Pricing Arithmetic Average Reset Options With Control Variates

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1 cn Ac Av R Opon W Conol V U-LA LAO Aoc ofo of nnc pn of on n Bnkn onl Cnc Unv 64 C-nn R.c. p 6 wn l : : El A:lol@nccu.u.w COU-WE WA An ofo of nnc pn of nnc onl oun Unv of cnc n cnolo Juou R. nz c oun 8 wn l : : El A: lov@pco.co.w : 7/4/

2 cn Ac Av R Opon W Conol V U-LA LAO A COU-WE WA B on clo-fo oluon of pl opon w v pc of nl opon w lvl n connuou. uo w pov o pcl ccc of cll n pu opon. W plo pnon of l up n fo cll n pu opon un n po wnv ock pc ouc. o pccl pplcon w u cll opon w connuou conol v o vlu pc of c v opon l on wn ock Ecn fo 998 o pnn opon wo poff nflunc p of pc of unln v co ncnl popul n cn. On of p-pnn opon opon. k pc of opon woul u onl on pcf f pc of unln low on of lvl. n pcc opon v n fo n. Cco Bo Opon Ecn CBOE n w Yok ock Ecn YE o nouc & 5 n pu wn w -on po n l 996. on nl u wn w n nl k pc of $44.73 n Jul 997. k pc woul u o $39.76 on Auu f pc of unln fll low $ n 998 n C cu n wn u opon k pc woul u f - v clon pc of Un colconc Copoon wo cu co n wn ock Ecn E 33 fll low 9% of nl k pc $58.5 un f on f wn w u. nc opon nw vv n fnncl k l lu W ful o n nonou f n o pn lwk o fo lpful con n uon. An nn o ou.

3 . n Wl 997] n pcn of & 5 k wn w poc n cop v-à-v n &5 pu n lvl n voll cn. n Wl 999] lo pov clo-fo oluon fo opon w nl. Cn n n ] u opon k pc wll o pvln ock pc f opon ou of on. A clo-fo pcn foul n of ulv nol uon v un k-nul fwok. nl poff of opon w n n nl k pc u Cn n n ] follow: C { n... ] } n n pcc nl poff of opon o ofn C ] ] R... n ] > f n fo... w... k pc n... lvl; > > L > >. opon c n n cull op n l wol. Lo n Wn ] pov c clo-fo oluon fo nl opon w lvl n n p-c un k-nul fwok. pnon of l n up co wll pop of l n wvn n lo nv n pp. vl kn opon w f on fo n pl non of ulv nol uon 9. ffcul o clcul ccu cuulv pol of ulv nol uon w non n nc ccu pc of opon w n lo ffcul o on. uo clo-fo oluon of c v opon o no

4 u o fc u of lonol vl no lonol. n o o ovco pol ovon fo pp o u l connn cl o ppo vlu of opon o o k l connn cl conol v n on Clo ulon o on pc of c v opon. l connn cl w u n pp ulpl pl opon w lvl n connuou po l n o u. Acconl un nl o w v clo-fo oluon fo nl- opon w onon po l n o u.. pl opon. foul ul of nn n 994] n u 997]. n w u pl opon uln lock o v clo-fo vluon of nl opon w lvl n connuou po wc l n o u. uo w conuc on Clo ulon un clo-fo oluon of opon w lvl n connuou po conol v o vlu pc of n c v opon w lvl n n. An ouln of pp follow. W l ol n v clo-fo vluon of nl- opon n nl opon w connuou n con Ⅰ. n con Ⅱ w pn o pop of opon w connuou n cop cul un pc of opon w ul pc. con Ⅲ pov concluon of pp. Ⅰ. RC ARAL REE OO W u nc of unln pc c follown occ ffnl quon: 3

5 Q W w n > conn n conn po n ov n po ]. Q W n fo on-nonl n Bownn oon fn on fl pol pc Ω Q w Q po nl u o k nul pol u. vn ccoun B fn B B L p n po nvl ] l of po w <. poff of nl- own-n-ou o own-n-n cll opon n own-n-ou o own-n-n pu opon p follow: OC O ] n ν > C ] n ν ν ν ] n ν > ] n ν ν w nco funcon; > n. ν ll poff p of nl- up-n-ou o up-n-n cll opon n up-n-ou o up-n-n pu opon vn follow: UOC UO ] ν < UC ] ν ν ν ] ν < U ] ν ν ν w < n. n o o v vlu of ll p of nl- cll n pu opon w po l n o u w f pov follown o: 4

6 5 o. Au W u n > n w v ] n u u u u u 3 ] n u u u u u 4 ] u u u u u 5 ] u u u u u 6 w cuulv pol of v nol uon w zo n n nnnou colon coffcn. W pov o n Appn A. Un o w cn fowl v pc of ll nl- opon own n o. o. L OC UC n U pc of own-n-ou cll own-n-n pu up-n-n cll n up-n-n pu w po wc l n -o-u pcvl. n w v OC 7

7 6 UC U 8 w n ln ln ln B pu-cll p w v follown lon: OC C C U UO UC C UOC O w ] ] C ] ] 9

8 7 W pov o n Appn B. Un fc ] ] l l n ε ε w ε n nol no vl w know wn ppoc pc of p of opon wll co opon w po n qul o o u. kn own-n-ou cll opon n pl wn ppoc w v ] ] l OC ] ] wc ul of Runn n Rn 99]. ow con cll opon w lvl n connuou w po l n o u. poff p follow: R C ] ] > n f fo... o quvlnl R C ] > > n n w... n > > > > L. Coponnl poff p of pu opon w lvl n connuou w po l n o u follow: R ] ] f > fo....

9 ll w v R ] w... n > > L > >. fo w cn plc cll opon w follown n :. uc on un of Euopn cll opon w k pc.. uc on un of Euopn own-n-ou cll opon w k pc... fo c. 3. o ll on un of Euopn own-n-ou cll opon w k pc... fo c. n w w cn plc pu opon w follown n :. uc on un of Euopn pu opon w k pc.. uc on un of Euopn up-n-n pu opon w k pc... fo c. 3. o ll on un of Euopn up-n-n pu opon w k pc... fo c. Conqunl w v follown ul: o 3. L R C n R no pc of opon w lvl n connuou w po l n o u pcvl. > > n fo.... > n > fo.... -f pc of nl opon R C ] ] OC OC R ] ] U U 8

10 w OC f o own-n-ou cll w k pc n lvl. U f o up-n-n pu w k pc n lvl. OC n U vn 7 n 8 pcvl. cll o pu opon w po l n o u cn v upp oun o conol v of cll o pu opon w c pcll wn n c n nc wn n wo cn ll. Ⅱ. UERCAL AALYE O REE OO Ccc of R Opon w Connuou R w cu pop of cll opon w connuou. Con on--u cll opon w nl k pc. k pc wll u f clon pc of unln ock fll low 8% 7% n 6% of nl k pc. W cop pc of opon w 3 lvl n po of on on n on o pln vnll cll opon. ul own n l Ⅰ. o l Ⅰ o ccc of cll opon l o n Euopn cll opon. vlu of cll opon ncn funcon of ock pc k-f n n voll of ock un. n on fv pop unqul n cll opon: vlu of cll opon ncn w uon of po. Un lvl low k pc wll v vlu fo cll opon. 3 u o o pocon ow ol of cll opon vlu of cll opon w connuou lw n vlu of pln vnll cll opon. 4 Bcu cll 9

11 opon ncn funcon of po opon w connuou cn v upp oun o conol v fo cll opon w c. 5 n c of vlu of ock pc n lvl n low voll of ock un pc oun fo cll opon w c wll ll. o pl wn ock pc k-f voll of ock un n k pc n 987 pcvl pc oun of cll opon w c c of on-on onon po ]. W now nv pop of pu opon w connuou. Con on--u pu opon w nl k pc. k pc wll u f clon pc of unln ock ov 5% % n 5% of nl k pc. W cop pc of opon w 3 lvl n po of on on n on o pln vnll pu opon. ul pn n l Ⅱ. l Ⅱ ow ccc of pu opon lo l o pln vnll Euopn pu opon; nl vlu of pu opon cn funcon of ock pc n k-f n ncn w voll of ock un. Aonll pu opon v follown pop: l o cll opon vlu of pu opon ncn w uon of po n n vlu of pln vnll pu opon u o o pocon ow ol. vlu cn v upp oun o conol v fo pu opon w c. Un lvl k pc wll nuc vlu of pu opon. 3 n c of low vlu of ock pc n lvl n low voll of ock un pc oun fo pu opon w c wll ll. o pl wn ock pc

12 k-f voll of ock un n k pc n 55 pcvl pc oun fo pu opon w c c of on-on onon po ]. l Jup of R Opon nc w cn plc opon w opon l o opon opon lo v pnonon of l up wnv ock pc ouc. Un cn ul of ffnon w cn v l of opon. ul pn n follown o. o 4. l of cll opon n pu opon follow: C R OC OC ] R U U ] w B A OC B A Y U Y ] ] ] p π A

13 ] ] p π ] ] p ] ] p π π B ] ] ] p π ] ] p π ] ] p ] ] p π π Y l up cn occu n wn po. f ock pc ouc fo l woul co follown pon: C R OC OC ] R U U ] Bcu k pc of OC n OC o U n U ffn OC OC n U U wll no qul o zo. A ul ] u l wll up f ock pc ouc.... o pl con wo c follow: On--u cll opon k pc wll u f clon pc of unln ock fll low 87.5% 75% 6.5% 5% n 37.5% of nl k pc 8. voll of ock un.5. On--u pu opon k pc wll u f clon pc of

14 unln ock ov n 6 7 of nl k pc 35. voll of ock un.3. Bo opon v po on. l plo n u n pcvl. pnon of l up n o c. l up ppn wnv ock pc ouc. o cll opon wn ock pc ov vo of l l o l of pln vnll cll opon. owv wn ock pc fll n ouc u o k pc n u o nw low lvl l of cll opon wll up o lvl. o pu opon wn ock pc low low vo of l l o l of pln vnll pu opon. vl wn ock pc o clo o u o fc k pc coul u o nw lvl f ock pc cull ouc pu opon o vlul. oc l of pu opon wll nc n vn co pov. Af ock pc cull ouc k pc u o nw lvl. Conqunl pu opon n v pln vnll pu opon n l up o low lvl. o kn cll opon n pl f nu ock pc ouc l of cll opon follow: C R l OC OC l ] ] owv l ] ] > ] Copn n w cp c ock pc 3

15 n l ffn fo l fo. A ul ovou l wll up n po ] wnv ock pc ouc low of cll opon. ucl Rul of c of Ac Av R Opon n pcc opon u c v of ock pc. owv u o pol u of lonol uon no lonol c clo-fo oluon of c v opon o no. o on w u opon w connuou po conol v fo on Clo ulon o on f pc of c v opon. o on Clo ulon w k cll opon n pl. W pon nvl ] no n unvl of qul ln n l pon pon... n. W n v ln ln ε... n w ε... n ncl n npnn n nol no vl. o ul p... w co nu vlu of ock pc un po. f nu vlu low l lvl fo l... n vlu of conol v pl cll opon ] C Conol ] ] p l o vlu of c v cll opon uppo n on n fo c pon pon. W v n. 3 n c v vlu of ock pc copu on pon pon... n. ll fo ul p w co nu c 4

16 v vlu of ock pc un po. f nu c v vlu low f lvl fo f... n vlu of c v cll opon ] C R p ] ] A ul vlu of c v cll opon w conol v R Conol R C C ] C R Ĉ f w R C copu on. W cn lo u nc v f popo Bol 977] n opon pcn cn pfcl nvl col. n w v p o copu opon vlu. B on ov o ou pcl pouc c v opon l on E fo 998 o 999. conon of c v opon own n l Ⅲ. o l w nu of c l fo o nn n nvl wn n wo cn onl on. Conqunl kn of opon w n c cn u opon w connuou po conol v n on Clo ulon o vlu f pc w ll n von. un 998 n 999 w u R Rpuc An po of k-f n wc wn 4% n 5%. nucl ul fo ulon uz n l Ⅳ. o l w cn n von of ulon ll un conol v. nwl cl kup of opon n wn wn % n %. 4 kup ul fo ncon co n ulon con.. no llow o u pu opon n wn. 5

17 Ⅲ. COCLUO Un nl o w pov clo-fo oluon fo p of pl opon n opon w connuou. W lo ow o pcl ccc of cll n pu opon. Bo of cll n pu opon ncn w uon of po n vlu of pln vnll opon u o o pocon ow ol n cn v upp oun fo cll n pu opon w c. W lo pov l of cll n pu opon n nv pnon of l up n n cll n pu opon un ll po wnv ock pc ouc. nll clo-fo oluon of opon w connuou po cn u conol v n on Clo ulon o copu f pc of c v opon. o nucl nl w u c v opon l on E fo 998 o 999. ulon ul ow kup of opon u n pcc wn % n %. kup cu ncon co n ulon con. 6

18 ABLE Ⅰ c of ln Vnll Cll Opon n R Cll w 3 R lvl n R o of On on n on 3% 5%.5. 7 R o R o On 3 On on on on on % % o pop unqul n cll opon: 3 vlu of cll opon ncn w uon of po. Un lvl low k pc v vlu fo cll opon. 3 u o o pocon ow ol of cll opon vlu of cll opon w connuou lw n vlu of pln vnll cll opon. 4 Bcu cll opon ncn funcon of po opon w connuou n pln vnll cll opon cn v upp n low oun fo cll opon w c pcvl. 5 n c of vlu of ock pc n lvl n low voll of ock un pc oun of cll opon w c wll ll. 7

19 ABLE Ⅱ c of ln Vnll u Opon n R u w 3 R lvl n R o of On on n on 3% 5%.5. 7 R o R o 3 On 3 On on on on on % % pop of pu opon : 3 vlu of pu opon ncn w uon of po n vlu of pln vnll pu opon u o o pocon ow ol n cn v upp oun fo pu opon w c. Un lvl k pc wll nuc vlu fo pu opon. 3 n c of low vlu of ock pc n lvl n low voll of ock un pc oun fo pu opon w c wll ll. 8

20 ABLE Ⅲ R Conon of R Opon on wn ock Ecn Co n E R Conon k pc woul u f - v clon pc of 33 on 57 E fll low 9% of nl k pc $58.5 un f on f wn w u. k pc woul u f - v clon pc of 33 on 5 E fll low 98% 96% 94% 9% 9% of nl k pc $8 un f on f wn w u. k pc woul u f - v clon pc of 33 on 53 E fll low 8% of nl k pc $57 un f on f wn w u. k pc woul u f - v clon pc of 3 on 57 E fll low 8% of nl k pc $95.5 un f on f wn w u. k pc woul u f - v clon pc of 373 on 58 E fll low 95% 9% 85 of nl k pc $58.5 un f on f wn w u. k pc woul u f - v clon pc of 33 on 538 E fll low 85% of nl k pc $8 un f on on f wn w u. l ow of opon l on E. opon u c v of ock pc n v ulpl lvl. 9

21 ABLE Ⅳ ucl Rul of Ac Av R Opon Co n u u ocl.4. 5 E c Voll c kup c kup // % % % 5 999/4/ % % % /5/ % % % /6/ % % % /6/ % % % // % % % nucl ul on of opon n l Ⅱ. n von of on Clo ulon. o u pon no 7 unvl of qul ln. kup coun u c - c / c%. o l cl kup of opon n wn wn % o %. kup ul fo ncon co n ulon con n wn no llow o u pu opon o.

22 URE l of R Cll w v R Lvl n -on R o l ock c 8... ] 76543]... ] 76543].5 v po on.

23 URE l of R u w v R Lvl n -on R o l ock c ] 45678]... ] 5 5 po on ].5 v. 5.

24 AE A oof of o Au u W c n > w W n fo on-nonl n Bownn oon on fl pol pc Ω w flon n. fn v W n W W u v. Acconl w v c E p w n { c }. L pol u quvln o uc Ron-ko vv v v p vw p v B nov o poc W fn W W v follow n Bownn oon un pol u. o flcon pncpl of Bownn oon w v c E p v v pv E p v v c nc c n { } { } { c } Conqunl c 3

25 4 p p c v v E v L pol u R on Ω fn Ron-ko vv p v v R. n poc R W fn v W W R n Bownn oon un pol u R. nc v W R n w on c p c v R v c v v p n w c v c v v p w cuulv v nol uon w n vco n nnnou colon coffcn. L ˆ n n n c c u u u c p ˆ ˆ ˆ A ul w v c c c c u u u c u u p Bcu follow

26 5 u u u u u p copl poof of 3. o pov 6 f no W v W v v v fo w v follown pon: u u u u u p ll u u u u u p A. n un A. w on u u u u u p copl poof of o.

27 6 AE B oof of o Un k-nul pol u Q -f pc of own-n-ou cll opon follow: ] > Q n E OC { } { } Q Q E E w > n n pcf n Appn A. Conqunl w v { } Q Q W W E E p L pol u fn Ron-ko vv p v vw Q. n W fn v W W n Bownn oon un pol u. fo { } { } Q E E > n > n > n ln ln w W n o n n npnn of

28 7 fl. Conqunl { } Q E > ln ln Un 3 w cn on { } Q E Coponnl w lo v { } Q E A ul OC. n follown pocu un 4 o 6 w cn v pc of UC n U. pu-cll p of OC n C follow: ] > Q n E OC ] Q n E ] { } Q E ] Q n E C C ll w v U UO UC C UOC O w C n fn n 9. copl poof of o.

29 EOE. nn n 994] u nl pon n u 997] u..e. ppoc o vn clo-fo oluon of pl opon; owv no v pc of opon. pp pov c w un pl vluon ppoc nl o n off n no occ pop of pc poc of unln.. l pnonon lo ppn fo pu opon. 3. w cn u o p o k n. 4. Alnvl w coul u pl voll o copu c v opon on E. owv w no on opon fo cu. u w onl u ocl voll o copu pc of opon. 8

30 REERECE Bol.. Opon: A on Clo ppoc. Jounl of nncl Econoc pp Cn W. Y. n. n. Anlc of R Opon. Jounl of vv pp n R. Wl. Vlun & 5 B k Wn w oc R. Jounl of vv 997 pp n R. Wl. R u Opon: Vluon Rk Ccc n n Applcon. Auln Jounl of nn pp. -. nn R. C. n... l B Opon. Jounl of nncl Ennn pp u C.. -pnn opon vlu. Jounl of uu k pp Lo.L n C. W. Wn. Vluon of R Opon w ulpl k R n R. Jounl of uu k ocon. Runn. n E. Rn. Bkn own B Rk 99 pp

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