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Andrea Pascucci • Wolfgang J.Runggaldier
Financial
Mathematics
Theory and Problems for
Multi-period Models
AndreaPascucci WolfgangJ.Runggaldier
DepartmentofMathematics DepartmentofPureand
UniversityofBologna AppliedMathematics
UniversityofPadova
TranslatedandextendedversionoftheoriginalItalianedition:
A.Pascucci,W.J.Runggaldier:FinanzaMatematica.Teoriaeproblemipermodelli
multiperiodali,©Springer-VerlagItalia2009
UNITEXT–LaMatematicaperil3+2
ISSNprintedition:2038-5722 ISSNelectronicedition:2038-5757
ISBN978-88-470-2537-0 ISBN978-88-470-2538-7(eBook)
DOI10.1007/978-88-470-2538-7
LibraryofCongressControlNumber:2011943827
SpringerMilanHeidelbergNewYorkDordrechtLondon
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Preface
Financial mathematics has recently undergone a considerable development,
due mainly to new financial instruments that have been introduced in order
to limit the risk in financial operations. The study of problems related to
such instruments requires mathematical techniques that occasionally may be
rather sophisticated and are to a great extent related to Probability.
Consequently, the financial institutions now offer job opportunities not
only to economists, but also to experts in scientific-technical disciplines, in
particular in mathematics. With the Bologna Accords the so-called 3+2+3
(bachelor-master-doctor)curriculumhasbeenintroducedinvariouscountries
withtheintentionthatstudentsmayenterthejobmarketalreadyatthebach-
elorlevel.Itthusturnsouttobeappropriate tohaveafinancialmathematics
course already at the bachelor level. Most mathematical techniques in use
in financial mathematics are related to continuous time models and require
therefore notions from stochastic analysis that are in general not familiar not
only to economists but neither to mathematicians at the bachelor level. It is
thus desirable to be able to transmit to bachelor students the basic notions
and methodologies in use in financial mathematics without the technicalities
from stochastic analysis that are inherent in continuous time models. This
canbe achievedbyusing discretetime(multi-period) modelsinstead. Onone
hand they generalize to a dynamic context the one-period models that are
still in wide use by economists, on the other hand they can also be seen as
possibleapproximationstocontinuoustimemodels.Multiperiodmodelshave
however also a genuine interest in their own and this also in view of possible
practical applications.
The present volume is intended as a possible textbook for a course as de-
scribed above and is the result of the teaching experience of the authors in
the area of financial mathematics. For multi-period models there do not exist
many textbooks (one of the best known is [18]) and so one of the purposes
of the present volume is to fill in this gap. Although conceived mainly for a
bachelor-level course in mathematics, the volume should also be appropriate
for quantitative finance courses for economics students.
VI Preface
Evidently, we could not take into account in this book all possible topics
in financial mathematics and so we have confined ourselves to those that one
mayconsiderasbasicones.Thestructureofthebookoriginatesfromtheidea
of teaching by examples and counterexamples. It has been expanded beyond
the examplestobecome acomplete textbook that includes alsothe necessary
theory. Consequently, and differently from other textbooks, this one includes
many examples and solved problems. In this context we want to mention
also [21] that contains examples for the specific binomial model and [19] that
includes problems both from discrete as well as continuous time models.
The majority of the solution methods for multi-period models is based on
recursive algorithms, for which the computational complexity increases con-
siderably with the number of periods. In practice one has therefore to use
computer programs to implement the algorithms. For problems in classrooms
and at exam sessions it is therefore appropriate to limit oneself to situations
where calculation can be performed “by hand”. For this reason, in the exam-
plesandproblemssuggestedinthebookweconsidersmallnumbersofperiods
and numerical data that may not correspond to realistic situations but allow
for easier calculations.
The book is divided into four chapters, in which we treat the following
topics:
• pricing and hedging of European derivatives;
• portfoliooptimization(dynamicprogrammingand“martingalemethod”);
• pricing, optimal exercise and hedging of American derivatives;
• multi-period models for the term structure of interest rates.
Each of the four chapters consists of two parts: a theoretical section and a
problem section. In the latter section we describe in detail the solution for
many possible problems.
Bologna/Padova, November 2011 Andrea Pascucci
Wolfgang J. Runggaldier
Contents
1 Pricing and hedging ....................................... 1
1.1 Primary securities and strategies .......................... 2
1.1.1 Discrete time markets.............................. 2
1.1.2 Self-financing and predictable portfolios .............. 3
1.1.3 Relative portfolio.................................. 5
1.1.4 Discounted market ................................ 6
1.2 Arbitrage and martingale measures ........................ 7
1.3 Pricing and hedging ..................................... 9
1.3.1 Derivative securities ............................... 9
1.3.2 Arbitrage pricing.................................. 10
1.3.3 Hedging.......................................... 13
1.3.4 Put-Call parity.................................... 13
1.4 Market models .......................................... 14
1.4.1 Binomial model ................................... 14
1.4.2 Trinomial model .................................. 18
1.5 On pricing and hedging in incomplete markets .............. 22
1.6 Change of numeraire..................................... 23
1.6.1 A particular case .................................. 23
1.6.2 General case...................................... 25
1.7 Solved problems......................................... 28
2 Portfolio optimization ..................................... 61
2.1 Maximization of expected utility .......................... 62
2.1.1 Strategies with consumption ........................ 62
2.1.2 Utility functions................................... 65
2.1.3 Expected utility of terminal wealth .................. 67
2.1.4 Expected utility from intermediate consumption and
terminal wealth ................................... 70
2.2 “Martingale” method .................................... 72
2.2.1 Complete market: terminal wealth................... 72
VIII Contents
2.2.2 Incomplete market: terminal wealth.................. 78
2.2.3 Complete market: intermediate consumption.......... 81
2.2.4 Complete market: intermediate consumption and
terminal wealth ................................... 86
2.3 Dynamic Programming Method ........................... 88
2.3.1 Recursive algorithm ............................... 88
2.3.2 Proof of Theorem 2.32 ............................. 92
2.4 Logarithmic utility: examples ............................. 94
2.4.1 Terminal utility in the binomial model: MG method ... 94
2.4.2 Terminal utility in the binomial model: DP method.... 96
2.4.3 Terminal utility in the completed trinomial model:
MG method ...................................... 99
2.4.4 Terminal utility in the completed trinomial model:
DP method.......................................101
2.4.5 Terminal utility in the standard trinomial model:
DP method ......................................103
2.4.6 Intermediate consumption in the binomial model:
MG method ......................................106
2.4.7 Intermediate consumption in the binomial model:
DP method.......................................109
2.4.8 Intermediate consumption in the completed trinomial
model: MG method................................113
2.4.9 Optimal consumption in the completed trinomial
model: DP method ................................114
2.4.10 Intermediate consumption in the standard trinomial
model: DP method ................................115
2.5 Solved problems.........................................118
3 American options..........................................165
3.1 American derivatives and early exercise strategies ...........166
3.1.1 Arbitrage pricing..................................167
3.1.2 Arbitrage price in a complete market ................169
3.1.3 Optimal exercise strategies .........................175
3.1.4 Hedging strategies.................................178
3.2 American and European options...........................180
3.3 Solved problems.........................................183
3.3.1 Preliminaries .....................................183
3.3.2 Solved problems...................................184
4 Interest rates ..............................................223
4.1 Bonds and interest rates..................................224
4.2 Market models for interest rates...........................227
4.3 Short models ...........................................230
4.3.1 Affine models .....................................232
4.3.2 Discrete time Hull-White model.....................233
Contents IX
4.4 Forward models .........................................237
4.4.1 Binomial forward model............................239
4.4.2 Multinomial forward model .........................241
4.5 Interest rate derivatives ..................................244
4.5.1 Caps and Floors...................................244
4.5.2 Interest Rate Swaps ...............................247
4.5.3 Swaptions and Swap Rate ..........................250
4.6 Solved problems.........................................252
4.6.1 Recalling the basic models..........................252
4.6.2 Options on T-bonds ...............................255
4.6.3 Caps and Floors...................................262
4.6.4 Swap Rates and Payer Forward Swaps ...............271
4.6.5 Swaptions ........................................276
References.....................................................287
