Finanza computazionale

Computational finance introduction basic tools

t ?What derivativethe of the Payofftheprice givenis( ))Assets St( quoted→ too v KY(Flt CallDerivatives ) European Flt St)St→ e.g St -=, . ,Goals Modellinge) assumptions: Price derivatives2)

Modelling assumptions

- )(wt{ tN onIf fixVK.tt ,(Process tWiemer You: )(Vhf,o Nwt Sto.- n ,µ+( )119001 Bachelier Vhftous St So 'TPME BECAN NEGATIVEtt: =119731 dstBlock Merton GBMScholes }dttrstdwtStµ:- - = .FI/dt+odvyaBm soVEN' 't(if dXt=)lofts Itt µ-- ( )logXo so-?)E)Them ( ( How thisthe Wt solutiont obtainobtain Stsolution So exp rt Iwe canpas -: = .!§I E) E)Flds ( VKdws (t VIXt t( ( )to (dies )to log logrx tp St rt= µ- so-- -. - =g-egfsttsf.tn/nNflp-E1stirst) ))( (Sto E)log ( t 'tan p r• - .; ?todoWhat this assumptionbeyondgowecansimmetnifutiom→ µE6 ) )-8 (( Vast ! model the tailsthe byIP estimateSs constructionto under= smalltoo→ -) discontinuous toSumps Leiy considerTwo triremesL processes ?÷gnaw→:ways : . !wwartstochasticStochastic doeVolatility2) processis: a= .. . →Instruments to Levywithworkorderim processes :Discontinuous17 processesdq.CHLA

Optional processes lag cadg-ca right continuous lag processes

No aµ oor•-oI I(Predictable ( ))/ /OPTIONALPROCESSES LAG CADG-CA RIGHT CONTINUOUSLAGPROCESSEScontinuousVE ft --FIE)FIH ) left) HttFlt right limitedcontinuous= -- -don't thiswe want thiswantwefunction characteristic functionIRDCharacteristic Let The by2) beX in is givenrva: :. . !) "Efeiu "×tf Rd that .IR 't e→ s: = =. abecausefade not !not always sometimes isNB renown. ]f( Xm) EXNOTATION mm =: )( IRD set { }I I open IC Eo,, T "ICHId ¥"] ?Proposition El )Klm 10xif andofthem neigh broadwith myIE or to a =: , , " HOz"d ¥"?Proposition if IIIcardthere forthememsetexists theatgo} EIopen Mr K isIan m=: :-. .,, ", HOz'T of"!YEfefunctionMoment where"generating CutA 10×1 inµH=mm→-: --- - I×"EE '10×67{ Ie= - oftx neighborhood. tohave in 0Characteristic3) exponent aif we : +i ¥ continuousis in 0" "' characteristictxtThere exponent ofthe the14exists 4xat ) e is rva =. .. )(4) blockftp.xe-mtqyo.ogforPoisson buildingprocesses constructing complexUsed processesSumnmoreaas { t'( )) cdf) E- (IP Y ye toI ey oe = - ,Exponential variable Crandom "Y ~→ hdf)()I( '-)I I dfMATLAB I eexp c.: = .( )Theorem Absence of memory : distribution(PITLet ) exponentialT)be hasThemHt =PStTso Itriv tts tot ana soo s. : > >. ..

Proof and exponential distribution

Proof :)( to⇐ f H Htt )- s( ) dyteTatts -IP (e *) )( hasT- TottsIP IT exponential=PtTatts se= an, =- == if)( TstIP to 'I ' dy'distribution -xeBaa t( ) P(T )(( (( )phot))) them =p ttsPttsl=PIP toTosT To tToie s => :. )PCT t>Define Phot) decreasingA) }ofg: = it satisfies the thesisthat- functionthe onlygltl ihocontinuous isisog e-.)Htsglslfltl if•*PC )them distributionthat- ofthe cdf the exponential BEBBTet s e in= - tPoisson distribution MN -) In)Poisson INNr timeINNE eru m =-.: -- !m1)""function -Moment Marcusgenerating e-. defineThen V) toodistribution CdKi )Let exponentialle of expset independent withProposition v.v :aies .: . (( )} )7¥; )Poissant{ Ntr Nt Poisson discreteimfNE PoissonAriot is v.vneo is rna a. .[ )teAssume NtTL 7123--1.2 STtTy =3kZz Ts Z2.3O2 33 t t =e.g = - =: -- . ?m yeem. . -± e.( ) !m s-lyavtnme )FIT ? (tty ofProof (G)Hipdf DTn Tn - HeoPentti isre expsum→ m--: m-- . ,.,,to ! b" ! !" Hm?÷ds !II fixsfhdx?Plants ) him ( ]{ }Them - #!by f' funguslxldxpartdst de It -I> - -= = . == 'f§ " HI' - ( CHI*?tsy.sk - e Pfm )Then PChove )e- -tyds tI Tntwe tt > e>- , -:. -= ! !m mi.)Pkn t> lmqI*Pfm ) !pftmts ) e-rt Poissonrt Nt variable therandomis- = a→)(IP NemRemarks : }(Reissf) )Ys isn (Yz) (YzReiss )Yy Reissda be tht nnYs 42I ( )InThem withInfinite divisibility BissYiYm2) independentReiss ) thIs YYrthis Ys isIN te ti -- ... ., ifmisPoissonDef ()( NE) AtenTmconsider Ti ii. Tid expprocess bi.e: i-;: . -..., ,where :inter timearrivalTi Ty 230.2 Te 2.3~ = =- .- . .Tm time Ts Tzfunny s0.2 2.~ ==theNe counting processincreasingis process~ -i valuesinteger process~Them ofNE (Es Poisson where)Aten parameter intensityis Ntprocess t ita . t some ↳ times whichof tbetweenthefnnocessi counts number random andcounting oown-ofnumber gumhs

Properties

Properties :fixIf )Poisslxtt Ntwe. ~, NzNt Nts Nzprocess Nzis NttCAD Ef 2 2LAG° I=a St: =s =s. . ..PC ) "publicity "stochastic continuity""continuityNt Nt =L in. - .- mtt)PINT -e° m. ... !m " )iv. ()f Nt )(ett e I characteristic function Poisson- theof process# e. =Independent increments Ntm NtzNimtr ton Ntste Ntmrt Ntn k I I. -rr -: --, . .. - .-,.a , .. }{frameworktheoreticalMore timesset of3 theTmlw) sunnyis smwe r = . . ..: ; ,Plt 5) ( butHt preference whichhave timeinso wezurnhe weoccurcan sono, ,)hrelohlity( )don't introduce aman oa oCompensated Poisson Process( ) We Poisson¢martingale martingale " of theNt# itit "want versionnotin= aa . NII(Ft martingaleIintroduce htNt ishwan o aso we -= -: -N't the sated processpoisson Chem )in com app NIhashemcounting becausenot that INcouldithwanCPP inN.rs a.Preferties (NtttaocalledThe deterministic of: the compensatorisexpressionn.rs .)El Ntl Ns his° = )El )Flat) his( I ]ElthisAt N' hisIDsNtl Ns Ns Ns INTE t- -s= == El )Nsoffor NIindependency increments aso→ - --" "settle iI (fnq.co charolais-- )function theoftic Poissoncompensated, process.Theorem with stationarycountingXt independent incrementsis process anda: Poisson Nt Ns NsXt Ntprocessis a⇐ STATIONARY ~B --N th th=. .

Levy processes and properties definitions

Levy law"'tprocessPROCESSES AND PROPERTIESDEFINITIONSi )leg WienerLEVY Poissonprocess . ,Def ( ) )C ifConsider LevyF A processisIP Xtr processcasino a :: too., ,Xo1) O=) independentincrements XtmXtn Itr rtm Xtn Yen2 orare r -i -. '' -. . ,. .as.(levy3) )the dehemdstationary doesn't ttfincrements Xt Xeare - onn: a, )Pllstochastic It) tecontinuity Xt E °u ⇐so -: e (For )considerdst just dt it Levyexample d becauseStr not processisWe Soto→ at=:Usually where Levy conditionXt of initialto theSte theproblemseewe is order ovoida imProperties (Let ThembeHt processa :: ., ..Infinite Indivisibility thro time sIN. : -., )(¥ () X.scm.nlXscm XsXt XXscm Xzt ttt- - -- ..- in ., ,a ... forA infinitelylad be divisibleprobability integerif there existstodistribution isif said anyon me ,XNNIN.MX?Ii.oYnYm has.hn nubYa distributiond variables that Yet Fii. random e.gtm . ... .. .. ., )(lowwhere Ya with armd Mmi. Ni.are . ,)Ef pi Xt I feivxs10¥ ]U 10×+4fu) )t in MULTIPLICATIVE. t ⇐> :characteristic= yEfeiuhtts )) ]) Efe Ef)("Hts H 'tpiEfe ) XtXt) int" txt ItsEfeiuttts tofuu ) f---4¥ go ,yes = =e ,== ,#function ) 'et "Nd " "Ele IRDthemI 10×+67=2continuous ItYr 4×101=0t theIRo s sa so- =: =. . ,functionthe ( ofthem not )is timeexponent isNB aUntil to the stock lovemodelorderim seemwenow :, ,Nt WIENERPoissonVlfNytHohhh+ tinindependentPoisson forprocess simulatingthey foodnot stockare a22 a !differentthe behaviourstockI isbecauseI • oo•

Compound Poisson process

Compound Poisson process Nt (4)We whereProcess Xt Yithe Compound E independentPoissonintroduce dii. v.vareis-: - , ..i I-from AtNt )forwhere Ntr Poissontand rvany .,Wt XtH compoundthe Levyharempoison processisNB a.! different hovehaveThanks differentthatthisto processwenow a canfwmhsite !of SunnsizesLDef with intensitycompound h booless andCOMPOUND poissonPoisson Process gumha: Etstochastic withprocessf Xtdistribution Yi where Yiin ii. dn'fumhs area ressize i ., )(INtlf fromdistribution independentPoisson Yiwith intensityprocessand is xa ios, LevyXtProposition with niecehoissomcompound wineprocessXt processis a⇐in ai tragediesconstant ?offunctionWhat characteristicthe Poisson processcompoundis aProposition function followingNd Itslet ( PoissonCompound characteristic hasbeHt theprocess: ona ., I }""( {the !µ "" ) Nd) "El ffdxlt ( wheretx thethe demotesrepresentation e eu exp xI= ---: ,distributionintensity f thesump sizeand gump .Proof i )Ei) )] )EfEf fEfe Efe" I)El ]Y Nt" " "Int " ("them 't IntNt .E#e e- ==- =) that" )f C Ii cnet.inf ] II *faint fineiner just it }alIcnet.in EI-## e-# itexpe - -= = -. . . !- E.ja ex, .fade ")flu "" "Efewhere "fld' =i I-. l ),µfldD densitybecause in=Lque ainside,± can)}I' !(El I"e' { "" ( e)" )10¥61 fade " { ' Ifflag Itit dy Dotaeexpexp e -= = =. ,For formfunctiondimensional characteristicthe has simplerPoissoncompound processesone a- :ToI( " " }{t "( e) fldxto10¥) " tuE ) IRee exp e= -.Introducing followsformulaflat theHat rewriteameasurenew wea ascan i= ,t.to]( " " }Old{ Cei¥1 e)"# tu) IRe exp. ex= -ofcalled (the )Levy but probabilitynotIRV positiveis process Xt ismeasure measure on ao atoo .foldyearsThe't formula of levy rehresemtatiomthe chinKhimX ¥ particularI issince previous= caseare a -. yf I fl( dy)dy )u b t= =NdInd 1Profs ( ) PoissonPoisson (let be compound withHtt intensityprocessJUMP OF COMPOUNDMEASURE andxA PROCESS a: so )Its Rdx (f with intensitydistribution PoissonTx is random coosump size measure measuresumn measureona ,.if) CdiI( dt)dxxdt dx )dtVµ =iThis Levyofalternative theproposition interpretation ofsuggests thePoissoncompound processan a asmeasurefactIn interpretation thethisof thangeneralmuch thethatnumber unit timeofher is moreavenge sumps one uses. ,It forto define allbe Levy onlythe Levydistribution Poissonnot compoundandprocessesused measurejump size can.followsones as : TheprofessM