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REDUCING THE STATE SPACE FOR MULTIDIMENSIONAL PDE CASE
I=Inserting the equation and reordering it, we obtain:
(f:÷s )t.E.fisiffsitt.E.a.siI " EsIs E.re. sir- -- .. ¥5,39off E.E.{ FF sif reIi osi t -t r - -; MULTIDIMENSIONALPDE CASEIN) Els IGtFIT ) SmSalt -.=, .. . ..REDVCINGTHESTATESPACEF financialthe derivativethe loosedoftothat compute hresult orderinprevious nicesee a onmom mwe tohas be donethisgeneralunderlying variableshave toassets solve with state inPose and casewe a m ,, dimensionsSometimes thenumerically it thishowever statetheofpossible to andis reduce shoe can. , ,,formulasanalyticallead to .Example : t)() ft)SIG)I (Options of)Salt SaltEXCHANGEg) ) ) thisparticular-= case: , +( ])(SPREAD2) E ( ) () lit Sy Sz )Sz(OPTION Ss tK Ktt --i =,,holder to the forthe right1) shareexchange timeatshare SsSgives one one t, .form( ) of theunderWe modelPconsider a :dss divaSe dt Ssde t q=dsz dinSz d t Sda t q= , .The ofaffiliation theoremthe thegives PDEus :{ Ifisifts raffi ¥II. fsiEri Erir ttt t r o-- -,, )f (ft ) Ssh
Salt)SaSs Oxmoe, -, , It sitInstead ¥1(kiss )Fwrite withZ zitiSzsSr isI -can = -- --: , -, ,)Flt thatGltitThem si satisfyshouldSzSs - :: -, ,site :¥:# Fiatsoof oasis Esi :÷¥*o- . .. -¥ioot¥E÷¥OF .. ) 9¥( rkooffEff ' ;sieHooft try aifraiseoff G te orse t ---- ,.Now hi obtainaidemote and'so tI :: { !,÷Itit !!!Hofreffett ":# !otin "off - .-+ .. ). T !This withBlack equationScholesis and LKr oa - ---Them )GHz Cdc)Nlds ) N2- -: = }{ ¥)¥(F ti ( )Nlds )G Cdcde( ) salt) N) NdaNSz SzSz= --- - - I Eat 'emAt'E'em -*da dawith --. .-. VTINTI rDIVIDENDSThe claimschapterof forofsect studypresent whichto contingentproblemsthe writtenpricingis onaredividend assetsunderlyingpaying . ?If dividends derivativehow theintroduce evaluatewewe can,Two dividendsofgroups :1) DiscreteContinuous2)" I TeTm TTmOr r r rry .. .- holderthese dividends the ofdeterministic time toAt points stockpaid outin are .followingdynamicsWe assume that the stock price, dividends, and publicity have a linear relationship. We assume that the stock price follows a continuous process and that dividends are paid at regular intervals. Before the payment of dividends, the stock price is assumed to be constant. After the payment of dividends, the stock price is adjusted by subtracting the dividend amount. To summarize the structure of the stock price: - Immediately before the dividend payment at time t, the stock price is denoted as St. - Immediately after the dividend payment at time t, the stock price is denoted as St-. - The dividend size is denoted as dt. Given the above assumptions, the stock price can be determined as follows: - Between time t and t+dt, the stock price is constant at St. - Immediately after the dividend payment at time t, the stock price is adjusted to St- = St - dt. This is the basic framework for modeling the dynamics of the stock price in relation to dividends.Il dhas ICsttime bydeterminedstock StStgwmhprice a° -= --,Pacino Wefour timewill this thenstartingsolve workingpreteens at wardsT and inwe: .for separatelythe each intervaldividendcompute processmice intra - .:{ EstaII. softer re. o-t r -trii. e Elst) )St =In )functionthe Ts Flparticular by Sisatpricing Tegivenis ,,Tz t TLre i ))(( S;)( F s1) Stif St TeTjJUMP tiCONDITION -=i ,,{ ¥7II FF ffor ort -t rs -c) )]l( d)( F StiStiF Sti TeTe --- - ,,theuntie end. . . .Particular dsBlock SdtScholes Stowdynamics a tocase -o: i = Esof ) IsformsDiscrete thedividends 8i =•We Eforfunction claimECSclaim the)consider demotelet Fett the)T pricingandou s- . , .modelfunctionIn Eparticular withforthat standardFo pricingobserve imis ourwe noadividends following holdsProposition )In ITItthis the relation lafelt ) Fo scase s .i: = -, ,Indeed : I I2s{ ¥7II FF ffor ort -t rs -tTe TT iE Elst'fast ) )= ))( ( )() F JfTz t TL 8 (TeJUMP TseTss FCONDITIONrE si s-i s.= n-, = -, ,{ ¥7II FFffor ort -t rs - ))( ( )() F df- 8. (Te TsTes Fss- s= n-, = -, , 2)) ((( ) d)F FSTs t f (Tz TsTzT sr sF ssi n--sump - =condition - --i , ,i{ ¥7II FF ffor ort -t rs - )) ( OT(FCT ) F F8. (isTz ss ss n- - -- --, , - ,, )ohh( laItis) Fo tFg s.= -o . . ,We the2) continuo shydividendswhen timeconsider paid out incase are .)CatDlt between) dividends distributedcumulative=In the continuousthe usualto dividendsobtain with proceedinorder pose case aswe,dividendstheconstructing the butusual portfolio alsoconsidering :dst Std WedSt tStock DYNAMICS a rt: -. dividend8 ratethewhereSdt) St isd DHDynamicsDIVIDEND : = dbWe theintroduce ds DD old(Wwegain G S t SdDprocess t s att t r: = = = derivativesportfolio thethetheconstruct with and usualprocessusual proceedandgain asthe bondthe dynamicsrisk imposingremoving and :{ }DV TEV us t or-- .From formulaHo DFthe dpfdtt dinF withor :-: - .}IIII.{{ soft forde f 'sat-- :Sso¥. .. .. { } dt { }DV din)late VorUs areV t ort rus ort.= .
..?÷÷I :S:L :::S :: ":we impose . .{ })DV dt(V dUs arofat t-= . II{ ¥ ft ' rfFs8)( os =t t -t -We )( dimpose Us ar PDEofat t r: =. )ElsFIT )st .=. ' ]" t (function therProposition Els ithas - whereThe )Flt - IIrepresentationpricing s e: =,dst d)Q by Crof dwtdynamics Std Stgivens t toare- : - --In martingalethedividendsdiscrete that dividendcontrast thewith inappropriatewe measureseefromdiffers dividendthat theofcase casemo(Proposition ) normalizedQthe martingale theneutral valuationRisk under gainmeasure:- ,process : ft ④GZA # martingalef d) DH is at -= ,callExample i lml¥)tl"dtFH""' "d "" )(Fft ) dcNlds) - N- daSte d VESt dywhereke-=, •. . , .=. THICURRENCY DERIVATIVESthe of thetorelated assets }riskrisk RISKSOFTYPES2productsQUAN