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