Asset Pricing: -Discrete Time Approach-

272 Pages·2003·9.78 MB·English

by  Takeaki Kariya

ASSET PRICING
-Discrete Time Approach-
ASSET PRICING
-Discrete Time Approach-
by
Takeaki Kariya
Kyoto University
Japan
Regina Y. Lin
Rutgers University
US.A.
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Asset Pricing-Discrete Time Approach-by Takeaki Kariya and Regina Liu
ISBN 978-1-4613-4849-8 ISBN 978-1-4419-9230-7 (eBook)
DOI 10.1007/978-1-4419-9230-7
Copyright © 2003 Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 2003
Softcover reprint of the hardcover 1s t edition 2003
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Printed on acid-free paper.
Contents
1. INTRODUCTION 1
1 Main Goals 1
2 The Importance of The No-Arbitrage Theory 2
3 The Discrete Time Approach and Some Key Features of
This Book 3
4 Comparisons with Other Textbooks 4
5 A BriefSummary of the Contents 6
2. OPTIONS, FUTURES AND OTHER DERIVATIVES 9
1 Overview 9
2 No-Arbitrage and Put-Call Parity 10
3 Exotic Options 19
4 Forward Contracts and Futures 22
3. BASIC PROBABILITY THEORY 27
1 Overview 27
2 Conditional Distributions and Conditional Expectations 28
3 Multivariate Normal Distribution and Normal Mixture
Distribution 35
4 Nonlinear Time Series Model 39
4. PRICING MODELS FOR FINANCIAL ASSETS 43
1 Overview 43
2 Stochastic Processes and Brownian Motion 45
3 Martingale and Product Process 51
4 log-DD Process and Change ofProbability Measures 56
VI ASSET PRICING
5. GENERAL NO-ARBITRAGE ASSET PRICE THEORY 65
1 Overview 65
2 Basic Framework of No-Arbitrage Price Theory 67
3 Condition for No-Arbitrage 70
4 PriceTheoryfor Derivativesand the Black-ScholesFormula 74
5 No-Arbitrage Binomial Process and Replicability of an
Option 80
6 Martingale Condition for log-DD Process 88
6. MODEL SPECIFICATIONS IN APPLICATIONS 97
1 Overview 97
2 Self-Consistency Tests for Models 100
3 Multi-Factor Model - Identifiability and Estimation 105
4 Model under Original Measure Q vs Risk Neutral Model
under Equivalent Measure Q* 107
7. VALUATION OFDERIVATIVES VIA MONTE CARLO
METHODS 111
1 Overview 111
2 Monte Carlo Method 112
3 Variance Reduction Methods 117
4 General Theory for CV Methods 133
8. STOCK OPTION THEORY AND ITS APPLICATIONS 139
1 Overview 139
2 General Price Theory for a Stock Option 140
3 Black-Scholes (BS) Formula 143
4 BS Option Portfolios 149
5 Valuation of Exotic Options 153
6 GARCH Model and Stochastic Volatility Model 157
7 Valuation of an American Put 162
9. CURRENCY OPTIONS 167
1 Overview 167
2 Pricing Currency Options 168
3 Currency Options Containing Stocks 172
4 A Condition for No-Arbitrage 175
Contents Vll
10. THE TERM STRUCTURE OF SPOT RATES 181
1 Overview 181
2 Spot Rate and No-Arbitrage Price of a Discount Bond 182
3 One Factor Term Structure Model for Spot Rates 188
4 Empirical Viewpoint on CIR Type Model 194
5 Interest Swaps 196
11. THE HJM MODEL FOR BONDS AND ITS APPLICATIONS 201
1 Overview 201
2 Forward Rates 202
3 The K-Factor HJM Model for Discount Bond Price 206
4 Specification Problems of HJM Model 213
5 Specification of Volatility Functions 217
6 Empirical Analyses of Interest Futures 222
12. PRICING DEFAULTABLE BONDS 229
1 Overview 229
2 Recovery Rate and Default Probability 230
3 Valuation of Corporate Discount Bond 231
4 Pricing a Coupon Bond 237
13. VALUATION OF CD WITH TRANSFER OPTION 239
1 Overview 239
2 Valuation of a CD with Transfer Option 239
3 Valuation of the Transfer Option 243
4 Valuation of the Closing Option 245
5 Ex Post Multiplier and Risk of the Bank 246
14. PRICING MORTGAGE-BACKED SECURITIES 251
1 Overview 251
2 Cashflow Function of an MBS 253
3 Valuation Formula for an MBS 254
4 Interest Incentive Function 258
5 Monte Carlo (MC) Valuation of an MBS 261
6 Estimation Procedure 267
viii ASSET PRICING
References 269
Index 273
Chapter 1
INTRODUCTION
1. Main Goals
The theory ofasset pricing has grown markedly more sophisticated in the
last two decades, with the application of powerful mathematical tools such
as probability theory, stochastic processes and numerical analysis. The main
goal of this book is to provide a systematic exposition, with practical appli
cations, of the no-arbitrage theory for asset pricing in financial engineering in
theframework ofadiscretetime approach.The bookshould also serve wellas
a textbook on financial asset pricing. It should be accessible to a broad audi
ence, inparticulartopractitioners infinancialand related industries, as wellas
to students in MBAor graduate/advanced undergraduate programs in finance,
financialengineering,financial econometrics,or financialinformation science.
The no-arbitrage asset pricing theory is based on the simple and well ac
cepted principle that financial asset prices are instantly adjusted at each mo
ment in time in order not to allow an arbitrage opportunity. Here an arbitrage
a
opportunity is an opportunity to have a portfolio of value at an initial time
lead to a positive terminal value with probability 1(equivalently, at no risk),
withmoney neitheradded norsubtracted from theportfolio inrebalancingdur
a
ingthe investment period. Itisnecessary for aportfolio of value to include a
short-sell position as wellas a long-buy position of some assets.
A set of financial asset prices which allow an arbitrage opportunity is not
in equilibrium. In other words, asset prices in equilibrium should be of "no
arbitrage". In this sense, the concept of no-arbitrage includes the concept of
equilibrium in economics. More importantly, the concept of no-arbitrage is at
the same time general enough toapply toany set of assets, and strong enough
to price each asset relative to a bank account asset. This allows a unified ap
proach to theories which had been originally separate. Thus both the deriva-
T. Kariya et al., Asset Pricing
© Kluwer Academic Publishers 2003
2 ASSETPRICING
tive theories for pricing stock options, interest swaps, creditrisk products, etc.
and the asset pricing theories for government and corporate bonds, mortgage
backed securities, can be developed simultaneously and systematically by the
discrete time no-arbitrage theory. In particular, the well-known Black-Scholes
pricing formula for stock options, developed in a continuous time setting, can
beexactly derived in ourdiscrete timeapproach.
In this book, we shall develop from scratch a general asset pricing theory
ofno-arbitrage together with practical examples. The presentation is gradual,
from basic concepts to dynamic price models (stochastic processes) for finan
cial assets, and finally to application methods. The basic ideas and techniques
are illustrated repeatedly. Fortunately, our discrete time approach circumvents
many mathematicaldetailsofthegeneral no-arbitragepricingtheory whichare
difficult to understand in a continuous time setting. In fact, our presentation
should be understandable to readers with a minimal background ofcalculus,
probability and statistics, and linear algebra at the level of an MBA, a first
year graduate course in social sciences or even of a senior undergraduate in
sciences.
2. The Importance of The No-Arbitrage Theory
Financialengineeringmay bedefinedasthe sciencefordevelopingthoughts,
knowledge, technologies and models for the functional efficiency offinance in
view ofan efficient use ofcapital relative to risk. It involves the following five
main areas ofapplication in practice.
(I) Pricing financial products;
(2) Investmentand assetmanagement, where portfoliotheory is usually ap-
plied;
(3) Risk management;
(4) Structured finance: Securitization;
(5) Real option: Valuation ofcorporate projectand enterprise risk.
The no-arbitrage theory is a fundamental part of (1). However, it also plays
an important role in (3), (4) and (5), where it can be used to value or de
sign financial products, and evaluate risks in various situations. What is more
important, understanding the no-arbitrage theory can help develop a unified
framework, from the perspective of an efficient use of capital, for designing
financial products, risk management schemes, securitization with special risk
transfer scheme, management decision making, and financial systems. In this
sense,a systematicunderstandingofthe no-arbitrage theory providesthe intel
lectual capability for developing innovative concepts in financial business and
financial policy.
A perhaps unforeseen benefit ofthe concept and the theory ofno-arbitrage
is their bringing to the fore the functional efficiency offinance, as a paradigm
Introduction 3
for valuing all risks fairly. For example, no-arbitrage promotes a tendency for
theconvergenceoffinance and insurance, sinceafair valuationofan insurance
risk tends tobe madesimilarlytothat ofacreditrisk. Even weatherderivatives
are now commonly traded in the financial industry and the insurance industry.
In fact, catastrophic risks such as hurricanes or earthquakes are securitized as
"cat" bonds.
It should be noted that a financial product is differentiated only by its three
attributes: return, risk and time. The paradigm of functional finance claims
that finance and risk transferis made only through financial products including
financial assets, notthroughfinancial institutions,and thataworldwideoptimal
allocationofcapitaland risk shouldbe made through various financial products
and instruments.
3. The Discrete Time Approach and Some Key
Features of This Book
As stated in Section I, the concept ofno-arbitrage excludes the possibility
that the initial value 0 in a portfolio can lead to a positive terminal value with
probability I, with no inflow and no outflow ofcapital in rebalancing. In for
mulating this concept as a mathematical expression and deriving a condition
for no-arbitrage, probabilistic arguments are required and random variables
and stochastic processes have to be introduced. For this purpose many intro
ductorytextbooksadoptacontinuoustime setting, partly inordertodiscussthe
Black-Scholesoptiontheory. Inthecontinuoustime setting,astockpriceisas
sumedtofollow astochasticdifferentialequation,and then apartialdifferential
equation whose solution is an option premium is derived. However, to under
stand completelysuch an argument, itisnecessaryto have astrongbackground
for measure-theoretic probability theory and continuous time stochastic pro
cesses. Naturally most introductory textbooks cannotprovidea self-contained
account, and must restrictthemselves tojustan outlineofthe argumentbehind
the theory. Such a treatment keeps the core ofthe theory away from the read
ers, and makes itdifficultfor them to develop the no-arbitrageargument which
they may need in a different problem in application.
Thediscretetime approachofthis book circumventsthese mathematicaldif
ficulties ofthe continuous time approach, and enables the readers to penetrate
more easilythe coreofthe no-arbitragetheory. In the discretetime approach,a
timeunit for analysis, say h, ischoseninadvanceinterms ofyear. Timepoints
are then denoted by n = 0,1,... ,N, where 0 is an initial time for analysis,
often regarded as present time, and N is a terminal time for analysis, such as
the maturities ofderivatives and bonds. Time N corresponds to Nh yearfrom
O. For example when the time unit is taken to be a second, that is,
h = 1/365 x 24 x 60 x 60year,