Algorithmic Differentiation in Finance Explained

112 Pages·2017·2.783 MB·English

Financial Engineering Explained
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Marc Henrard
Algorithmic Differentiation
in Finance Explained
MarcHenrard
AdvisoryPartner,
OpenGamma
London,UnitedKingdom
ISBN978-3-319-53978-2 ISBN978-3-319-53979-9(eBook)
DOI10.1007/978-3-319-53979-9
LibraryofCongressControlNumber:2017941212
TheEditor(s)(ifapplicable)andTheAuthor(s)2017
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Contents
1 Introduction .............................................. 1
2 ThePrinciplesofAlgorithmic
Differentiation ............................................ 15
3 ApplicationtoFinance ...................................... 31
4 AutomaticAlgorithmicDifferentiation ......................... 49
5 DerivativestoNon-inputsandNon-derivativestoInputs ........... 67
6 Calibration ............................................... 77
AppendixA MathematicalResults ............................. 99
Index ...................................................... 101
List of Figures
Fig. 1.1 Comparison of the convergence for different finite
differenceapproximationofthefirstorderderivative .......... 6
Fig. 1.2 Comparison of the convergence for different finite
differenceapproximationsofthefirstorderderivative.
Includingthe“verysmallshifts”partofthegraph ............ 9
Fig. 6.1 VisualrepresentationofthetransitionmatrixoftheEUR
discountingforcollateralinUSDwithFedFundrates ......... 82
Fig. 6.2 Computationtimeratios(presentvalueandsensitivities
timetopresentvaluetime)forthefinitedifferenceand
AADmethods.Thevegarepresentsthederivativeswith
respecttotheSABRparameters;thedeltarepresentsthe
derivativeswithrespecttothecurves.TheAADmethod
usestheimplicitfunctionapproach.Figuresforannually
amortizedswaptionsinaLMMcalibratedtovanilla
swaptionsinvaluedwiththeSABRmodel ................. 90
Fig. 6.3 Representation of LMM calibration for a 2 years
amortised swaption. Each yearly volatility block is
multipliedbyacommonmultiplicativefactorandeach
yearlydisplacementblockcontainsthesamenumber.......... 94
Fig. 6.4 Computationtimeratios(priceandsensitivitiestimeto
pricetime)forthefinitedifferenceandAADmethods.
Thevegarepresentsthederivativeswithrespecttothe
SABRparameters;thedeltarepresentsthederivatives
with respect to the curves. The AAD method uses
theimplicitfunctionapproach.Figuresforannually
amortisedswaptionsinaLMMcalibratedyearlytothree
vanillaswaptionsinSABR ............................. 95
vii
viii ListofFigures
Fig. 6.5 Computationtimeratios(priceandsensitivitiestimeto
pricetime)forthefinitedifferenceandAADmethods.
Thevegarepresentsthederivativeswithrespecttothe
SABRparameters;thedeltarepresentsthederivatives
with respect to the curves. The AAD method uses
theimplicitfunctionapproach.Figuresforannually
amortisedswaptionsinaLMMcalibratedyearlytothe
givennumberofvanillaswaptionswithdifferentstrikes
inSABR........................................... 96
List of Tables
Table.2.1 The generic single assignment code for a function
computingasinglevalue .............................. 19
Table.2.2 Thegenericcodeforafunctioncomputingasingle
value and its standard algorithmic differentiation
pseudo-code ....................................... 20
Table.2.3 Thegenericcodeforafunctioncomputingasingle
valueanditsadjointalgorithmicdifferentiationcode ......... 23
Table.2.4 Computationtimefortheexamplefunctionandfour
derivatives ......................................... 25
Table.2.5 Thegenericcodeforafunctioncomputingasingle
valuewithanif statement ............................. 27
Table.2.6 Thegenericcodeforafunctioncomputingasingle
valueanditsalgorithmicdifferentiationcode ............... 27
Table.3.1 Computationtimefordifferentimplementationsofthe
Blackfunctionandfivederivatives(withrespectto
forward,volatility,numeraire,strike,andexpiry) ............ 35
Table.3.2 Computation time for the SABR price and eight
derivatives(withrespecttoforward,alpha,beta,rho,
nu,numeraire,strike,andexpiry) ....................... 40
Table.3.3 Computationtimeforthepresentvalueanddifferent
typeofsensitivities.Themulti-curvesethastwocurves
andatotalof30nodes ................................ 47
Table.4.1 TapeforthesimplefunctiondescribedinListing2.1 ......... 58
Table.4.2 Computationtimefortheexamplefunctionandfour
derivatives ......................................... 60
Table.4.3 Computation time for the Black function and five
derivatives ......................................... 61
Table.4.4 ComputationtimefortheSABRvolatilityfunctionand
sevenderivatives .................................... 61
ix
x ListofTables
Table.4.5 ComputationtimefortheSABRoptionpricefunction
andeightderivatives ................................. 62
Table.4.6 Computation time for the Black function and five
derivatives ......................................... 64
Table.4.7 ComputationtimefortheSABRpricefunctionand
eightderivatives .................................... 65
Table.6.1 Summarizedrepresentationofthedependencyofthe
EURwithcollateralinUSDFedFunddiscounting
curvetotheothercurvesoftheexample ................... 82
Table.6.2 Performancefordifferentapproachestoderivatives
computations:cashsettledswaptioninHull-Whiteone
factormodel........................................ 88
Table.6.3 Performancefordifferentapproachestoderivatives
computations:amortizedswaptionintheLMM ............. 89