Since σ, an 6, we mus have σ < 4 So, σ 5

16 8: Answe: A Le S α S σs Z Then S Sexp[ α 5σ σz ] Thus, fo sock, σ an α 5 Fo sock, σ 3 an α α Because of he no-abiage consain, a each poin of ime he Shape aios of σ he wo socks mus be equal: Answe: D The quesion asks fo he pu-opion vesion of fomula 5 on page 38 As poine ou in he las senence of he fis paagaph on page 38, σ is he volailiy of he pepai fowa The fomula fo he unconiional vaiance in iii means ha σ The ime- pepai fowa pice fo ime- elivey of he sock is F 75, S S V, Div 5 5e The pepai fowa pice of he sike is is iscoune value, F, K 45e 3994 Thus, ln[ F, S / F, K] ½σ T σ T ln45433/ 3994 ½ σ T The pice of he one uni of he pu opion is F K N F S N ,, The pice of unis of he pu opion is

17 Remaks To eive 5, one assumes ha he pepai fowa pice pocess, { FT, S ; T}, is a geomeic Bownian moion wih volailiy σ, ie, one assumes ha FT, S μ σz, T, FT, S o FT, S F, T S exp[μ ½σ σz], T Thus, Va[ln FT, S ] Va[σZ] σ, T, which is coniion iii in he quesion Fo a eivaion of 5, see oposiion 63 in he book Maingale Mehos in Financial Moelling by M Musiela an M Rukowski 997 The exbook eas wo cases of ivien paymens: i The iviens ae eeminisic Tha is, hei amouns an when hey ae pai ae known an fixe ii The sock pays iviens coninuously a a ae popoional o is pice Because he sock pice is sochasic, he iviens ae sochasic Case i: Wih eeminisic iviens, he sock pice is S FT, S V,T Div, T, which is equivalen o fomula 53 on page 3 of McDonal 6 Diffeeniaing he equaion wih espec o yiels S FT, S V,T Div FT, S [μ σz] V,T Div If is no a ivien-paymen ae, hen V,T Div V,T Div If is a ivien-paymen ae, hen he iffeenial V,T Div is he negaive of he amoun of ivien pai a ha ime Because of he sock pice jumps ownwa a each ivienpaymen ae, he sock pice pocess {S} oes no have coninuous sample pahs an hence canno be a geomeic Bownian moion I follows fom S FT, S[ μ σ Z ] V T, Div S S FT, S μ V T, Div FT, S σ Z S S FT, S ha he volailiy of he sock is σ, which is a funcion of, no a consan S FT, S The expession σ gives a moivaion fo he appoximae coecion fomula a S he op of page 365 in McDonal 6 7

18 Case ii: The ime- pepai fowa pice is FT, S e δt S, T I follows fom Iô s Lemma ha FT, S e δt Sδ e δt S, o FT, S δ S FT, S S Hence, S α δ σz S if an only if FT, S α σz FT, S This means ha he pepai fowa pice pocess, { FT, S ; T}, is a geomeic Bownian moion if an only if he sock pice pocess, {S}, is a geomeic Bownian moion; boh sochasic pocesses have he same paamee σ In case i, he ime- pice of he eeminisic iviens pai beween an T is S FT, S V,T Div In case ii, he ime- pice of he sochasic iviens pai beween an T is S FT, S S e δt S S[ e δt ] Sδ a T, whee he annuiy-ceain a is calculae using he ivien yiel δ, no he isk-fee T ae, as he foce of inees Answe: C Accoing o equaion 35 o Taylo seies expansion, fo a small move of size ε in he sock pice, V S ε V S V S ε V S ε V S Δ S ε Γ S ε!! Wih VS 34, ΔS 8, an ΓS 35, he equaion above becomes 34 8 ε 35 ε, o 75ε 8ε 3, whose soluions ae b ± b 4ac 8± ε o a 75 The fis soluion ε is no a small move in he sock pice Thus, ε an S ε 86 S