Simulation and Estimation of Diffusion Processes : Applications in Finance
(2019) In Doctoral Theses in Mathematical Sciences 2019(4).- Abstract
- Diffusion processes are the most commonly used models in mathematical finance, and are used extensively not only by academics but also practitioners. Nowadays a wide range of models, that can capture many of the effects observed in financial markets, are available. A very important task is to calibrate the models to observed market data and to achieve a good fit, since a slight misspecification can have large monetary consequences. The focus of this thesis is to investigate both theoretical and computational aspects of parameter estimation for diffusion processes.
In the first paper we consider adaptive calibration where the model parameters are considered to be part of a hidden dynamic state. We then use filtering techniques to... (More) - Diffusion processes are the most commonly used models in mathematical finance, and are used extensively not only by academics but also practitioners. Nowadays a wide range of models, that can capture many of the effects observed in financial markets, are available. A very important task is to calibrate the models to observed market data and to achieve a good fit, since a slight misspecification can have large monetary consequences. The focus of this thesis is to investigate both theoretical and computational aspects of parameter estimation for diffusion processes.
In the first paper we consider adaptive calibration where the model parameters are considered to be part of a hidden dynamic state. We then use filtering techniques to estimate the parameter paths. An optimal method for tuning the hyperparameters using the expectation maximization algorithm is presented. The method is evaluated on both simulated and real data, where it is shown to be robust.
The second and third paper cover simulation-based methods for density estimation of diffusion processes using multilevel Monte Carlo estimation. This is a technique that uses simulation on a hierarchy of discretization levels in order to reduce computational complexity. In the second paper we provide an improvement to existing multilevel kernel density estimation by proposing a bandwidth choice that takes model-specific information into account. The third paper extends a simulated maximum likelihood algorithm to the multilevel Monte Carlo framework. Both methods are evaluated on simulated data, where they are shown to provide improvements to the compared methods.
The fourth paper introduces a software package for high-performance simulation of diffusion processes in the Julia programming language. Specific features of Julia are utilized in order to create a simulation library that performs significantly better in terms of computational speed compared to other available libraries, while allowing models to be defined using mathematical notation instead of code. (Less) - Abstract (Swedish)
- Kvantitativ analys är en teknik som syftar till att förstå komplexa system genom att använda matematisk och statistisk modellering. Det är en viktig del av dagens finansiella system och innebär bland annat att modellera den slumpmässiga utvecklingen av finansiella tillgångar, och för att förutspå verkliga händelser som till exempel förändring av riksbankens styrränta. En mycket vanlig metod som används för detta är så kallad Monte Carlo-simulering. Enkelt uttryckt innebär detta att ett stort antal slumpmässiga utfall från en matematisk modell simuleras för att sedan användas för att beräkna ett förväntad värde av de eftersökta kvantiteterna.
Det finns idag ett överflöd av matematiska modeller som kan fånga många av de egenskaper... (More) - Kvantitativ analys är en teknik som syftar till att förstå komplexa system genom att använda matematisk och statistisk modellering. Det är en viktig del av dagens finansiella system och innebär bland annat att modellera den slumpmässiga utvecklingen av finansiella tillgångar, och för att förutspå verkliga händelser som till exempel förändring av riksbankens styrränta. En mycket vanlig metod som används för detta är så kallad Monte Carlo-simulering. Enkelt uttryckt innebär detta att ett stort antal slumpmässiga utfall från en matematisk modell simuleras för att sedan användas för att beräkna ett förväntad värde av de eftersökta kvantiteterna.
Det finns idag ett överflöd av matematiska modeller som kan fånga många av de egenskaper som observeras i verkligheten. Problemet ligger i att alla modeller styrs av parametrar som måste anpassas till historisk data för att modellerna ska vara praktiskt användbara. Även om en komplex modell i teorin är mer kapabel än en enkel modell, kan den i praktiken prestera sämre på grund av att den är svårare att kalibrera.
Denna avhandling syftar till att utveckla och förbättra metoder för att kalibrera diffusionsprocesser, som är den vanligaste typen av modeller som används inom finansiell matematik. I den första artikeln studeras en metod som tillåter parametrarna att fluktuera med tiden. Artikel två och tre studerar simuleringsbaserade metoder för att skatta fördelningen av observationer från diffusionsprocesser. Den fjärde artikeln beskriver ett mjukvarupaket för att definiera och simulera diffusionsprocesser mycket snabbt i programmeringsspråket Julia. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/c3bd102f-aa6b-405c-be05-d79ea5c6e53c
- author
- Åkerlindh, Carl LU
- supervisor
- opponent
-
- Docent Blomvall, Jörgen, Linköpings Universitet, Linköping
- organization
- publishing date
- 2019-09-27
- type
- Thesis
- publication status
- published
- subject
- keywords
- Diffusion processes, Kalman filter, Uncented Kalman filter, EM algorithm, Kernel estimation, Bandwidth selection, Multilevel Monte Carlo, Simulated maximum likelihood estimation, Julia language
- in
- Doctoral Theses in Mathematical Sciences
- volume
- 2019
- issue
- 4
- pages
- 145 pages
- publisher
- Mathematical Statistics, Centre for Mathematical Sciences, Lund University
- defense location
- Lecture hall MH:Riesz, Centre for Mathematical Sciences, Sölvegatan 18A, Lund
- defense date
- 2019-09-27 13:15:00
- ISSN
- 1404-0034
- ISBN
- 978-91-7895-239-7
- 978-91-7895-240-3
- language
- English
- LU publication?
- yes
- id
- c3bd102f-aa6b-405c-be05-d79ea5c6e53c
- date added to LUP
- 2019-08-30 14:32:57
- date last changed
- 2022-04-08 07:22:42
@phdthesis{c3bd102f-aa6b-405c-be05-d79ea5c6e53c,
abstract = {{Diffusion processes are the most commonly used models in mathematical finance, and are used extensively not only by academics but also practitioners. Nowadays a wide range of models, that can capture many of the effects observed in financial markets, are available. A very important task is to calibrate the models to observed market data and to achieve a good fit, since a slight misspecification can have large monetary consequences. The focus of this thesis is to investigate both theoretical and computational aspects of parameter estimation for diffusion processes.<br/><br/>In the first paper we consider adaptive calibration where the model parameters are considered to be part of a hidden dynamic state. We then use filtering techniques to estimate the parameter paths. An optimal method for tuning the hyperparameters using the expectation maximization algorithm is presented. The method is evaluated on both simulated and real data, where it is shown to be robust.<br/><br/>The second and third paper cover simulation-based methods for density estimation of diffusion processes using multilevel Monte Carlo estimation. This is a technique that uses simulation on a hierarchy of discretization levels in order to reduce computational complexity. In the second paper we provide an improvement to existing multilevel kernel density estimation by proposing a bandwidth choice that takes model-specific information into account. The third paper extends a simulated maximum likelihood algorithm to the multilevel Monte Carlo framework. Both methods are evaluated on simulated data, where they are shown to provide improvements to the compared methods.<br/><br/>The fourth paper introduces a software package for high-performance simulation of diffusion processes in the Julia programming language. Specific features of Julia are utilized in order to create a simulation library that performs significantly better in terms of computational speed compared to other available libraries, while allowing models to be defined using mathematical notation instead of code.}},
author = {{Åkerlindh, Carl}},
isbn = {{978-91-7895-239-7}},
issn = {{1404-0034}},
keywords = {{Diffusion processes; Kalman filter; Uncented Kalman filter; EM algorithm; Kernel estimation; Bandwidth selection; Multilevel Monte Carlo; Simulated maximum likelihood estimation; Julia language}},
language = {{eng}},
month = {{09}},
number = {{4}},
publisher = {{Mathematical Statistics, Centre for Mathematical Sciences, Lund University}},
school = {{Lund University}},
series = {{Doctoral Theses in Mathematical Sciences}},
title = {{Simulation and Estimation of Diffusion Processes : Applications in Finance}},
url = {{https://lup.lub.lu.se/search/files/68963973/Thesis_introduction.pdf}},
volume = {{2019}},
year = {{2019}},
}