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IS-1 Financial Primer
Stochastic ModelingSymposium
By
Thomas S.Y. Ho PhD
Thomas Ho Company, Ltd
April 3, 200
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Purpose !"#r"i#$ o% th# &asic principl#s in th#
r#lati"# "al'ation mod#ls
!"#r"i#$ o% th# &asic t#rminolo(i#s )*'ity d#ri"ati"#s +i#d incom# s#c'riti#s
Practical impl#m#ntation o% th# mod#ls
)ampl#s o% applications
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Traditional Valuation -#t pr#s#nt "al'#
)p#ct#d cash%lo$s
Cost o% capital as oppos#d to cost o%
%'ndin(
Capital ass#t pricin( mod#l
Cost o% capital o% a %irm as oppos#d to cost
o% capital o% a pro#ct /or s#c'rity
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Relative Valuation La$ o% on# pric#1 #t#ndin( to non
trada&l# %inancial instr'm#nts
Applica&ility to ins'ranc# prod'cts and
ann'iti#s /loans and 4Cs Ar&itra(# proc#ss and r#lati"# pricin(
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Stoc !ption Model 5od#lin( approach1 sp#ci%yin( th#
ass'mptions, typ#s o% ass'mptions
D#scription o% an option
)conomic ass'mptions1 Constant ris6 %r## rat#
Constant "olatility
Stoc6 r#t'rn distri&'tion )%%ici#nt capital mar6#ts
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"inomial #attice Model #n#rality o% th# mod#l in d#scri&in( th#
#*'ity r#t'rn distri&'tion
5ar6#t lattic# and ris6 n#'tral lattic#
Dynamic h#d(in( and "al'ation 4nt'iti"# #planation o% th# mod#l r#s'lts
Comparin( th# r#lati"# "al'ation approach
and th# traditional approach 7 th# cas# o%a lon( dat#d #*'ity p't option
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!ne-Period "inomial Model S'8S 9 #p/rT9 Sd8S
4n th# a&s#nc# o% ar&itra(# opport'niti#s, th#r##ist positi"# stat# pric#s s'ch that th# pric# o%any s#c'rity is th# s'm across th# stat#s o% th#$orld o% its payo%% m'ltipli#d &y th# stat# pric#.
:/C'7 Cd8/S'Sd
;' :/S #p/rT Sd8/S' Sd
C :
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$umerical %&le' (all!ption Pricing
Stock Price($) S 100
Strike Price ($) X 100
Stock Volatility S 0.2
Time to e!iratio" (year) T 1
#ik%&ree rate r 0.05
'ii'e"' yiel' ' *+
t,e "-mer o& !erio' " 6 't / T*"
-!ar' moeme"t - 1.0851 / e!('t)'o"ar' moeme"t ' 0.216 / 1*-
rik%"e-tral !roaility o& - ! 0.5308 / (e!(r't)%')*(-%')
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Stoc lattice163.21465
150.41805
138.62447
138.62447
127.755612
117.73805
127.755612
117.73805
108.50756 100
117.73805
108.50756 100
2.154775
84.3363
108.50
756100 2.15
4775
84.33
63
78.274
4477
72.137
3221
tock lattice 1002.154775
84.3363
78.2744477
72.1373221
66.481371
61.268817
time 0 1 2 3 4 5 6
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(all !ption #attice
63.21465
51.24730 38.62447
40.277352 28.585483 17.73805
30.224621 1.3175 .337430 0.000000
21.723634 12.44533 4.15050 0.000000 0.000000
15.055460 7.780762 2.58711 0.000000 0.000000 0.000000
10.125573 4.72344 1.36184 0.000000 0.000000 0.000000 0.000000
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Martingale Processes) p and *measures C8? : p'C'8?'= pdCd8?d
S8? : p'S'8?'= pdSd8?d
> : p' = pd
C8S : *'C'8S'= *dCd8Sd
?8S : *'?'8S'= *d?d8Sd
> : *' = *d Pro&a&ility m#as'r#1 assi(nin( pro&
D#nominator1 n'm#rair#
5artin(al#1 #p#ct#d "al'#: c'rr#nt "al'#
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(ontinuous Time Modeling 4to proc#ss
d/t : /tdt = /tdB/t /dt2:0
/dt/dB:0 /dB2:dt
E : (/ t, dE : (tdt = (d = >82 (/d2
#om#tric Bro$nian motion dS8S :dt = dB/t
S/t : S/0#p /t 2t82 = B/t
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$umeraires and
Pro+a+ilities dS8S : sdt = sdBs/t di"id#nd payin( dF8F : *dt = dS8S di"id#nd r#in"#st#d dY8Y : Gdt = GdBG/t any ass#t
?/t : int#(ral o% r/s stochastic rat#s ?is6 n#'tral m#as'r#
E/t : F/t8?/t dS8S : /r * dt = sdB/t
F as n'm#rair# E/t : ?/t8F/t dS8S : /r 7 * = s2dt = sdB
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$umeraire ,eneral (ase Y as n'm#rair#
E/t : F/t8Y/t
dS8S : /r 7 * = Is ydt = sdB
Folatility in"ariant
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Ris $eutral Measure 5artin(al# proc#ss
)ampl#s o% m#as'r#s p m#as'r#, %or$ard m#as'r#, mar6#t m#as'r#
#n#raliJation o% th# Blac6Schol#s 5od#l Applications in th# capital mar6#ts
Applications to th# ins'ranc# prod'cts
Li%# prod'cts +i#d ann'iti#s
Faria&l# ann'iti#s
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Sensitivity Measures D#lta , S
amma K,
Th#ta /tim# d#cay t
F#(a " m#as'r# ?ho , r
?#lationships o% th# s#nsiti"ity m#as'r#s
4nt'iti"# #planation o% th# (r##6s )'rop#an, Am#rican, B#rm'dian, Asian p't8call options
Comparin( $ith th# #*'ili&ri'm mod#ls Contin'al ad'stm#nt o% th# impli#d "olatility
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??
Stock Price (S) 100
Strike Price () 100
Time to e!iratio" (T) 1
Stock olitility () 0.2
#ik%&ree rate (r) 0.04
ii'e"' yiel' () 0
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$umerical %&le o the
,reesall P-t
Price .2505 6.00400
(elta) 0.6171 %0.3820
(9amma) 0.0107 0.0107
(Ve:a) 38.13878 38.13878
;(T,eta) %5.88852 %2.04536
< (#,o) 51.8660 %44.21286
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Interest Rate Modeling Lattic# mod#ls
Yi#ld c'r"# #stimation
Yi#ld c'r"# mo"#m#nts
Dynamic h#d(in( o% &onds
T#rm str'ct'r# o% "olatiliti#s
S#nsiti"ity m#as'r#s D'ration, 6#y rat# d'ration, con"#ity
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Interest Rate Model'
Setting .pyear 0 1 2 3 4 5
i"itial yiel' c-re 0.060 0.060 0.065 0.070 0.075 0.080
i"itial 'ico-"t &-"ctio" !(") 1.0000000.4176
50.8780
50.81058
40.74081
80.67032
0
o"e !erio' &orar' c-re 0.060 0.060 0.070 0.080 0.00 0.100
lo:"ormal !ot olatility (S) 0 0.0775 0.0775 0.0775 0.0775 0.0775
lo:"ormal &orar' olatility (&) 0 0.0775 0.0775 0.0775 0.0775 0.0775
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/o 0#ee +asic2 Model
( 1)(1) 2
( ) (1 )
i
n
i n
P nP
P n
+= +
0.861241
0.87647
0.87465
0.8651
0.82004
0.888358
0.13105
0.08143
0.04534
0.02235
0.28058
0.23066
0.1474
0.17241
0.16328
ico-"t &-"ctio" lattice0.4176
50.3672
0.3313
60.304
60.3012
60.3064
2
year 0 1 2 3 4 5
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/o-#ee !ne Period Rates
ln ( )( )
n
n i
i
P Tr T
T
=
0.143806
0.12823
510.13388
06
0.10875380.11428
510.11838
06
0.000470.0635
380.10033
510.10288
06
0.0746608 0.0800547 0.083538 0.0863851 0.0873806
="teret rate lattice 0.060.06536
080.0620
470.07155
380.07243
510.07188
06
year 0 1 2 3 4 5
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( ) ( )
( ) ( ) ( )1 2 1 1
1 2
1 1 2( 1)(1)
( ) 1 1 1
n n nn i
i n
n n n
P nP
P n
+ + +
= + + +
0.86673
10.8806
30.87807
2
0.8580
0.82668
0.88562
0.
1135
0.0775 0.04530 0.01201
0.26800
0.23044
0.1765
0.1654
0.124
0.41760.34 0.318 0.2872 0.244
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( ) ( )
( ) ( ) ( )1 2 1 1
1 2
1 1 2( 1)(1)
( ) 1 1 1
n n nn i
i n
n n n
P nP
P n
+ + +
= + + +
L L
L L L
>o%?eemo'el rate
it, termtr-ct-re o&olatilitie
0.1430264
0.1267402 0.1300264
0.108368 0.1135402 0.1170264
0.027784 0.067368 0.1003402 0.1040264
0.076018 0.0800784 0.0836368 0.0871402 0.010264
0.06 0.064018 0.0673784 0.0705368 0.073402 0.0780264
0 1 2 3 4 5
lo:"ormal !ot olatility (S) 0 0.1 0.05 0.0 0.085 0.08
lo:"ormal &orar' olatility (&) 0 0.1 0.007143 0.081875 0.0733333 0.065
1
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3lternative 3r+itrage-reeInterest Rate ModelingTechni*ues Th#s# ar# not #conomic mod#ls &'t
t#chni*'#s
Spot rat# mod#l
-%actor mod#l Lattic# mod#l
Contin'o's tim# mod#l
Cali&rations
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3lternative Valuation
3lgorithms Disco'ntin( alon( th# spot c'r"# Bac6$ard s'&stit'tion
Path$is# "al'ation mont#carlo
Antith#tic, control "ariat#
Str'ct'r#d samplin(
+init# di%%#r#nc# m#thods
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%&le o Interest Rate
Models HoL##, Blac6D#rmanToy, H'llMhit# H#athNarro$5orton mod#l
Brac#atar#65'si#la8Namshidian mod#l
/5ar6#t 5od#l
Strin( mod#l
A%%in# mod#l
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%&les o 3pplications Corporat# &onds /li*'idity and cr#dit ris6s
!ption ad'st#d spr#ads
5ort(a(#&ac6#d s#c'riti#s
Pr#paym#nt mod#ls C5!s
Capital str'ct'r# ar&itra(# "al'ation
4ns'ranc# prod'cts
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(onclusions Comparin( r#lati"# "al'ation and th# -PF
mod#l
4ma(in# th# $orld $itho't r#lati"#
"al'ation B#yond th# Prim#r1
4mportanc# o% %inancial #n(in##rin(
4d#nti%yin( th# #conomics o% th# mod#ls
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Reerences Ho and L## /200O Th# !%ord 'id# to
+inancial 5od#lin( !%ord ni"#rsity Pr#ss
)c#l mod#ls />QO mod#ls
$$$.thomasho.com )mail1 [email protected]
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