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John F. Welch College of Business

Sacred Heart UniversityMFIM 64! "#$I%&'I%#S &(" $IS) M&(&*#M#('+ S,ring -6

8. ('/o01eriod Bino2ial Model) a.

44.559)45(1.10)(.Sud36.4545(.9)Sd40.5,45(.9)Sd

54.4545(1.10)Su49.5,45(1.10)Su45,S

22

22

==

====

=====

 b.

040)5Max(0,36.4C

4.5540)5Max(0,44.5C

14.4540)5Max(0,54.4C

2

2

d

ud

u

=−=

=−=

=−=

c.

3.25050(.25))/1.(4.55(.75)

r) p))/(1(1C p(CC

11.40/1.054.55(.25)))(14.45(.75

r) p))/(1(1C p(CC

.25.751 p1

.75.9).9)/(1.10(1.05d)d)/(ur)((1 p

2

2

dudd

uduu

=+=

+−+=

=+=

+−+=

=−=−

=−−=−−+=

d.

8.92/1.053.25(.25)))(11.40(.75

r) p))/(1(1C p(CC du

=+=

+−+=

e.

9056.)5.405.49/()25.340.11(

Sd))/(SuC(Ch du

=−−=

−−=

f. If he !"c# $"e! up

1)55.4445.54/()55.445.14(

Sud))/(SuC(Ch  2

uduu   2

=−−=

−−=

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h. *u+ 906 !hare! ad r'e 1,000 ca--!

a-ue "f p"rf"-'" "da+

906(45) 1,000(8.92) 31,850

a-ue "f p"rf"-'" "e per'"d -aer

  If !"c# $"e! up,

906(49.50) 1,000(11.40) 33,447

  If he !"c# $"e! d",

906(40.50) 1,000(3.25) 33,443

he!e " a"u! are e!!e'a--+ eu'a-e.

eur "er "e per'"d (33,447/31,850) 1 0.05

If !"c# $"e! up " 49.5, hu  1. he bu+ 94 ca--! a 11.40 f"r 1,072. *"rr" he "e+ a her'!#free rae. " +"u hae 906 !hare! ad 906 ca--!, h'ch '! a hed$e ra'" "f 1. :"ur p"rf"-'"'!

 906 !hare! a 49.50 44,847 906 ca--! a 11.40 10,328 -"a 1,072

 33,447

a-ue "f p"rf"-'" "e per'"d -aer

If !"c# $"e! up,

906(54.45) 906(14.45) 1,072(1.05) 35,114

If !"c# $"e! d",

906(44.55) 906(4.55) 1,072(1.05) 35,114

If he !"c# had $"e d" ' he f'r! per'"d " 40.50, he h d  0.562. he !e-- 344 !hare! a40.50. a#e he pr"ceed! "f 13,932 ad 'e! h'! a"u f"r he ex per'"d ' r'!#free b"d!ear'$ 5 perce. " +"u hae 562 !hare! ad 1,000 ca--!, h'ch '! a hed$e ra'" "f 0.562. :"ur  p"rf"-'" '!

MS%IM &!!'$e 2 Spr'$ 2016

$. If he !"c# $"e! d"

.561736.45)0)/(44.55(4.55

)Sd)/(SudC(Ch  2

dudd  2

=−−=

−−=

22

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562 !hare! a 40.50 22,761 1,000 ca--! a 3.25 3,250 b"d! 13,932

  33,443

a-ue "f he p"rf"-'" a he ed "f he !ec"d per'"d

If !"c# $"e! up,

562(44.55) 1,000(4.55) ; 13,932(1.05) 35,116

If !"c# $"e! d",

562(36.45) ; 13,932(1.05) 35,114

he d'fferece beee 35,114 ad 35,116 '! due " r"ud'$.

eur "er "e per'"d 850,31/114,35   1 0.05

If ' ere "erpr'ced, he 'e!"r !h"u-d e!ab-'!h he !ae r'!#-e!! hed$e b+ bu+'$ 906 !hare!ad r''$ 1,000 ca--!. If ' ere uderpr'ced, he 'e!"r !h"u-d bu+ 1,000 ca--! ad !e-- !h"r906 !hare!. h'! "u-d creae a +pe "f -"a ' h'ch "e+ '! rece'ed "da+ ad pa'd bac# -aer. he effec'e rae " he -"a "u-d be -e!! ha he r'!#free rae.

9. ('/o01eriod Bino2ial Model)

2

2 2

ud

u ud

Su =30(1.15)(1- 0.06) =32.43, Sd =30(.9)(1 - 0.06) =25.38

Su =32.43(1.15) =37.29, Sd =25.38(0.9) =22.84

 =32.43(0.9) or 25.38(1.15) 29.19S

 =Max(0, 37.29 - 25) =12.29, C =Max(0, 29.C

=

2d

u

19 - 25) =4.19

 =Max(0, 22.84 - 25) =0Cp =(1.05 - 0.9)/(1.15 - 0.9) =0.6

0.6(12.29) +0.4(4.19) = =8.62C

1.05

but at t!" 1 # t$" top %tat", t$" %to&' % at 30(1.15) 34.50 b"or" t o"% "x-d=

u

*d"#d. So

"x"r&%" t$" &a or 34.50 - 25 9.50. $u%, C 9.50

0.6(4.19) 0.4(0)2.39

1.05

0.6(9.50) 0.4(2.39)6.34

1.05

d C

C

= =

+= =

+= =

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10. ('/o01eriod Bino2ial Model)

64.7968.20(.95)Sud

55.9658.90(.95)Sd75.02)68.20(1.10Su

58.9062(.95)Sd

68.2062(1.10)Su

2

2

==

==

==

==

==

.63,1.08

.13(5.21);.87(0) <

14.0455.96)70Max(0,<

5.2164.79)70Max(0,<

075.02)70Max(0,<

.8667.95).95)/(1.10(1.08 p

u

d

ud

u

2

2

 bu "rh Max(0,70 68.2) 1.80 'f exerc'!ed !"

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18.871.10

.143(4.05)).857(23.54

C

4.051.10

.143(0.0).857(5.20)C

23.541.10

.143(5.20)).857(29.35C

d

u

=

+

=

=+

=

=+

=

.309538.4055.20

0.05.20h

1.0055.2079.35

5.2029.35h

.9284869

4.0523.54h

d

u

=

−

−

=

=

−

−

=

=

−

−

=

& 'e 0, h 0.928. =e u! bu+ 928 !hare! a 60 ad !e-- 1,000 ca--! a 18.87. he he a-ue '!

928(60) 1,000(18.87) 36,810

& 'e 1 he he !"c# '! 69, he p"rf"-'" '! "rh

928(69) 1,000(23.54) 40,492

he e hed$e ra'" '! 1.0. =e u! bu+ 72 !hare! a 69, c"!'$ 4,968, h'ch e b"rr". >ur p"!''" '!" 1000 !hare!, 1000 !h"r ca--!, ad a -"a "f 4,968.

& 'e 2 he he !"c# $"e! fr" 69 " 79.35, he p"rf"-'" '! "rh

1000(79.35) 1000(29.35) 4,968(1.10) 44,535

& 'e 2 he he !"c# $"e! fr" 69 " 55.20, he p"rf"-'" '! "rh

1000(55.20) 1000(5.20) 4,968(1.10) 44,535

& 'e 1 he he !"c# '! 48, he p"rf"-'" '! "rh

928(48) 1,000(4.05) 40,494

he e hed$e ra'" '! 0.310. =e u! !e-- he !hare! " $eerae 618(48) 29,664 ad 'e! h'! ' b"d!.>ur p"!''" '! " 310 !hare!, 1000 !h"r ca--! ad 29,664 'e!ed ' b"d!.

& 'e 2 he he !"c# $"e! fr" 48 " 55.20, he p"rf"-'" '! "rh

310(55.20) 1,000(5.20) ; 29,664(1.10) 44,542

& 'e 2 he he !"c# $"e! fr" 48 " 38.40, he p"rf"-'" '! "rh

310(38.40) 1,000(0.0) ; 29,664(1.10) 44,534.

hu!, a 'e 1 he 36,810 $re " 40,492 ("r 40,494, a r"ud "ff d'fferece), h'ch '! 10 ?. %r" 'e 1,he 40,492 $re " 44,542 ("r 44,535 "r 44,534, r"ud "ff d'fferece!), a reur "f 10?.

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12. (Behavior of the Bino2ial Model for 3arge n and Fied 5,tion 3ife)

eca--, , d1/u, ad r (1;r aua-)/. hu! he ab-e "u-d be

u d r1 1.7333 0.5769 0.07

5 1.2789 0.7819 0.013610 1.1900 0.8404 0.006850 1.0809 0.9252 0.0014100 1.0565 0.9465 0.0007

13. (#tending the Bino2ial Model to n 1eriods)I!er'$ he pr"per a-ue! '" he !pread!hee $'e! he f"--"'$

C1 10.46035 9.058510 8.536525 8.772050 8.6721

17. (5ne0 1eriod Bino2ial Model) eca-- ha he a-ue "f p (1 ; r d)/(u d) h'ch ' h'! ca!e eua-!51.44?. he hed$e ra'" ar'e! b+ he !r'#e pr'ce ad '! 0.743 (@90), 0.571 (@100), ad 0.400 (@110).hu! he hed$e ra'" dec-'e! a! he !r'#e pr'ce 'crea!e!, bu he pr"bab'-'+ p d"e! " cha$e.

MS%IM &!!'$e 2 Spr'$ 201626