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Efficient Barrier Option Greeks using Automatic Differentation

Hedin, Gustav (2019) In LUTFMS-3362-2019 FMSM01 20191
Mathematical Statistics
Abstract
Automatic Differentiation (AD) is an effective method for calculation of derivatives.
It can evaluate an unlimited number of derivatives to a fixed cost relative to the
computing time of the original function. The AD technique is used in many fields
for large and complex calculations in order to get accurate values of derivatives fast.
Banks and other financial institutions handle calculations of portfolio values which
could depend on a large number of input variables. AD enables fast calculation and
sensitivity analysis of these complex functions such as risks of day to day trading
as well as XVA’s. The accuracy of AD is also better than currently used methods
for derivative calculations as it is not based on approximations. In... (More)
Automatic Differentiation (AD) is an effective method for calculation of derivatives.
It can evaluate an unlimited number of derivatives to a fixed cost relative to the
computing time of the original function. The AD technique is used in many fields
for large and complex calculations in order to get accurate values of derivatives fast.
Banks and other financial institutions handle calculations of portfolio values which
could depend on a large number of input variables. AD enables fast calculation and
sensitivity analysis of these complex functions such as risks of day to day trading
as well as XVA’s. The accuracy of AD is also better than currently used methods
for derivative calculations as it is not based on approximations. In this paper, AD
in a Monte Carlo setting is studied and one way of implementing AD by operator
overloading in its reverse mode is presented. Different ways to handle discontinuous
payoffs are tested and applied on calculations of price and derivatives of Barrier
Options. It is shown that AD combined with sigmoid smoothing of discontinuous
payoffs gives accurate values for option price and Greeks with fast convergence rate. (Less)
Popular Abstract (Swedish)
K¨anslighetsanalys av barri¨aroptioner med
algoritmisk derivering
F¨or att f˚a ut optionspris och derivator p˚a priset med avseende p˚a tex r¨anta och
volatilitet, utf¨ors vanligtvis ett antal simuleringar varp˚a ett medelv¨arde r¨aknas ut.
D¨arefter r¨aknas derivatorna ut med en finit differensmetod. F¨or att erh˚alla tillf¨orlitliga resultat med denna metod kr¨avs ett stort antal simuleringar vilket g¨or metoden
ineffektiv. Med bakgrund av detta har det i detta projekt studerats om algoritmisk
derivering g˚ar att anv¨anda f¨or att ber¨akna dervator p˚a finansiella kontrakt och i
synnerhet barri¨aroptioner.
Algoritmisk derivering(AD) ¨ar en konceptuell bek¨akninsteknik som bygger p˚a kedjeregeln och det faktum att l˚anga och... (More)
K¨anslighetsanalys av barri¨aroptioner med
algoritmisk derivering
F¨or att f˚a ut optionspris och derivator p˚a priset med avseende p˚a tex r¨anta och
volatilitet, utf¨ors vanligtvis ett antal simuleringar varp˚a ett medelv¨arde r¨aknas ut.
D¨arefter r¨aknas derivatorna ut med en finit differensmetod. F¨or att erh˚alla tillf¨orlitliga resultat med denna metod kr¨avs ett stort antal simuleringar vilket g¨or metoden
ineffektiv. Med bakgrund av detta har det i detta projekt studerats om algoritmisk
derivering g˚ar att anv¨anda f¨or att ber¨akna dervator p˚a finansiella kontrakt och i
synnerhet barri¨aroptioner.
Algoritmisk derivering(AD) ¨ar en konceptuell bek¨akninsteknik som bygger p˚a kedjeregeln och det faktum att l˚anga och komplexa algoritmer kan brytas ned till en
f¨oljd av enkla r¨akneoperationer s˚a som addition och multiplikation. Dessa r¨akneoperationer ¨ar var f¨or sig enkla att derivera. Det ˚aterst˚ar sedan att l¨anka samman alla
derivator i ett ber¨akniningstr¨ad och slutligen samla ihop dem. I detta projekt har
Reverse Mode AD eller ibland kallad Adjoint Mode AD implementerats genom operator¨overlagring. Det inneb¨ar att koden som k¨ors vi kommandon som tex addition
eller multiplikation skrivs om s˚a att nya algoritmer utf¨ors ist¨allet. Ber¨akningar av
derivator med reverse mode AD kan ses som en genoml¨opning av ett ber¨akningstr¨ad
d¨ar varje nod best˚ar av en enkel r¨akneoperation. Funktionsv¨arden r¨aknas ut i en
f¨orsta genoml¨opning av ber¨akningstr¨adet. Under tiden sparas ocks˚a information
om hur tr¨adet ser ut. I en andra genoml¨opning r¨aknas de partiella derivatorna ut
rekursivt och skickas ned˚at i tr¨adet, till de initiala variablerna. F¨ordelen med operator¨overlagring j¨amf¨ort med andra implementeringsmetoder ¨ar att det inte beh¨ovs
g¨oras n˚agra ¨andringar i den kod man vill exekvera med AD.
I detta projekt har AD applicerats p˚a ber¨akningar av derivator p˚a barri¨aroptioner. Dessa finansiella kontrakt inneh˚aller diskontinuiteter som inte g˚ar att derivera i klassisk mening. Olika metoder f¨or att hantera dessa diskontinuiteter har
testast och j¨amf¨orts, b˚ade med och utan AD.
Det visar sig att den implementerade versionen av AD ¨ar fullt kompatibel med
optionsprisber¨akningar p˚a kontrakt inneh˚allandes diskontinuiteter. Inom ramen
f¨or Vibrato Monte Carlo kan ¨aven k¨anligheter ber¨aknas f¨or kontrakt med diskret
barri¨ar¨overvakning, s.k. bermudan barrier. Vibrato Monte Carlo bygger p˚a att
f¨ordelningsfunktionen ¨ar k¨and vid diskontinuiteten. Sedan utf¨ors nya Monte Carlosimuleringar av den deriverade f¨ordelningsfunktionen, ¨over barri¨aren. B˚ade Vibrato Mote Carlo och AD kr¨aver f¨arre simuleringar f¨or att konvergera ¨an befintlig
finit-differensmetodik. (Less)
Please use this url to cite or link to this publication:
author
Hedin, Gustav
supervisor
organization
course
FMSM01 20191
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Automatic Differentiation, Greeks, Barrier Option, Vibrato Monte Carlo
publication/series
LUTFMS-3362-2019
report number
2019:E5
ISSN
1404-6342
language
English
id
8969681
date added to LUP
2019-03-18 08:16:05
date last changed
2019-04-03 12:51:00
@misc{8969681,
  abstract     = {Automatic Differentiation (AD) is an effective method for calculation of derivatives.
It can evaluate an unlimited number of derivatives to a fixed cost relative to the
computing time of the original function. The AD technique is used in many fields
for large and complex calculations in order to get accurate values of derivatives fast.
Banks and other financial institutions handle calculations of portfolio values which
could depend on a large number of input variables. AD enables fast calculation and
sensitivity analysis of these complex functions such as risks of day to day trading
as well as XVA’s. The accuracy of AD is also better than currently used methods
for derivative calculations as it is not based on approximations. In this paper, AD
in a Monte Carlo setting is studied and one way of implementing AD by operator
overloading in its reverse mode is presented. Different ways to handle discontinuous
payoffs are tested and applied on calculations of price and derivatives of Barrier
Options. It is shown that AD combined with sigmoid smoothing of discontinuous
payoffs gives accurate values for option price and Greeks with fast convergence rate.},
  author       = {Hedin, Gustav},
  issn         = {1404-6342},
  keyword      = {Automatic Differentiation,Greeks,Barrier Option,Vibrato Monte Carlo},
  language     = {eng},
  note         = {Student Paper},
  series       = {LUTFMS-3362-2019},
  title        = {Efficient Barrier Option Greeks using Automatic Differentation},
  year         = {2019},
}