DEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES

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1 DEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES The ulti-bioil odel d pplictios by Ti Kyg Reserch Pper No. 005/03 July 005 Divisio of Ecooic d Ficil Studies Mcqurie Uiversity Sydey NSW 09 Austrli

2 The Mcqurie Uiversity Acturil Studies Reserch Ppers re writte by ebers or ffilites of the Deprtet of Acturil Studies, Mcqurie Uiversity. Although urefereed, the ppers re uder the review d supervisio of editoril bord. Editoril Bord: Piet de Jog Leoie Tickle Copies of the reserch ppers re vilble fro the World Wide Web t: Views expressed i this pper re those of the uthor(s) d ot ecessrily those of the Deprtet of Acturil Studies.

3 THE MULTI-BINOMIAL MODEL AND APPLICATIONS By Tiothy Kyg, M Stts, M Ec, FIAA Lecturer i Acturil Studies, Mcqurie Uiversity, Sydey This pper develops ethod for the vlutio of ultivrite cotiget clis d is extesio of the well kow bioil optio pricig odel. I refer to it s the Multi-Bioil Model. Structure of the pper: ) Itroductio ) The proble the ethod is desiged to ddress 3) exples of rel world ficil probles / ultivrite cotiget clis 4) lytic solutios d uericl solutios: the diesiol cse 5) the ulti-bioil ethod for vlutio of cotiget clis 6) exples / uericl results 7) coclusios 3

4 ) Itroductio The ethod preseted is uericl lgorith for the vlutio of cotiget clis ivolvig ultiple, correlted, orlly distributed, or logorlly distributed, sources of risk. The ethod costructs discrete pproxitio to correlted ultidiesiol geoetric Browi otio process s the drivig process which geertes the outs pid uder the cotiget cli beig vlued. It is ultidiesiol tree pproch to the vlutio of cotiget clis. My such cotiget clis re itrctble whe it coes to derivig or usig lytic pproch. The ethodology developed here c be pplied to wide rge of optio pricig d cotiget clis vlutio probles. I prticulr it is pplicble to vlutio of executive shre optio schees d vlutio of coplex exotic derivtives ivolvig ore th oe sset, such s executive shre optio schees, spred optios d ribow optios. This uericl ethod is esy to progr d is efficiet d prcticl, t lest for low diesiol probles (up to bout 5 ssets). Whe the uber of ssets ivolved gets too big, it is better to use Mote Crlo Siultio. Vrious exples re icluded to illustrte the ethod. 4

5 ) The proble We hve soe ficil cotrct or istruet or cshflow with pyoff / cshflow de t tie T, d (,..., ) (,..., ) the pyoff f f S ( T) S ( T) vribles S ( T) S ( T ). = depeds o the vlues of differet These vribles could be the vlues of differet sset prices observed t tie T. Assue these re either orlly or logorlly distributed Wht is the expected vlue of the pyoff / cshflow? Wht is the preset vlue of this expecttio? c we express the pv or expected vlue s soe fuctio g of the iitil vlues ( S ( 0 ),..., S ( 0)) d soe other vrible? Is lytic solutio vilble? If so is it esy to uericlly copute it? Nuericl solutios (e.g. Mote Crlo Siultio) 5

6 3) Exples of rel world ficil probles iclude:. exchge optio (optio to exchge oe risky sset for other) optio to exchge oz of gold (G) for 00 oz of silver (S). Pyoff = x S G,0. ( ) T T b. spred optio (optio o the differece betwee the prices of risky ssets) Pyoff = x ST GT K,0 where K is soe costt. Eve differece if both S d G re logorl, the the differece will hve soe other distributio (ot logorl) so BS type forul will ot pply. Altertively isted of beig optio cotrct S ight be the ssets d G the libilities of isurce copy, so the pyoff is the extet to which the surplus exceeds the level K. c. 3-sset ribow optio (optio o portfolio of 3 ssets R, S & V) Pyoff = x ( R + S + V K,0). T T T d. Optio defied o sset S t 3 differet ties (,, 3) S if( S>S d S>S 3) pyoff de t tie T d is VT = 0if( S S or S S ) where S i = vlue of S t tie T i T T T d with the 3 e. A 4-istruet pyoff (optio o portfolio of log d short positios i 4 ssets R, E, A &L or the differece betwee portfolio of ssets d libilities d soe costt K. Pyoff = x R E + A L K,0. ([ T T T T] ) 6

7 4) Alytic d Nuericl Solutios: The uivrite situtio ( = ) Alytic solutios: For soe types of pyoff, there is lytic forul. The pyoff fro Europe cll optio cotrct over the stock S, with pyoff = x( S X,0) = S X + exercise price X, pid t tie T is ( ) The pyoff fro put optio cotrct over the stock S, with exercise pyoff = x( X S,0) = X S + price X, pid t tie T is ( ) Stdrd cll d put optios over sigle sset c be vlued usig the well kow Blck d Scholes forul. This lytic forul pplies oly to optios with Europe style exercise rights (those tht c be exercised oly o the turity dte of the cotrct). This forul expresses the vlue of the optio i ters of the cuultive desity fuctio of the stdrd orl distributio. Blck Scholes Forule for Europe Optios yt rt c = Se N( d) Xe N( d) rt yt p = Xe N( d) Se N( d) where S = "spot price" (i.e. the curret price) of the stock X = the exercise price of the optio r = the risk free iterest rte per u y = divided yield (divided pid p.. s proportio of the stock price) T = ter to turity of the optio σ = voltility of the stock c = vlue of Europe cll optio p = vlue of Europe put optio S σ T X d d = d σ T x z N( x) = e dz = Pr { z < x z ~ N( 0,) } π d = loge + r y + σ T 7

8 Fro sttisticl poit of view,. the pyoff / cshflow fro optio is rdo vrible, d. the vlue of the optio is the expected vlue of tht cshflow, discouted to preset vlue. The pyoff depeds o the distributio of the uderlyig sset ( stock). Uder the Blck Scholes Model, the logrith of the price reltive hs orl distributio: S T It is ssued tht loge S0 follows Browi Motio process with drift d hs orl distributio with. e = r σ T. vrice = σ T These preters defie the so clled risk eutrl distributio of the sset price t y tie T. This distributio is the oe tht kes the expected retur o the sset S equl to the risk free iterest rte r. Liittios of the BS forul: it is ot pplicble to optios tht hve erly exercise fetures (c be exercised before the turity dte) such s Aeric optios d Berud optios. it is ot pplicble to y of the exotic optios (those tht re differet fro stdrd cll or put optio). 8

9 The oe diesiol bioil odel: Oe wy to overcoe soe of the liittios of the BS odel is to use the bioil odel isted. I prticulr it llows us to vlue eric optios d other optios with ore coplex pyoffs th the stdrd cll d put optios, i the cse where the pyoff depeds o oly oe sset. This is discrete tie, discrete stte spce pproxitio to the vlue of optio. There re severl wys to ipleet it. The ost coo is clled the Cox Ross Rubistei (CRR) versio. We look t this first. Aother wy to ipleet it is the equl probbility versio of the odel. The iputs to the odel re (S, X, r, y, σ, T, ) Model ssues tht the sset price c chge oly t discrete poits i tie. We brek up the tie itervl [ 0,T ] ito eqully spced tie itervls of legth T = T. Tie goes i steps of legth T. At ech tie step the sset price c either go up or it c go dow. The proportiote icrese or decrese is costt: Either S = t u S + t or S = S t+ d t The u d d fctors re costt t ech tie step. u = exp( σ T ) d = exp( σ T ) l( ) l( ) u = d = σ T The probbility of up step is exp (. ) ( ) ( ) p= r T d u d 9

10 For Europe cll optio: The vlue of the optio is E x S X,0. e rt ( ) { ( T )} which is discouted coditiol expecttio: (.. r T) k k k e x ( ) ( ) ( k ) vlue = S0u d X,0 p p discout i= 0 k fctor pyoff probbility this forultio of the optio price s discouted expecttio shows tht we c vlue other ore coplicted pyoffs just replce the pyoff fuctio show here with soe other pyoff fuctio It c be show tht s the uber of tie steps gets bigger, the vlue of cll optio give by this forul coverges to the blck scholes vlue Lookig t the odel sttisticlly: the stock price t tie T (= tie step ) fter j up jups i price is j j ST S0u d k k Sud 0 : k= 0,,.., =. This tkes vlues i the set { } there re ( + ) = ( + ) differet vlues i this set The probbility distributio of the sset vlue t tie step is k k k ) Pr( ) ( ) ( k S = S0u d = p p k for k = 0,,..., The probbility distributio of the log price reltive is k Pr( l( ) ( ) ( )) ( ) ( k S ) T S0 = l d + k.l u d = p p k ( l( ST S0 ) l( d) ) k = hs bioil distributio. It represets the l( ud) uber of up jups out of tie steps. The log of the price reltive hs trsfored bioil distributio, it is log bioil isted of log orl. The trsfortio is ffie trsfortio ( lier trsfortio plus costt) 0

11 MOMENT MATCHING The crr versio of the odel fixes the up d dow oveets (u,d) d coputes the probbility p fro the u d d. Preters u, d d p re chose i such wy tht the probbility distributio of the price reltive hs the right oets. I prticulr, the expected retur o the sset is the risk free rte d the stdrd devitio of the log of the retur is the voltility of the sset. This is the risk eutrl distributio I order for the expected retur o the sset to be the risk free rte, the rte of price growth ust be equl to the differece betwee the risk free rte d the divided yield o the sset. The totl retur is coprised of the divided yield d the cpitl gi retur. The equl probbility versio of the odel fixes the probbility p to be 0.50 d coputes the fctors u d d so s to chieve oet tchig to produce the risk eutrl distributio of the sset price. I optio pricig there re vrious lytic d uericl ethods used. These uericl ethods ivolve coputig the expected vlue of the pyoff uder the so clled risk eutrl distributio d the discoutig this expecttio t the risk free rte. This is true of the bs odel, the bioil odel d vrious ote crlo siultio ethods for vlutio of optios.

12 5) The Multi-Bioil Method for vluig Exotic Optios For soe ulti-sset ulti period exotic optios, it is possible to derive lytic vlutio forul, i Blck-Scholes frework. These ofte ivolve the cdf of the ultivrite orl distributio isted of the uivrite orl distributio. The k diesiol ultivrite orl cdf is the fuctio (,..., ; ρ,,..., ρ, ) { Z Zk b corr( Zi Z j) ρij Zi N( )} N k k k k = Pr <,..., <, =, ~ 0, It is ofte quite difficult to derive lytic forul for such ulti sset optio. Soeties it is ot possible to fid oe. Eve if lytic forul exists, it y ot be prticulrly useful becuse it requires uericl coputtio of the ultivrite orl cuultive distributio fuctio. There exists good uericl pproxitio for the diesiol orl cdf (kow s drezer s pproxitio). However for higher diesios, there is o good uericl pproxitio d i prctice ote crlo siultio is used to evlute it.

13 Exples: ) For the exchge optio, lytic forul exists. It is clled the rgrbe forul d is relly the bs forul i disguise. This is ST possible becuse the rtio is logorl if both the uertor d GT deoitor re. ) For the spred optio ( Pyoff = x ST GT K,0 where K is soe differece costt. Eve if both S d G re logorl, the the differece will hve soe other distributio (ot logorl) so BS type forul will ot pply. 3) Optios o portfolios such s 3-sset ribow optio with Pyoff = x R + S + V K,0 d 4-istruet pyoff such s ( T T T ) ([ T T T T] Pyoff = x R E + A L K,0). The su or differece of logorlly distributed vribles is ot logorl so BS type forul wo t pply 4) Optio defied o sset S t 3 differet ties (,, ) is de t tie T d is S if S>S d S>S VT = 0if S S or S S ( 3) ( ) 3 where S i T T T d the pyoff 3 = vlue of S t tie T This oe hs lytic forul rt ( T ) µ µ T T vlue = Se N T T, T3 T, σ σ T3 T N, b, ρ = Pr Z <, Z < b corr Z, Z = ρ, Z ~ N 0, { i )} where ( ) ( ) ( this ivolves the cdf of the diesiol orl distributio i 3

14 UTILIT OF MULTI -BINOMIAL MODEL It is these types of probles which the ulti-bioil odel c vlue uericlly. It is ltertive to ote crlo siultio whe the diesio of the proble (the uber of ssets) is low (sy less th 6). For higher diesiol probles, ote crlo siultio is ore efficiet. This ethod is pplicble to wide rge of optio pricig d other ficil probles, such s ) Executive shre optio schees with perforce hurdles. These re optios where the pyoff depeds o the shre price of the fir t the turity dte, d the bsolute perforce (rte of retur) of the fir up to soe tie or set of ties the reltive perforce of the fir (reltive to other stocks, or to soe stock rket idexes) up to soe tie or set of ties. other hurdles ) spred optios, ribow optios 3) sset libility studies where ech of the ssets d ech of the libilities re logorlly distributed 4) ulti sset optios with erly exercise fetures (these re ore difficult to vlue usig ote crlo siultio) 4

15 The i result: Assue tht we hve (,,..., ) ssets, vlue t tie t give by the vector: S ( t) S ( t) S ( t ) These re ssued to be logorl. vector of returs () ( ) ( l ( S t S 0 ),...,l ( S ( ) ( 0 t S ))) this returs vector follows correlted -di Browi otio with dsi () t drift = µ i. dt + σi. dwi() t : i =,..., S t i () The e vector of the returs is ssued to be µ = r q σ, r q σ,..., r q σ T = T µ The covrice trix of the returs is ssued to be σ σσρ... σσρ σσ ρ σ... σσ ρ Σ= T =Σ T σσ ρ σσ ρ... σ Let R be -diesiol iid bioil rdo vrible with preters d probbility p = 0.5 so tht the copoets e vector E ( ) of the vector re iid ~ bi(,0.50) i = µ = =... R covrice trix S = I 4 i 5

16 The there exists trix X = A + b hs ( ) ( ) ( ) diesio E X A d vector b ( ) such tht the vector ( ) = µ T e vector ( ) covrice trix cov( X) = T Σ The trix A d the vector b re : T A = ( ) ( Σ ) b = µ A ( ) Where ( Σ ) is squre root of the trix Σ, defied to be trix L such tht ' L L =Σ. Oe type of squre root is the cholesky squre root. = (,...,) is vector of s It c be show tht li X ( ) is ultivrite orlly distributed with the bove e vector d covrice trix. The vector X ( ) is pproxitio to the vector of log price reltives. ( ) ( ) ( ) ( ) The vector W ( ) = exp X ( ) = exp X,...,exp X is pproxitio to the vector of price reltives ( ) The vector S( )( T) S ( 0) exp ( X ),..., S ( 0) exp( X ) = is pproxitio to the vector of prices t tie T. This is pproxitely logorlly distributed. We c write this ore copctly s S T = S 0.exp X ( ) ( ) ( ) Note tht r is the risk free rte of iterest d sset i. q i is the divided yield o 6

17 Algorith for pricig Europe style optio prices For the -diesiol, step bioil odel, we ssue tht: the preters rtµ,,,, Σ re kow the iitil vlues ( 0 ), ( 0 ),..., ( 0) ( ) S S S re kow, the pyoff (P) we re vluig hppes t tie T d is soe fuctio = f S ( T), S ( T),..., S ( T of the sset vlues t tie T P ( )) T Copute A = chol ( Σ ) Copute b= µ T A Expecttio = 0 i = 0to + For ( ) Copute diesiol vector y Copute x= Ay+ b Copute w= exp( x) i.e. w = exp ( x), w = exp ( x),..., w = exp( x ) Copute ( ),..., ( ) 0,..., Copute Pyoff = f S ( T) S ( T) S ( T) ( S T S T ) = ( WS ( ) W S ( T) ) (,,..., ) Pr {... =... } =... y y Copute ( ) ( y y ) expecttio = expecttio + Pyoff probbility Next i price = expecttio ( ) exp -rt This lgorith gives the price of the cotiget cli s discouted expected cshflow. 7

18 The vector b for rbitrge freeess: The vector b defied bove kes the distributio of x hve the right e vector d covrice trix. However the expecttio of the vector w (which represets the price reltives) will ot be wht it should be for ( r qi ) T rbitrge freeess. We require tht E( wi ) = e so tht the expected rte of retur o sset i is the risk free rte., i = Let e j be the diesiol vector defied by e ji = δ ji = 0, i This vector hs i positio j, d 0 everywhere else. j. j Defie the vector s by copoet of row j of the trix A. the choice j ( j) ' s = Ae j. The i-th copoet of this vector is the ith A e ij + b = r q T l for the copoets of i= l EW = r q Td hece tht the vector b gurtees tht ( j) r ( qj ) T EW ( j ) = e for ll j = This vrible ( ) ( j) W j is the price reltive for the jth sset i our odel. The uericl differece betwee the vlue of the vector b coputed i this wy d the vlue coputed the other wy is usully sll d s gets bigger the coverge. 8

19 Coets o the bove result ) This is -diesiol extesio of the well kow bioil optio pricig ethod. It is extesio of the equl probbility versio of the bioil odel, ot of the Cox Ross Rubistei versio. The CRR versio fixes the size of the up d dow oveets (u d d) d the solves for the probbility p of up step. This versio of the odel is ore difficult to geerlise to ore th oe diesio. It c be geerlised to diesios but it is difficult to go beyod. The equl probbility versio of the odel fixes the probbility of up step to be 0.50 d the solves for the up d dow oveets per tie step. This is uch esier to geerlise to diesios. We solve for either the probbilities or the up d dow oveets by doig oet tchig. ) The bove result pplies to Europe type optios (those oly exercisble t turity d where the pyoff depeds o sset prices observed t turity). However it c be exteded to pply to optios with erly exercise pyoffs d optios where the pyoff t tie T depeds o prices observed t soe erlier tie. 3) The b vector i the i result does ot gurtee tht the distributio of sset prices is rbitrge free but it is pproxitely rbitrge free d the pproxitio iproves s icreses. The other forul for b does gurtee the odel will tch the oets of the rbitrge free distributio. This is slightly ore coplicted expressio but the uericl vlues it produces re very close to the vlues give by the first b-vector forul. 4) This uses ffie trsfortio of the diesiol bioil distributio to odel the vector of log price reltives. This is siilr to wht we do i the -diesiol bioil odel. 9

20 6) Nuericl Results / Testig the odel The lgorith ws tested by pplyig it to vrious ulti sset optios where it is possible to perfor vlutio lyticlly. ) We shll price exchge optio to exchge 00 oz of silver i pyoff = x G S,0 retur for oz of gold. The pyoff t turity is ( ) The vlutio ssuptios re: 5 yer ter = 60 tie steps risk free rte = 0% p.. sset div yield voltility drift rte iitil price gold 0% 0% 4.00% $ silver 0% 0% 4.00% $ correltios 00% 70% 70% 00% Results: expected pyoff $ optio vlue usig bioil $ 44.5 optio vlue usig lytic forul $ 44. Rtio of lytic price to bioil price ) We shll price spred optio o the differece betwee the vlues of 00 oz of silver d oz of gold, with exercise price of K= 0. The pyoff t turity is pyoff = x ( GT ST K,0). This c t be vlued lyticlly by the BS forul. The vlutio ssuptios re s bove. Results: optio vlue usig bioil $ 38. T T 0

21 3) we shll price optio with pyoff pyoff = x G 380,0 + x S 400,0 ( ) ( ) T This pyoff c be vlued lyticlly s the su of stdrd cll optios. Usig the se vlutio ssuptios s bove, the results re: bioil optio vlue $34.66 lytic vlue $ ) we shll price optio with pyoff pyoff = x G + S 780,0 ( ) T T This is sset ribow optio. We vlue it usig the se vlutio ssuptios s bove, except tht we vry the correltio ssuptio. This pyoff c t be vlued lyticlly by the bs forul. Whe the correltio is very high (99%) the sset portfolio behves like sigle sset so the optio vlue ($34.53) is close to tht i ex 3. Whe we chge the correltio to 0% the optio vlue drops to $3.9. T 5) we price 3 sset optio with pyoff pyoff = x( ST S 0,0) + x( ST S 0,0) + x( S3T S3 0,0) This pyoff is the su of the pyoffs o 3 stdrd cll optios, where the exercise price is equl to the iitil sset price ( t the oey clls ). This c be vlued lyticlly, so it llows us to check the ccurcy of the bioil ethod. The vlutio ssuptios were r = 6%, T = 0.5, sset (i) div yld voltility correl trix % 90.0% 60.0% % 00.0% 80.0% % 80.0% 00.0% The lytic vlue is The bioil odel vlue (usig 30 tie steps) is $ The rtio of these vlues is 99.77%. If we chge the pyoff to the su of 3 put optio pyoffs, the the lytic vlue is $ while the bioil vlue is The rtio is 99.73%

22 7) Cocludig rerks: This pper hs show how to crete discrete pproxitio to the ultivrite orl distributio d the ultivrite logorl distributio. This result is useful for optio pricig pplictios d other ficil pplictios ivolvig correlted orl or logorl vribles. Becuse the uber of clcultios required is proportiol to ( + ) (where = uber of tie steps d = diesio), d this grows very quickly, the ethod is prcticl oly for up to bout 4. It is ltertive to ote crlo siultio for low diesiol probles. The ethod covered here for optio pricig c be exteded to cover Aeric style ulti sset optios d other optios with erly exercise fetures. The results show lso hve pplictios to sttistics d thetics, s wy to uericlly copute expected vlues of fuctios of ultivrite orl or ultivrite logorl rdo vrible. I prticulr for >, expecttios d probbilities fro these distributios hve to be coputed uericlly ywy. This is ltertive wy to do it. The results show here lso hve cdeic pplictios, for geertig discrete ultivrite rdo vribles with specified e vector d specified covrice trix. This could be useful for cretig exples for studets to exie d work o. It hs lso show how to copute expecttios of fuctios of rdo vribles with these distributios. Speed of coputtio is issue with this ethod. However for y pplictios speed is ot the oly cosidertio. For optio deler doig del o the phoe, istt coputtio y be required but for y other situtios it is ot such big issue.

23 Appedix : Detiled Nuericl Exple: The best wy to uderstd the bove result d lgorith is by workig through uericl exple. We shll do 4 step vlutio of 3 sset ribow optio with ter to turity of 3 oths. Assue we hve t the oey put optio over the vlue of portfolio of 3 ssets. The vlutio detils re: T T = 0.5, = 4, = 6 = T = 0.5 r = 6% sset div yield voltility drift rte iitil price 4% 0% 0.00% $5.00 % 40% -3.00% $ % 0% 3.50% $.00 $0.00 µ = Correltio trix 00% 90% 60% 90% 00% 80% 60% 80% 00% Covrice trix (ulised) Covrice trix Σ= T. Σ Σ 3

24 Cholesky squre root of ul cov trix chol ( Σ ) Coputig the A trix A T = chol ( Σ ) Coputig the b vector: We c copute the b vector so tht the distributio of the vector X the vector of log price reltives, hs the correct e d covrice structure. This gives us b= µ T A b = b = Altertively we c copute the b the b vector so tht the distributio of the vector X hs the right covrice structure but the distributio of the vector W = exp( X ) hs the correct e vector (for the rbitrge free joit distributio of the sset prices) 4

25 s e i + bj = ( r qj) T l where i= ' s = Ae j ' For j = we hve e = ( 0 0) d ( ) j ( ) ( ) s = Ae j = d s e i + b = ( r q) T l i= = = ' For j = we hve e = ( 0 0) d ( ) j ( ) ( ) s = Ae j = d s e i + b = ( r q) T l i= = = For j = 3 we hve e j = ( 0 0 ) d ' ( ) s = Ae j = d s e i + b3 = ( r q3) T l i= = = ( ) ( ) Note tht the uericl vlues i the b vector coputed this wy b = re very close to the uericl vlues i the b vector coputed i the other wy. b = Note lso tht the 3 s-vectors re the 3 rows of the trix A 5

26 Sple Clcultio t tie step =4 Cosider tie step =4: the set of vectors / odes t tie step 4 is of size 5 = ( 4 + ) 3 The 7 th y vector is y = (,3, 4) The ffie trsfortio x= Ay+ b ps this vector to x = ( , , ) The expoetitio of x is w = ( ) The vector of sset prices t tie step =4 is thus S = ( ) The vlue of the portfolio is $ = x ,0 = The put optio pyoff is ( ) The probbility of this pyoff is The Optio Price for Europe Optio = = 3 4 The expected pyoff t turity (tie step 4) is $0.44. This is the su of the products of pyoff ties probbility cross ll 5 possible odes i the tree t tie step 4, i.e. the expecttio t tie step 0 of the pyoff t tie step 4. The vlue of the optio usig 4 tie steps is $0.45. This is the expected pyoff t tie step 4, discouted bck to the vlutio dte (tie step 0). If we icrese the uber of tie steps to 0 the optio price is $ If we icrese the uber of tie steps to 30 the optio price is $

27 Appedix : Proofs It is well kow tht! y. ~ bi(, p) Pr( = y) = p p y! y! yt t. M () t = E( e ) = pe + ( p) ( ) ( ) y for y = 0,,..., 3. if,..., re iid bi(, p) the! yi Pr ( = y,..., = y) = p p i= yi! ( yi)! ' t ti (,..., ) = ( e ) = + ( ) M t t E pe p,..., i= ( ) y i ti ( M ( t t) ) pe ( p) ( ) l,..., = l,..., + i= t i e + l ( M (,..., )) l,..., t t = i= whe p = Result fro ultivrite sttistics (bout oet geertig fuctios ' ' x= Ay+ b M t = exp t b M A t d lier trsfortios): x( ) ( ) y( ) 5 Defie the vector e j by e () i, i j j = = d ( ) 0, j ' follows tht ( ) ( ) exp( ) ( ) x ' E e Mx ej ej b M y Aej ' j e b= b d j ' e i = i j. The it = =. We lso hve A e = jth colu of the trspose of A = jth row of A. j j 7

28 6. Let R be rdo vrible with e vector E ( ) µ R covrice trix S = cov( ) = d Let A be soe costt trix d let b R be soe costt vector The the vector rdo vrible X = A + b hs diesio e vector ( ) covrice trix cov( ) E X = A µ + b X = A S A 7. Let S be positive defiite, syetric covrice trix. The there exists uique lower trigulr trix L such tht T T L L = S. Also the trix L is ivertible. This trix L is kow s the cholesky squre root of the trix S = cov( ) Lij = 0 for i > The ottio L chol( S) Le. Beig lower trigulr es tht jso tht the etries bove the i digol re ll zero. = es L is the cholesky squre root of S L= chol( S) T L= chol( T S) for costtt R 8

29 8. if,..., re iid (, the we sy tht the vector =,..., hs the bioil bi(,, p ) distributio. The e vector d covrice trix of this distributio re µ = d Σ = I 4 bi p) ( ) 9. Affie Trsfortios d Moet Mtchig Let Σ be specified positive defiite syetric covrice trix Let Let µ R be soe specified costt vector R be rdo vrible with e vector E ( ) µ R covrice trix S = cov( ) The there exists trix A d vector b R = d Such tht the ffie trsfortio of the vector defied by X = A +b hs diesio e vector E( X ) = µ covrice trix cov( X ) = Σ R The trix Ais defied by ( ) ( ) A = L L where L = chol Σ d L = chol S The vector b is defied by b= µ Aµ 9

30 0. Affie Trsfortio for equl probbility bioil odel Let R be -diesiol bioil rdo vrible with preters d eqully likely probbilities so tht e vector E ( ) = µ = =... R covrice trix S = I the copoets ( ) ( y y ) 4 of the vector re iid ~ bi(,0.50) i Pr {... =... } =... y y The L = chol( S ) = I i L = I A = L L = L = chol Σ ( ) ( ) b= µ Aµ = µ chol Σ = µ L i = i j= The ith copoet of this vector is b µ L ( i, j) 30

31 Exple: diesiol cse σ ρσσ σ 0 Σ= L = chol ( Σ ) = ρσσ σ ρσ σ ρ 0 σ L = ρ σ ρ σ ρ 0 S = L = chol ( S ) = 0 L = 0 σ 0 A= L L = ρσ σ ρ µ µ = µ µ = µ σ 0 b= µ Aµ = µ ρσ σ ρ µ σ = µ σ ( ρ ρ ) + σ µ b= µ σ ( ρ + ρ ) 3

32 Exple: 3 diesiol cse σ ρσσ ρ3σσ 3 Σ= ρσσ σ ρ3σσ 3 ρ3σσ3 ρ3σσ3 σ 3 L ( ) = chol Σ σ 0 0 = σρ σ ρ 0 ( ρ ρ ρ ) ρ ρ ρ σρ σ σ ( ρ ) ( ) ρ ( ρ ) 0 0 S = L = chol( S) = L =

33 A= L L = σ 0 0 σρ σ ρ 0 ( ρ ρ ρ ) ρ ρ ρ σρ σ σ ( ρ ) ( ) ρ ( ρ ) µ µ = µ µ 3 µ = b= µ Aµ = µ L where is vector of s µ The djustet vector L is subtrcted fro the vector µ = µ µ 3 σ This djustet is L = σ α σ 3β where α = ρ + ρ ( ) ( ρ ) ( ) ( ) ρ ρ ρ ρ ρ ρ β = ρ + + ρ ( ρ ) Note tht the su of the squres of the copoets of α & β is.0 33

34 . Algorith for pricig Europe style optio prices For the -diesiol, step bioil odel, we ssue tht: the preters rtµ,,,, Σ re kow the iitil vlues ( 0 ), ( 0 ),..., ( 0) ( ) S S S re kow, the pyoff (P) we re vluig hppes t tie T d is soe fuctio = f S ( T), S ( T),..., S ( T of the sset vlues t tie T P ( )) T Copute A = chol ( Σ ) Copute b= µ T A Expecttio = 0 For i = 0to( + ) Copute diesiol vector y Copute x= Ay+ b Copute w= exp( x) i.e. w = exp ( x), w = exp ( x),..., w = exp( x ) Copute ( ),..., ( ) 0,..., Copute Pyoff = f S ( T) S ( T) S ( T) ( S T S T ) = ( WS ( ) W S ( T) ) (,,..., ) Pr {... =... } =... y y Copute ( ) ( y y ) expecttio = expecttio + Pyoff probbility Next i price = expecttio ( ) exp -rt This lgorith gives the price of the cotiget cli. The ter expecttio here is the risk eutrl expected pyoff 34

35 . The vector rdo vrible X = A + b is discrete odel for the vector of logriths of price reltives of the ssets. As the distributio of X = A + b coverges to ultivrite orl distributio with e vector µ = r q σ, r q σ,..., r q σ T = T µ σ σσρ... σσρ σσ ρ σ... σσ ρ covrice trix Σ = T = Σ T σσ ρ σσ ρ... σ This is true becuse of the cetrl liit theore. Ech copoet of coverges to orl distributio, so coverges to ultivrite orl. X beig ffie trsfortio of lso coverges to ultivrite orl. It follows tht syptoticlly ( ), the vector w= exp( x) i.e. wi = exp ( xi), i =... coverges to ultivrite logorl distributio d tht E ( wi) = E( exp( xi) ) = exp( ( r qi) T) This is the rbitrge free joit distributio of the ssets. However, beig discrete rdo vrible, for fiite we hve E w = E exp x exp( r q T but the pproxitio iproves i i ( i ) ( i) ) ( ) ( ) ccurcy s gets bigger. We c use differet b vector here, which will gurtee tht E ( wi) = E( exp( xi )) = exp( ( r qi) T) for ll vlues of, so tht the odel is rbitrge free for ll vlues of. The forul for the b vector is ore coplicted however. 35

36 The vector b for rbitrge freeess: Let e j be the diesiol vector defied by Defie the vector s by ' s = Ae j X j bj By result 5 bove we hve ( ) = ( s ) X j by result 4 we hve l ( ) E e e M y ( E e ) bj i= e ji, i = = δ ji = 0, i Si e + = + l j j s e i + It follows tht the choice bj = ( r qj) T l i= for the copoets of the vector b EW = r q gurtees tht ( ) d hece tht EW ( j ) e This vrible ( ) ( ) l j j r ( q j ) = for ll j = W j is the price reltive for the jth sset i our odel. Copute b fro j ( j) s e i + b = r q T l i= for ll j = 36