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(11)

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(12)

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(13)

Estimated Implied Distribution

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(14)

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(15)

Estimated Implied Distribution

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(16)

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MXFhjaD^Yhj{TlVc{s^YnzX¡ij{DTM{GVYsaDi)ZRnzaDVRaZ\^YXFVch=b^chjZ]baZFZF{VR/hjTMt^Y{^YnMX]h;VbvuGXFTMX[bYb¢aDTz

u~MVY^Y{bvh=b]¡ nMX/±² X[b^ch;sa^ch;{GT{Gx/^cah;TMX[BlVc{s ^cnMX^{¨S stX\^cnM{/MbyZ\{Gh;TzZFhjMX]b

h;^Yn^YnMX|X]bv^Yhjsa^ch;{GT8{x/^RaDhjTMX]lVY{Gs ^cnMX{T^YXF¤aVYij{mbYh;s`~MijaD^Yhj{T8XFijidG^cnMXMh©ÍPXFVY

X]TzZ\XyxQX\^XFX]T^cnMXyX]bv^Yhjsa^ch;{GTzb {x/^RaDhjTMX]lVc{sÇ^cnMX^{S b h=b TM{D^GXFVc_ijaVYGX

(17)

TW^YnzX{^YnMX]V¡nzaTzJd}XZ]aDTaijbY{BX]bv^Yhjsa^cX8^YnzXhjstfMi;hjX]i;{/Z]aDi{i=a^ch;ijh;^_5bY~MVYl£aZ\X

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h;T^YnzXzVcbv^ ^{`zG~MVYX[bFdMbY{stX|X]/GXai;~zX]bZFaTMTM{^ xQX{x/^RaDhjTMX]¨/hjVYX[Zª^ci;_lVc{s,^YnMX

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h;TgZ\VcX]abYX|h©^cn^ch;stXG

Implied Local Volatility Surface

s

1.75 3.00 4.25 5.50

0.07 0.14 0.22

50.00 75.00 100.00 125.00

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bvfQ{Tg/h;Tz8^Y{^YnMX[bvXb^cVYhjuX¡fMVYh=Z\XaDTz^YhjsX`^Y{sa^Y~zVYh;^_i;X]XFi=b]Ôeh;G~MVcX`qMjo*bvnM{b

^YnzaD^^cnMXBhjstfMi;hjX]{GijaD^Yhji;h;^_bv~zVvl£aGZ\X{Dl{G~MVX\MaDstfMijX`VcX\¬gX]Zª^Rb^YnMXZRnzaVcaGZª^YX]VYh=bv

^Yh=ZFb^YnzaD^¢^YnzX`¨h;stfMijhjX]G{i=a^Yhjijh©^_¨/X]ZFVYX[abYXyh©^cn^YnMXb^cVYhjuXyfzVYh=Z\XG~M^¢^Ynzhjb

h;stfMijhjX]5{GijaD^Yhji;h;^_bY~MVvl£aGZ\X/{X]bTz{D^¡ZRnzaTMXh©^cnk^YhjstX8aij{D^[dwnMh=ZRnkhjbmfMVY{GxzaDxzi;_

/~MX¢^Y{{~MV abcbv~zsfM^Yhj{Taxg{G~/^^cnMXij{/ZFaighjTzbv^caDT^RaDTMX]{~zb{i=a^ch;ijh;^_`l~zTzZª^ch;{GT dnMhjZRn

{TMij_ZRnzaTMX[bh©^cn^cnMXmbv^Y{/ZRu8fMVchjZFX

(18)

Implied Local Volatility Surface

v

1.75 3.00 4.25 5.50

0.07 0.13 0.20

50.00 75.00 100.00 125.00

eh;G~MVYXqMqM SUstfMijh;X[Wi;{/Z]aDi{i=a^ch;ijh;^_bY~MVvl£aGZ\X8{l^YnMX¨bvhjs¡~zijaD^YX]$s{//X]i«d

ZFaijZF~MijaD^YX[8lVc{sÌë ¤ S

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saVYuGX\^Ma^RaMÆÄX|~zbYXy² Ï hjTz/X\8{Gf/^Yhj{TfMVchjZFX]b Ma^Ra`aD^-aDT~zaVY_ÊzdPo[pppzdGnzhjZRn

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za^caGbvXF^}l{V/XFst{Tgb^cVcaD^Yhj{TtfM~MVcfg{bvX[bFeSUTt^YnzXl{iji;{hjTMyfMVc{GVcasdDX¢X]bv^YhjstaD^YX^cnMX

8h;stfMijh;X[8G{i=a^Yhjijh©^_bY~MVvl£aGZ\X¢zVcbv^]dnMhji;X^YnzX|¯~zaTG^ci;XF^

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d,zXFTMGi;X]V]d "aVcMi;X

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fzVYh=Z\X[bFdg^YnMX]TZ\{GTzbv^YVc~zZª^^cnMX¡S 3~zbYh;Tz²¢X]VYsaTBaDTzBÉyaTMh stX\^cnM{/BaDTgBZFaijZF~MijaD^YX

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l~zTzZª^ch;{GT{Dl§{f/^ch;{GTfMVYh=Z\X[b/hjVYX[Zª^ci;_lVY{Gs,^YnzXmsaDVcuX\^{Gf/^Yhj{TfMVch=Z\X]bhjbngaDVR/i;_XFGXFV

aD^v^cXFstf/^YX[bYh;TgZ\X8^cnMX8l~zTzZª^ch;{GTaDfMfzVY{aZRnMX[bmaBaDij~MX8{lÅFX]VY{l{GV¡GXFVc_nMhjnb^cVYhjuX

(19)

Implied Local Volatility Surface

1.62 2.75 3.88 5.00

0.13 0.20 0.28

50.00 75.00 100.00 125.00

eh;G~MVcXqMpM SUstfMijh;X[$ij{/ZFaDi{GijaD^Yhji;h;^_5bY~MVYl£aZFX8{Dl^cnMXbYh;s`~Mi=a^YX[kst{MXFid

ZFaijZF~Mi=a^YX[8lVc{s#{T^cX\v¤aDVci;{bvhjs¡~Mi=a^ch;{GT

fMVchjZFX]baDTg{Gf/^Yhj{T¨fMVchjZFX]baDVcXxg{G~MTz/X[¨x_TM{T/UaDVcxMh;^YVRaDGXZ\{GTz/h;^Yhj{Tzb]

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¤aDuh=Z\hÆS dJl{Vyê5ì * _GX]aV]xhj{~gbvij_dP^cnMXX]bv^Yhjsa^YX[5M±²Çx_^cnMX8²¢X]VYsaT

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^YnMXtX[b^ch;sa^cX]fMVchjZFX/h=bv^YVch;xM~M^Yhj{T5{x/^RaDhjTMX]x_^cnMXtS stX\^cnM{/Jd XtZF{stfzaVYX`h©^

^Y{8~gbvX²

Ï

MaDhji;_¨fMVchjZFX]b¢xgXF^X]XFTU-aDT~zaDVc_oGd)o[pprdzaTz=-GaT~zaDVc_Êgyo]ppGpM nMX

nMh=b^c{VchjZ]aDiD^ch;stX bvX]VYhjX]b)/XFTgbvh;^_yX]bv^Yhjsa^Yhj{T¡sXF^YnM{/`/X]bcZ\Vch;xQX]mh;T

"$

^v/aDngaDijhjazd]ÄkaDTM

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{Dl/±²ÌbvnM{G~Mi=xQX8X[¯G~gaDidwnMh=b^c{VchjZ]aDiÆ^ch;stX8bYXFVch;X[bm/±²ÌhjbyhjT5l£aZ\^y^YnMXZF{TzMh©^ch;{GTzaDi

/XFTgbvh;^_¨l~MTzZª^ch;{GT{l}^YnMXt/h;ÍQ~gbvhj{TfMVY{/ZFX]bcbF|ÄX{xM^caDhjTB^cnMXnMhjbv^Y{GVYh=ZFai ^ch;stXtbvX]VYhjX]b

/±²X[b^ch;saD^Yhj{Tx_^YnMXl{Gi;ij{hjTMtfMVY{/ZFX]/~MVcX

o¢¤{Gi;ijX]Z\^¢b^c{ZRufMVYh=Z\X[b^YhjsX`bvX]VYhjX]b

(20)

Volatility Surface

93.8107.5121.2 1.4

2.6

3.7 0.1 0.1 0.1

(80.0,0.2,0.1)

(135.0,0.2,0.1)

(80.0,4.9,0.1)

(80.0,0.2,0.1)

Moneyness

Maturity Vola

eh;G~MVYXmqz;o*MÇh;stfMijh;X[{GijaD^Yhji;h;^_bY~MVvl£aGZ\XX]bv^Yhjsa^ch;{GT¨x_¨{T^YXF¤aVYij{

bvhjs¡~zijaD^Yhj{T

î/

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. åPæ

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(21)

Estimated Implied Distribution

5000 10000

stock price

0 0 0.01 0.01 0.02 0.03 0.03

probability*E-2

)hj~zVYXyqMjooG! nMXX]bv^Yhjsa^cX]b^c{/ZRu8fMVchjZFXMhjbv^YVch;xz~/^Yhj{T dzê8ìÝ* `_X]aV]

nMX]VYX .3 æhjbaÄ hjXFTMX]VfMVc{/Z\X]bcbh©^cnsX[aDTÅFX]VY{taDTgaDVch=aDTzZFX|X[¯~zaDiP^c{ . ª

ÎMÈb^ch;sa^cX/h©ÍP~zbYh;{GTÃl~MTzZª^ch;{GT ë§Óç Óth;T^YnzXB/h©ÍP~zbYh;{GT fMVY{/ZFX]bcb`st{//X]i~zbYh;TM

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ÊzBauX {T^cX\v¤aDVci;{|bvhjs¡~Mi=a^ch;{GTml{GV)^cnMX/h;ÍQ~zbYhj{TfMVY{/ZFX]bcbwh;^Yn/Vch©lI^Æl~MTgZª^Yhj{T

h=bhjTG^cXFVcX]bv^VcaD^YXyaDTzX]bv^Yhjsa^cX]/h;ÍQ~zbYhj{T¨l~MTzZ\^Yhj{T

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ä

å è å æ ç7/§çtë tlVc{s {GTG^cX\

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{T5bvuGXFTMX[bYb|aDTzu~MVv^c{GbYhjb]SUTl£aGZª^|^YnMX/XF^YX[Zª^Yhj{T{Dl}^cnMX/h;ÍQX]VYX]TzZ\XxgXF^X]XFT^YnMX

nMh=b^c{VchjZ]aDi/^ch;stXbYXFVchjX]b/±²°X[b^ch;saD^Yhj{T8aDTg^YnMX/±²&VYX[Z\{GXFVcX]¡lVY{Gs Mah;ij_{f/^ch;{GT

fMVchjZFX]b|sa_xgX~zbYX]5ab|^cVcaG/hjTMVc~MijX]b]dwbYXFXt)axMi;Xqz;otaDTzW¤nzaf/^YX]Vo*M`SUTW)axMi;X

qMjod/±²3X]bv^Yhjsa^cX]tlVc{sÇMah;ij_`{Gf/^Yhj{TfMVYh=Z\X[bza^ca¡bvXF^h=bXFfzVYX[bYbYX]tx_aDTz^YnMX

^YhjstXbYXFVch;X[b/±²Ìh=bQ

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TMX]aVù| Z]aDiji¢fM~/^hjbMX\zTMX[aGb {TMXnM{bvX|X\/X]VcZFhjbYXfzVYh=Z\Xh=b6 §± nMhjnzXFVyij{X]VR

(22)

State Price Density Estimation

5000

£

10000 stock price

0 0.05 0.1 0.15 0.2 0.25 0.3

probability*E-3

¤

eh;G~MVYXmqz;oî/ /±² X]bv^Yhjsa^ch;{GTx_^YnzVYX]XmstX\^Ynz{zbFdgx_¨nMh=b^c{VchjZ]aDi X[b^ch;saD

^Yhj{T dgaDTz¨h;^cbp' §± ZF{T/gMXFTzZFXyxzaDTzk«MabYnMX]gªdMx_Õë ¤S dzaDTz¨x_²

ë!É S !I^cnMhjTgªdgê8ìÝ* `_X]aV]

xz~/^mo*P± i;{XFVnzh;GnMXFVª^cnzaDT^YnMX|l~/^c~MVYXyfzVYh=Z\XG§Ä nMXFTbYuXFm¥J6ø3bvuGXFm/\dMaDX]T^cb

afMfzaVYX]TG^ci;_8aGbYbYhjTaij{X]V fMVY{GxzaDxzh;ijh©^_t^c{tnMh;Gn{G~/^cZF{stX]b {lw^cnMX~MTz/X]VYij_h;TM^cnzaDT

{~zijxQX

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i=a^Y^YX]Veh=b§bYstai;ijXFV)^YngaDT`^YnMXfMVchjZFXhjstfMi;hjX]`x_m/h;ÍQ~zbYhj{TfMVY{/ZFX]bcbw~zbYhjTM¢^cnMX^YhjstX bYXFVchjX]b

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)VRa/hjTM¥~MijX]b^c{X\/fMij{h;^M±² /h;ÍPXFVcXFTzZFX]b

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Wd

bvX]i;iwTMX]aVÔ|¦{Gf/^Yhj{Tzb

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^cVYX]Xh=b}aDTX[abY_ma_¡^Y{maGbYbYX]bcbÆ^YnzXl~/^Y~zVYX¢bv^Y{/ZRu`fMVYh=Z\X[bFdZFaf/^Y~MVcX ^YnzX^YX]VYs b^cVY~gZª^Y~zVYX

(23)

Skewness Trade

1

^ª

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