European Exotic Options

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1 Hado # for B9.38 rg lecre dae: 4/3/ * Rsk-Neral Valao Eroea Exoc Oos e. Prce rocess of he derlyg secry. e. Payoff of he dervave. e 3. Execao of dscoed ayoff der RNPM.. Chooser Oo oo o oo A me : rchase chooser oo wh exrao me, srke rce e, ad decso me < <. A me : decde wheher o se he chooser oo as a call or a. A me : decde wheher o exercse he oo. ee samle roblem a ed.. Look-Back Oo Payoff of he oo s he dfferece bewee he srke ad he maxmm or mmm secry rce drg he lfe f he oo. For examle, he ayoff of maxmm rce look-back call s max { Y e,} where Y max {,,..., }. Recall refleco rcle Assme /d. he call oo me zero rce becomes: E r Y e, Y,,,,...,. Obvosly we eed he dsrbo of he maxmm. Le s ge some o abo hs maxmm. We showed:

ee samle roblem a ed.. Look-Back Oo Payoff of he oo s he dfferece bewee he srke ad he maxmm or mmm secry rce drg he lfe f he oo.

2 Le, * smalles eger greaer ha or eqal o / whch we wre as / he, Pr{Y } > / /. > / / ee examle a ed. How wll yo do a ormal aroxmao? Wha haes whe /?

3 Barrer Oo Kock-o oo ayoff s he same as a valla call or oly f he secry rce ever reaches a re-secfed barrer drg he lfeme of he oo. Kock- oo ayoff s he same as a valla call or oly f he secry reaches a re secfed barrer drg he lfeme of he oo. * Prce of a valla call or oo s eqal o he sm of rces of he barrer call or oos a arg. Examle. N,,,., d.8, r.4. Wha s he rce rocess of a chooser oo wh srke e ad decso me? Wha s he rce rocess of a -ad-o oo wh srke ad wh srke e barrer k.? Is ayoff s X max {,} f max{ f max{,, } < k,, } k 3 Wha s he rce rocess of a -ad- oo wh srke e ad barrer k.? Is ayoff s X max {,} f max{ f max{,, } k,, } < k 3

* Prce of a valla call or oo s eqal o he sm of rces of he barrer call or oos a arg. Examle. N,,,., d.8, r.4.

4 heorecal Comao of a barrer oo rce Cosder /d, r, e, k, ad o call. where k,,,,...,. Le τ be he frs me he secry rce hs he level k. we se τ f he rce remas below k l me. Wha dsrbo do we eed? r E e I{ τ } e { ; τ } r where : a ms be greaer ha - ms b ms have same ary as recollec ha me rce s mles: N d -N -- b d /, hs, N - N > N - > ad have same ary ** c for he call o ay moey we eed > e, > loge/ /log. e Noce ha we kow he robably ha τ! {τ } {τ } - {τ -} {Y } - { Y - }. 4

r E e I{ τ } e { ; τ } r where : a ms be greaer ha - ms b ms have same ary as recollec ha me rce s mles: N d -N -- b

5 e Now oce ha f we kew ha he rce h level k a me he he robably ha eqals a me : { τ } { - }? Le N be he mber of heads eeded o ge from o or N d - -N or N - N or N - - / heads - osses ossble oly f N s a eger. From ** ad ms have same ary becase we kow ad have same ary. e a e 3 { ad τ } { τ } { τ } If we add he robably se a over all ossble fe vales ha τ ca ake we oba: { ad τ s fe } hs, f we sbrac hs from { } we shold oba? 5

s a eger. From ** ad ms have same ary becase we kow ad have same ary.

6 6 Wha are all ossble vales of τ? say he smalles vale. For he larges vale choose bgges sch ha - - Or: hs, ; { } { } ; { < τ τ ; { } { {Y {Y q q q q τ

7 Hado # 3 for B.9.38 lecre dae: 4/3/ P-Call Pary Porfolo of Oos ad Oos o Porfolo Idex Relaosh bewee a ar of Eroea call ad wh he same exrao ad srke rce e C [ B B ] P ee, for,,..., where C ad P are rces for he call ad he a me. hs s becase X c X e Examle. N,,,., d.8, r.4. Cosder a ar of Eroea call ad oos, boh wh srke e. a Do he wo oos sasfy he -call ary? b ose ad d are kow. he me marke rce for he Eroea call s $.3, ad for he Eroea s $.5. Are here arbrage oores? If yes, fd oe. 7

exrao ad srke rce e C [ B B ] P ee, for,,..., where C ad P are rces for he call ad he a me. hs s becase X c X e Examle.

8 Lear Prcle ose he ayoff of a C.C. ca be wre as L X Y Y Y m X X X L where X, K, X are ayoffs of Eroea calls ad Y, K,Ym are ayoffs of m Eroea s wh dffere exrao mes ad srke rced. he he rce of hs C.C. s gve by x L C P P P C C L P where C s he rce for he h call, K, ad P s he rce for he h, K,m. Examle. For each of he followg ayoff srcre, fd a orfolo of Eroea calls, s, derlyg sock, ad bak acco ha gves he ayoff srcre. How wold yo calclae he far rce of each orfolo? Add wo fgres by had m 8

exrao mes ad srke rced. he he rce of hs C.

9 Irodco o Mahemacal Face, by R Plska, age 9 Examle 4.4 ose K9, N,, r, ad he rce rocess ad formao srcre are as dslayed fgre 4.. here s a qe rsk eral robably measre for hs model; s also dslayed fgre 4.. Now cosder a call oo wh exercse rce e3 o he me vale of he sock dex. I oher words, hs coge clam X 3 ad s dslayed fgre 4.. As wll be exlaed seco 4.4 X s markeable. Hece s me rce s easly comed o be E q X. ω X ω 7, 7 /9 8, 5 9, 5 /9 8, 3 /9 6, 8 /6 7, 6 7, 8 6, 9 /, 7 / 3 3, 8 / 6, 5 6, 3 /6 9, 6 / Fgre 4. Daa for examle 4.4 9

. Now cosder a call oo wh exercse rce e3 o he me vale of he sock dex. I oher words, hs coge clam X 3 ad s dslayed fgre 4.

REVISTA INVESTIGACION OPERACIONAL Vol. 25, No. 1, 2004. k n ),

REVISTA INVESTIGACION OPERACIONAL Vol 25, No, 24 RECURRENCE AND DIRECT FORMULAS FOR TE AL & LA NUMBERS Eduardo Pza Volo Cero de Ivesgacó e Maemáca Pura y Aplcada (CIMPA), Uversdad de Cosa Rca ABSTRACT