Completed notes of the course

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ConPUTATIONALPART FINANCIALPART • MC }Pricing ° LEVY• Poe §, s'meato leva⊥ (oFFT) MCsimulazione [Bdsenvironment] De PDE(finitediff) . clraracherisiticfunction → FFTcalibrato Levy• STOCHASTICVOLATILITY HISTORICALBACKROUND ° t' R Se= µ-1 one WeWeiner processohtt>0 WE r.ir/phmtt:possipb?oejegat~(.w...o° Wen N(0,E)° WI- WII Wèoctet° NE+h- Wèi WeHh>o ° 1979BLACKRSCHOLESKMERtfgzt-mdt.iodwt GBM Sogiven 㱺 Se= Soexp((µ- Eit +owt} BlackandsoholesfarnuolapipoeyoffEuropeancall option µC.= SoNcd,)- ke- rtNcdz) _ di= ln(f)+(raffi K S - GTT da= di- ott FIRSTLIMIT: The formulaimplicathat log(§)- NIM- Est.cat) - incleed µ- E)ti GW' Se= Soe |logfjt.ir- {ritienennin- fait, ott) but in realitythedistribution is not symmetric→ the lefttail is bighethenthe rightone ④ Plllargeinacaseor longedcorease)is undenestinated byBds + IanMATLABEXAMPLEon real appleprices A= 1day= I 252= # warningdayperyeon252 log/¥)n n(intE)a.ora) cylogreturn Wedo an Ristagnavaof the log. return to check theRealdistribution → similetonormal distribution We use QQ plottochecknormale tyassumptronskewnessamdkurrtosi.siare not normal SECONDLIMIT: Wecan computertheo in relation to K , we expecta Constant 0 , vrcorrelated.to K. On the appositewe see theVOLOTILITY SMILE Call optionon theRome optionand with thesaune timetoNatural ]shouldhavethesaune volatilihg 1)suaelywe andthequaterimplicavolatiliher not thepsicosi NOWCANI use BAS TO PRICEANOPTION? Wehaveto calbrate.tl#d= findthe best 0(Constant) sit Bls priceis clase (as cose as possibile) to the real price | " III.ÌÌthemodel withoption,whichare closet to thematuritaneeoled and theOmar which I are more liquid LA TTICEMETHODOLOGY BinomiaTree Lattice: setofpartiallyordered points MODEL o discretemonitoring(time) E= 0, A, 2A, . . . . T ÷ tote ti T M-11time stepa and A= I M ti=…