Session 22 4 Black Scholes Equation and Finite Difference Schemes Nov 16,2018 Mmmmm 4 Derivation of Black Scholes Equation mmmm let the stock price process begeon BM dS µSdtt6SdW solution SCH Soe la Elt 16Wh let cls.tl be the price of an option Ito's comma gives dC f t µStI 6 sYdtt osdW recall Ito dt fdttgdwthendfk.tk 9ItffItIg'fIE dt f gdw Merton's trick consider the night portfolio to eliminate risk value of portfolio D Ct S rfCtff DM dct ds rMdt such that A noert T such a portfolio grows ft't Mst's is dt ssdw with risklessrater µSdt ffoSdW deterministic.lk foc z0oI.EsYdt s
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Session224 Black Scholes Equation and Finite Difference Schemes Nov16,2018Mmmmm
4 DerivationofBlack Scholes Equationmmmm
let thestockprice processbegeon BM dS µSdtt6SdWsolution SCH Soela Elt 16Wh
let cls.tl be theprice of anoption
Ito's comma gives dC f t µStI 6 sYdtt osdW
recall Ito dt fdttgdwthendfk.tk 9ItffItIg'fIE dt
f gdwMerton's trick consider thenightportfolio toeliminate risk
value ofportfolio D Ct S rfCtffDM dct ds rMdt suchthat A noert
Tsuchaportfoliogrows
ft't Mst's is dt ssdw
with risklessrater µSdt ffoSdWdeterministic.lk
foc z0oI.EsYdts
1625 o trs rc Black Scholes Equation
Notesindependent of µ optionpriceonly dependsonvolatility
secondorder PDE
by achange ofvariables this eq can be transformedinto a heatequation
cOo z or c Su forhigherdin
backward drift diffusionequation specify SF solve for us 01Europeancalls strikeprice