R/RmetricsWorkshop Singapore 2010 · Portfolio Optimization with R/Rmetrics (2010), Diethelm Würtz, William Chen, Yohan Chalabi, Andrew Ellis Basic R for Finance (2010), Diethelm

R/Rmetrics Workshop Singapore 2010

Diethelm WürtzMahendra MehtaDavid ScottJuri Hinz

Rmetrics Association & Finance Online

R/Rmetrics eBook Series

R/Rmetrics eBooks is a series of electronic books and user guides aimedat students and practitioner who use R/Rmetrics to analyze financialmarkets.

A Discussion of Time Series Objects for R in Finance (2009)Diethelm Würtz, Yohan Chalabi, Andrew Ellis

Portfolio Optimization with R/Rmetrics (2010),Diethelm Würtz, William Chen, Yohan Chalabi, Andrew Ellis

Basic R for Finance (2010),Diethelm Würtz, Yohan Chalabi, Longhow Lam, Andrew EllisEarly Bird Edition

Financial Market Data for R/Rmetrics (2010)Diethelm Würtz, Andrew Ellis, Yohan ChalabiEarly Bird Edition

Indian Financial Market Data for R/Rmetrics (2010)Diethelm Würtz, Mahendra Mehta, Andrew Ellis, Yohan Chalabi

Presentations from the R/Rmetrics Singapore Workshop (2010)Diethelm Würtz, Mahendra Mehta, Juri Hinz, David Scott

PRESENTATIONS FROM THE

R/RMETRICS

SINGAPORE WORKSHOP 2010

EDITORS:

DIETHELM WÜRTZ

MAHENDRA MEHTA

JURI HINZ

DAVID SCOTT

RMETRICS ASSOCIATION & FINANCE ONLINE

Series Editors:Prof. Dr. Diethelm WürtzInstitute of Theoretical Physics andCurriculum for Computational ScienceSwiss Federal Institute of TechnologyHönggerberg, HIT G 32.38093 Zurich

Dr. Martin HanfFinance Online GmbHZeltweg 78032 Zurich

Contact Address:Rmetrics AssociationZeltweg 78032 [email protected]

Publisher:Finance Online GmbHSwiss Information TechnologiesZeltweg 78032 Zurich

Authors:Diethelm Würtz, Swiss Federal Institute of Technology ZurichMahendra Mehta, NeuralSoft Technologies MumbaiJuri Hinz, National University of Singapore, SingaporeDavid Scott, University of Auckland, Auckland

ISBN: 978-3-906041-08-7

© 2009, Finance Online GmbH, ZurichPermission is granted to make and distribute verbatim copies of this manual provided thecopyright notice and this permission notice are preserved on all copies.Permission is granted to copy and distribute modified versions of this manual under the con-ditions for verbatim copying, provided that the entire resulting derived work is distributedunder the terms of a permission notice identical to this one.Permission is granted to copy and distribute translations of this manual into another lan-guage, under the above conditions for modified versions, except that this permission noticemay be stated in a translation approved by the Rmetrics Association, Zurich.

Limit of Liability/Disclaimer of Warranty: While the publisher and authors have used theirbest efforts in preparing this book, they make no representations or warranties with respectto the accuracy or completeness of the contents of this book and specifically disclaim anyimplied warranties of merchantability or fitness for a particular purpose. No warranty maybe created or extended by sales representatives or written sales materials. The advice andstrategies contained herein may not be suitable for your situation. You should consult with aprofessional where appropriate. Neither the publisher nor authors shall be liable for any lossof profit or any other commercial damages, including but not limited to special, incidental,consequential, or other damages.

Trademark notice: Product or corporate names may be trademarks or registered trademarks,and are used only for identification and explanation, without intent to infringe.

WELCOME

Welcome to the first R/Rmetrics Singapore Conference on “ComputationalTopics in Finance”. We are very glad that you found the time to come tothe Singapore, and for the many of you traveling from the U.S., Europeand various places in Asia, we hope that your journey was not too arduous.

With the R/Rmetrics Singapore Conference, we want to create a new fo-rum where fund and/or risk managers from banks and insurance firms,decision makers, researchers from industry and academia, and studentscan exchange ideas and engage in stimulating discussions.

The environment for this workshop should be a place a little bit asidefrom the mainstream conference of venues, and we are happy to havefound this at the Risk Management Institute of the National University ofSingapore.

About 40 participants are attending the conference, and the mixture, asplanned, is quite heterogeneous. About half are from academia, and theother half from the software and financial industries, including banks.

Last but not least, we want to thank the organizing committee and oursponsors.

We wish you an interesting conference with many inspiring and stimulat-ing discussions.

Diethelm WürtzSingapore, February 2010

v

CONTENTS

WELCOME V

CONTENTS VII

I Friday Morning 1

1 STEFANO IACUS 2

2 JURI HINZ 40

3 DAVID SCOTT 56

4 MARC PAOLELLA 82

II Friday Afternoon 85

5 VIKRAM KURIYAN 86

6 BERNARD LEE 120

7 KAM FONG CHAN 122

8 ANDREW ELLIS 134

9 ANMOL SETHY 148

10 KARIM CHINE 164

III Saturday Morning 167

11 DEFENG SUN 168

12 YOHAN CHALABI 190

vii

CONTENTS VIII

13 JOEL YU 192

14 LEONG CHEE KIA 202

15 DIETHELM WÜRTZ 204

16 PRATAP SONDHI 214

IV Appendix 231

SPONSORS 232

RMETRICS ASSOCIATION 234

PART I

FRIDAY MORNING

1

CHAPTER 1

STEFANO IACUS

2

The "yuima" package: An R framework for simulation and inference

of stochastic differential equations

Stefano M. Iacuson behalf of Yuima Project Team

Department of Economics, Business and StatisticsUniversità degli Studi di Milano, Italy

Most of the theoretical results in modern finance rely on the assumption that the underlying dynamics of asset prices, currencies exchange rates, interest rates, etc are continuous time stochastic proces-ses driven by stochastic differential equations. Continuous time models are also at the basis of option pricing and option pricing often requires Monte Carlo methods. In turn, the Monte Carlo method re-quires a preliminary good model to simulate whose parameters has to be estimated from historical data. Most ready-to-use tools in computational finance relies on pure discrete time models, like arch, garch, etc. and very few examples of software handling continuous time processes in a general fashion are available also in the R community. There still exists a gap between what is going on in mathematical finance and applied finance. The "yuima" package is intended to help in filling this gap.

The Yuima Project is an open source and collaborative effort of several mathematicians and sta-tisticians aimed at developing the R package named "yuima" for simulation and inference of sto-chastic differential equations. The "yuima" package is an environment that follows the paradigm of methods and classes of the S4 system for the R language.

In the "yuima" package stochastic differential equations can be of very abstract type, e.g. uni or multi-dimensional, driven by Wiener process of fractional Brownian motion with general Hurst parameter, with or without jumps specified as Lévy noise. Lévy processes can be specified via compound Poisson description, by the specification of the Lévy measure or via increments and stable laws.

The "yuima" package is intended to offer the basic infrastructure on which complex models and inference procedures can be built on. In particular, the basic set of functions includes the following: 1) Simulation schemes for all types of stochastic differential equations (Wiener, fBm, Lévy). 2) Different subsampling schemes including random sampling with user specified random times distribution, space discretization, tick times, etc. 3) Automatic asymptotic expansion for the approximation and estimation of functionals of diffusion processes with small noise via Malliavin calculus, useful in option pricing. 4) Efficient quasi-likelihood inference for diffusion processes and diffusion processes with jumps.

All simulation schemes, subsampling and inference are designed to work on both regular or irregular grid times (i.e. regular or irregular time series). In special cases also asynchronous data and sampling schemes can be handled. As proof-of-concept (but fully operational) examples of statistical procedures have been implemented like change point analysis in volatility of stochastic differential equations, asynchronous covariance estimation, divergence test statistics.

The Yuima Project was partly supported by Japan Science and Technology Agency, Basic Research Programs PRESTO, Grants-in-Aid for Scientific Research No. 19340021.

1 / 56

The Yuima Project

Stefano M. Iacus (University of Milan & R Core Team)on behalf of Yuima Core Team

Computational Topics in Finance, 1st R/Rmetrics Workshop, February 19/20, 2010, Singapore

Overview of the Yuima Project

Overview of the YuimaProject

Overview of the yuimapackage

What contains a yuimaobject ?

What is possible to dowith a yuima object inhands?

How it is supposed towork?

Inference

Change-point Analysis

LASSO estimation

Asymptotic Expansion

Roadmap

2 / 56

The Yuima Project Team






Inference


LASSO estimation


Roadmap

3 / 56

A. Brouste (Univ. Le Mans, FR)M. Fukasawa (Osaka Univ. JP)H. Hino (Waseda Univ., Tokyo, JP)S.M. Iacus (Milan Univ., IT)K. Kengo (Tokyo Univ., JP)H. Masuda (Kyushu Univ., JP)Y. Shimitzu (Osaka Univ., JP)M. Uchida (Osaka Univ., JP)N. Yoshida (Tokyo Univ., JP). . .more to come

The yuima package1 is written by people working in mathematical statisticsand finance, who actively publish results in the field, have some knowledge ofR, and have the feeling on “what’s next” in the field.

Aims at filling the gap between theory and practice!1The Yuima Project is funded by the Japan Science Technology (JST) Basic Research Programs PRESTO, Grants-in-Aid for Scientific

Research No. 19340021.

The yuima package goal: fill the gap between theory and practice






Inference


LASSO estimation


Roadmap

4 / 56

The Yuima Project aims at implementing, via the yuima package, a veryabstract framework to describe probabilistic and statistical properties ofstochastic processes in a way which is the closest as possible to theirmathematical counterparts but also computationally efficient.

⌅ it is an R package, using S4 classes and methods, where the basic classextends to SDE’s with jumps (simple Poisson, Levy), SDE’s driven byfBM, Markov switching regime processes, HMM, etc.

⌅ separates the data description from the inference tools and simulationschemes

⌅ the design allows for multidimensional, multi-noise processesspecification

⌅ it includes a variety of tools useful in finance, like asymptotic expansionof functionals of stochastic processes via Malliavin calculus

Overview of the yuima package






Inference


LASSO estimation


Roadmap

5 / 56

The yuima object






Inference


LASSO estimation


Roadmap

6 / 56

The main object is the yuima object which allows to describe the model in amathematically sound way.

Then the data and the sampling structure can be included as well or, just thesampling scheme from which data can be generated according to the model.

The package exposes very few generic functions like simulate, qmle, plot,etc. and some other specific functions for special tasks.

Before looking at the details, let us see an overview of the main object.

What contains a yuima object ?






Inference


LASSO estimation


Roadmap

7 / 56

YuimaSampling

randomdeterministic

tick timesspace disc.

...

regularirregular

multigridasynch.

Dataunivariatemultivariate

ts, xts, ...

Model

diffusionLevy

fractionalBM

MarkovSwitching HMM

YuimaSampling

randomdeterministic


...

regularirregular

multigridasynch.


ts, xts, ...

Model

diffusionLevy

fractionalBM

MarkovSwitching HMM

YuimaSampling

randomdeterministic


...

regularirregular

multigridasynch.


ts, xts, ...

Model

diffusionLevy

fractionalBM

MarkovSwitching HMM

YuimaSampling

randomdeterministic


...

regularirregular

multigridasynch.


ts, xts, ...

Model

diffusionLevy

fractionalBM

MarkovSwitching HMM

What is possible to do with a yuima

object in hands?






Inference


LASSO estimation


Roadmap

12 / 56

Yuima

SimulationExact

Euler-Maruyama

Space discr.

NonparametricsCovariation p-variation

ParametricInference

High freq.Low freq.

Quasi MLEDiff, Jumps,

fBM

AdaptiveBayesMCMC

Changepoint

Modelselection

Akaike’s

LASSO-typeHypothesesTesting

Optionpricing

Asymptoticexpansion

Monte Carlo

Yuima

SimulationExact

Euler-Maruyama

Space discr.


ParametricInference

High freq.Low freq.


fBM

AdaptiveBayesMCMC

Changepoint

Modelselection

Akaike’s


Optionpricing

Asymptoticexpansion

Monte Carlo

Yuima

SimulationExact

Euler-Maruyama

Space discr.


ParametricInference

High freq.Low freq.


fBM

AdaptiveBayesMCMC

Changepoint

Modelselection

Akaike’s


Optionpricing

Asymptoticexpansion

Monte Carlo

Yuima

SimulationExact

Euler-Maruyama

Space discr.


ParametricInference

High freq.Low freq.


fBM

AdaptiveBayesMCMC

Changepoint

Modelselection

Akaike’s


Optionpricing

Asymptoticexpansion

Monte Carlo

Yuima

SimulationExact

Euler-Maruyama

Space discr.


ParametricInference

High freq.Low freq.


fBM

AdaptiveBayesMCMC

Changepoint

Modelselection

Akaike’s


Optionpricing

Asymptoticexpansion

Monte Carlo

Yuima

SimulationExact

Euler-Maruyama

Space discr.


ParametricInference

High freq.Low freq.


fBM

AdaptiveBayesMCMC

Changepoint

Modelselection

Akaike’s


Optionpricing

Asymptoticexpansion

Monte Carlo

How it is supposed to work?






Inference


LASSO estimation


Roadmap

19 / 56

The model specification






Inference


LASSO estimation


Roadmap

20 / 56

We consider here the three main classes of SDE’s which can be easilyspecified. All multidimensional and eventually parametric models.

R/RmetricsWorkshop Singapore 2010 · Portfolio Optimization with R/Rmetrics (2010), Diethelm Würtz, William Chen, Yohan Chalabi, Andrew Ellis Basic R for Finance (2010), Diethelm

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