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