Table Of ContentMathematical Modeling
and
Computation in Finance
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Mathematical Modeling
and
Computation in Finance
with Exercises and Python and MATLAB computer codes
Cornelis W Oosterlee
Centrum Wiskunde & Informatica (CWI) &
Delft University of Technology, The Netherlands
Lech A Grzelak
Delft University of Technology, The Netherlands
World Scientific
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Head office: 5 Toh Tuck Link, Singapore 596224
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Library of Congress Control Number: 2019950785
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
MATHEMATICAL MODELING AND COMPUTATION IN FINANCE
With Exercises and Python and MATLAB Computer Codes
Copyright © 2020 by Cornelis W. Oosterlee and Lech A. Grzelak
All rights reserved.
ISBN 978-1-78634-794-7
ISBN 978-1-78634-805-0 (pbk)
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/Q0236#t=suppl
Desk Editor: Shreya Gopi
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore
Dedicated to Anasja, Wim, Mathijs, Wim en Agnes (Kees)
Dedicated to my mum, brother, Anna and my whole family (Lech)
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Preface
This book is discussing the interplay of stochastics (applied probability theory)
and numerical analysis in the field of quantitative finance. The contents will be
usefulforpeople workinginthe financialindustry, forthose aimingtowork there
one day, and for anyone interested in quantitative finance.
Stochastic processes, and stochastic differential equations of increasing com-
plexity, are discussed for the various asset classes, reaching to the models that
are in use at financial institutions. Only in exceptional cases, solutions to these
stochastic differential equations are available in closed form.
The typical models in use at financial institutions have changed over time.
Basically, each time when the behavior of participants in financial markets
changes, the corresponding stochastic mathematical models describing the prices
maychangeaswell. Alsofinancialregulationwillplayitsroleinsuchchanges. In
the book we therefore discuss a variety of models for stock prices, interest rates
as well as foreign-exchange rates. A basic notion in such a diverse and varying
field is “don’t fall in love with your favorite model”.
Financial derivatives are products that are based on the performance of
another, uncertain, underlying asset, like on stock, interest rate or FX prices.
Next to the modeling of these products, they also have to be priced, and the
risk related to selling these products needs to be assessed. Option valuation is
also encountered in the financial industry during the calibration of the stochastic
models of the asset prices (fitting the model parameters of the governing SDEs
so that model and market values of options match), and also in risk management
when dealing with counterparty credit risk.
Advanced risk management consists nowadays of taking into account the risk
thatacounterpartyofafinancialcontractmaydefault(CCR,CounterpartyCredit
Risk). Because of this risk, fair values of option prices are adjusted, by means of
the so-called Valuation Adjustments. We will also discuss the Credit Valuation
Adjustment (CVA) in the context of risk management and derive the governing
equations.
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viii Mathematical Modeling and Computation in Finance
Option values are governed by partial differential equations, however, they
can also be defined as expectations that need to be computed in an efficient,
accurate and robust way. We are particularly interested in stochastic volatility
basedmodels,withthewell-knownHestonmodelservingasthepointofreference.
As the computational methods to value these financial derivatives, we present
a Fourier-based pricing technique as well as the Monte Carlo pricing method.
Whereas Fourier techniques are useful when pricing basic option contracts, like
European options, within the calibration procedure, Monte Carlo methods are
often used when more involved option contracts, or more involved asset price
dynamics are being considered.
Bygraduallyincreasingthecomplexityofthestochasticmodelsinthedifferent
chapters of the book, we aim to present the mathematical tools for defining
appropriatemodels, aswellasfortheefficientpricingofEuropeanoptions. From
the equity models in the first 10 chapters, we move to short-rate and market
interest rate models. We cast these models for the interest rate into the Heath-
Jarrow-Morton framework, show relations between the different models, and we
explain a few interest rate products and their pricing as well.
It is sometimes useful to combine SDEs from different asset classes, like stock
and interest rate, into a correlated set of SDEs, or, in other words, into a system
ofSDEs. Wediscussthehybridassetpricemodelswithastochasticequitymodel
and a stochastic interest rate model.
Summarizing, the reader may encounter a variety of stochastic models,
numerical valuation techniques, computational aspects, financial products and
riskmanagementapplicationswhilereadingthisbook. Theaimistohelpreaders
progress in the challenging field of computational finance.
ThetopicsthatarediscussedarerelevantforMScandPhDstudents,academic
researchers as well as for quants in the financial industry. We expect knowledge
of applied probability theory (Brownian motion, Poisson process, martingales,
Girsanov theorem,...), partial differential equations (heat equation, boundary
conditions),familiaritywithiterativesolutionmethods,liketheNewton-Raphson
method, and a basic notion of finance, assets, prices, options.
Acknowledgment
Here, wewouldliketoacknowledgedifferentpeoplefortheirhelpinbringingthis
book project to a successful end.
Firstofall,wewouldliketothankouremployersfortheirsupport,ourgroups
at CWI — Center for Mathematics & Computer Science, in Amsterdam and at
the Delft Institute of Applied Mathematics (DIAM), from the Delft University
of Technology in the Netherlands. We thank our colleagues at the CWI, at
DIAM, and at Rabobank for their friendliness. Particularly, Nada Mitrovic is
acknowledged for all the help. Vital for us were the many fruitful discussions,
cooperationsandinputfromourgroupmembers,likefromourPhDstudents,the
post-docs and also the several guests in our groups. In particular, we thank
our dear colleagues Peter Forsyth, Luis Ortiz Gracia, Mike Staunton, Carlos
Vazquez, Andrea Pascucci, Yuying Li, and Karel in’t Hout for their insight and
the discussions.
Proofreadingwithdetailedpointersandsuggestionsforimprovementshasbeen
very valuable for us and for this we would like to thank in particular Natalia
Borovykh, TimDijkstra, ClarissaElli, IrfanIlgin, MarkoIskra, FabienLeFloc’h,
Patrik Karlsson, Erik van Raaij, Sacha van Weeren, Felix Wolf and Thomas van
der Zwaard.
Wegotinspiredbythegroup’sPhDandpost-doctoralstudents, inalphabetic
order, Kristoffer Andersson, Anastasia Borovykh, Ki Wai Chau, Bin Chen, Fei
Cong, Fang Fang, Qian Feng, Andrea Fontanari, Xinzheng Huang, Shashi Jain,
Prashant Kumar, Coen Leentvaar, Shuaiqiang Liu, Peiyao Luo, Marta Pou,
Marjon Ruijter, Beatriz Salvador Mancho, Luis Souto, Anton van der Stoep,
Maria Suarez, Bowen Zhang, Jing Zhao, and Hisham bin Zubair. We would like
to thank Shreya Gopi, her team and Jane Sayers at World Scientific Publishing
for the great cooperation.
Our gratefulness to our families cannot be described in words.
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