Numerical Partial Differential Equations in Finance еxplained. An Introduction to Computational Finance

131 Pages·2017·2.699 MB·english

by  Karel in ’t Hout

Karelin’tHout
Numerical Partial
Differential Equations
in Finance Explained
An Introduction to Computational Finance
Karelin’tHout
DepartmentofMathematicsand
ComputerScience
UniversityofAntwerp
Antwerp
Belgium
FinancialEngineeringExplained
ISBN978-1-137-43568-2 ISBN978-1-137-43569-9(eBook)
DOI10.1057/978-1-137-43569-9
LibraryofCongressControlNumber:2017934655
©TheEditor(s)(ifapplicable)andTheAuthor(s)2017
ThisPalgraveMacmillanimprintispublishedbySpringerNature
TheregisteredcompanyisMacmillanPublishersLtd.
Theregisteredcompanyaddressis:
TheCampus,4CrinanStreet,London,N19XW,UnitedKingdom
Preface
A few years after Black and Scholes [5] derived their famous par-
tial differential equation (PDE) for the fair values of European call
and put options, Schwartz [78] considered a finite difference discret-
ization for its approximate solution. Today, the numerical solution
of time-dependent PDEs forms one of the pillars of computational
finance. Efficient, accurate and stable numerical methods are imper-
ative for financial institutions and companies worldwide. Extensive
researchisperformed,bothinacademiaandindustry,intotheirdevel-
opment, analysis and application. This book is intended as a concise,
gentle introduction into this interesting and dynamic field. Its aim is
to provide students and practitioners with an easily accessible, prac-
ticaltextexplainingmainconcepts,models,methodsandresults.The
text is organized through a sequence of short chapters. The style is
moredescriptivethan(mathematically)rigorous.Numerousexamples
and numerical experiments are given to illustrate results. Only some
elementary knowledge of mathematics, notably calculus and linear
algebra, is assumed.
The numerical solution processes in this book are obtained fol-
lowing the popular method of lines (MOL) approach. Here a given
time-dependent PDE is semidiscretized on a grid by finite difference
formulas, which yields a large system of ordinary differential equa-
tions (ODEs). Subsequently, a suitable temporal discretization method
is applied, which defines the full discretization.
Chapters1and2introducefinancialoptionvaluationandpartialdif-
ferentialequations.Next,theMOLapproachiselaboratedinChapters
3–8. Much attention is paid to studying stability and convergence of
the various discretizations. Important special topics, such as bound-
ary conditions, nonuniform grids, the treatment of nonsmooth initial
data and approximation of the so-called Greeks, are included in the
discussion. In this part the Black–Scholes PDE serves as the prototype
equation for the numerical experiments. Examining numerical meth-
odsintheirapplicationtothisequationprovideskeyinsightintotheir
properties and performance when applied to many advanced PDEs in
contemporary financial mathematics.
After having considered European call and put options as an ex-
ample, we move on to explore the numerical valuation of more
challenging modern types of options: cash-or-nothing options in
Chapter 9, barrier options in Chapter 10 and American options in
Chapter 11. The latter type of options leads to partial differential in-
equalities and an additional step in the numerical solution process is
required,whereso-calledlinearcomplementarityproblemsaresolved.
Chapter 12 is devoted to option valuation in the presence of jumps
in the underlying asset price evolution. This gives rise to partial
integro-differential equations. These equations can be viewed as PDEs
with an extra integral term. For their effective numerical solution,
operator splitting methods of the implicit-explicit (IMEX) kind are
introduced.
Chapter 13 extends the MOL approach to two-dimensional PDEs in
finance.SemidiscretizationthenresultsinverylargesystemsofODEs.
For the efficient temporal discretization, operator splitting methods
of the Alternating Direction Implicit (ADI) kind are discussed. As an
example, the numerical valuation of a two-asset option under the
Black–Scholes framework is considered.
Mostofthechaptersconcludewithashortsectionwherenotesand
referencestotheliteraturearegiven.Theseareintendedaspointersto
readers who wish to broaden their knowledge or deepen their under-
standingofthetopicsunderconsideration.Supplementarymaterialto
this book will be provided on my website.
I am grateful to Peter Forsyth, Sven Foulon, Willem Hundsdorfer,
Wim Schoutens, Jari Toivanen and Maarten Wyns for their genuine
interest and their valuable suggestions and comments on prelimin-
ary versions of this book. Last but not least, I wish to thank Palgrave
Macmillan for the pleasant cooperation.
Antwerp, July 2016 Karel in ’t Hout
Contents
1 FinancialOptionValuation........................................................ 1
1.1 FinancialOptions................................................................ 1
1.2 TheBlack–ScholesPDE ......................................................... 3
2 PartialDifferentialEquations.................................................... 9
2.1 Convection-Diffusion-ReactionEquations...................................... 9
2.2 TheModelEquation............................................................. 10
2.3 BoundaryConditions............................................................ 12
2.4 NotesandReferences ........................................................... 14
3 SpatialDiscretizationI............................................................. 15
3.1 MethodofLines.................................................................. 15
3.2 FiniteDifferenceFormulas...................................................... 17
3.3 Stability........................................................................... 21
3.4 NotesandReferences ........................................................... 23
4 SpatialDiscretizationII............................................................ 25
4.1 BoundaryConditions............................................................ 25
4.2 NonuniformGrids............................................................... 29
4.3 NonsmoothInitialData.......................................................... 32
4.4 MixedCentral/UpwindDiscretization.......................................... 33
4.5 NotesandReferences ........................................................... 35
5 NumericalStudy:Space............................................................ 37
5.1 CellAveraging.................................................................... 38
5.2 NonuniformGrids............................................................... 41
5.3 BoundaryConditions............................................................ 42
6 TheGreeks............................................................................. 45
6.1 TheGreeks....................................................................... 45
6.2 NumericalStudy................................................................. 47
6.3 NotesandReferences ........................................................... 50
7 TemporalDiscretization........................................................... 51
7.1 Theθ-Methods................................................................... 51
7.2 StabilityandConvergence....................................................... 52
7.3 MaximumNormandPositivity.................................................. 58
7.4 NotesandReferences ........................................................... 60
8 NumericalStudy:Time............................................................. 61
8.1 ExplicitMethod.................................................................. 61
8.2 ImplicitMethods................................................................. 63
8.3 NotesandReferences ........................................................... 68
9 Cash-or-NothingOptions.......................................................... 69
10 BarrierOptions....................................................................... 75
11 American-StyleOptions............................................................ 81
11.1 American-StyleOptions......................................................... 81
11.2 LCPSolutionMethods........................................................... 84
11.3 NumericalStudy................................................................. 86
11.4 NotesandReferences ........................................................... 90
12 MertonModel......................................................................... 91
12.1 MertonModel.................................................................... 91
12.2 SpatialDiscretization............................................................ 93
12.3 IMEXSchemes................................................................... 95
12.4 NumericalStudy................................................................. 96
12.5 NotesandReferences ........................................................... 97
13 Two-AssetOptions................................................................... 99
13.1 Two-AssetOptions............................................................... 99
13.2 SpatialDiscretization............................................................ 101
13.3 ADISchemes..................................................................... 106
13.4 NumericalStudy................................................................. 108
13.5 NotesandReferences ........................................................... 111
AppendixA:WienerProcess........................................................... 113
AppendixB:Feynman–KacTheorem............................................... 115
AppendixC:Down-and-OutPutOptionValue.................................... 117
AppendixD:Max-of-Two-AssetsCallOptionValue ............................. 119
Bibliography................................................................................ 121
Index .......................................................................................... 127
Figures
Figure1.1 Payofffunctionsforcallandputoptionson[0,3K]............... 4
Figure1.2 Exactcallandputoptionvaluefunctionson[0,3K]
fort=T andparameterset(1.8)....................................... 7
Figure3.1 Samplegridinthe(s,t)-domain,indicatedbycircles.............. 16
Figure3.2 Geometricinterpretationofthefinitedifference
formulas(3.3),(3.7),(3.10)forthefirstderivativef(cid:2)(s).......... 17
Figure4.1 Mappingϕ definedinExample4.2.1.................................. 30
Figure5.1 Spatialerrorε(m)versuss form = 50(top)and
i i
m = 51 (bottom). Semidiscretization onuniform
gridbysecond-ordercentralformulas.Nocellaveraging........ 39
Figure5.2 Spatialerrore(m)versus1/mforall10≤m≤100.
Semidiscretizationonuniformgridbysecond-order
centralformulas.Nocellaveraging.................................... 40
Figure5.3 Spatialerrore(m)versus1/mforall10≤m≤100.
Semidiscretizationonuniformgridbysecond-order
centralformulas.Withcellaveraging.................................. 40
Figure5.4 SpatialgridpointscorrespondingtoExample4.2.1
ifm=50...................................................................... 41
Figure5.5 Spatialerrore(m)versus1/mforall10≤m≤100.
Semidiscretization on nonuniform grid by
second-order central formulas. Formula A for
convection: bullets. Formula B for convection:
squares.Withcellaveraging............................................. 42
Figure5.6 Spatial errors e(m) (dark squares) and eROI(m)
(light squares) versus 1/m for 100 ≤ m ≤ 1000.
Semidiscretization on nonuniform grid by
second-order central formulas. Formula B for
convection.Linearboundaryconditionats=S .
max
Withcellaveraging ........................................................ 43
Figure6.1 Greeksforacalloptionfort=T andparameterset(1.8)........ 46
Figure6.2 Delta spatial error ed(m) versus 1/m for all
10≤m≤100.Semidiscretizationbysecond-order
centralformulas.FormulaBforconvection.With
cellaveraging.Uniformgrid:bullets.Nonuniform
grid:squares................................................................. 48
Figure6.3 Gamma spatial error eg(m) versus 1/m for all
10≤m≤100.Semidiscretizationbysecond-order
centralformulas.FormulaBforconvection.With
cellaveraging.Uniformgrid:bullets.Nonuniform
grid:squares................................................................. 49
Figure6.4 Vega spatial error ev(m) versus 1/m for all
10≤m≤100.Semidiscretizationbysecond-order
centralformulas.FormulaBforconvection.With
cellaveraging.Uniformgrid:bullets.Nonuniform
grid:squares................................................................. 49
Figure6.5 Rho spatial error er(m) versus 1/m for all
10≤m≤100.Semidiscretizationbysecond-order
centralformulas.FormulaBforconvection.With
cellaveraging.Uniformgrid:bullets.Nonuniform
grid:squares................................................................. 50
Figure7.1 Stabilityregionθ-methodwithθ =0(shaded)...................... 54
Figure7.2 Stabilityregionθ-methodwithθ = 1 (shaded)...................... 55
2
Figure7.3 Stabilityregionθ-methodwithθ =1(shaded)...................... 55
Figure8.1 Fullydiscreteapproximationofcalloptionvalue
functionfort =T obtainedwiththeforwardEuler
methodifN =75(top)andN =80(bottom)........................ 62
Figure8.2 Temporal error(cid:2)e((cid:5)t;50) versus (cid:5)t for all
1 ≤ N ≤ 100. Backward Euler (dark bullets),
Crank–Nicolson(lightsquares),Crank–Nicolson
withdamping(darksquares)............................................ 64
Figure8.3 Temporal error(cid:2)e((cid:5)t;200) versus (cid:5)t for all
1 ≤ N ≤ 100. Backward Euler (dark bullets),
Crank–Nicolson(lightsquares),Crank–Nicolson
withdamping(darksquares)............................................ 65
Figure8.4 Total error E((cid:5)t;m) versus 1/m with N = (cid:4)m/5(cid:5)
for10 ≤m≤ 1000.BackwardEuler(darkbullets),
Crank–Nicolson(lightsquares),Crank–Nicolson
withdamping(darksquares)............................................ 67
Figure9.1 Exactcash-or-nothingcalloptionvaluefunctionon
[0,3K]×[0,T]withparameterset(9.2).............................. 70
Figure9.2 Cash-or-nothing call option with parameter set
(9.2). Total error EROI((cid:5)t;m) versus 1/m with
N = (cid:4)m/5(cid:5) for 10 ≤ m ≤ 1000. Crank–Nicolson
method. Cell averaging without damping (dark
bullets).Cellaveragingwithdampingusingtwo
substeps(darksquares).Nocellaveragingbutwith
dampingusingtwosubsteps(lightsquares)......................... 71
Figure9.3 Cash-or-nothingcalloptiondeltawithparameter
set(9.2).TotalerrorEd,ROI((cid:5)t;m)versus1/mwith
N = (cid:4)m/5(cid:5) for 10 ≤ m ≤ 1000. Crank–Nicolson
method. Cell averaging and: no damping (dark
bullets), damping using two substeps (dark
squares)anddampingusingfoursubsteps(dark
triangles). No cell averaging but with damping
usingfoursubsteps(lighttriangles)................................... 72
Figure9.4 Cash-or-nothingcalloptiongammawithparameter
set(9.2).TotalerrorEg,ROI((cid:5)t;m)versus1/mwith
N = (cid:4)m/5(cid:5) for 10 ≤ m ≤ 1000. Crank–Nicolson
method. Cell averaging and: no damping (dark
bullets), damping using two substeps (dark
squares)anddampingusingfoursubsteps(dark
triangles). No cell averaging but with damping
usingfoursubsteps(lighttriangles)................................... 73
Figure10.1 Exactdown-and-outputoptionvaluefunctionon
[H,3K]×[0,T]withparameterset(10.1)........................... 76
Figure10.2 Down-and-out put option with parameter set
(10.1). Total error EROI((cid:5)t;m) versus 1/m with
N = (cid:4)m/5(cid:5) for 10 ≤ m ≤ 1000. Crank–Nicolson
method. Cell averaging without damping (dark
bullets).Cellaveragingwithdampingusingtwo
substeps(darksquares).Nocellaveragingbutwith
dampingusingtwosubsteps(lightsquares)......................... 77
Figure10.3 Numericallyapproximateddiscretedown-and-out
put option value function on [0,3K]×[0,T]
with parameter set(10.1) and monitoring times
τ =jT/5(1≤j≤5)....................................................... 78
j
Figure11.1 Dark: numerically approximated American put
option valuefunctionon [1K,3K] fort = T and
2 2
parameterset(11.10).Light:payofffunction....................... 86
Figure11.2 Numericallyapproximatedearlyexerciseboundary
fortheAmericanputoptionandparameterset(11.10).......... 87
Figure11.3 Americanputoptionwithparameterset(11.10).
Temporal error(cid:2)eROI((cid:5)t;m) versus 1/m with
N = (cid:4)m/2(cid:5) for 10 ≤ m ≤ 1000. Constant step
sizes.BackwardEuler:light.Crank–Nicolson:dark.
Method (11.5): bullets. Method (11.6): squares.
Method(11.7):triangles.................................................. 88
Figure11.4 Americanputoptionwithparameterset(11.10).
Temporal error(cid:2)eROI((cid:5)t;m) versus 1/m with