Computational finance using C and C#

372 Pages·2008·1.504 MB·English

by  George Levy

Computational Finance
Using C and C#
George Levy
AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK
OXFORD•PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE
SYDNEY•TOKYO
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LibraryofCongressCataloging-in-PublicationData
Levy,George.
ComputationalFinanceUsingCandC#/GeorgeLevy.
p.cm.–(Quantitativefinance)
Includesbibliographicalreferencesandindex.
ISBN-13:978-0-7506-6919-1(alk.paper)1.Finance-Mathematicalmodels.I.Title.
HG106.L4842008
332.0285’5133-dc22
2008000470
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PrintedintheUnitedStatesofAmerica
08 09 10 11 9 8 7 6 5 4 3 2 1
Contents
1 Overviewoffinancialderivatives 1
2 Introductiontostochasticprocesses 5
2.1 Brownianmotion 5
2.2 ABrownianmodelofassetpricemovements 9
2.3 Ito’sformula(orlemma) 10
2.4 Girsanov’stheorem 12
2.5 Ito’slemmaformultiassetgeometricBrownianmotion 13
2.6 Itoproductandquotientrulesintwodimensions 15
2.7 Itoproductinndimensions 18
2.8 TheBrownianbridge 19
2.9 Time-transformedBrownianmotion 21
2.10 Ornstein–Uhlenbeckprocess 24
2.11 TheOrnstein–Uhlenbeckbridge 27
2.12 Otherusefulresults 31
2.13 Selectedproblems 33
3 Generationofrandomvariates 37
3.1 Introduction 37
3.2 Pseudo-randomandquasi-randomsequences 38
3.3 Generationofmultivariatedistributions:independentvariates 41
3.4 Generationofmultivariatedistributions:correlatedvariates 47
4 Europeanoptions 59
4.1 Introduction 59
4.2 Pricingderivativesusingamartingalemeasure 59
4.3 Putcallparity 60
4.4 VanillaoptionsandtheBlack–Scholesmodel 62
4.5 Barrieroptions 85
5 SingleassetAmericanoptions 97
5.1 Introduction 97
5.2 ApproximationsforvanillaAmericanoptions 97
5.3 Latticemethodsforvanillaoptions 114
5.4 Gridmethodsforvanillaoptions 135
5.5 PricingAmericanoptionsusingastochasticlattice 172
6 Multiassetoptions 181
6.1 Introduction 181
6.2 ThemultiassetBlack–Scholesequation 181
6.3 MultidimensionalMonteCarlomethods 183
6.4 Introductiontomultidimensionallatticemethods 185
6.5 Twoassetoptions 190
6.6 Threeassetoptions 201
6.7 Fourassetoptions 205
7 Otherfinancialderivatives 209
7.1 Introduction 209
7.2 Interestratederivatives 209
7.3 Foreignexchangederivatives 228
7.4 Creditderivatives 232
7.5 Equityderivatives 237
8 C#portfoliopricingapplication 245
8.1 Introduction 245
8.2 Storingandretrievingthemarketdata 254
8.3 ThePricingUtilsclassandtheAnalytics_MathLib 262
8.4 Equitydealclasses 267
8.5 FXdealclasses 280
AppendixA: TheGreeksforvanillaEuropeanoptions 289
A.1 Introduction 289
A.2 Gamma 290
A.3 Delta 291
A.4 Theta 292
A.5 Rho 293
A.6 Vega 294
AppendixB: Barrieroptionintegrals 295
B.1 Thedownandoutcall 295
B.2 Theupandoutcall 298
AppendixC: Standardstatisticalresults 303
C.1 Thelawoflargenumbers 303
C.2 Thecentrallimittheorem 303
C.3 Thevarianceandcovarianceofrandomvariables 305
C.4 Conditionalmeanandcovarianceofnormaldistributions 310
C.5 Momentgeneratingfunctions 311
AppendixD: Statisticaldistributionfunctions 313
D.1 Thenormal(Gaussian)distribution 313
D.2 Thelognormaldistribution 315
D.3 TheStudent’st distribution 317
D.4 Thegeneralerrordistribution 319
AppendixE: Mathematicalreference 321
E.1 Standardintegrals 321
E.2 Gammafunction 321
E.3 Thecumulativenormaldistributionfunction 322
E.4 Arithmeticandgeometricprogressions 323
AppendixF: Black–Scholesfinite-differenceschemes 325
F.1 Thegeneralcase 325
F.2 Thelogtransformationandauniformgrid 325
AppendixG: TheBrownianbridge:alternativederivation 329
AppendixH: Brownianmotion:moreresults 333
H.1 SomeresultsconcerningBrownianmotion 333
H.2 ProofofEq.(H.1.2) 334
H.3 ProofofEq.(H.1.4) 335
H.4 ProofofEq.(H.1.5) 335
H.5 ProofofEq.(H.1.6) 335
H.6 ProofofEq.(H.1.7) 338
H.7 ProofofEq.(H.1.8) 338
H.8 ProofofEq.(H.1.9) 338
H.9 ProofofEq.(H.1.10) 339
AppendixI: TheFeynman–Kacformula 341
AppendixJ: Answerstoproblems 343
J.1 Problem1 343
J.2 Problem2 344
J.3 Problem3 345
J.4 Problem4 346
J.5 Problem5 346
J.6 Problem6 347
J.7 Problem7 348
J.8 Problem8 350
J.9 Problem9 350
J.10 Problem10 352
J.11 Problem11 354
References 355
Index 361
1
Overview of financial derivatives
A financial derivative is a contract between two counterparties (here referred
to as A and B) which derives its value from the state of underlying financial
quantities.Wecanfurtherdividederivativesintothosethatcarryafutureoblig-
ation and those that don’t. In the financial world a derivative which gives the
ownertherightbutnottheobligationtoparticipateinagivenfinancialcontract
is called an option. We will now illustrate this using both a Foreign Exchange
ForwardcontractandaForeignExchangeoption.
Foreign Exchange Forward—a contract with an obligation
In a Foreign Exchange Forward contract a certain amount of foreign currency
will be bought (or sold) at a future date using a prearranged foreign exchange
rate.
For instance, counterparty A may own a Foreign Exchange Forward which,
inoneyear’stime,contractuallyobligesAtopurchasefromB thesumof$200
for£100.Attheendofoneyearseveralthingsmayhavehappened.
(i) Thevalueofthepoundmayhavedecreasedwithrespecttothedollar
(ii) Thevalueofthepoundmayhaveincreasedwithrespecttothedollar
(iii) CounterpartyB mayrefusetohonorthecontract—B mayhavegonebust,
etc.
(iv) CounterpartyAmayrefusetohonorthecontract—Amayhavegonebust,
etc.
Wewillnowconsiderevents(i)–(iv)fromA’sperspective.
Firstly,if(i)occursthenAwillbeabletoobtain$200forlessthanthecurrent
market rate, say £120. In this case the $200 can be bought for £100 and then
immediatelysoldfor£120,givingaprofitof£20.However,thisprofitcanonly
berealizedifB honorsthecontract—thatis,event(iii)doesnothappen.
Secondly, when (ii) occurs then A is obliged to purchase $200 for more than
thecurrentmarketrate,say£90.Inthiscasethe$200areboughtfor£100but
couldhavebeenboughtforonly£90,givingalossof£10.
Theprobabilityofevents(iii)and(iv)occurringarerelatedtotheCreditRisk
associated with counterparty B. The value of the contract to A is not affected
by (iv), although A may be sued if both (ii) and (iv) occur. Counterparty A
shouldonlybeconcernedwiththepossibilityofevents(i)and(iii)occurring—
thatis,theprobabilitythatthecontractisworthapositiveamountinoneyear
2 ComputationalFinanceUsingCandC#
and the probability that B will honor the contract (which is one minus the
probabilitythatevent(iii)willhappen).
From B’s point of view the important Credit Risk is when both (ii) and (iv)
occur—thatis,whenthecontracthaspositivevaluebutcounterpartyAdefaults.
Foreign Exchange option—a contract without an obligation
A Foreign Exchange option is similar to the Foreign Exchange Forward, the
difference being that if event (ii) occurs then A is not obliged to buy dollars
at an unfavorable exchange rate. To have this flexibility A needs to buy a For-
eignExchangeoptionfromB,whichherecanberegardedasinsuranceagainst
unexpectedexchangeratefluctuations.
For instance, counterparty A may own a Foreign Exchange option which, in
oneyear,contractuallyallowsAtopurchasefromB thesumof$200for£100.
Asbefore,attheendofoneyearthefollowingmayhavehappened:
(i) Thevalueofthepoundmayhavedecreasedwithrespecttothedollar
(ii) Thevalueofthepoundmayhaveincreasedwithrespecttothedollar
(iii) CounterpartyB mayrefusetohonorthecontract—B mayhavegonebust,
etc.
(iv) CounterpartyAmayhavegonebust,etc.
Wewillnowconsiderevents(i)–(iv)fromA’sperspective.
Firstly,if(i)occursthenAwillbeabletoobtain$200forlessthanthecurrent
market rate, say £120. In this case the $200 can be bought for £100 and then
immediatelysoldfor£120,givingaprofitof£20.However,thisprofitcanonly
berealizedifB honorsthecontract—thatis,event(iii)doesnothappen.
Secondly, when (ii) occurs then A will decide not to purchase $200 for more
thanthecurrentmarketrate;inthiscasetheoptionisworthless.
We can thus see that A is still concerned with the Credit Risk when events
(i)and(iii)occursimultaneously.
The Credit Risk from counterparty B’s point of view is different. B has sold
to A a Foreign Exchange option, which matures in one year, and has already
received the money—the current fair price for the option. Counterparty B has
no Credit Risk associated with A. This is because if event (iv) occurs, and A
goes bust, it doesn’t matter to B since the money for the option has already
been received. On the other hand, if event (iii) occurs B may be sued by A but
B stillhasnoCreditRiskassociatedwithA.
This book considers the valuation of financial derivatives that carry obliga-
tionsandalsofinancialoptions.
Chapters 1–7 deal with both the theory of stochastic processes and the pric-
ing of financial instruments. In Chapter 8 this information is then applied to a
C# portfolio valuer. The application is easy to use (the portfolios and current
market rates are defined in text files) and can also be extended to include new
tradetypes.
Overviewoffinancialderivatives 3
The book has been written so that (as far as possible) financial mathematics
resultsarederivedfromfirstprinciples.
Finally, the appendices contain various information, which we hope the
readerwillfinduseful.
2
Introduction to stochastic
processes
2.1 Brownian motion
BrownianmotionisnamedafterthebotanistRobertBrownwhousedamicro-
scopetostudythefertilizationmechanismoffloweringplants.Hefirstobserved
the random motion of pollen particles (obtained from the American species
Clarkiapulchella)suspendedinwater,andwrote:
Thefovillaorgranulesfillthewholeorbiculardiskbutdonotextendtothe
projectingangles.Theyarenotsphaericalbutoblongornearlycylindrical,
and the particles have manifest motion. This motion is only visible to my
lenswhichmagnifies370times.Themotionisobscureyetcertain...
RobertBrown,12thJune1827;seeRamsbottom(1932)
ItappearsthatBrownconsideredthismotionnomorethanacuriosity(hebe-
lieved that the particles were alive) and continued undistracted with his botan-
ical research. The full significance of his observations only became apparent
about eighty years later when it was shown (Einstein, 1905) that the motion
is caused by the collisions that occur between the pollen grains and the water
molecules. In 1908 Perrin (1909) was finally able to confirm Einstein’s predic-
tions experimentally. His work was made possible by the development of the
ultramicroscopebyRichardZsigmondyandHenrySiedentopfin1903.Hewas
able to work out from his experimental results and Einstein’s formula the size
of the water molecule and a precise value for Avogadro’s number. His work
established the physical theory of Brownian motion and ended the skepticism
abouttheexistenceofatomsandmoleculesasactualphysicalentities.Manyof
the fundamental properties of Brownian motion were discovered by Paul Levy
(Levy, 1939, 1948), and the first mathematically rigorous treatment was pro-
vided by Norbert Wiener (Wiener, 1923, 1924). Karatzas and Shreve (1991) is
an excellent textbook on the theoretical properties of Brownian motion, while
Shreve,Chalasani,andJha(1997)providesmuchusefulinformationconcerning
theuseofBrownianprocesseswithinfinance.
Brownian motion is also called a random walk, a Wiener process, or some-
times(morepoetically)thedrunkard’swalk.Wewillnowpresentthethreefun-
damentalpropertiesofBrownianmotion.
6 ComputationalFinanceUsingCandC#
2.1.1 The properties of Brownian motion
In formal terms a process W = (W :t (cid:2) 0) is (one-dimensional) Brownian
t
motionif:
(i) W iscontinuous,andW =0,
t 0
(ii) W ∼N(0,t),
t
(iii) TheincrementdWt =Wt+dt−Wt isnormallydistributedasdWt ∼N(0,dt),
soE[dW ] = 0andVar[dW ] = dt.TheincrementdW isalsoindependent
t t t
ofthehistoryoftheprocessuptotimet.
From(iii)wecanfurtherstatethat,sincetheincrementsdW areindependent
t
ofpastvaluesW ,aBrownianprocessisalsoaMarkovprocess.Inadditionwe
t
shallnowshowthataBrownianprocessisalsoamartingaleprocess.
In a martingale process Pt,t (cid:2) 0, the conditional expectation E[Pt+dt|Ft] =
P , where F is called the filtration generated by the process and contains the
t t
information learned by observing the process up to time t. Since for Brownian
motionwehave
(cid:2) (cid:3)
E[Wt+dt|Ft]=E (Wt+dt −Wt)+Wt|Ft =E[Wt+dt −Wt]+Wt
=E[dW ]+W =W
t t t
where we have used the fact that E[dWt] =0. Since E[Wt+dt|Ft] = Wt the
BrownianmotionW isamartingaleprocess.
Using property (iii) we can also derive an expression for the covariance of
Brownian motion. The independent increment requirement means that for the
ntimes0 (cid:3)t < t < t <··· < t <∞therandomvariablesW −W ,W −
0 1 2 n t1 t0 t2
W ,...,W −W areindependent.So
t1 tn tn−1
Cov[W −W ,W −W ]=0, i (cid:4)=j (2.1.1)
ti ti−1 tj tj−1
WewillshowthatCov[W ,W ]=s ∧t.
s t
Proof. UsingW =0,andassumingt (cid:2)s wehave
t0 (cid:2) (cid:3)
Cov[W −W ,W −W ]=Cov[W ,W ]=Cov W ,W +(W −W )
s t0 t t0 s t s s t s
FromAppendixC.3.2wehave
(cid:2) (cid:3)
Cov W ,W +(W −W ) =Cov[W ,W ]+Cov[W ,W −W ]
s s t s s s s t s
=Var[W ]+Cov[W ,W −W ]
s s t s
Therefore
Cov[W ,W ]=s +Cov[W ,W −W ]
s t s t s
Now
Cov[W ,W −W ]=Cov[W −W ,W −W ]=0
s t s s t0 t s
wherewehaveusedEq.(2.1.1)withn=2,t =s andt =t.
1 2