Is-1 Financial Primer

7/25/2019 Is-1 Financial Primer

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IS-1 Financial Primer

Stochastic ModelingSymposium

By

Thomas S.Y. Ho PhD

Thomas Ho Company, Ltd

[email protected]

April 3, 200
mailto:[email protected]:[email protected]


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2

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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3

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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4

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 %&ample' (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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16

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 %&ample 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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1

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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%&ample 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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%&amples 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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