M AR K ET MO DE LS W ITH ST O CHASTIC V O LA TILIT Y 20 19 ISBN 978-91-7485-433-6 ISSN 1651-4238
Address: P.O. Box 883, SE-721 23 Västerås. Sweden Address: P.O. Box 325, SE-631 05 Eskilstuna. Sweden E-mail: info@mdh.se Web: www.mdh.se
Jean-Paul Murara 0 50 100 150 200 250 300 −0.6 −0.4 −0.2 0 0.2 0.4 0.6
0.8 Example of a Stochastic Process
Number of Days (T=300)
Daily change in price
0 0.05 0.1 0.15 0.2 0.25 −1.5 −1 −0.5 0 0.5 1 1.5 Time (Years) Brownian State
Standard Brownian Motions
0 10 20 30 40 50 60 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78
0.8 An example of Implied Volatility Shape
Implied Volatility (Max=1)
MARKET MODELS WITH STOCHASTIC VOLATILITY
Jean-Paul Murara 2019
School of Education, Culture and Communication
MARKET MODELS WITH STOCHASTIC VOLATILITY
Jean-Paul Murara 2019
Copyright © Jean-Paul Murara, 2019 ISBN 978-91-7485-433-6
ISSN 1651-4238
Printed by E-Print AB, Stockholm, Sweden
Copyright © Jean-Paul Murara, 2019 ISBN 978-91-7485-433-6
ISSN 1651-4238
No. 294
MARKET MODELS WITH STOCHASTIC VOLATILITY
Jean-Paul Murara
Akademisk avhandling
som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik vid Akademin för utbildning, kultur och kommunikation kommer att offentligen försvaras fredagen den 4 oktober 2019, 13.15 i Lambda, Mälardalens högskola, Västerås.
Fakultetsopponent: Professor Guglielmo D’Amico, University G. D'Annunzio of Chieti-Pescara
Akademin för utbildning, kultur och kommunikation
No. 294
MARKET MODELS WITH STOCHASTIC VOLATILITY
Jean-Paul Murara
Akademisk avhandling
som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik vid Akademin för utbildning, kultur och kommunikation kommer att offentligen försvaras fredagen den 4 oktober 2019, 13.15 i Lambda, Mälardalens högskola, Västerås.
Fakultetsopponent: Professor Guglielmo D’Amico, University G. D'Annunzio of Chieti-Pescara
Abstract
Financial Markets is an interesting wide range area of research in Financial Engineering. In this thesis, which consists of an introduction, six papers and appendices, we deal with market models with stochastic volatility in order to understand some financial derivatives, mainly European options. Stochastic volatility models appear as a response to the weakness of the constant volatility models. Paper
A is presented as a survey of different models where the volatility is itself a stochastic process and we
present the techniques of pricing European options. Comparing single factor stochastic volatility models to constant factor volatility models, it seems evident that the stochastic volatility models represent nicely the movement of the asset price and its relations with changes in the risk. However, these models fail to explain the large fluctuations in the volatility levels and slope. We propose also a new model which is a variation of the Chiarella and Ziveyi model and we use the first order asymptotic expansion methods to determine the price of European options. Multiscale stochastic volatility models can capture the smile and skew of volatilities and therefore describe more accurately the movements of the trading prices. In paper B , we present an asymptotic expansion for the option price. We provide experimental and numerical studies on investigating the accuracy of the approximation formulae given by this asymptotic expansion. We present also a procedure for calibrating the parameters produced by our first-order asymptotic approximation formulae. Our approximated option prices are compared to the approximation obtained by Chiarella and Ziveyi. In paper C , we implement and analyze the Regime-Switching GARCH model using real NordPool Electricity spot data. We allow the model parameters to switch between a regular regime and a non-regular regime, which is justified by the so-called structural break behaviour of electricity price series. In splitting the two regimes we consider three criteria, namely the intercountry price difference criterion, the capacity/flow difference criterion and the spikes-in-Finland criterion. We study the correlation relationships among these criteria using the mean-square contingency coefficient and the co-occurrence measure. We also estimate our model parameters and present empirical validity of the model. In paper D , we consider a market model with four correlated factors and two stochastic volatilities which is the same model as the one introduced in paper A and used in paper B . An advanced Monte Carlo method is used to find the no-arbitrage price of the European call option in the considered model. In paper E , we forecast the stochastic volatility for exchange rates using Exponential Weighted Moving Average (EWMA) model and study the effect of the out-of-sample periods and also the effect of the decay factor on the forecasts. In Paper F , considering a two-dimensional Black-Scholes equation, we compare the performances between the Crank-Nicolson scheme and the lognormality condition when pricing the European options. We do this by studying the effects of different parameters.
ISBN 978-91-7485-433-6 ISSN 1651-4238
Abstract
Financial Markets is an interesting wide range area of research in Financial Engineering. In this thesis, which consists of an introduction, six papers and appendices, we deal with market models with stochastic volatility in order to understand some financial derivatives, mainly European options. Stochastic volatility models appear as a response to the weakness of the constant volatility models. Paper
A is presented as a survey of different models where the volatility is itself a stochastic process and we
present the techniques of pricing European options. Comparing single factor stochastic volatility models to constant factor volatility models, it seems evident that the stochastic volatility models represent nicely the movement of the asset price and its relations with changes in the risk. However, these models fail to explain the large fluctuations in the volatility levels and slope. We propose also a new model which is a variation of the Chiarella and Ziveyi model and we use the first order asymptotic expansion methods to determine the price of European options. Multiscale stochastic volatility models can capture the smile and skew of volatilities and therefore describe more accurately the movements of the trading prices. In paper B , we present an asymptotic expansion for the option price. We provide experimental and numerical studies on investigating the accuracy of the approximation formulae given by this asymptotic expansion. We present also a procedure for calibrating the parameters produced by our first-order asymptotic approximation formulae. Our approximated option prices are compared to the approximation obtained by Chiarella and Ziveyi. In paper C , we implement and analyze the Regime-Switching GARCH model using real NordPool Electricity spot data. We allow the model parameters to switch between a regular regime and a non-regular regime, which is justified by the so-called structural break behaviour of electricity price series. In splitting the two regimes we consider three criteria, namely the intercountry price difference criterion, the capacity/flow difference criterion and the spikes-in-Finland criterion. We study the correlation relationships among these criteria using the mean-square contingency coefficient and the co-occurrence measure. We also estimate our model parameters and present empirical validity of the model. In paper D , we consider a market model with four correlated factors and two stochastic volatilities which is the same model as the one introduced in paper A and used in paper B . An advanced Monte Carlo method is used to find the no-arbitrage price of the European call option in the considered model. In paper E , we forecast the stochastic volatility for exchange rates using Exponential Weighted Moving Average (EWMA) model and study the effect of the out-of-sample periods and also the effect of the decay factor on the forecasts. In Paper F , considering a two-dimensional Black-Scholes equation, we compare the performances between the Crank-Nicolson scheme and the lognormality condition when pricing the European options. We do this by studying the effects of different parameters.
ISBN 978-91-7485-433-6 ISSN 1651-4238
Sammanfattning
Finansmarknader ¨ar ett intressant och brett forskningsomr˚ade inom finans-matematik. I denna avhandling, som best˚ar av en kappa och sex artiklar, be-handlas marknadsmodeller med stokastisk med syfte att f¨orst˚a vissa finsiella derivat, fr¨amst europeiska optioner. Modeller med stokastisk volatilitet har introducerats f¨or att grund av svagheter i modeller med konstant volatilitet. Artikel A inneh˚aller en ¨oversikt av olika modeller d¨ar volatiliteten sj¨alv ¨ar en stokastisk process och vi beskriver metoder f¨or att priss¨atta europeiska optioner i detta sammanhang. Vid j¨amf¨orelse av enfaktorsmodeller med stokastik volatilitet och modeller med konstant volatilitet, s˚a ¨ar det up-penbart att de stoskastika modellerna v¨al beskriver prisf¨or¨andringarna hos tillg˚angarna i f¨orh˚allande till f¨or¨andringarna i riskniv˚a. Dessa modeller kan dock inte f¨orklara stora f¨or¨andringar i volatilitetsniv˚a och lutning. Vi in-troducerar ocks˚a en ny modell som ¨ar en variant av en modell utvecklad av Chirella och Ziveyi och vi anv¨ander en asymptotisk utvecklingsmetod av f¨orsta ordningen f¨or att priss¨atta europeiska optioner. Flerskaliga stokastiska volatilitetsmodeller kan beskriva volatilitetsleendet och volatilitetsskevheten och d¨armed beskriva prisf¨or¨andringar mer noggrannt. I Artikel B , beskriver vi en asymptotisk utveckling av optionspriset. Vi utf¨or experimentella och numeriska unders¨okningar av noggrannheten hos de approximationsformler som denna asymptotiska utveckling ger. Vi beskriver ocks˚a en metod f¨or att kalibrera de parametrar som ges av v˚ara formler f¨or asymptotisk utveckling av f¨orsta ordningen. V˚ara approximativa optionspriser j¨amf¨ors med de ap-proximationer som Chiarella och Ziveyi fick. I Artikel C , implementerar och analyseras den regim-v¨axlande GARCH modellen baserad p˚a data fr˚an Nord-Pool Electricity. Vi l˚ater modellens parameterar v¨axla mellan en regulj¨ar- och en icke-regulj¨ar regim, vilket motiveras av den s˚a kallade strukturella bryt-ningsegenskapen hos serier med elektricitetspriser. Vid uppdelningen mellan de tv˚a regimerna s˚a tar vi h¨ansyn till tre kriterier, n¨amligen prisskillnaden mellan l¨ander, skillnader i fl¨ode/kapacitet och det s˚a kallade toppar-i-Finland kriteriet. Vi unders¨oker korrelationsf¨orh˚allandena mellan dessa kriterier med hj¨alp av koefficienten f¨or medelkvadratisk beredskap och m˚attet f¨or samtidig f¨orekomst. Vi uppskattar ocks˚a v˚ar modell och beskriver empiriska argument f¨or modellens korrekthet. I Artikel D , diskuterar vi en marknadsmodell med fyra korellerade faktorer och tv˚a stokastiska volatiliter, detta ¨ar samma mod-ell som introducerades i Artikel A och anv¨andes i Artikel B . En avancerad Monte Carlo metod anv¨ands f¨or att hitta det arbitrage-fria priset f¨or en
eu-Sammanfattning
Finansmarknader ¨ar ett intressant och brett forskningsomr˚ade inom finans-matematik. I denna avhandling, som best˚ar av en kappa och sex artiklar, be-handlas marknadsmodeller med stokastisk med syfte att f¨orst˚a vissa finsiella derivat, fr¨amst europeiska optioner. Modeller med stokastisk volatilitet har introducerats f¨or att grund av svagheter i modeller med konstant volatilitet. Artikel A inneh˚aller en ¨oversikt av olika modeller d¨ar volatiliteten sj¨alv ¨ar en stokastisk process och vi beskriver metoder f¨or att priss¨atta europeiska optioner i detta sammanhang. Vid j¨amf¨orelse av enfaktorsmodeller med stokastik volatilitet och modeller med konstant volatilitet, s˚a ¨ar det up-penbart att de stoskastika modellerna v¨al beskriver prisf¨or¨andringarna hos tillg˚angarna i f¨orh˚allande till f¨or¨andringarna i riskniv˚a. Dessa modeller kan dock inte f¨orklara stora f¨or¨andringar i volatilitetsniv˚a och lutning. Vi in-troducerar ocks˚a en ny modell som ¨ar en variant av en modell utvecklad av Chirella och Ziveyi och vi anv¨ander en asymptotisk utvecklingsmetod av f¨orsta ordningen f¨or att priss¨atta europeiska optioner. Flerskaliga stokastiska volatilitetsmodeller kan beskriva volatilitetsleendet och volatilitetsskevheten och d¨armed beskriva prisf¨or¨andringar mer noggrannt. I Artikel B , beskriver vi en asymptotisk utveckling av optionspriset. Vi utf¨or experimentella och numeriska unders¨okningar av noggrannheten hos de approximationsformler som denna asymptotiska utveckling ger. Vi beskriver ocks˚a en metod f¨or att kalibrera de parametrar som ges av v˚ara formler f¨or asymptotisk utveckling av f¨orsta ordningen. V˚ara approximativa optionspriser j¨amf¨ors med de ap-proximationer som Chiarella och Ziveyi fick. I Artikel C , implementerar och analyseras den regim-v¨axlande GARCH modellen baserad p˚a data fr˚an Nord-Pool Electricity. Vi l˚ater modellens parameterar v¨axla mellan en regulj¨ar- och en icke-regulj¨ar regim, vilket motiveras av den s˚a kallade strukturella bryt-ningsegenskapen hos serier med elektricitetspriser. Vid uppdelningen mellan de tv˚a regimerna s˚a tar vi h¨ansyn till tre kriterier, n¨amligen prisskillnaden mellan l¨ander, skillnader i fl¨ode/kapacitet och det s˚a kallade toppar-i-Finland kriteriet. Vi unders¨oker korrelationsf¨orh˚allandena mellan dessa kriterier med hj¨alp av koefficienten f¨or medelkvadratisk beredskap och m˚attet f¨or samtidig f¨orekomst. Vi uppskattar ocks˚a v˚ar modell och beskriver empiriska argument f¨or modellens korrekthet. I Artikel D , diskuterar vi en marknadsmodell med fyra korellerade faktorer och tv˚a stokastiska volatiliter, detta ¨ar samma mod-ell som introducerades i Artikel A och anv¨andes i Artikel B . En avancerad Monte Carlo metod anv¨ands f¨or att hitta det arbitrage-fria priset f¨or en
eu-Market Models with Stochastic Volatility
ropeisk k¨opoption enligt den unders¨okta modellen. I Artikel E , f¨oruts¨ager vi den stokastiska volatiliteten f¨or v¨axelkursen enligt modellen f¨or exponen-tiellt viktat glidande medelv¨arde och studerar effekterna av f¨oruts¨agelser utanf¨or det uppm¨atta omr˚adet och effekten av avklingningskoefficienten p˚a f¨oruts¨agelserna. I Artikel F , diskuteras en tv˚a-dimensionell Black–Scholes ekvation, vi j¨amf¨or prestandan f¨or resultat med Crank–Nicholson metoden och med long-normalitets villkoret f¨or priss¨attning av europeiska optioner. Vi g¨or detta genom att studera effekten av att ¨andra lika parametrar.
ISBN 978-91-7485-433-6 ISSN 1651-4238
6
Market Models with Stochastic Volatility
ropeisk k¨opoption enligt den unders¨okta modellen. I Artikel E , f¨oruts¨ager vi den stokastiska volatiliteten f¨or v¨axelkursen enligt modellen f¨or exponen-tiellt viktat glidande medelv¨arde och studerar effekterna av f¨oruts¨agelser utanf¨or det uppm¨atta omr˚adet och effekten av avklingningskoefficienten p˚a f¨oruts¨agelserna. I Artikel F , diskuteras en tv˚a-dimensionell Black–Scholes ekvation, vi j¨amf¨or prestandan f¨or resultat med Crank–Nicholson metoden och med long-normalitets villkoret f¨or priss¨attning av europeiska optioner. Vi g¨or detta genom att studera effekten av att ¨andra lika parametrar.
ISBN 978-91-7485-433-6 ISSN 1651-4238
Acknowledgements
Dear ALL, allow me to start by expressing my sincere gratitude to my main supervisor Professor Sergei Silvestrov for his continuous guidance of my stu-dies and especially this thesis, for his motivation, for his remarks and many advices. His guidance helped me in all the time of research and writing of this doctoral thesis. I could not have imagined having a better supervisor for my Ph.D studies. My sincere thanks also goes to my co-supervisors Pro-fessor Anatoliy Malyarenko and Dr. Ying Ni for their insightful comments, remarks and encouragement, but also for the hard question which incented me to widen my research from various perspectives. Without their precious support it would not be possible to conduct this research.
I am thankful to the staff and all my colleagues at the Division of Applied Mathematics at M¨alardalen University for being supportive and helpful. Of course I wish to thank the International Science Programme (ISP - Uppsala) for providing the financial support, which enables me to conduct this research work in the first place. In particular, I am grateful to Prof. Leif Abrahams-son, Mrs Pravina Gajjar, Mrs Kristina Hassel Konpan, Mr. Michael Gahiri-ma and Dr. D´esir´e Karangwa.
Last but not the least, I would like to thank my wife Marie-Claire Musabwa-mana and our three boys Evan Bonheur Murara Mpano, Luc Kyllian Murara Sano, Franz Markus Murara Nganzo for allowing me to leave their incom-parable company during the time of this research. This goes also to my parents Vincent Murara and C´ecile Mukantabana who always wanted me to learn something new since I was a kid and to my brothers, sisters, cousins and family-in-law for supporting me spiritually throughout writing this the-sis and my life in general.
I dedicate my efforts in this work to Dr. Alex Behakanira Tumwesigye, Dr. Betuel Jesus Canhanga, Dr. Milica Ranˇci´c, Dr. Karl Lundeng˚ard, Henrik Bladh, Caroline Kyhlb¨ack, Andr´e Brun, Arne Vauhkola’s family and all my friends that I have met in Sweden during my stay in V¨aster˚as.
V¨aster˚as, October 04, 2019 Jean-Paul Murara
Acknowledgements
Dear ALL, allow me to start by expressing my sincere gratitude to my main supervisor Professor Sergei Silvestrov for his continuous guidance of my stu-dies and especially this thesis, for his motivation, for his remarks and many advices. His guidance helped me in all the time of research and writing of this doctoral thesis. I could not have imagined having a better supervisor for my Ph.D studies. My sincere thanks also goes to my co-supervisors Pro-fessor Anatoliy Malyarenko and Dr. Ying Ni for their insightful comments, remarks and encouragement, but also for the hard question which incented me to widen my research from various perspectives. Without their precious support it would not be possible to conduct this research.
I am thankful to the staff and all my colleagues at the Division of Applied Mathematics at M¨alardalen University for being supportive and helpful. Of course I wish to thank the International Science Programme (ISP - Uppsala) for providing the financial support, which enables me to conduct this research work in the first place. In particular, I am grateful to Prof. Leif Abrahams-son, Mrs Pravina Gajjar, Mrs Kristina Hassel Konpan, Mr. Michael Gahiri-ma and Dr. D´esir´e Karangwa.
Last but not the least, I would like to thank my wife Marie-Claire Musabwa-mana and our three boys Evan Bonheur Murara Mpano, Luc Kyllian Murara Sano, Franz Markus Murara Nganzo for allowing me to leave their incom-parable company during the time of this research. This goes also to my parents Vincent Murara and C´ecile Mukantabana who always wanted me to learn something new since I was a kid and to my brothers, sisters, cousins and family-in-law for supporting me spiritually throughout writing this the-sis and my life in general.
I dedicate my efforts in this work to Dr. Alex Behakanira Tumwesigye, Dr. Betuel Jesus Canhanga, Dr. Milica Ranˇci´c, Dr. Karl Lundeng˚ard, Henrik Bladh, Caroline Kyhlb¨ack, Andr´e Brun, Arne Vauhkola’s family and all my friends that I have met in Sweden during my stay in V¨aster˚as.
V¨aster˚as, October 04, 2019 Jean-Paul Murara
Market Models with Stochastic Volatility
This work was funded by
the International Science
Program
(ISP - Uppsala University)
in cooperation with
Eastern Africa Universities
Mathematics Programme
(EAUMP).
8
Market Models with Stochastic Volatility
This work was funded by
the International Science
Program
(ISP - Uppsala University)
in cooperation with
Eastern Africa Universities
Mathematics Programme
(EAUMP).
List of Papers
The present thesis is based on the following papers:
Paper A. Betuel Canhanga, Anatoliy Malyarenko, Jean-Paul Murara, and Sergei Silvestrov. ”Pricing European options under stochastic volatilities models”. In: Engineering mathematics. I. Electromagnetics, Fluid Mechanics, Mate-rial Physics and Financial Engineering. Ed. by Sergei Silvestrov and Milica
Ranˇci´c. Vol. 178. Springer Proc. Math. Stat. Springer, Cham, 2016, pp.
315 – 338.
Paper B. Betuel Canhanga, Anatoliy Malyarenko, Jean-Paul Murara, Ying Ni, and Sergei Silvestrov. ”Numerical studies on asymptotics of European Option under Multiscale Stochastic Volatility”. Methodol. Comput. Appl. Probab. 19.4 (2017), pp. 1075 – 1087.
Paper C. Jean-Paul Murara, Anatoliy Malyarenko, and Sergei Silvestrov. ”Model-ling Electricity Price Series using Regime-Switching GARCH Model”. In: Stitistical, Stochastic and Data Analysis Methods and Applications. Ed. by Alex Karagrigorion, Teresa Oliveira, and Christos H. Skiadas. ISAST: International Society for the Advancement of Science and Technology, 2015, pp. 457–469.
Paper D. Anatoliy Malyrenko, Betuel Canhanga, Jean-Paul Murara, Ying Ni, and Sergei Silvestrov. ”Advanced Monte Carlo pricing of European options in a market model with two stochastic volatilities”. Accepted for publication in:
Algebraic structures and Applications. SPAS2017, V¨aster˚as and Stockholm,
Sweden, October 4–6, 2017, Sergei Silvestrov, Anatoliy Malyarenko, Milica
Ranˇci´c (Eds), Springer International Publishing, 2019.
Paper E. Jean-Paul Murara, Anatoliy Malyarenko, Milica Ranˇci´c, Sergei Silvestrov.
”Forecasting Stochastic Volatility for Exchange Rates using EWMA”. Sub-mitted for publication.
List of Papers
The present thesis is based on the following papers:
Paper A. Betuel Canhanga, Anatoliy Malyarenko, Jean-Paul Murara, and Sergei Silvestrov. ”Pricing European options under stochastic volatilities models”. In: Engineering mathematics. I. Electromagnetics, Fluid Mechanics, Mate-rial Physics and Financial Engineering. Ed. by Sergei Silvestrov and Milica
Ranˇci´c. Vol. 178. Springer Proc. Math. Stat. Springer, Cham, 2016, pp.
315 – 338.
Paper B. Betuel Canhanga, Anatoliy Malyarenko, Jean-Paul Murara, Ying Ni, and Sergei Silvestrov. ”Numerical studies on asymptotics of European Option under Multiscale Stochastic Volatility”. Methodol. Comput. Appl. Probab. 19.4 (2017), pp. 1075 – 1087.
Paper C. Jean-Paul Murara, Anatoliy Malyarenko, and Sergei Silvestrov. ”Model-ling Electricity Price Series using Regime-Switching GARCH Model”. In: Stitistical, Stochastic and Data Analysis Methods and Applications. Ed. by Alex Karagrigorion, Teresa Oliveira, and Christos H. Skiadas. ISAST: International Society for the Advancement of Science and Technology, 2015, pp. 457–469.
Paper D. Anatoliy Malyrenko, Betuel Canhanga, Jean-Paul Murara, Ying Ni, and Sergei Silvestrov. ”Advanced Monte Carlo pricing of European options in a market model with two stochastic volatilities”. Accepted for publication in:
Algebraic structures and Applications. SPAS2017, V¨aster˚as and Stockholm,
Sweden, October 4–6, 2017, Sergei Silvestrov, Anatoliy Malyarenko, Milica
Ranˇci´c (Eds), Springer International Publishing, 2019.
Paper E. Jean-Paul Murara, Anatoliy Malyarenko, Milica Ranˇci´c, Sergei Silvestrov.
”Forecasting Stochastic Volatility for Exchange Rates using EWMA”. Sub-mitted for publication.
Market Models with Stochastic Volatility
Paper F. Jean-Paul Murara, Anatoliy Malyarenko, Betuel Canhanga, Sergei Sil-vestrov. ”Pricing European Options under two-dimensional Black-Scholes Equation by two different approaches”. Submitted for publication.
Parts of this thesis have been presented at the following international conferences: 1. The second and third papers in this thesis have been presented during the
16th Conference of the Applied Stochastic Models and Data Analysis
(AS-MDA 2015) International Society and Demographics 2015 Workshop, De-partment of Statistics and Insurance Science, University of Piraeus – Greece
( 30th June – 4th July 2015).
2. The fifth and sixth papers in this thesis have been presented during the 18th
Conference of the Applied Stochastic Models and Data Analysis (ASMDA 2019) International Society and Demographics 2019 Workshop, Florence –
Italy ( 11th – 14th June 2019).
10
Market Models with Stochastic Volatility
Paper F. Jean-Paul Murara, Anatoliy Malyarenko, Betuel Canhanga, Sergei Sil-vestrov. ”Pricing European Options under two-dimensional Black-Scholes Equation by two different approaches”. Submitted for publication.
Parts of this thesis have been presented at the following international conferences: 1. The second and third papers in this thesis have been presented during the
16th Conference of the Applied Stochastic Models and Data Analysis
(AS-MDA 2015) International Society and Demographics 2015 Workshop, De-partment of Statistics and Insurance Science, University of Piraeus – Greece
( 30thJune – 4th July 2015).
2. The fifth and sixth papers in this thesis have been presented during the 18th
Conference of the Applied Stochastic Models and Data Analysis (ASMDA 2019) International Society and Demographics 2019 Workshop, Florence –
Italy ( 11th – 14th June 2019).
Contents
Sammanfattning 5 Acknowledgement 7 List of Papers 9 1 Introduction 21 1.1 Preliminaries . . . 21 1.1.1 Financial market . . . 21 1.1.2 Option Pricing . . . 22 1.1.3 Stochastic Processes . . . 25 1.1.4 Risk-neutral measure . . . 271.2 Black-Scholes model for asset evolution and option pricing . . . 30
1.2.1 Black-Scholes Option Pricing Model . . . 30
1.3 Stochastic volatility models . . . 34
1.3.1 Volatility quid ? . . . 34
1.3.1.1 Historical Volatility . . . 34
1.3.1.2 Instantaneous volatility . . . 35
1.3.1.3 Implied volatility . . . 35
1.3.2 Stochastic Volatility Models . . . 35
Contents
Sammanfattning 5 Acknowledgement 7 List of Papers 9 1 Introduction 21 1.1 Preliminaries . . . 21 1.1.1 Financial market . . . 21 1.1.2 Option Pricing . . . 22 1.1.3 Stochastic Processes . . . 25 1.1.4 Risk-neutral measure . . . 271.2 Black-Scholes model for asset evolution and option pricing . . . 30
1.2.1 Black-Scholes Option Pricing Model . . . 30
1.3 Stochastic volatility models . . . 34
1.3.1 Volatility quid ? . . . 34
1.3.1.1 Historical Volatility . . . 34
1.3.1.2 Instantaneous volatility . . . 35
1.3.1.3 Implied volatility . . . 35
Market Models with Stochastic Volatility
1.3.3 Pricing and Hedging in Stochastic Volatility
Mo-dels . . . 36
1.3.4 Implied Volatility Smiles . . . 37
1.4 Multiscale stochastic volatility models . . . 39
1.4.1 Multiscale Stochastic Volatility Models . . . 39
1.4.2 Asymptotic expansion . . . 41
1.4.3 Regular-Singular Perturbations . . . 41
1.5 Markov Regime-Switching model for asset price . . . 43
1.6 Numerical Methods for Option Pricing . . . 46
1.6.1 Finite Difference Methods . . . 46
1.6.2 Monte Carlo Method . . . 48
2 Summary of the papers 53 2.1 Novelty in this thesis . . . 53
2.2 Paper A . . . 54 2.3 Paper B . . . 55 2.4 Paper C . . . 56 2.5 Paper D . . . 56 2.6 Paper E . . . 57 2.7 Paper F . . . 57 2.8 Appendices . . . 57 References 57 3 Pricing European Options Under Stochastic Volatilities Models 65 3.1 Introduction . . . 66
3.2 The constant elasticity of variance (CEV) model . . . 67
3.2.1 European option pricing formulae under the CEV model . . 68
12 Market Models with Stochastic Volatility 1.3.3 Pricing and Hedging in Stochastic Volatility Mo-dels . . . 36
1.3.4 Implied Volatility Smiles . . . 37
1.4 Multiscale stochastic volatility models . . . 39
1.4.1 Multiscale Stochastic Volatility Models . . . 39
1.4.2 Asymptotic expansion . . . 41
1.4.3 Regular-Singular Perturbations . . . 41
1.5 Markov Regime-Switching model for asset price . . . 43
1.6 Numerical Methods for Option Pricing . . . 46
1.6.1 Finite Difference Methods . . . 46
1.6.2 Monte Carlo Method . . . 48
2 Summary of the papers 53 2.1 Novelty in this thesis . . . 53
2.2 Paper A . . . 54 2.3 Paper B . . . 55 2.4 Paper C . . . 56 2.5 Paper D . . . 56 2.6 Paper E . . . 57 2.7 Paper F . . . 57 2.8 Appendices . . . 57 References 57 3 Pricing European Options Under Stochastic Volatilities Models 65 3.1 Introduction . . . 66
3.2 The constant elasticity of variance (CEV) model . . . 67
3.2.1 European option pricing formulae under the CEV model . . 68
3.2.2 Implied volatility smile in the CEV model . . . 69
3.3 The Stochastic αβρ (SABR) model . . . 70
3.3.1 European option pricing formulae under the SABR model . 71 3.3.2 Implied volatility smile in the SABR model . . . 71
3.4 The Detemple–Tian model (DTM) . . . 71
3.5 Grzelak–Oosterlee–Van Veeren (GOVV) Model . . . 72
3.5.1 Pricing European options for the GOVV type models . . . 74
3.5.1.1 Pricing Method . . . 74
3.5.1.2 Sch¨obel–Zhu–Hull–White (SZHW) Model . . . 77
3.6 Jourdain–Sbai Model (JSM) . . . 79
3.6.1 The weak scheme of second order . . . 80
3.7 Ilhan–Sircar model (ISM) . . . 81
3.8 Two stochastic volatilities model . . . 84
4 Numerical studies on Asymptotics of European Option under Multiscale Stochastic Volatility 91 4.1 Introduction . . . 92
4.2 Accuracy of First Order Approximation . . . 95
4.2.1 Procedure on Studying the Accuracy . . . 95
4.3 Adjustment to volatilities and calibration . . . 98
4.3.1 The calibration procedure . . . 99
4.3.2 Calibration of the first order approximation . . . 100
4.3.3 Numerical Analysis . . . 103
4.4 Conclusions and future work . . . 105
3.2.2 Implied volatility smile in the CEV model . . . 69
3.3 The Stochastic αβρ (SABR) model . . . 70
3.3.1 European option pricing formulae under the SABR model . 71 3.3.2 Implied volatility smile in the SABR model . . . 71
3.4 The Detemple–Tian model (DTM) . . . 71
3.5 Grzelak–Oosterlee–Van Veeren (GOVV) Model . . . 72
3.5.1 Pricing European options for the GOVV type models . . . 74
3.5.1.1 Pricing Method . . . 74
3.5.1.2 Sch¨obel–Zhu–Hull–White (SZHW) Model . . . 77
3.6 Jourdain–Sbai Model (JSM) . . . 79
3.6.1 The weak scheme of second order . . . 80
3.7 Ilhan–Sircar model (ISM) . . . 81
3.8 Two stochastic volatilities model . . . 84
4 Numerical studies on Asymptotics of European Option under Multiscale Stochastic Volatility 91 4.1 Introduction . . . 92
4.2 Accuracy of First Order Approximation . . . 95
4.2.1 Procedure on Studying the Accuracy . . . 95
4.3 Adjustment to volatilities and calibration . . . 98
4.3.1 The calibration procedure . . . 99
4.3.2 Calibration of the first order approximation . . . 100
4.3.3 Numerical Analysis . . . 103
Market Models with Stochastic Volatility
5 Modelling electricity price series using Regime-Switching GARCH
model 111
5.1 Introduction . . . 111
5.2 Nord Pool Spot data overview . . . 113
5.3 The Regime-Switching GARCH model . . . 113
5.4 Regime Splitting . . . 114
5.4.1 Non-regular regime by intercountry price diffe-rence criterion . . . 114
5.4.2 Non-regular regime by capacity/flow difference criterion . . 116
5.4.3 Non-regular regime by spikes-in-Finland criterion . . . 118
5.4.4 Correlation and Co-occurence relationship between the three criteria . . . 118
5.5 Estimation of Parameters and validity of the model . . . 122
5.5.1 Model Parameters Estimation . . . 122
5.5.2 Validity of our RS-GARCH Model . . . 124
5.6 Conclusions . . . 125
6 Advanced Monte Carlo pricing of European options in a market model with two stochastic volatilities 131 6.1 Introduction . . . 131
6.2 Stochastic cubature formulae . . . 132
6.3 The simulation algorithm . . . 137
6.4 Numerical results . . . 139
6.4.1 Rate of Convergence . . . 141
6.4.2 The effect of mean-reversion rates . . . 143
6.4.3 The effect of moneyness . . . 144
6.4.4 The effect of correlation coefficient . . . 145
6.5 Conclusions and further remarks . . . 146
6.6 Cubature on a tensor algebra . . . 146
14 Market Models with Stochastic Volatility 5 Modelling electricity price series using Regime-Switching GARCH model 111 5.1 Introduction . . . 111
5.2 Nord Pool Spot data overview . . . 113
5.3 The Regime-Switching GARCH model . . . 113
5.4 Regime Splitting . . . 114
5.4.1 Non-regular regime by intercountry price diffe-rence criterion . . . 114
5.4.2 Non-regular regime by capacity/flow difference criterion . . 116
5.4.3 Non-regular regime by spikes-in-Finland criterion . . . 118
5.4.4 Correlation and Co-occurence relationship between the three criteria . . . 118
5.5 Estimation of Parameters and validity of the model . . . 122
5.5.1 Model Parameters Estimation . . . 122
5.5.2 Validity of our RS-GARCH Model . . . 124
5.6 Conclusions . . . 125
6 Advanced Monte Carlo pricing of European options in a market model with two stochastic volatilities 131 6.1 Introduction . . . 131
6.2 Stochastic cubature formulae . . . 132
6.3 The simulation algorithm . . . 137
6.4 Numerical results . . . 139
6.4.1 Rate of Convergence . . . 141
6.4.2 The effect of mean-reversion rates . . . 143
6.4.3 The effect of moneyness . . . 144
6.4.4 The effect of correlation coefficient . . . 145
6.5 Conclusions and further remarks . . . 146
6.6 Cubature on a tensor algebra . . . 146
7 Forecasting Stochastic Volatility for Exchange Rates using EWMA153
7.1 Introduction . . . 153
7.2 Data . . . 154
7.3 Empirical Model . . . 155
7.4 Exchange Rate Volatility Forecasting . . . 157
7.5 Conclusion . . . 162
8 Pricing European Options under two-dimensional Black-Scholes Equation by two different approaches 167 8.1 Introduction . . . 167
8.2 The Crank-Nicolson f.d. method . . . 169
8.3 Lognormal approximation . . . 171
8.4 Comparisons of European basket option prices . . . 172
8.4.1 Effect of the weights: θ1 and θ2. . . 173
8.4.2 Effect of the strikes: K1 and K2. . . 174
8.4.3 Effect of the correlation . . . 175
8.5 Conclusions . . . 177
9 Appendices 181 9.1 Appendix 1: Statistics and Probability . . . 181
9.1.1 Descriptive statistics . . . 181
9.1.2 Dictionary of probabilists’ dialect . . . 182
9.2 Appendix 2: Feynman-Kac Theorem . . . 182
9.3 Appendix 3: Girsanov Theorem . . . 183
9.4 Appendix 4: The fundamental theorem of perturbation theory . . 184
9.5 Appendix 5: Itˆo versus Stratonovich SDEs . . . 185
Index 186 7 Forecasting Stochastic Volatility for Exchange Rates using EWMA153 7.1 Introduction . . . 153
7.2 Data . . . 154
7.3 Empirical Model . . . 155
7.4 Exchange Rate Volatility Forecasting . . . 157
7.5 Conclusion . . . 162
8 Pricing European Options under two-dimensional Black-Scholes Equation by two different approaches 167 8.1 Introduction . . . 167
8.2 The Crank-Nicolson f.d. method . . . 169
8.3 Lognormal approximation . . . 171
8.4 Comparisons of European basket option prices . . . 172
8.4.1 Effect of the weights: θ1 and θ2. . . 173
8.4.2 Effect of the strikes: K1 and K2. . . 174
8.4.3 Effect of the correlation . . . 175
8.5 Conclusions . . . 177
9 Appendices 181 9.1 Appendix 1: Statistics and Probability . . . 181
9.1.1 Descriptive statistics . . . 181
9.1.2 Dictionary of probabilists’ dialect . . . 182
9.2 Appendix 2: Feynman-Kac Theorem . . . 182
9.3 Appendix 3: Girsanov Theorem . . . 183
9.4 Appendix 4: The fundamental theorem of perturbation theory . . 184
9.5 Appendix 5: Itˆo versus Stratonovich SDEs . . . 185
List of Figures
1 Example of a Classification of Financial Markets . . . 22
2 Option Pricing Payoff Functions . . . 23
3 Example of a path of a Stochastic Process . . . 26
4 Examples of 10 paths Standard Brownian Motions . . . 27
5 Stochastic Volatility Example . . . 37
6 Example of an Implied Volatility Smile . . . 38
7 Returns split into 2 Regimes following a GARCH(1,1) . . . 44
8 Relationship between continuous and discrete problems . . . 46
9 Computational molecules for FDM . . . 47
10 The steps of Monte Carlo method . . . 49
1 Optional caption for list of figures . . . 104
2 Option prices given by Chiarella–Ziveyi and by our approximation 105 1 Finnish and Swedish prices combined. . . 115
2 Price scatter plots for regimes split by price difference. . . 115
3 Regimes split by intercountry price difference. . . 116
4 Regimes split by Finnish capacity and flow. . . 117
5 Price scatter plots for regimes split by capacity-flow difference. . . 117
6 Regimes split by capacity-flow difference. . . 118
7 Regimes split by spikes-in-finland criterion. . . 119
8 Price scatter plots for regimes split by spikes-in-finland criterion. . 119
9 Regimes split by spikes-in-finland criterion. . . 120
10 Time instances when the non-regular regime is ON with respect to different criteria. . . 121
11 Finnish returns which correspond to regular and non-regular regimes by capacity/flow difference criterion. . . 123
12 Finnish returns fitted to RS-GARCH(1,1). . . 125
List of Figures
1 Example of a Classification of Financial Markets . . . 222 Option Pricing Payoff Functions . . . 23
3 Example of a path of a Stochastic Process . . . 26
4 Examples of 10 paths Standard Brownian Motions . . . 27
5 Stochastic Volatility Example . . . 37
6 Example of an Implied Volatility Smile . . . 38
7 Returns split into 2 Regimes following a GARCH(1,1) . . . 44
8 Relationship between continuous and discrete problems . . . 46
9 Computational molecules for FDM . . . 47
10 The steps of Monte Carlo method . . . 49
1 Optional caption for list of figures . . . 104
2 Option prices given by Chiarella–Ziveyi and by our approximation 105 1 Finnish and Swedish prices combined. . . 115
2 Price scatter plots for regimes split by price difference. . . 115
3 Regimes split by intercountry price difference. . . 116
4 Regimes split by Finnish capacity and flow. . . 117
5 Price scatter plots for regimes split by capacity-flow difference. . . 117
6 Regimes split by capacity-flow difference. . . 118
7 Regimes split by spikes-in-finland criterion. . . 119
8 Price scatter plots for regimes split by spikes-in-finland criterion. . 119
9 Regimes split by spikes-in-finland criterion. . . 120
10 Time instances when the non-regular regime is ON with respect to different criteria. . . 121
11 Finnish returns which correspond to regular and non-regular regimes by capacity/flow difference criterion. . . 123
Market Models with Stochastic Volatility
1 20 simulated stock price pathes for multi-scale stochastic volatility
model : (left) original pathes; (right) “advanced MC” scheme . . . 139
2 Relative errors under Advanced MC and Euler–Maruyama scheme for different strikes . . . 142
3 Increasing N improves the accuracy under large mean-reversion rate (ε = 0.0001) . . . 144
1 Exchange rates data. Note that SEK/RWF (×10) and KES/RWF (×100). . . 156
2 Logarithmic returns. . . 158
3 Rolling volatility. . . 159
4 One-week volatility forecasts λ = 0.25 . . . 160
5 One-month volatility forecasts λ = 0.25 . . . 161
6 One-quater volatility forecasts λ = 0.25 . . . 161
1 Effects of the weights θ1 and θ2 for both approaches. . . 174
2 Effects of the strike price K for both approaches. . . 175
3 Effects of the correlation for both approaches. . . 176
18 Market Models with Stochastic Volatility 1 20 simulated stock price pathes for multi-scale stochastic volatility model : (left) original pathes; (right) “advanced MC” scheme . . . 139
2 Relative errors under Advanced MC and Euler–Maruyama scheme for different strikes . . . 142
3 Increasing N improves the accuracy under large mean-reversion rate (ε = 0.0001) . . . 144
1 Exchange rates data. Note that SEK/RWF (×10) and KES/RWF (×100). . . 156
2 Logarithmic returns. . . 158
3 Rolling volatility. . . 159
4 One-week volatility forecasts λ = 0.25 . . . 160
5 One-month volatility forecasts λ = 0.25 . . . 161
6 One-quater volatility forecasts λ = 0.25 . . . 161
1 Effects of the weights θ1 and θ2 for both approaches. . . 174
2 Effects of the strike price K for both approaches. . . 175
3 Effects of the correlation for both approaches. . . 176
List of Tables
1.1 Models use in Volatility Forecasting . . . 51
4.1 A part of the used data on ABB call options prices, S0= 148.3 . . 99
4.2 Illustration of the deviation obtained for different values of ε and δ 106 5.1 Mean-square contingency coefficients (Φ) for occurrences of ON states and Co-occurrences (Ψ) of regimes with respect to the three criteria. . . 121
5.2 Model Parameters Estimations. . . 123
5.3 Transition probabilities . . . 123
5.4 Estimations of simulated parameters. . . 124
5.5 Estimations of simulated transition probabilities. . . 124
6.1 Model parameters (shortened as para) used in numerical experiments140 6.2 Relative errors under Advanced MC and Euler–Maruyama scheme (ε = 0.01) . . . 141
6.3 Relative errors under Advanced MC and Euler–Maruyama scheme (ε = 0.001) . . . 142
6.4 Relative errors for ATM call options under various ε . . . 143
6.5 Relative errors for ATM call options under various pairs of ε and δ. 144 6.6 Relative errors under various strike price K . . . 145
7.1 Descriptive statistics of raw data. . . 155
7.2 Descriptive statistics of logarithmic returns. . . 157
7.3 Week-errors (RMSE and MAPE) for different decay factors λi. . . 158
7.4 Month-errors (RMSE and MAPE) for different decay factors λi. . . 159
7.5 Term-errors (RMSE and MAPE) for different decay factors λi. . . 160
8.1 Effects of the weights. . . 173
8.2 Effects of the strikes. . . 174
8.3 Effects of the correlation. . . 176
List of Tables
1.1 Models use in Volatility Forecasting . . . 514.1 A part of the used data on ABB call options prices, S0= 148.3 . . 99
4.2 Illustration of the deviation obtained for different values of ε and δ 106 5.1 Mean-square contingency coefficients (Φ) for occurrences of ON states and Co-occurrences (Ψ) of regimes with respect to the three criteria. . . 121
5.2 Model Parameters Estimations. . . 123
5.3 Transition probabilities . . . 123
5.4 Estimations of simulated parameters. . . 124
5.5 Estimations of simulated transition probabilities. . . 124
6.1 Model parameters (shortened as para) used in numerical experiments140 6.2 Relative errors under Advanced MC and Euler–Maruyama scheme (ε = 0.01) . . . 141
6.3 Relative errors under Advanced MC and Euler–Maruyama scheme (ε = 0.001) . . . 142
6.4 Relative errors for ATM call options under various ε . . . 143
6.5 Relative errors for ATM call options under various pairs of ε and δ. 144 6.6 Relative errors under various strike price K . . . 145
7.1 Descriptive statistics of raw data. . . 155
7.2 Descriptive statistics of logarithmic returns. . . 157
7.3 Week-errors (RMSE and MAPE) for different decay factors λi. . . 158
7.4 Month-errors (RMSE and MAPE) for different decay factors λi. . . 159
7.5 Term-errors (RMSE and MAPE) for different decay factors λi. . . 160
8.1 Effects of the weights. . . 173
8.2 Effects of the strikes. . . 174
Chapter 1
Introduction
1.1
Preliminaries
1.1.1
Financial market
A financial market is any market where people are involved in sales and purchases of financial securities and derivatives. In Fig. 1, we have an example of a classifi-cation of financial markets. We can add insurance markets and foreign exchange markets aside money markets and capital markets. Our thesis is mainly related to derivatives found in capital markets.
Definition 1. Derivative securities
A financial contract is a derivative security, or a contingent claim, if its value is determined exactly by the market price of the underlying cash instrument (e.g. bonds, commodities, currencies, interest rates, market indexes and stocks) at ex-piration time date T (maturity)[38].
We have five main groups of underlying assets namely: stocks, currencies, inte-rest rates, indexes (S&P -500, FT-SE500, ...) and commodities (gold, oil, electri-city, grain, . . .). We can group derivative securities into three main groups:
• Futures and forwards, • Options and
• Swaps.
Futures and options are actively traded in exchange markets and swaps and many exotic options are traded outside of exchanges, called the over-the-counter (OTC) markets [44]. In this thesis, we will focus mainly on option pricing problems.
Chapter 1
Introduction
1.1
Preliminaries
1.1.1
Financial market
A financial market is any market where people are involved in sales and purchases of financial securities and derivatives. In Fig. 1, we have an example of a classifi-cation of financial markets. We can add insurance markets and foreign exchange markets aside money markets and capital markets. Our thesis is mainly related to derivatives found in capital markets.
Definition 1. Derivative securities
A financial contract is a derivative security, or a contingent claim, if its value is determined exactly by the market price of the underlying cash instrument (e.g. bonds, commodities, currencies, interest rates, market indexes and stocks) at ex-piration time date T (maturity)[38].
We have five main groups of underlying assets namely: stocks, currencies, inte-rest rates, indexes (S&P -500, FT-SE500, ...) and commodities (gold, oil, electri-city, grain, . . .). We can group derivative securities into three main groups:
• Futures and forwards, • Options and
• Swaps.
Futures and options are actively traded in exchange markets and swaps and many exotic options are traded outside of exchanges, called the over-the-counter (OTC) markets [44]. In this thesis, we will focus mainly on option pricing problems.
Market Models with Stochastic Volatility
Figure 1: Example of a Classification of Financial Markets
For the next, we will denote S(t) the price of the relevant underlying asset at time t. Thus, the price of a derivative asset at the time of expiration is S(T ) and after that date, the security has no any influence.
Naturally, prices of one specified asset change depending on different reasons, but essentially, they change due to shifts related to an economic theory known as the law of supply and demand [29]. The problem arises when we want to comprehend what makes people buy more or sell more what they own.
Financial engineers are interested in the way these changes occur. They try to design different mathematical models which can help to determine the optimal selling or buying price for a given financial derivative. They also study different models which can forecast the price’s behaviour minimizing risk and maximizing profits/returns.
1.1.2
Option Pricing
In order to determine whether the prices will go down or up in the future, many tools have been introduced. In our papers A , B , D and F , we are interested in option pricing models [26]. Options have two types, namely call option and put option.
22
Market Models with Stochastic Volatility
Figure 1: Example of a Classification of Financial Markets
For the next, we will denote S(t) the price of the relevant underlying asset at time t. Thus, the price of a derivative asset at the time of expiration is S(T ) and after that date, the security has no any influence.
Naturally, prices of one specified asset change depending on different reasons, but essentially, they change due to shifts related to an economic theory known as the law of supply and demand [29]. The problem arises when we want to comprehend what makes people buy more or sell more what they own.
Financial engineers are interested in the way these changes occur. They try to design different mathematical models which can help to determine the optimal selling or buying price for a given financial derivative. They also study different models which can forecast the price’s behaviour minimizing risk and maximizing profits/returns.
1.1.2
Option Pricing
In order to determine whether the prices will go down or up in the future, many tools have been introduced. In our papers A , B , D and F , we are interested in option pricing models [26]. Options have two types, namely call option and put option.
Definition 2. A Call/Put option is any financial contract between two parties where its holder has the right but not the obligation to either buy/sell the un-derlying asset at a given price known as strike/exercise price and also within a mentioned period or date which depends on the nature of the option.
• A call option gives the right to buy at an agreed price and here the buyer is interested if the price goes up in the future.
• A put option gives the right to sell at an agreed price and here the buyer wishes that the price goes down in the future.
Payoff functions
We will denote the strike price by K and the maturity date by T . In Fig. 2, we
Figure 2: Option Pricing Payoff Functions
0 1 2 3 4 0 0.5 1 1.5 2
Stock Price (Strike Price: K=2)
Payoff: h(t,s)
Payoff from a Purchased Call Option
0 1 2 3 4 −2 −1.5 −1 −0.5 0
Stock Price (Strike Price: K=2)
Payoff: h(t,s)
Payoff from a Written Call Option
0 1 2 3 4 0 0.5 1 1.5 2
Stock Price (Strike Price: K=2)
Payoff: h(t,s)
Payoff from a Purchased Put Option
0 1 2 3 4 4 4.5 5 5.5 6
Stock Price (Strike Price: K=2)
Payoff: h(t,s)
Payoff from a Written Put Option
present examples of payoff functions with K = 2 for purchased or written calls and puts. The terms purchased and written are respectively referred to buying and selling an option. We remark that these European-style derivatives are path-independent because the payoff functions h(S(T )) are only functions of the value of the stock price at maturity time T .
Definition 2. A Call/Put option is any financial contract between two parties where its holder has the right but not the obligation to either buy/sell the un-derlying asset at a given price known as strike/exercise price and also within a mentioned period or date which depends on the nature of the option.
• A call option gives the right to buy at an agreed price and here the buyer is interested if the price goes up in the future.
• A put option gives the right to sell at an agreed price and here the buyer wishes that the price goes down in the future.
Payoff functions
We will denote the strike price by K and the maturity date by T . In Fig. 2, we
Figure 2: Option Pricing Payoff Functions
0 1 2 3 4 0 0.5 1 1.5 2
Stock Price (Strike Price: K=2)
Payoff: h(t,s)
Payoff from a Purchased Call Option
0 1 2 3 4 −2 −1.5 −1 −0.5 0
Stock Price (Strike Price: K=2)
Payoff: h(t,s)
Payoff from a Written Call Option
0 1 2 3 4 0 0.5 1 1.5 2
Stock Price (Strike Price: K=2)
Payoff: h(t,s)
Payoff from a Purchased Put Option
0 1 2 3 4 4 4.5 5 5.5 6
Stock Price (Strike Price: K=2)
Payoff: h(t,s)
Payoff from a Written Put Option
present examples of payoff functions with K = 2 for purchased or written calls and puts. The terms purchased and written are respectively referred to buying and selling an option. We remark that these European-style derivatives are path-independent because the payoff functions h(S(T )) are only functions of the value of the stock price at maturity time T .
Market Models with Stochastic Volatility
Definition 3. European Call and Put Options
• For an European Call Option, considering S(T ) as the price of the underly-ing asset at expiration/maturity, the value of this contract is given by the following payoff function:
h(S(T )) = (S(T ) − K)+=
(
S(T ) − K if S(T ) > K,
0 if S(T ) ≤ K. (1.1)
The holder will exercise this option once the first case happens and will make a profit of S(T ) − K by buying the stock for K and selling it at S(T ). • For an European put option, the payoff function is as follows:
h(S(T )) = (K − S(T ))+=
(
K − S(T ) if S(T ) < K,
0 if S(T ) ≥ K. (1.2)
The holder can exercise this put option only for the first case if he is intere-sted in the profit of K − S(T ).
At time t ≤ T , this contract has a value known as the premium or the price of the option, which will vary with t and the observed stock price S(t). This option price at time t and for a stock price S(t) is denoted by C(t, S(t)) or P (t, S(t)). The problem of derivative pricing is determining this pricing function.
Definition 4. Filtration
Let us denote the information about security prices available in the market at
time t by Ft. A filtration on a probability space (Ω, F , P) is an increasing family
(Ft)t≥0 of sub-σ-algebras of F . In other words, Ft includes any information as far
as the time-t security prices is known based on the information and
Fs ⊆ Ft⊆ F if s ≤ t. (1.3)
The sequence of information {Ft; t ∈ [0, T ]} (or {Ft} for short) satisfying (1.3) is
called a filtration [44].
Other styles of options exist such as American or exotic [8]. An American option is a contract in which the holder decides whether to exercise the option or not at any time of his choice before the option’s expiration date T . The time τ at which the option is exercised is called the exercise time. Because the market cannot be anticipated, the holder of the option has to decide to exercise or not
at time t ≤ T with information up to time t contained in the σ-algebra Ft. In
other words, τ which is called a stopping time with respect to the filtration Ft is
a random time such that the event {τ ≤ t} belongs to Ft for any t ≤ T .
24
Market Models with Stochastic Volatility
Definition 3. European Call and Put Options
• For an European Call Option, considering S(T ) as the price of the underly-ing asset at expiration/maturity, the value of this contract is given by the following payoff function:
h(S(T )) = (S(T ) − K)+=
(
S(T ) − K if S(T ) > K,
0 if S(T ) ≤ K. (1.1)
The holder will exercise this option once the first case happens and will make a profit of S(T ) − K by buying the stock for K and selling it at S(T ). • For an European put option, the payoff function is as follows:
h(S(T )) = (K − S(T ))+=
(
K − S(T ) if S(T ) < K,
0 if S(T ) ≥ K. (1.2)
The holder can exercise this put option only for the first case if he is intere-sted in the profit of K − S(T ).
At time t ≤ T , this contract has a value known as the premium or the price of the option, which will vary with t and the observed stock price S(t). This option price at time t and for a stock price S(t) is denoted by C(t, S(t)) or P (t, S(t)). The problem of derivative pricing is determining this pricing function.
Definition 4. Filtration
Let us denote the information about security prices available in the market at
time t by Ft. A filtration on a probability space (Ω, F , P) is an increasing family
(Ft)t≥0 of sub-σ-algebras of F . In other words, Ft includes any information as far
as the time-t security prices is known based on the information and
Fs ⊆ Ft⊆ F if s ≤ t. (1.3)
The sequence of information {Ft; t ∈ [0, T ]} (or {Ft} for short) satisfying (1.3) is
called a filtration [44].
Other styles of options exist such as American or exotic [8]. An American option is a contract in which the holder decides whether to exercise the option or not at any time of his choice before the option’s expiration date T . The time τ at which the option is exercised is called the exercise time. Because the market cannot be anticipated, the holder of the option has to decide to exercise or not
at time t ≤ T with information up to time t contained in the σ-algebra Ft. In
other words, τ which is called a stopping time with respect to the filtration Ft is
a random time such that the event {τ ≤ t} belongs to Ft for any t ≤ T .
1.1.3
Stochastic Processes
The prices of underlying assets, for example: stock prices, interest rates, foreign exchange rates and commodity prices evolve in a stochastic way [39]. Mathemati-cally, to deal with them we have recourse to stochastic processes.
Definition 5. Stochastic process
Let T be a subset of [0, ∞). We call a stochastic process or a random process any family composed by random variables {S(t); t ∈ T } parameterized by time
t ∈ T [44]. When T = N (or T = N0), then {S(t); t ∈ T } is called a discrete-time
stochastic process, and when T = [0, ∞), we call it a continuous-time stochastic process.
For a random variable S(t), we know how to compute the probability that its value belongs to some subset of R, even if we don’t know exactly which value it will take in the future. In this thesis, t = 0 represents the current time (: today). Considering a probability space (Ω, F , P), a random variable S(t), is a measura-ble function S(t) : Ω −→ R. We denote a stochastic process by:
{S(t) : t ∈ T }
and the value of the random variable S(t) at a certain ω ∈ Ω by S(t, ω). In Fig. 3, we give an example of a path of a stochastic process made by 300 simulated daily changes in price. A stochastic process S(t) on the same time set T is said to be adapted to the filtration if:
∀t ∈ T , S(t) is Ft-measurable.
Definition 6. Standard Brownian Motion
A Standard Brownian Motion (SBM) called also a Wiener Process (WP) is a family of random variables {W (t) : t ≥ 0} on a probability space (Ω, F , P) such that:
1. W (0) = 0 with probability 1.
2. W (t) − W (s) is normally distributed with mean 0 and variance t − s for 0 ≤ s < t.
3. The increments W (t1) − W (t0), W (t2) − W (t1), · · · , W (tn) − W (tn−1) are
stationary, independent random variables for 0 ≤ t1< t2< · · · < tn.
4. The function t 7 −→ W (t, ω) is continuous with probability 1.
1.1.3
Stochastic Processes
The prices of underlying assets, for example: stock prices, interest rates, foreign exchange rates and commodity prices evolve in a stochastic way [39]. Mathemati-cally, to deal with them we have recourse to stochastic processes.
Definition 5. Stochastic process
Let T be a subset of [0, ∞). We call a stochastic process or a random process any family composed by random variables {S(t); t ∈ T } parameterized by time
t ∈ T [44]. When T = N (or T = N0), then {S(t); t ∈ T } is called a discrete-time
stochastic process, and when T = [0, ∞), we call it a continuous-time stochastic process.
For a random variable S(t), we know how to compute the probability that its value belongs to some subset of R, even if we don’t know exactly which value it will take in the future. In this thesis, t = 0 represents the current time (: today). Considering a probability space (Ω, F , P), a random variable S(t), is a measura-ble function S(t) : Ω −→ R. We denote a stochastic process by:
{S(t) : t ∈ T }
and the value of the random variable S(t) at a certain ω ∈ Ω by S(t, ω). In Fig. 3, we give an example of a path of a stochastic process made by 300 simulated daily changes in price. A stochastic process S(t) on the same time set T is said to be adapted to the filtration if:
∀t ∈ T , S(t) is Ft-measurable.
Definition 6. Standard Brownian Motion
A Standard Brownian Motion (SBM) called also a Wiener Process (WP) is a family of random variables {W (t) : t ≥ 0} on a probability space (Ω, F , P) such that:
1. W (0) = 0 with probability 1.
2. W (t) − W (s) is normally distributed with mean 0 and variance t − s for 0 ≤ s < t.
3. The increments W (t1) − W (t0), W (t2) − W (t1), · · · , W (tn) − W (tn−1) are
stationary, independent random variables for 0 ≤ t1< t2< · · · < tn.
Market Models with Stochastic Volatility
Figure 3: Example of a path of a Stochastic Process
0 50 100 150 200 250 300 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
Example of a Stochastic Process
Number of Days (T=300)
Daily change in price
In Fig. 4, we present examples of 10 simulated paths of standard Brownian motions.
Definition 7. Martingale
• Given a filtration {Ft : t ∈ T } and an integrable stochastic process {S(t) :
t ∈ T }, the collection {(S(t), Ft) : t ∈ T } with t, u ∈ T is called a martingale
if:
1. {S(t) : t ∈ T } is adapted to {Ft: t ∈ T },
2. E[S(t)] < ∞ for all t ∈ T , and
E(S(u)|Ft) = S(t); t < u. (1.4)
• The collection {(S(t), Ft) : t ∈ T } is called a sub-martingale if
E(S(u)|Ft) ≥ S(t), t < u (1.5)
and a super-martingale if
E(S(u)|Ft) ≤ S(t), t < u. (1.6)
• The Standard Brownian Motion is an example of a martingale.
26
Market Models with Stochastic Volatility
Figure 3: Example of a path of a Stochastic Process
0 50 100 150 200 250 300 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
Example of a Stochastic Process
Number of Days (T=300)
Daily change in price
In Fig. 4, we present examples of 10 simulated paths of standard Brownian motions.
Definition 7. Martingale
• Given a filtration {Ft : t ∈ T } and an integrable stochastic process {S(t) :
t ∈ T }, the collection {(S(t), Ft) : t ∈ T } with t, u ∈ T is called a martingale
if:
1. {S(t) : t ∈ T } is adapted to {Ft: t ∈ T },
2. E[S(t)] < ∞ for all t ∈ T , and
E(S(u)|Ft) = S(t); t < u. (1.4)
• The collection {(S(t), Ft) : t ∈ T } is called a sub-martingale if
E(S(u)|Ft) ≥ S(t), t < u (1.5)
and a super-martingale if
E(S(u)|Ft) ≤ S(t), t < u. (1.6)
• The Standard Brownian Motion is an example of a martingale.
Figure 4: Examples of 10 paths Standard Brownian Motions 0 0.05 0.1 0.15 0.2 0.25 −1.5 −1 −0.5 0 0.5 1 1.5 Time (Years) Brownian State
Standard Brownian Motions
1.1.4
Risk-neutral measure
Considering a risk-free asset such as a bond with price B(t) at time t. It can be described by the ordinary differential equation:
dB(t) = rB(t)dt (1.7)
where r > 0 is the instantaneous interest rate for lending or borrowing money and
for t ≥ 0 we have B(t) = B(0)ert.
Using the Brownian motion {W (t), t ≥ 0} with filtration {Ft} on a probability
space (Ω, F , P), let us start with some basic notions:
• Stock price: For a risky asset like stock or stock index, the price S(t) evolves according to the stochastic differential equation:
dS(t) = µ(t)S(t)dt + σ(t)S(t)dW (t) (1.8)
The processes µ(t) and σ(t) are adapted to the filtration and W (t) is a standard Brownian motion.
• Interest rate: We have the process r(t), 0 ≤ t ≤ T also adapted to the
filtration.
Figure 4: Examples of 10 paths Standard Brownian Motions
0 0.05 0.1 0.15 0.2 0.25 −1.5 −1 −0.5 0 0.5 1 1.5 Time (Years) Brownian State
Standard Brownian Motions
1.1.4
Risk-neutral measure
Considering a risk-free asset such as a bond with price B(t) at time t. It can be described by the ordinary differential equation:
dB(t) = rB(t)dt (1.7)
where r > 0 is the instantaneous interest rate for lending or borrowing money and
for t ≥ 0 we have B(t) = B(0)ert.
Using the Brownian motion {W (t), t ≥ 0} with filtration {Ft} on a probability
space (Ω, F , P), let us start with some basic notions:
• Stock price: For a risky asset like stock or stock index, the price S(t) evolves according to the stochastic differential equation:
dS(t) = µ(t)S(t)dt + σ(t)S(t)dW (t) (1.8)
The processes µ(t) and σ(t) are adapted to the filtration and W (t) is a standard Brownian motion.
• Interest rate: We have the process r(t), 0 ≤ t ≤ T also adapted to the
Market Models with Stochastic Volatility
• Wealth of an agent: With X(0) = x, we can write the wealth process in many forms:
dX(t) = ∆(t)dS(t)
| {z }
Capital gains from Stock
+ r(t)[X(t) − ∆(t)S(t)]dt | {z } Interest earnings = r(t)X(t)dt + ∆(t)[dS(t) − r(t)S(t)dt] = r(t)X(t)dt + ∆(t) (µ(t) − r(t)) | {z } Risk premium S(t)dt + ∆(t)σ(t)S(t)dW (t) = r(t)X(t)dt + ∆(t)σ(t)S(t) µ(t) − r(t) σ(t) | {z }
Market price of risk = θ(t)
dt + dW (t) • Discounted processes: de− Rt 0r(u)duS(t) = e− Rt 0r(u)du[−r(t)S(t)dt + dS(t)] d e− Rt 0r(u)duX(t) = e− Rt 0r(u)du[−r(t)X(t)dt + dX(t)] = ∆(t)de−R0tr(u)duS(t)
• Changing the measure: Let b(t) = (b1(t), b2(t), . . . , bd(t)) be a d-dimensional
adapted process, W (t) a d-dimensional Brownian motion, and define:
f
W (t) = W (t) +
Z t
0
b(u)du; Let now Z(t) be the process defined by:
Z(t) = exp − Z t 0 b(u)dW (u) −1 2 Z t 0 |b(u)|2du , and define a new measure Q by
Q(A) = Z
A
Z(T )dP, A ∈ F .
If Z(t) is a martingale then fW (t) is a Brownian motion under the measure
Q up to time T .
28
Market Models with Stochastic Volatility
• Wealth of an agent: With X(0) = x, we can write the wealth process in many forms:
dX(t) = ∆(t)dS(t)
| {z }
Capital gains from Stock
+ r(t)[X(t) − ∆(t)S(t)]dt | {z } Interest earnings = r(t)X(t)dt + ∆(t)[dS(t) − r(t)S(t)dt] = r(t)X(t)dt + ∆(t) (µ(t) − r(t)) | {z } Risk premium S(t)dt + ∆(t)σ(t)S(t)dW (t) = r(t)X(t)dt + ∆(t)σ(t)S(t) µ(t) − r(t) σ(t) | {z }
Market price of risk = θ(t)
dt + dW (t) • Discounted processes: de− Rt 0r(u)duS(t) = e− Rt 0r(u)du[−r(t)S(t)dt + dS(t)] d e− Rt 0r(u)duX(t) = e− Rt 0r(u)du[−r(t)X(t)dt + dX(t)] = ∆(t)de−R0tr(u)duS(t)
• Changing the measure: Let b(t) = (b1(t), b2(t), . . . , bd(t)) be a d-dimensional
adapted process, W (t) a d-dimensional Brownian motion, and define:
f
W (t) = W (t) +
Z t
0
b(u)du; Let now Z(t) be the process defined by:
Z(t) = exp − Z t 0 b(u)dW (u) −1 2 Z t 0 |b(u)|2du , and define a new measure Q by
Q(A) = Z
A
Z(T )dP, A ∈ F .
If Z(t) is a martingale then fW (t) is a Brownian motion under the measure
Q up to time T .
Definition 8. Risk-Neutral Probability Measure
A risk-neutral measure also called a martingale measure is any probability measure Q, equivalent to the market measure P, which makes all discounted asset prices
martingales i.e. given a probability space (Ω, F , P) with filtration {Ft, t ∈ T },
a probability measure Q is said to be risk-neutral if:
• Q is equivalent to P, that is P(A) > 0 ⇐⇒ Q(A) > 0 ∀ A ∈ F, and • Equation
EQt [Si∗(u) + di∗(u)] = Si∗(t), t, u ∈ T and t < u; i = 1, 2, . . . , n (1.9)
holds for all i and t with S0(t) = B(t), the money-market account [44].
If, in particular, the securities pay no dividends, then from (1.9) the denominated
price processes {Si∗(t)}, where S∗i(t) = Si(t)/B(t), are martingales under Q. By
this reason, the risk-neutral probability measure is often called a martingale mea-sure [44].
Definition 8. Risk-Neutral Probability Measure
A risk-neutral measure also called a martingale measure is any probability measure Q, equivalent to the market measure P, which makes all discounted asset prices
martingales i.e. given a probability space (Ω, F , P) with filtration {Ft, t ∈ T },
a probability measure Q is said to be risk-neutral if:
• Q is equivalent to P, that is P(A) > 0 ⇐⇒ Q(A) > 0 ∀ A ∈ F, and • Equation
EQt [Si∗(u) + di∗(u)] = Si∗(t), t, u ∈ T and t < u; i = 1, 2, . . . , n (1.9)
holds for all i and t with S0(t) = B(t), the money-market account [44].
If, in particular, the securities pay no dividends, then from (1.9) the denominated
price processes {Si∗(t)}, where Si∗(t) = Si(t)/B(t), are martingales under Q. By
this reason, the risk-neutral probability measure is often called a martingale mea-sure [44].
