Mathematical Engineering - Computation finance

Completed notes of the course

Complete course

ConPUTATIONALPARTFINANCIALPART • MC } Pricing ° LEVY • Poe § , s'meato leva "¥ ( oFFT)MCsimulazione[Bdsenvironment]DePDE(finitediff) . clraracherisiticfunction→FFTcalibratoLevy • STOCHASTICVOLATILITY HISTORICALBACKROUND ° t' R Se = µ -1 oneWeWeiner processo htt >0 WE r.ir/phmtt:possipb?oejegat~(.w...o ° We n N( 0, E) ° WI - WIIWèoctet ° NE + h - Wè i WeHh >o° 1979BLACKRSCHOLESKMERtfgzt-mdt.iodwtGBMSogiven 1> d>0MODEL( Recornbiniug tree) THEMODELSupposethat i loco( §;)unità - { ' it , cit) ÷ A = IMWhat happens.wh.eu M →oo?Frombinomialetree , thepossibilevaeues one S , utd "-JJ =0,...M Ta k i n gtheexploitationandvariano Eflog ( ° )) = Mplogl;) t' Meogdvon(log t ) ) = Mp in - pillogl:)) 'ifyouchooseottii.e deficit p . ttffraè weobtainthat Ellog ( 1) " % va (log ( sont ))è G.venthevalueat maturitahowconI computertheprice ?ASSETOPTION IS 28Son(Sona - K) '/¨ % :{ ÷ :/ 's.ua - Kr ) ?\ >(sodi _ test/ ' Wenametogobaciavano -o- Son%/f. : ? { fu = (son - kit\5.dfd . (Sori - KÌGiovanAthenumberofHooksbouglrtte o testIT .→In -- Ita(aitestIT .- Tle12) u from(atwe havea = fu.FI⑦IT .= Tl "e- testsala-d)Neobtainf. =e- est(qfu + (1 - g)fd)q = ersi q . riskneutrale u- d probabilita(donotdepend on p) COMPUTATIONALPARTMONTEC.HR LOMETHOD ' ComputerEXPECTEDVA LU EInfinance wecanusaMontecarloto ° PRICEDERIVATNESPriceof = e"(e-rt payoff) aT- alain-discountFactor /¨ BCOT) = ben ¥È e- rtlpayoff );Nato ° ComputerthevalueOfaPORTFOLIO "¥ Va r - O - tramenata(1,7 , PI)Xr. roPLX e a) = (nel : Xiwiex) 1¥ : computaO = EEXIÒn = fÈ XiXiudXi - X THR(Lawof large number)Let(Xiii ». iidrv EI7=0tiDefine % = ¥ÌÈ XiThenno #aP (LÒ . -01 > E) - ote > o THR(Centrallimittheorem)Let(Xi! >o iid , a-EEXIJ0!Varchi)tiThen Òm Nostro✓ È I neo.es#convergereindistributionAssumeNis large enoughts.to "- o - NNCO , 1)• EThenIZ , Z~NCO.at se2- = ÒN-0so Ò = O+ZEa- nrnrtthen ònnnco , § , III. ÷.IE ' → DiracdeltaDEF§n istheMCstandarderrore EXAMPLE : Office - 2019:Wecomputatheaverageage : 32,7standarddeviatore152020 : 25randomeneploee p →a.(p - Xlt))eosineIt'sconstantlndeedtheEllenschemaisexactforcrritmeticBrownianimotionFORTHEGEOMETRIEBrownianiMOTION Xlt-idtl-Xlttxjexlhat-o-XLHTEZ-1.gr Xlttatlz ' -1) In reality thetuoschemaoneaemostthesaunelndeedE OCV(continuatorevalue)doIhavetoexercise? [ IV(intrrinsicvalue)IV = (K-sci.FI)) " value =Max(CV,Iv)Whatisthecontinuatorevalue? °AtzatT=/, MCisdiscretetimeMonitoring + I •@- IfIdon'texercise in J ,thenexttimesci ,T)I can exerciseistxtattimeJIhavetuapossibilita :1)ExercisetheoptionIV= (K- sci , tilt2)NaifuntiltimeItsandactoptimaelyCV = E [vitale - rdtlsctos-sli.tt] Vlt ,-+ a) = valueoftheoptionattimejitl ThisalgorithnrequiresnestedMC's→ Nunericaeeyveryhard "¥ WeapprossimatetheCVwith a leastSquareapproaclnAssumethatwesimulateStiltimeJ ° We can computaNo ° WecannotcomputaCVJsince Vo te is KnownattimejTa r t i n e)1W is computer corre ct.lyCVisapproximatedwithpolynomials(usingonlythevalueSt) cvi.IE/e-nstV-....lStI--?aan.,lSi.t ) ""tTerrore termSimulatoriottimeJ| " " II !÷ ::[ '"" " "→"""" """LINEARREGRESSIONIGOINGBACKWARDINTIME Step - by - step ° attimeM : tm = MAt.at- rit"mi =e lk-sm.it e- È . c'È . siii? +ei-Known-(k) set)e( r- E) (t-t'+ ott - g) =±.-si Negeigible F-(t.SI =e-rit-tie ' [set) -K]=Wilhoutthepositivepost(itis positivewithprobalmost1 ) =e-e't-"e ' [set)] . µe-rit-ti= 5 - K e-rct-t)if5 »S.-te(T-t)Flt , 5) = 5 -treFLT,S)vswhen5→+a-rct-ti /lineF(t, si = limS-ke= limS=+aS→+as→+aS→to Then :^ It ,s ) -1 RS3¥(t.SI t § ' s' È :(t,S)-rFlt,s ) =o75E RtttE [a.TIFLT , 5) = (S -te) " ( f,e,o,=og,, Flt ,s ) =SuberPUTEUROPEANOPTIONS È +rs+ E ' s'II. -RF-°" pittima ..+ FIT,si-(K-s)so.-K-_.....- th . I:*:÷: "" ¥ ! -r(T-t) FIt, 01 =eE' [(K-sett)" /set)-0 ] -r(T-t).-sel'T-ti= ke - siti = K e BARRIEROPTIONS( OUTOPTIONs )DOWNXOUTUP&OUTKNOCK&OUT :c : : :p :L : :L " :/ "io° % , ÷:÷ .. EuropeanoptionExample(UPandOUTcalloption)U ! So È +rs¥5 t { s' II. rf=0Htt[a.TItseio,u) I:c :: "" ÷::: F-(t,U)=0 LOGPRICETRANSFORMATIONX.log(§ . )vlt ,×) = F(t.se " )F-(t.SI : priceof a EUcalloptionintheBolsframeworkThen v satisfies : ff -1(r- f)È + {¥; -no-ov-teio.ttHERNLT ,×)= (sol"-K)' { no ,± , ±, ✓(t,× ) → so e'×→+ao Weusethisframework inondatohaveonlyConstantsbecomederivative DISCRETIZATIONSCHEME①PDEon boundarycondition(EUCALL) ^ È + rs Est{ S' È - rf.cottfto.ttHSE#[ 0, Smax] | ÷ :: :c .FCT , 5) = IS - KÌPllSCT) > Sma , Iso) e 10 -8 F( 0, 5) =OF(t, Snax) - Snax or Fit , Snow) . Sma ,- Ka -eCt-t)-Weuse this one ②Discretize→ FD(finitedifferenze)F-(t , Stds) = Flt ,s) + Asaf (t.si ¥2§Ì it , siti .-ri F(t , S - AstFit , S) - asoff(t ,s) + ¥ ' ajfct.SI -i..-iF(t.AT , si = Fit.SI +ItElt.SItl ...Iat TAY L O REXPANSIONNCX -1 4) =vcx) + Lv '(×)-1 ¥ v' 'lx)+ fa >✓'"(× ) + TI v"exit... 11)24vlx - b) =vcx) - hvixit { n' ' cx) - % ✓"' exit ...(2)lei → viene nè + O'ch) (Ff ' %! E =,*o- h -so L(2) → v'cx). Nix)+ ÒCh)h(1) - (2) → vcx + h) -✓ ( ×- L) = NY - 4) + hv 'exithvix)+ 1¥ i- h ex)t...~(Xth)-v ( ×- L) '= { "" te -'' dy = RIT >s )e-at ( t.is/Tstl=PlTss) = gia) Il= s ;÷ [ È : scusa,g(ti = PICT , ti→getisrightcontinuavaanddecreasiug Since1 - gis a distributionfunction , gisdecreasingand rightcontinuavatogetherwiththemoltiplicativapropertythisimplicathatget) = expc - it)for sane ) >oo s.tgctt-e.tt (T>a) =e-'a (Ta s) = 1 - e-da [ ColfofEipci) 1 :ai> tg ( NtPoissonprocessof intensityXPROPERTIES1-Ntiscadla.gg(rightcontinuavaandleftlimitedfunction)lineNs = Nes→ ti2-Ne = Ne - withprobabilita1Ne × 3-Neiscontinuavainprobabilita ← | "÷⑤-0 {I.Te .... Tu}setofdiscontinuiTr e st,T, T , tel ...) GwenteptDItetti.to .... .tn}) =0 SupposeEtsitPCIC- (te .... .tn)) = ¥thenyou( IEEE a pointwithmore probabilitatheatre)STOCHASTICCONTINUITY→continuavawithprob . 14 - blu) = e[einnt] = ètlei " -11¥ ,Nt5 - NthasindependentincrementoV-t.at , et,...< tre ... Nen - Ntn , INtn ..- Ntn , I ... IN ,- No6 . NehastheMarkovpteopertyhtt >sIEIFCNEIINnusa I = Elfine)1ns]7-E[Ne] -- atNot a martingale÷Negenerate anewVersionoftheprocessi . COMPENSATEDPoissonPROCESS COMPENSATEDPoissonPROCESSCenteredversionofthePoisson processione = Ne - ht . NÌ is a martingale -n compensato ° NehasIndependentincrementi ° ④Necu) = ettlei "-1-in)REMhtt isNOT a CountingprocessoancehasnotOrlyIntegervaluesLEMMANtisa CountingprocesswithiidincrementoIffNtis a PoissonprocessNOTE : Countingpteocess →#ofrandomTimesoccuiring in lottaiftherandomTimes onecon stirvcted as pontiac sons of a Sequenceofi.i.dexponentialrandomvcniablesthen weobtaina Poissonpteocess ④ LEVYPROCESS - DEFLet(Xe)e>. be a cadlagprocesso on (1,7 , R)haringvalicain Bd , sit . X. =0(Xe)isa LevyProcessifHo1-Incrementaare independent : Xe - XsIXs2-Incrementoarestationany : Xt.in - Xe - Xe7h >03 . Stochasticcontinuity : Ht >oHE >0RIIXT .ie- XH ?E EREM . TheWeinerprocessis Levy CWttts.o.NO UNIQUECONTINUOUSLEVYPROCÉSSESIf we sample a LevypirocessatregulartimeInterval weobtànearandomwalk°(NtIt », (N'e)t>oone levyprocesso -UNIQUELEVYCOUNTINGPROCESS PROPERTIES1-INFINITEDNISIBILITY DEFAprobabili]distributionFissocialtobeinfinilelydivisibileIfthenItYe ,... Yasit È YihasdistributionFXtLevyprocessothenftXehasInfinitydivisibiledistributioni.e. htt . treni , eYe ...- YniidXe -_ È , YiProofA = EYi e- XE.i-XE.ci -aiR2 . MULTIPLICATNECHARACTERISTICFUNCTION Dèi E [ ein# ] neRldpiedi ) = paia )txscu)htt .stoProoflo# +, lui = E[ einXtts ] = e[ einlXt.is - Xe) + inXt) = #[ein(Xtts - Xe) e. in# ) = # [einHas - Xe) )#[ e.in#f=tEfeiuXtfeEeiuXsf = 10¥ ' 0¥ 'l 3-EXISTANCEOFCHARACTERISTICEXPONENTtuxcu)7=4 ,: È → RI , 4¥e.(Rd)lo>lui =e Exampletn , lui =etale ""-=,( e'"- 1)COMPOUNDPOISSONPROCESSXeSe = SoeXt ' pt-10Wttnt - 1-BlackandScholar È . I.Igiene; normanna B)MIJic(AxB) = K) =e(µ(AXB)) " ÷JDXt = ut + one + / ×Jxc(dsxdx) [0,f)×dGENERALLEVYPROCESS -° We can alwaysdegnetheLevyueeasureoUCAI = #[ #{te[0,1] : Axe #0: DXTEA))AXT = Xt - Xe -= Xt - III. Xs ^° Asudthatd.1> A c padI È a compactset .2 ° UCAI is limitedHAAssumethatv(A) =+a1°SnailJurnps →1×1 tz " fenicei.Proof(1)Stisa Lévyse = I.f'#s 'AX ,= Xs - III. XnAXS70° fcx)= O(1×12)ina neighborhoodof0'stia» = (FÉI can- / ecan ) =° ÷ =ate[ e# ] =1inTe t-① B. ÷ ... IÌE can = è inlive1) ÷ ivrtc. ÷ .IE ,e /; e'""" s.ca»- | = f. [ ein'+ g ,cdx)-Elei "-i'" I=p,iv.i)iortea }Lockbackcalloption → payeoff : (Mt - k) " FIXEDSTRIKEpayoff : (St - Mt)FLOATINGSTRIKE → Neneedthejointdistributionof(St , Mt)and(St . Me)IntheexponenhalLevyFrameworkForsimpliciter , take1=0,so=L Se =e#Me =Max Xsme = minXsasset oesce Example : UPdOUTCALLOPTION(St - KI ' 11MtCUb. = logbarrier = logcu)Uf.Upperbarrier) .> So = 1K = logStrike = log(It)((T , e.b) = E" ((e"-e" )"#{µ, .bg? ="¥ (ex -e" ) " I{gas,ptix.y)dxdye-. Jointpdfof(Xt , Mt) < (1)applytheFourierTr a n s f e rex@"@bin bolhlogstrikeandlogbarrierspace "¥, eiuh +ivbCCt,h, b)diedb = %t'"" nv is +in) "¥ Thisis a closedformulabutwe havetoXp , isthecharacterstiafun . Usanumericol←ofthejointprobabilita(Xtimt)~,@thoseandIt's Unknown(2)applytheLaplacetransformer •n qfètfp , e """"cit . ii.b)dxdbdt = qfèot di 0o - _-° Laplacetransformer on T → g poi( v+n- i)ta( n-i) =-° FT on k →aunCsxin) ° FT onv→ bC'T. i e , b) = 7a& ; f- taciturnità) uv(1+ In)PROPLetq >0. Thent; is a functioninIR ' and dj inB - St . q -= patchto; Cz)4=4 ,e chanacterist-icexpone.atq - 4lz)ofXWIENER - HORFDECOMPOSITIONandpoilz) = exp {{ " t'e-9' [ è "-s side)dt } WIENER -HORF tilt ) = exp { [ t - re-9' [ e' O→ f( ×'-×,T) -incrementiIme(( o,× ) -e-rt §C(T,×' )dx ' density in the④ueeasureofhaving unincremento×'-×ina timeIntervalT|g.[f, . ,» ,.,> µ ,., qq.gg!§( z, ti = fc-z.tt → 7-[f.t.tocuie-rt"¥ fb( x.×')CIT .×' )dx' 1- CONVOLUTION I → { c( o,× )) ' vi =e- rtf >→, ( "¥ fbcx-xittcct.x.idx.gov )-} = % "" ["""" " ""* ? """) """= È cv )7-[CCT..)) (v) I*= 7- ' [ e_rtto >+ivi7-[cct .. )) cv) ) CONVOLUTIONALGORITHMFOREUROPEANOPTIONSALGORITHMTHEORY(1)ComputaFTofthepayoffFTofaconvolutronintegra 'is (2)Multiplaitbye-attotheproductoftheF.tof(3)lnverttheFourierTr a n s f e rtheintegrandoREMThisistrueatt → t.at once-raa cit .× ) =e E[cct-ia.xct.at/Xcti=xI +a=e-^" / fb(x.×',a)ccx',tiA)dx'-a=e-ma a -i fpj.lv )7-[citta ,. ))) oisanX-stableLevyprocess ,Oa×o""e'% .. IÌÌÌI °X=I→ St isaninvero gaussianaprocessg (×)=o parametri zatons Levy neasuire / ofthesane \(FI) "'e-'" " /¨ (an) manna SCH =- 1) ×>o×1-1aK:varianteofthesubordinatoattime1 exploitingthesecondpaytowriteLaymanovreWeareabletoperformertheexpectedvalueandvariano>Forsufficientelegis ofÙlt)wecan writeÙlt +A)=a (b + Z " ) " QUADRATICulnare aandto areConstantatobedeterminanobynnoment - nnatelurg( aandbwithdepend onthetimestepaandÙlt)) > FareliceoofÙlt) weusean approximate.cldensityforÙitxa)PIÙltta)) - Ipsia )-11311 - p)e- P " )dxEXPONENTIAL ° WhensfnouldIuseexpomentalandwhenquadrate?esponentialquadratico -)C-\,Vit) 0to Weintroduce4 = m%(todeterminowhichschemato use ) ° if4E2 → Thequadratoschemaconbeuaed ° If471 → Theexponentialschema can beusedquadraticoFROMÙlt)toYLt)-YLt) o12+a -e> ponentino\ No more emptyspaceSUMMARY[GwenVit)wenametosimulateUltra) ) (alcomputer m→ E[vetta)IVCHIsi → Var(vetta)IVCH](2)Complete4 . I n2 (3)IfY s, 1,5QUADRATICSCHEME :o Compiutea.b o VlttA) =aCb + randn)elsaEXPONENTIALSCHEME :o Computapandfa a vetta) -- 4- ' (randa , p.pt "° WhataboutXCH?luiXlt + Di) = lnlxch) - { viti - A + VTVTT + ZxtaHourcanwesimulateZ , inardertoneautaincorrelatore? "¥ NotpossibilewithHestonmodel STOCHASTICVOLATILITYMODELWITHJUMPS⑦ BatesModeldst = use At + a.Sed.Wes + Sedztce = tedvt = T(y - Veidt + Ottant' ° gdt = dwesdwev ° (Zt)e>o compoundpoisson(finiteactivity)withintensitadand lognornaljump - sizes.tkisitsJump -sizeandlog( sen ) ~ N(log( str) - fa ', sa ) 2-IWs , Z#W " Log - pricetransformer : Xe = log(St)andµ = r. I E( no- arbitragecondition)d.Xe = (µ - { Vt)dt.ttd.We stdèe ° (Èe! » compoundpoissonwithinlensitydandnormalJump - sizeknN(logistici , 5)REMThismodel can be seanasa generalizzazioneof ° Mertonmodelwithstochasticvolatilitra ° HestonmodelwithJuniper OPTIONPRICINGE[ einXt ) = # [ einl + Et' ] = #f e.in#eiuzt } = f(x.v.t) Efei " È) --TOBECONPUTEDKnown→ character.estrefunctionofIfd ←0,fisthe a compoundpoissoncharactertristefunctionoftheHestonmodel XI : continuavapart → DXÌ = (r- XK - Ita )citaTv t d w t sZe : Jumppost(omogeneo vs andIndependentfromthecontinuavapost)0Thecharacterstiafunctionofthe log - priceisKnowninclosedformCHARACTERISTICFUNCTIONF( ×,e, f) = E [ ei" #/XÌ =×, Ve =o } OFTHELOGPrice dffxe ', Vt) If 4- È + sovtff-au-foh.FI + (fix.v.tt =/¨x.vi) + ( r- th - f)¥ ,+ Tfr - Vt) + Ef) cit + ti ¥ , dw: + ott¥ , dw:f-(x.v.ti = Ee[ e' " ×!)isa martingale lndeedforOcset | E Ifcx.v.tt/=EsfEe-ei "" )) = EsLei " " ) = fcx.v.esWesetthedrifttermtozeroandobtainthefollowingPDE : { II + ( r- th - f)aItri 2- vtiff-ifh-II.gov?ao-iEoh-È .=o| [ iuxf-(x.v.t) = E,ei/XI = × , Vt = c) = cio ' Neguessthefunctionalformoff :c (T- t) +v D(T- t) + in × f-(x.v.f) =e ⑦finalcondition : ((o ) +o Dio) + in>in × F(×,~, T) = e =eIo LevywithC8, olio)andt ,non gaussianoIfii.one revertbacktoaWeinerprocessWehavethat [ e'"-t)dzs = si [ e''s-t'da +o fate >ca-t'dw ,+ [ ( ×,'a-" Jldsxdx) + ( " [ "¥ ne """Jzldsxdx)(^'isequivalentetoy ,= yo - 4%9sds + Zt → itslocalbehaviorisdescribed.by : dye =- → e dt + dzt = )(o- gelatidei -→menu revetelngprocesswithspeedof mediereversione=Xmeoni(long - term neon ) =0 Q : WhathappensifZtisa finiteactivityLevy? iii.È .> 8=0Betueen one gumpandtheotherdue =- IVtdtVe = e- 't"¥ reretetto0exponentially > Stodvt =- due + sedetargetdrift L AssumeZE → Finiteactivity → subordinata } (0,0 ,U) → NodrifttVo '0a Levy rneasureofa compound" § ! !:\! ! ! Poissonunepositive-TocomputerthecharacleiristicfunctionoftheprocessCs) we headthefollowingLEMMALetf : [ 0,T] → TRbe a leftcontinuavafunctionand(Ztl eroa LevyprocessiThen /¨ [ ei [ flttdzt } =e [ Ylfchldturlare4isthecharacterstiaexponentof(Ztle >o Proof(sketch) o fpiacervi seConstant(then we generalizzotoanyfunction) I feti = fi11lei ...ti]"I we have /¨ ( ei [ fczsdzct) ) = IÌ , e[eifi(Zei - Zaia)) = !Ì @ Ullfi)(ti-ti- a) =@ [ Ylflt))cit1 E[eifzt ) =etutti° Thepteoof can beextendedtoanyfunctionThislemmaistheKeyingredientetocomputerthecharacterstiafunctionoftheprocess(1) Efeiutt) = eiuy.ee -*+ [ Ylue ""-t' ids (ye . g.e-* + { 'e''s-t'dze) 1 PROPLet(Zt)be a Levy(8 , A.v) t7, 0 Then(Ya l e ,. hav.mg(Zt) ».asa backgrounddon'ringLevyprocessisareINFINITEDivisiBLEPROCESShttandithastriplet(se ', ai ,u !) "¥sumoffiniteprocessowithof = ¥(1-e-' e) + g. e-* AI = È ii. e-2" ) ite v' eio) =/ 3'da313shortforLEx.x EB} ^Ed1 (ve ' (B)te,isNot a Levy incasinabecavseitdepenols on timeHowevcr , if we Freezetime , ofisa Levy rincasare It' s not a Levybuthasallthepropertiesofa Levy [ z' ne ' (dz) (n)isthecharacterstiaexponentofY ° Yv(t," )istheLaplaceexponentofwei.e. Holt , a) = log(E[e"" ))ExampleY ~kouModel → VGNe → positiveon dye =- dyedt-id.ttZtLevysubordinatoZt→a-stablesubordinato(with -utdrift) 2D + timePDE INOEXACBasfz.in - E ' '+ ¥Fa .- rv .-onon10 - 12→OndatHeston È + In - E)¥ ,+ È-3¥ .- ev +alo - v' ¥ ,e EfII. + graffa:-ONowtonumericallytreatthis?1345Frameworks .= s.co,e'r- F) t.to . We iSz = 5 , co)e( r-ti← Nel { dwf.dwi-gdtdsn-rsndttc.sn dwt ' BASKETOPTIONpayoff = (s . (T) + Sa(t) - K) + X. = log(s . )×,= log(sa)✓(T ,×.. xa) =max(e"+e"- k ,o)È II +in -