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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
Wespecify C Sit payoff I maxlo S K
4.2DiscreteFiniteDifferences
qmmsmaxbomdaycoudii.io initial conditionCcsit payoff
boundary conditionsIt
t Clt 01 0
to T CH Sua Sway Ke r t
actuallyaaol.to BS afor Clt Smail Sma since Suratshould
bedoosenmuchbiggerthanK
partition at into Mstepsof size At In tj jstpartition o Sma into N steps ofsize Is STI Si its
call CCtj sit C
oof outAt
one could choose Olds
alternatively Ifix j
h CI Sitts Cls It Isil Ast I 8 Is Is't oIs701k
14 US Is CI Sil Es si Sst III si Ss 9 As'tolds
I 121 CCSit Is I C Si Is1 2 si Is Olds3
centralized derivative zcfsi OCST bettererror
secondderivative 111 1121
ccsitdsltclsi dsl 2CH.lt o Si Ss't Olds41
0z i 2 C
t O Sse 1alsoSiemerIs 2
4 3 Stability of Time stepping Methodsmmmm
E If _Xy solution yltl yoe.tt
consider Xc 0 say also 1 1 71
Explicit Euler methodY jt Xy nhs evaluatedat j
3 yitl lyist yi ll lt.tl yjyJ It Stliyo
here if yj sfiyyoett.IOO then scheme is stable
we need 11 711 1 1
It Xftcl and 1 HtcX co Stc condition on smallnessof Sf
for scheme tobestable
ImplicitEulermethod Y t Xy r.h.sevahatedatj.it
I Xstlyitkyi 3 yit
µ yJ
yi Fist
now stabilitycondition is I I c l i this alwaysholdshere since toounconditionallystable
4 4 Application to HeatEquation
murreousider8 initial value Vfx 01 wanttoknow Vlxit 11for BS backwards
0fj
U iltIt'l VHi outSt
0 f E VHitntl 2VK.it Vi tocse
1 2
f Ej explicitschemeC tj i implicitscheme
denote VK.at ViiVii Vii
explicitvieizvi.tv
1 2
vii It vi it II Wi't tat viiinote need
g zc 1 for stability
visit Vii Viii 2Vii ViiiimplicitIt 1 2
vis vii tat visit z ViiiWa
e it it
0harmmatrix A vector v
WiVector V
need to solve tidiagonal systemof equationstoget T fromVT
whathappens at theboundary
Vo andVu are givenby fixed bonday conditions
wehave V a V H2 a V aVo
Vai aVII't lH2a V aUnit
so withboudayconditions the tridiagonalsystem is
i
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