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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book provides a detailed study of Financial Mathematics. In addition to the extraordinary depth the book provides, it offers a study of the axiomatic approach that is ideally suited for analyzing financial problems. This book is addressed to MBA's, Financial Engineers, Applied Mathematicians, Banks, Insurance Companies, and Students of Business School, of Economics, of Applied Mathematics, of Financial Engineering, Banks, and more.
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Veröffentlichungsjahr: 2013
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Table of Contents
Preface
Part I. Deterministic Models
Chapter 1. Introductory Elements to Financial Mathematics
1.1. The object of traditional financial mathematics
1.2. Financial supplies. Preference and indifference relations
1.3. The dimensional viewpoint of financial quantities
Chapter 2. Theory of Financial Laws
2.1. Indifference relations and exchange laws for simple financial operations
2.2. Two variable laws and exchange factors
2.3. Derived quantities in the accumulation and discount laws
2.4. Decomposable financial lawas
2.5. Uniform financial laws: mean evaluations
2.6. Uniform decomposable financial laws: exponential regime
Chapter 3. Uniform Regimes in Financial Practice
3.1. Preliminary comments
3.2. The regime of simple delayed interest (SDI)
3.3. The regime of rational discount (RD)
3.4. The regime of simple discount (SD)
3.5. The regime of simple advance interest (SAI)
3.6. Comments on the SDI, RD, SD and SAI uniform regimes
3.7. The compound interest regime
3.8. The regime of continuously comound discount (CCD)
3.9. Complements and exercises on compound regimes
3.10. Comparison of laws of different regimes
Chapter 4. Financial Operations and their Evaluation: Decisional Criteria
4.1. Calculation of capital values: fairness
4.2. Retrospective and prospective reserve
4.3. Usufruct and bare ownership in “discrete” and “continuous” cases
4.4. Methods and models for financial decisions and choices
4.5. Appendix: outline on numberical methods for the solution of equations
Chapter 5. Annuities-Certain and their Value at Fixed Rate
5.1. General aspects
5.2. Evaluation of constant installment annuities in the compound regime
5.3. Evaluation of constant installment annuities according to linear laws
5.4. Evaluation of varying installment annuities in the compound regime
5.5. Evaluation of varying installment annuities according to linear laws
Chapter 6. Loan Amortization and Funding Methods
6.1. General features of loan amortization
6.2. General loan amortization at fixed rate
6.3. Life amortization
6.4. Periodic funcing at fixed rate
6.5. Amortizations with adjustment of rates and values
6.6. Valuation of reserves in unshared loans
6.7. Leasing operation
6.8. Amortizations of loans shared in securities
6.9. Valuation in shared loans
Chapter 7. Exchanges and Prices on the Financial Market
7.1. A reinterpretation of the financial quantities in a market and price logic: the perfect market
7.2. Spot contracts, price and rates. Yield rate
7.3. Forward contracts, prices and rates
7.4. The implicit structure of prices, rates and intensities
7.5. Term structures
Chapter 8. Annuities, Amortizations and Funding in the Case of Term Structures
8.1. Capital value of annuities in the case of term structures
8.2. Amortizations in the case of term structures
8.3. Updating of valuations during amortization
8.4. Funding in term structure environments
8.5. Valuations referred to shared loans in term structure environments
Chapter 9. Time and Variability Indicators, Classical Immunization
9.1. Main time indicators
9.2. Variability and dispersion indicators
9.3. Rate risk and classical immunization
Part II. Stochastic Models
Chapter 10. Basic Probabilistic Tools for Finance
10.1. The sample space
10.2. Probability space
10.3. Random variables
10.4. Expectation and independence
10.5. Main distribution probabilities
10.6. Conditioning
10.7. Stochastic processes
10.8. Martingales
10.9. Brownian motion
Chapter 11. Markov Chains
11.1. Definitions
11.2. State classification
11.3. Occupation times
11.4. Absorption probabilities
11.5 Asymptotic behavior
11.6 Examples
Chapter 12. Semi-Markov Processes
12.1. Positive (J-X) processes
12.2. Semi-Markov and extended semi-Markov chains
12.3. Primary properties
12.4. Examples
12.5. Markov renewal processes, semi-Markov and associated counting processes
12.6. Particular cases of MRP
12.7. Markov renewal functions
12.8. The Markov renewal equation
12.9. Asymptotic behavior of an MRP
12.10. Asymptotic behavior of SMP
12.11. Non-homogenous Markov and semi-Markov processes
Chapter 13. Stochastic or ItÔ Calculus
13.1. Problem of stochastic integration
13.2. Stochastic integration of simple predictable processes and semi-martingales
13.3. General definition of the stochastic integral
13.4. ItÔ’s formula
13.5. Stochastic integral with standard Brownian motion as integrator process
13.6. Stochastic differentiation
13.7. Back to ItÔ’s formula
13.8. Stochastic differential equations
13.9. Diffusion processes
Chapter 14. Option Theory
14.1. Introduction
14.2. The Cox, Ross, Rubinstein (CRR) or binomial model
14.3. The Black-Scholes formula as the limit of the binomial model
14.4. The Black-Scholes continuous time model
14.5. Exercises on option pricing
14.6. The Greek parameters
14.7. The impact of dividend repartition
14.8. Estimation of the volatility
14.9. Black-Scholes on the market
14.10. Exotic options
14.11. The formula of Barone-Adesi and Whaley (1987): formula for American options
Chapter 15. Markov and Semi-Markov Option Models
15.1. The Janssen-Manca model
15.2. The extension of the Black-Scholes pricing formula with a semi-Markov environment: the Janssen-Manca-Volpe formula
15.3. Markov and semi-Markov option pricing models with arbitrage possibility
Chapter 16. Interest Rate Stochastic Models — Application to the Bond Pricing Problem
16.1. The bond investments
16.2. Dynamic deterministic continuous time model for instantaneous interest rate
16.3. Stochastic continuous time dynamic model for instantaneous interest rate.
16.4. Zero-coupon pricing under the assumption of no arbitrage
16.5. Appendix (solution of the OUV equation)
Chapter 17. Portfolio Theory
17.1. Quantitative portfolio management
17.2. Notion of efficiency
17.3. Exercises
17.4. Markowitz theory for two assets
17.5. Case of one risky asset and one non-risky asset
Chapter 18. Value at Risk (VaR) Methods and Simulation
18.1. VaR of one asset
18.2. Coherence and VaR extensions
18.3. VaR of an asset portfolio
18.4. VaR for one plain vanilla option
18.5. VaR and Monte Carlo simulation methods
Chapter 19. Credit Risk or Default Risk
19.1. Introduction
19.2. The Merton model
19.3. The Longstaff and Schwartz model (1995)
19.4. Construction of a rating with Merton’s model for the firm
19.5. Discrete time semi-Markov processes
19.6. Semi-Markov credit risk models
19.7. NHSMP with backward conditioning time
19.8. Examples
Chapter 20. Markov and Semi-Markov Reward Processes and Stochastic Annuities
20.1. Reward processes
20.2. Homogenous and non-homogenous DTMRWP
20.3. Homogenous and non-homogenous DTSMRWP
20.4. MRWP and stochastic annuities
20.5. DTSMRWP and generalized stochastic annuities (GSA)
References
Index
First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
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© ISTE Ltd, 2009
The rights of Jacques Janssen, Raimondo Manca and Ernesto Volpe to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Cataloging-in-Publication Data
Janssen, Jacques, 1939-
Mathematical finance : deterministic and stochastic models / Jacques Janssen, Raimondo Manca, Ernesto Volpe.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-081-3
1. Finance--Mathematical models. 2. Stochastic processes. 3. Investments--Mathematics. I. Manca, Raimondo. II. Volpe, Ernesto. III. Title.
HG106.J33 2008
32.01'51922--dc22
2008025117
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN: 978-1-84821-081-3
ISBN 13: 978-1-905209-54-5
Preface
This book, written as a treatise on mathematical finance, has two parts: deterministic and stochastic models.
The first part of the book, managed by Ernesto Volpe di Prignano, aims to give a complete presentation of the concepts and models of classical and modern mathematical finance in a mainly deterministic environment. Theoretical aspects and economic, bank and firm applications are developed.
The most important models are presented in detail after the formalization of an axiomatic theory of preferences. This performs the definition of “interest” and the financial regimes, which are the basis of financial evaluation and control the models. They are applied by means of clarifying examples with the solutions often obtained by Excel spreadsheet.
Chapter 1 shows how the fundamental definitions of the classical financial theory come from the microeconomic theory of subjective preferences, which afterwards become objective on the basis of the market agreements. In addition, the concepts of interest such as the price of other people’s money availability, of financial supply and the indifference curve are introduced.
Chapter 2 develops a strict mathematical formalization on the financial laws of interest and discount, which come from the postulates defined in Chapter 1. The main properties, i.e. decomposability and uniformity in time, are shown.
Chapter 3 shows the most often used financial law in practice. The most important parametric elements, such as interest rates, intensities and their relations, are defined. Particular attention is given to the compound interest and discount laws in different ways. They find wide application in all the pluriennial financial operations.
Chapter 4 gives the concept of discrete time financial operation as a set of financial supplies, of operation value, of fair operation, of retrospective and prospective reserve at a given time, of the usufruct and bare ownership. In addition, a detailed classification of the financial projects based on their features is given. The decision and choice methods among projects are deeply developed. In the appendix to this chapter, a short summary of simple numerical methods, particularly useful to find the project internal rate, is reported.
Chapter 5 discusses all versions of the annuity operations in detail, as a particular case of financial movement with the same sign. The annuity evaluations are given using the compound or linear regime.
Chapter 6 is devoted to management mathematical procedures of financial operations, such as loan amortizations in different usual cases, the funding, the returns and the redemption of the bonds. Many Excel examples are developed. the final section is devoted to bond evaluations depending on a given rate or on the other hand to the calculus of return rates on bond investments.
In Chapters 7 and 8, the financial theory is reconsidered assuming variable interest rates following a given term structure. Thus, Chapter 7 defines spot and forward structures and contracts, the implicit relations among the parameters and the transforming formulae as well. Such developments are carried out with parameters referred to real and integer times following the market custom. Chapter 8 discusses the methods developed in Chapters 5 and 6 using term structures.
Chapter 9 is devoted to definition and calculus of the main duration indexes with examples. In particular, the importance of the so-called “duration” is shown for the approximate calculus of the relative variation of the value depending on the rate. However, the most relevant “duration” application is given in the classical immunization theory, which is developed in detail, calculating the optimal time of realization and showing in great detail the Fisher-Weil and Redington theorems.
The second part of the book, managed by Jacques Janssen and Raimondo Manca, aims to give a modern and self-contained presentation of the main models used in so-called stochastic finance starting with the seminal development of Black, Scholes and Merton at the beginning of the 1970s. Thus, it provides the necessary follow up of our first part only dedicated to the deterministic financial models.
However, to help in assuring the self-containment of the book, the first four chapters of the second part provide a summary of the basic tools on probability and stochastic processes, semi-Markov theory and Itô’s calculus that the reader will need in order to understand our presentation.
Chapter 10 briefly presents the basic tools of probability and stochastic processes useful for finance using the concept of trajectory or sample path often representing the time evolution of asset values in stock exchanges.
Chapters 11 and 12 summarize the main aspects of Markov and semi-Markov processes useful for the following chapters and Chapter 13 gives a strong introduction to stochastic or Itô’s calculus, being fundamental for building stochastic models in finance and their understanding.
With Chapter 14, we really enter into the field of stochastic finance with the full development of classical models for option theory including a presentation of the Black and Scholes results and also more recent models for exotic options.
Chapter 15 extends some of these results in a semi-Markov modeling as developed in Janssen and Manca (2007).
With Chapter 16, we present another type of problem in finance, related to interest rate stochastic models and their application to bond pricing. Classical models such as the Ornstein-Uhlenbeck-Vasicek, Cox-Ingersoll-Ross and Heath-Jarrow-Morton models are fully developed .
Chapter 17 presents a short but complete presentation of Markowitz theory in portfolio management and some other useful models.
Chapter 18 is one of the most important in relation to Basel II and Solvency II rules as it gives a full presentation of the value at risk, called VaR, methodology and its extensions with practical illustrations.
Chapter 19 concerns one of the most critical risks encountered by banks: credit or default risk problems. Classical models by Merton, Longstaff and Schwartz but also more recent ones such as homogenous and non-homogenous semi-Markov models are presented and used for building ratings and following the time evolution.
Finally, Chapter 20 is entirely devoted to the presentation of Markov and semi-Markov reward processes and their application in an important subject in finance, called stochastic annuity.
As this book is written as a treatise in mathematical finance, it is clear that it can be read in sections in a variety of sequences, depending on the main interest of the reader.
This book addresses a very large public as it includes undergraduate and graduate students in mathematical finance, in economics and business studies, actuaries, financial intermediaries, engineers but also researchers in universities and RD departments of banking, insurance and industry.
Readers who have mastered the material in this book will be able to manage the most important stochastic financial tools particularly useful in the application of the rules of governance in the spirit of Basel II for banks and financial intermediaries and Solvency II for insurance companies.
Many parts of this book have been taught by the three authors in several universities: Université Libre de Bruxelles, Vrije Universiteit Brussel, University of West Brittany (EURIA) (Brest), Télécom-Bretagne (Brest), Paris 1 (La Sorbonne) and Paris VI (ISUP) Universities, ENST-Bretagne, University of Strasbourg, Universities of Rome (La Sapienza), Napoli, Florence and Pescara.
Our common experience in the field of solving financial problems has been our main motivation in writing this treatise taking into account the remarks of colleagues, practitioners and students in our various lectures.
We hope that this work will be useful for all our potential readers to improve their method of dealing with financial problems, which always are fascinating.
Part I
Deterministic Models
Chapter 1
Introductory Elements to Financial Mathematics
1.1. The object of traditional financial mathematics
The object of traditional financial mathematics is the formalization of the exchange between monetary amounts that are payable at different times and of the calculations related to the evaluation of the obligations of financial operations regarding a set of monetary movements.
The reasons for such movements vary and are connected to: personal or corporate reasons, patrimonial reasons (i.e. changes of assets or liabilities) or economic reasons (i.e. costs or revenues). These reasons can be related to initiatives regarding any kind of goods or services, but this branch of applied mathematics considers only the monetary counterpart for cash or assimilated values1.
The evaluations are founded on equivalences between different amounts, paid at different times in certain or uncertain conditions. In the first part of this book we will cover financial mathematics in a deterministic context, assuming that the monetary income and outcome movements (to which we will refer as “payment” with no distinction) will happen and in the prefixed amount. We will not consider in this context decision theory in uncertain conditions, which contains actuarial mathematics and more generally the theory of random financial operations2 .
We suppose that from now, unless otherwise specified, the deterministic hypotheses are valid, assuming then – in harmony with the rules of commonly accepted economic behavior – that:
a) the ownership of a capital (a monetary) amount is advantageous, and everyone will prefer to have it instead of not having it, whatever the amount is;
b) the temporary availability of someone else’s capital or of your own capital is a favorable service and has a cost; it is then fair that whoever has this availability (useful for purchase of capital or consumer goods, for reserve funds, etc.) pays a price, proportional to the amount of capital and to the time element (the starting and closing dates of use, or only its time length).
The amount for the aforementioned price is called interest. The parameters used for its calculation are calculated using the rules of economic theory.
1.2. Financial supplies. Preference and indifference relations
1.2.1. The subjective aspect of preferences
Referring to one of the contracting parties, S has an algebraic sign which refers to the cash flow; it is positive if it is an income and negative if it is an outcome, and the unit measure depends on the chosen currency. Furthermore, the time (or instant) can be represented as abscissas on an oriented temporal axis so as to have chronological order. The time origin is an instant fixed in a completely discretionary way and the measure unit is usually a year (but another time measure can be used). Therefore, even the times X, Y, etc., have an algebraic sign, which is negative or positive according to their position with respect to the time origin. It follows that “X<Y” means “time X before time Y”.
From a geometric viewpoint, we introduce in the plane Σ(2) the Cartesian orthogonal reference system OXS (with abscissas X and ordinate S). Σ(2) is then made of the points P ≡ [X,S] that represent the supply (X,S), that is the amount S dated in X.
As a consequence of the postulates a) and b), the following operative criteria can be derived:
c) given two financial supplies (X,S1) and (X,S2) at the same maturity date X, the one with the higher (algebraically speaking) amount is preferred;
d) given two financial supplies (X,S) and (Y,S) with the same amount S and valued at instant Z before both X and Y, if S>0 (that is, from the cashing viewpoint) the supply for which the future maturity is closer to Z is preferred; if S<0 (that is, from the paying viewpoint) the supply with future maturity farther from Z is preferred. More generally ∀Z3, with two supplies having the same amount, the person who cashes (who pays) prefers the supply with prior (with later) time of payment.
Formulations c) and d) express criteria of absolute preference in the financial choices and clarify the meaning of interest. In fact, referring to a loan, where the lender gives to the borrower the availability of part of his capital and its possible use for the duration of the loan, the lender would perform a disadvantageous operation (according to postulate a) and b) and criteria c) if, when the borrower gives back the borrowed capital at the fixed maturity date, he would not add a generally positive amount to the lender, which we called interest, as a payment for the financial service.
The decision maker’s behavior is then based on preference or indifference criteria, which is subjective, in the sense that for them there is indifference between two supplies if neither is preferred.
To provide a better understanding of these points, we can observe that:
– the decision maker expresses a judgment of strong preference, indicated with , of the supply (X1,S1) compared to (X2,S2) if he considers the first one more advantageous than the second; we then have (X1,S1)(X2,S2);
– the decision maker expresses a judgment of weak preference, indicated with , of the supply (X1,S1) compared to (X2,S2), if he does not consider the second one more advantageous than the first; we then have (X1,S1)(X2,S2)4 .
The amplitude of the set of supplies comparable with a given supply for a preference judgment depends on the criteria on which the judgment is based.
Criteria c) and d) make it possible to establish a preference or no preference of (X0,S0), but only with respect to a subset of all possible supplies, as we show below.
From a geometric point of view, let us represent the given supply (X0,S0) on the plane Σ(2), with reference system OXS, by the point P0 ≡[X0,S0]. Then, considering the four quadrants adjacent to P0, based only on criteria c) and d), it turns out that:
1) Comparing S0>0 to supplies with a positive amount, identified by the points Pi (i=1,…,4) (see Figure 1.1), being incomes, it is convenient to anticipate their collection. Therefore, all points P2 ≡[X2,S2] in the 2nd quadrant (NW) are preferred to P0 because they have income S2 greater than S0 and are available at time X2 previous to time X0; whereas P0 is preferred to all points P4 ≡[X4,S4] in the 4th quadrant (SE) because they have income S4 smaller than S0 and are available at time X4 later than X0; it is not possible to conclude anything about the preference between P0 and points P1 ≡[X1,S1] in the 1st quadrant (NE) or points P3 ≡[X3,S3] in the 3rd quadrant (SW).
Figure 1.1.Preferences with positive amounts
2) Comparing S0<0 to supplies with a negative amount, identified by the points Pi (i=1,…,4) (see Figure 1.2), being outcomes, it is convenient to postpone their time of payment. Therefore all points P1 ≡[X1,S1] in the 1st quadrant (NE) are preferred to P0 because they have outcome S smaller than S0 and are payable at time X later than X0; whereas P0 is preferred to all points P3 ≡[X3,S3] in the 3rd quadrant (SW) because they have outcome S3 greater than S0 and are payable at time X3, which is later than X0. Nothing can be concluded on the preference between P0 and all points P2 ≡[X2,S2] of the 2nd quadrant (NW) or all points P4 ≡[X4,S4] of the 4th quadrant (SE).
Briefly, on the non-shaded area in Figures 1.1 and 1.2 it is possible to establish whether or not there is a strong preference with respect to P0, while on the shaded area this is not possible.
To summarize, indicating the generic supply (X,S) also with point P ≡[X,S] in the plane OXS, we observe that an operator, who follows only criteria c) and d) for his valuation and comparison of financial supplies, can select some supplies P’ with dominance on P0 (we have dominance of P′ on P0 when the operator prefers P’ to P0) and other supplies P″ dominated by P0 (when he prefers P0 to P″), but the comparability with P0 is incomplete because there are infinite supplies P'" not comparable with P0 based on criteria c) and d). To make the comparability of P0 with the set of all financial supplies complete, corresponding to all points in the plane referred to OXS, it is necessary to add to criteria c) and d) – which follow from general behavior on the ownership of wealth and the earning of interest – rules which make use of subjective parameters. The search and application of such rules – to fix them external factors must be taken into account, summarized in the “market”, making it possible to decide for each supply if it is dominant on P0, indifferent on P0 or dominated by P0 – is the aim of the following discussion.
Figure 1.2.Preferences with positive amounts
To achieve this aim it is convenient to proceed in two phases:
1) the first phase is to select, in the zone of no dominance (shaded in Figures 1.1 and 1.2), the supplies P* ≡[X*,S*] with different times of payment from that of P0 and in indifference relation with P0;
2) the second phase, according to the transitivity of preferences, is to select the advantageous and disadvantageous preferences with respect to P0, with any maturity.
In the first phase, we can suppose an opinion poll on the financial operator to estimate the amount B payable in Y that the same operator evaluates in indifference relation, indicated through the symbol ≈, with the amount A payable in X. For such an operator we will use:
(1.1)
Given the supply (X,A), on varying Y the curve obtained by the points that indicate the supplies (Y,B) indifferent to (X,A), or satisfying (1.1), is called the indifference curve characterized by point [X,A].
From an operative viewpoint, if two points P' ≡[X,A] and P" ≡[Y,B] are located on the same indifference curve, the corresponding supplies (X,A) and (Y,B) are exchangeable without adjustment by the contract parties.
If (1.1) holds, according to criteria c) and d), the amounts A and B have the same sign and |B|-|A| has the same sign of Y-X. The fixation of the indifferent amounts can proceed as follows, as a consequence of the previous geometric results (see Figures 1.1 and 1.2).
Let us denote by P0 ≡ [X0,S0] the point representing the supply for which the indifference is searched. Then:
– if S0>0 (see Figure 1.3), with X=X0 , Y=X1>X0, the rightward movement from P0 to A1 ≡ [X1,S0] is disadvantageous because of the income delay; to remove such disadvantage the amount of the supply must be increased. The survey, using continuous increasing variations, fixes the amount S1>S0 which gives the compensation, where P0 and P1 ≡[X1,S1], obtained from A1 moving upwards, and represents indifferent supply (or, in brief, P1 and P0 are indifferent points). Instead, if Y=X3<X0, the leftwards movement from P0 to A3 ≡[X3,S0] is advantageous for the income anticipation; therefore, in order to have indifference, there needs to be a decrease in the income from S0 to S3, obtained through a survey with downward movement of the indifference point P3 ≡ [X3,S3] with S3<S0;
Figure 1.3.Indifference curve assessment – positive amounts
Figure 1.4.Indifference curve assessment – negative amounts
– if S0<0 (see Figure 1.4), since the delay of outcome is advantageous and its anticipation is disadvantageous, proceeding in a similar way starting from A2 ≡[X2,S0] and A4 ≡[X4,S0], the points (indifferent to P0) P2 ≡[X2,S2], with X2<X0, S2>S0, are obtained through leftwards and then upwards movement, or P4 ≡[X4,S4], with X4>X0, S4<S0, through rightwards and then downwards movement.
Continuously increasing or decreasing the abscissas Xi (i=1,3), we obtain, if S0>0, a continuous curve with increasing ordinate in the plane OXS, resulting from P0 and the points of type P1 and P3, all indifferent to P0. If S0<0, the continuous curve resulting by P0 and the points of type P4 and P2 , all indifferent to P0, obtained by continuously varying Xi (i=2,4), have a decreasing ordinate5. However, if P0 is fixed, these curves of indifference are individualized from P0 by definition.
We can now define, in general terms, the interest defined in section 1.1, considering only the positive amount. If (1.1) holds with X<Y, the exchange between indifferent supplies implies that giving away the availability of amount A from X to Y is fairly compensated by the payment of the amount
(1.2)
We will say that A is the invested principal, I is the interest, and B is the accumulated value, in an operation of lending or investment.
If (1.1) holds with X>Y, the anticipation of the income of A from X to Y is fairly compensated by the payment in Y of the amount
(1.3)
We will say that A is the capital at maturity, D is the discount and B is the present value or discounted value, in an operation of discounting or anticipation.
The second phase is applied in an easy way. It is enough to add, referring to (1.1) in the case A>0, that if a generic P ≡[Y,B] is indifferent to a fixed P0 ≡[X,A] then all the points P’ ≡[Y,B’] where B’>B are preferred to P0, while P0 is preferred to all points P” ≡[Y,B”] where B”< B. This leads to the conclusion that, once the indifference curve through P0 is built, all the supplies of the type (Y,B') are preferred to the supply (X,A), while the opposite occurs for all supplies of type (Y,B").
1.2.2. Objective aspects of financial laws. The equivalence principle
The previous considerations enable us to give a first empirical formulation of the fundamental “principle of financial equivalence”, which is that it is equivalent6 to a
cash (pay) amount today or to cash (pay) at a later time if there is the cashing (payment) of the interest for such deferment.
In Chapter 2, the indifference curves and the principle of financial indifference will be formalized in objective terms, defining financial factors, rates and intensities for lending and discounting operations, in relation to the possible distribution of interest payments in the deferment period. The equivalence principle will then become objective, assuming the hypothesis that different parties to a financial contract agree in fixing a rule, valid for them, to calculate the equivalent amount B, according to the amount A and the times X, Y.
1.3. The dimensional viewpoint of financial quantities
In financial mathematics, as in physics, it is necessary to introduce, together with numerical measures, a dimensional viewpoint distinguishing between fundamental quantities and derived quantities.
To describe the laws of mechanics, the oldest of the physical sciences, the following fundamental quantities are introduced: length l, time t, mass m, with their units (meter, second, mass-kilogram) and the derived quantities are deduced, such as volume l3, velocity l/t, acceleration l/t2, force ml/t2, etc. Their units are derived from those of the fundamental quantity. We then speak about the physical dimension of different quantities, which are completely defined when they are given the dimensions and the numbers which represent the measurement of the given quantity in the unit system.
In financial mathematics we also make a distinction between fundamental quantities and derived quantities.
The fundamental quantities are:
1) monetary amount (m), to measure the value of financial transaction in a given unit (i.e., dollar, euro, etc.);
2) time (t), to measure the length of the operation and the delay of its maturity in a given unit (i.e. year).
The derived quantities, relating to the fundamental quantities based on dimensions, are:
1) flow, defined as amount over time (then with dimension m1t-1);
2) rate, defined as amount over amount (thus a “pure number”, with dimension m0t0);
3) intensity, defined as amount over the product of amount multiplied by time (then with dimension m0t-1).
To clarify:
– flow relates the monetary amount to the time interval in which it is produced; a typical flow is the monetary income (i.e.: wages, fees, etc.) expressed as the monetary amount matured in a unit of time as a consequence of the considered operation;
– rate relates two amounts which are connected and thus is a “pure number” without dimension; for example, the rate is the ratio between matured interest and invested principal;
– intensity, obtained as the ratio between rate and time or flow and amount, takes into account the time needed for the formation of an amount due to another amount; for example, the ratio between interest and invested principal time length of the investment.
This is all summarized in the following dimensional table where we go from left to right, dividing by a “time” and from top to bottom, dividing by an “amount”.
Table 1.1.Financial dimensions
1 The reader familiar with book-keeping concepts and related rules knows that each monetary movement has a real counterpart of opposite movement: a payment at time x (negative financial amount) finds the counterpart in the opening of a credit or in the extinction of a debt. In the same way, a cashing (positive financial amount) corresponds to a negative patrimonial variation or an income for a received service. The position considered here, in financial mathematics, looks to the undertaken relations and the economic reasons for financial payments.
2 In real situations, which are considered as deterministic, the stochastic component is present as a pathologic element. This component can be taken into account throughout the increase of some earning parameter or other artifices rather than introducing probabilistic elements. These elements have to be considered explicitly when uncertainty is a fundamental aspect of the problem (for example, in the theory of stochastic decision making and in actuarial mathematics). We stress that in the recent development of this subject, the aforementioned distinction, as well as the distinction between “actuarial” and “financial” mathematics, is becoming less important, given the increasing consideration of the stochastic aspect of financial problems.
3 It is known that the symbol ∀ has the meaning “for all”.
4 The judgment of weak preference is equivalent to the merging of strong preference of (X1,S1) with respect to (X2,S2) and of (X2,S2) with respect to (X1,S1). In other words:
The economic logic behind the postulates a), b), from which the criteria c), d) follow, implies that the amounts for indifferent supply have the same sign (or are both zero).
5 If criteria d is removed, supplies with same amount and different time become indifferent and the indifference curves have constant ordinate. All loans without interest made for free are contained in this category.
6 “Equivalent” is often used instead of “indifferent”; if this does not make sense then imagine that P’ equivalent to P” means that these supplies are in the same equivalence class as in the set theory meaning. For this to be true, other conditions are needed. which we will discuss later.
Chapter 2
Theory of Financial Laws
2.1. Indifference relations and exchange laws for simple financial operations
Let us consider again the indifference relation, indicated by ≈ in (1.1), which depends on the judgment of an economic operator which gives rise to indifferent supplies with the process described in section 1.2.
In a loan operation of the amount S at time T the economic operator can calculate the repayment value S′ in T′> T such that (T′,S′)≈(T,S). Therefore, S′ ≥ S is calculated according to a function (subjective) of S, T, T′ and it is written as
(2.1)
where fc is the accumulation function (given that in S′ the repayment of S and the incorporation of the possible interest is included) that realizes indifference.
In a discounting operation, at time T″ <T′, of amount S′ with maturity T′, let S″ ≤ S′ be the discounted value so that subjectively (T″,S″) ≈ (T′,S′). We then have
(2.2)
where fa is the discounting function (because S′ is discounted at time T″ with a possible reduction due to anticipation of availability) that realizes indifference.
It is obvious that if two operators, one at each side of a loan or discounting contract, want to realize an advantageous contract according to their preference scale, it is not always possible for them to do so.
It can be the case that, in a loan in T′ of the principal S′, indicating by S″a the indifferent accumulated amount (= min acceptable) for the lender to cash in T″ and by S″b the indifferent accumulated amount (= max acceptable) for the borrower to pay out in T″, if S″b<S″a the contract is not stipulated. In the same way, we can prove that, in a discounting operation of the capital S′ at maturity T′, indicating by S″a the present indifferent value (= max acceptable) for the lender to pay out in T″ < T ′ and by S″b the present indifferent value (= min acceptable) for the borrower to cash in T″ < T ′, if S″a < S″b the contract is not stipulated.
EXAMPLE 2.1.– Let us suppose that Mr. Robert, who is lending the amount S′ at time T′ for the period (T′,T″), wants to cash in T″ at least 1.09.S′. At the same time Mr. George, who is borrowing S′ for the same time interval, wants to pay back in T″ no more than 1.07.S′. It is obvious that in this way they will not proceed with the loan contract. Indeed:
– with S″ < 1.07.S′, the lender prefers not to lend;
– with 1.07.S′ <S″ < 1.09.S′, the lender prefers not to lend and the borrower prefers not to borrow;
– with S″> 1.09.S′, the borrower prefers not to borrow.
EXAMPLE 2.2.– Let us suppose that Mr. John wants to discount a bill from Mr. Tom, which is amount S′ for the time from T′ to T″<T′ offering a discounted value not greater than 0.92.S′, while Mr. Tom wants to offer this discount for an amount not lower than 0.94.S′. It is clear that the contract cannot be reached, because each discounted amount is considered disadvantageous by at least one of the parties.
To further consider the economic theory of market prices, we carry on our analysis using objective logic and supposing that the operators, in a specific market, want a fair contract between two supplies (T,S) and (T′,S′) in a loan, if their fundamental quantities satisfy equation (2.1); and in the same way, for a discount, which is a type of loan, if equation (2.2) is satisfied. We will now talk about a fair contract if equation (2.1) or equation (2.2) is satisfied, but as favorable (or unfavorable) for one of the parties if the equations are not satisfied. Trade contracts between two supplies (T′,S′) and (T″,S″) give rise to simple financial operations. As already mentioned in Chapter 1:
– if T″ > T′ (= loan or investment), the parties consider fair the interest S″-S′ as the payment for the lending of S′ from T′ to T″, as delayed payment in T″; then S″ is called accumulated amount in T″ of the amount S′ lent in T′;
– if T″ < T′ (= discount or anticipation), both parties consider fair the interest S′-S″ for the discount of S′ from T′ to T″, as advance payment in T″; then S″ is called discounted value from time T″ of the amount S′ to maturity T′1.
The indifference relation thus assumes a collective value. The function fc defined in equation (2.1) is an accumulation law (or interest law), while the function fa defined in equation (2.2) is a discount law. Referring now to the case of positive interest and fixing S and T in equation (2.1), the value S′ is an increasing function of T′; fixing S′ and T′ in equation (2.2), and the value S″ is also an increasing function of T″, because it decreases when T" decreases.
Applying equation (2.1) and then equation (2.2) with T″ =T, we obtain the present value in T of the accumulated amount in T′ of S invested in T ≤ T′, given by
(2.3)
Unifying the cases T ≤ T′ and T>T′, we can talk of an exchange law given by a function f that gives the amount S′ payable in T′ and exchangeable2 with S payable in T. It follows that
(2.4)
Let us consider some properties of the indifference relation ≈:
1) reflexive property
If ∀(T,S) we have (T,S ) ≈ (T,S), we will say that ≈ satisfies the reflexive property3;
2) symmetric property
If ∀(S,T,T′), from (T,S) ≈ (T’,S′) follows (T’,S′) ≈ (T,S), we will say that ≈ satisfies the symmetric property4;
3) property of proportional amounts
If ∀(S,T,T′), ∀k>0, from (T,S )≈ (T′,S′) follows (T,kS) ≈ (T′,kS′), we will say that ≈ satisfies the property of proportional amounts.
Because of criteria c) and d), if T′-T the amount in T′ exchangeable with S in T is the same as S. Therefore in the set of financial supplies the relation ≈ always satisfies the reflexive law. We can then define the exchange law for all three variables as
(2.5)
If the symmetric law holds in the considered set P, then
(2.6) 5
In this case, recalling (2.3), the laws fc and fa are conjugated, and because of (2.4), (2.5) can be written in the form
which remains valid with the same f if the primed values are changed with the unprimed values and vice versa6.
If, in the considered set , the property of proportional amounts holds, f as defined in (2.4) is linear homogenous compared to the amount7.
2.2. Two variable laws and exchange factors
Let us continue the analysis of exchange laws the reflexive and proportional amount properties assumed to be valid for ≈. Due to the second property, it is possible to transform (2.1) in the multiplicative form
(2.1′)
where m(T,T′), increasing with respect to T′, is called the accumulation factor and expresses the accumulation law only as a function of the two temporal variables; in the same way it is possible to transform (2.2) in the form
(2.2′)
where a(T′,T"), increasing with respect to T", is called the discounted factor and expresses the discounting law only as a function of the two temporal variables. We will now address the two variables laws.
The reflexive law for ≈ is now equivalent to
(2.7)
Furthermore if, using T″ =T in systems (2.1′) and (2.2′), we obtain S″ =S, i.e. the symmetric property is valid for ≈, the laws m(.) and a(.) satisfy
(2.8)
Equation (2.8) shows that conjugated laws for the same time interval give rise to reciprocal factors.
When describing (2.4) in detail, we consider the exchange law of two variables characterized by the exchange factor z(X,Y), a pure number increasing with respect to Y, defined using
(2.5′)
(2.5′) being a particular case of (2.5).
We will now assume that
(2.5″)
(considering, if needed, only the part of the definition set for the function z where such a condition holds) in order that it cannot be possible that an encashment (payment) can never be indifferent to a payment (encashment) with different time maturity.
If relation ≈ expressed by z(X,Y) satisfies the symmetric property, as a particular case of (2.6′) the below condition follows:
(2.9)
If z(X,Y) satisfies (2.9), then it defines a couple of two-variable financial interest and discount laws which are conjugated.
2.3. Derived quantities in the accumulation and discount laws
In light of the laws defined in (2.1′) and (2.2′), we can deduce the following derived quantities11.
2.3.1. Accumulation
As a function of the initial accumulation factor
(2.10)
(:= means “equal by definition”) – which measures the multiplicative increment from X to Y>X of the invested capital in X. The factor is “initial” because the date X of investment coincides with the beginning of the time interval (X,Y) on which such an increment is measured. We can also define (see Figure 2.1):
– the initial interest (per period) rate (= interest on the unitary invested capital in the time interval from X to Y>X) is expressed by
(2.11)
– the initial interest (per period) intensity, expressed by
(2.12)
Alternatively, still using X as the investment time and imposing X<Y<z, the capital increment is measured on a time interval (Y,Z) subsequent to X, then continuing with respect to interval (X,Y) without disinvesting in Y, we can then generalize and define continuing factors, rates and intensities in the following way:
– the continuing accumulation factor from Y to Z (= accumulated amount in Z=Y+u, u>0, of the unitary accumulated amount in Y=X+t, t>0, for the investment started in X) is expressed by
(2.13)
– the continuing interest (per period) rate from Y to Z (= interest for unitary accumulated amount in Y passing from Y to ZY+u, u>0, for the investment started in X) is expressed by
(2.14)
(2.15)
Figure 2.1.Times in accumulation
Intensity (2.15) is obtained by dividing the partial incremental ratio of function m(ξ,η), considered with ξ=X and respect to η from Y to Y +u, by m(X,Y). In the hypothesis that m(ξ,η) is partially differentiable with respect to η with a continuous derivative in the interesting interval, the right limit of (2.15) then exists when u→0, which represents the instantaneous interest intensity12(implying: continuing) in Y of an investment started in X, indicated by δ(X,Y). Using symbols, where “loge” is indicated with “ln”:
(2.16)
Working on the variables ξ, η , with ξ < η, it can be concluded that δ(ξ,η) is the logarithmic derivative (partial with respect to η) of m(ξ,η).
Inverting function δ and the derivative operator in (2.16), the important relationship is obtained for continuing accumulated amount (2.13) as a function of the instantaneous intensity13:
(2.16′)
2.3.2. Discounting
Let X be the final time of a financial operation (for example, the maturity of a credit). Analogously to accumulation, as a function of the initial discounting factor
(2.17)
we can also define (see Figure 2.2):
– the initial per period discounting rate (= discount for unitary capital at maturity for the anticipation from X to Y<x), given by
(2.18)
– the initial per period discounting intensity, which can be expressed by:
(2.19)
– the continuing discounting factor from Y to Z (= present value in Z<Y of the present unitary value in Y<x of the capital at maturity in X, then of amount 1/a(X,Y)), expressed by:
(2.20)
– the continuing discounting rate from Y to Z (= discount for the anticipation from Y to Z of the present unitary value in Y<x of a capital with maturity in X, then of amount 1/a(X,Y)), expressed by:
(2.21)
– the continuing discounting intensity from Y to Z, expressed by:
(2.22)
Figure 2.2.Times in discounting
Considering the limit as already calculated for the instantaneous interest intensity, it is possible to obtain:
– the instantaneous discounting intensity in Y, indicated by θ(X,Y) and given by:
(2.23)
As θ(X,Y) is the logarithmic derivative (partial with respect to Y ≤ X) of a(X,Y), by inverting the process we obtain, ∀Z < Y,
(2.24)
2.4. Decomposable financial laws
2.4.1. Weak and strong decomposability properties: equivalence relations
In the case of the financial law of two variables, we consider the meaning and the consequences of the decomposability property, which was introduced by Cantelli.
We have decomposability in an accumulation (or discounting) operation when investing (or discounting) a given capital available at time X, we have the same accumulated amount (or present value) in Z, both if we realize and reinvest immediately the obtained value in a intermediate time Y, or if we continue the financial operation. To summarize, decomposability means invariance of the result with respect to interruptions of the financial operation.
With reference to the interest law m(X,Y), which follows from relation ≈, and to the three times X, Y, Z, with X<Y<z, let S2 be the realized accumulated amount in Y of S1 invested in X; moreover, let S3 be the accumulated amount in Z of S2 immediately reinvested in Y. Instead S′3 is the accumulated amount Z after only one accumulation of S1 from X to Z. Due to (2.1′)
(2.25)
If
(2.26)
the interest law is decomposable. It follows from (2.25) that (2.26) is equivalent to
(2.27)
which expresses the decomposability condition for an interest law in terms of accumulation factors.
In the same way, referring to the discount law a(X,Y) following ≈ and recalling (2.2′), if X> Y> Z we can define the following discounted values starting from S1, payable in X:
(2.28)
If
(2.29)
the discount law is decomposable and because of (2.28) the decomposability condition for this law can be written as
(2.30)
Until now, we have defined in weak form the decomposability of single laws in accumulation or discounting operations, considering the times X, Y, Z in increasing or decreasing order. This signifies that we require the prospective transitivity or respectively the retrospective transitivity to the indifference relations, which give rise to the laws.14 In this case we will talk of weak decomposability.
If instead the previous considerations are related to an exchange law following an indifference relation ≈ and expressed by the factors z(X,Y) defined in (2.5′), we can think of extending the decomposability relation in (2.25) and (2.26) for any order of payment times. So the relation ≈ satisfies the strong decomposability property, which bi-implies
(2.31)
and then the following condition on the exchange factors:
(2.32)
THEOREM A.– If and only if for the exchange law the strong decomposability is valid, the relation ≈ is reflexive, symmetric and transitive, then it is an equivalence relation, which we denote by .
Proof
Note: this argument could be developed, in a more formally complicated, but equipollent way, based on relation (2.31).
Considering the relation between weak decomposability (WD) and strong decomposability (SD), it is obvious that the condition of SD implies WD, when X, Y, Z are in increasing or decreasing order from which there are only accumulation or discounting respectively. However, the WD does not imply SD in other cases, when both an accumulation and a discounting occur together. Then, if SD holds, the properties of an equivalence are immediately verified. In fact, considering X < Z < Y (analogously we could consider X> Z> Y), the SD expressed by (2.32) gives rise to
(2.33)
(2.34)
Let us summarize as follows. Given an indifference relation ≈ in the hypothesis of proportional amount, the strong decomposability, expressed by (2.32) for the exchange factor z(X,Y), implies that ≈ is reflexive, symmetric and transitive, and then it is an equivalence indicated by . In this case, the derived interest and discount laws are decomposable and conjugated to each other.
2.4.2. Equivalence classes: characteristic properties of decomposable laws
Based on theorem A, if an indifference relation ≈ gives rise to a strongly decomposable exchange law, it is an equivalence relation 15 between all elements (T,S) of the set of supplies, which makes it possible to separate such supplies into equivalence classes. Each class is made up of financially equivalent supplies, but which are indifferent. However, two supplies in different classes are not equivalent because it is possible to express a judgment of strong preference. Each class is characterized by an abstract, made up of the intrinsic financial value of its supplies.
By geometrically representing the supplies (T,S) on the plane OTS, a class of equivalent supplies is identified by a curve, a locus of points P ≡ [T,S], corresponding to equivalent supplies. The infinite curves do not have common points. In addition:
1) for each point in the plane there is one and only one curve, which is a locus of equivalent points;
The classes of equivalent supplies on the basis of an SD, i.e. the elements of the quotient set /, form a totally ordered set, because the elements of each couple are comparable for a weak preference judgment , using the meaning specified in section 1.2. Moving monotonically towards the classes (= curve in the plane OTS), the intrinsic financial value of the supplies improves in one sense (but gets worse in the other sense)16. It follows that the SD laws, on the basis of stratification in equivalence classes, allow a global, rather than just local, comparison between supplies, i.e. due to transitivity they make it possible to extend to any number of supplies on the plane OTS the preference or indifference relations introduced in Chapter 1 with respect to a given supply.
It is easy to give a method for such a comparison, verifying the existence of total order in . It is enough to identify the classes using supplies that have the same maturity T0; then class α identified by (T0,S′0) is preferred to class β identified by (T0,S″0) if S′0 > S″0; β is preferred to α if S′0 < S″0; α and β are equivalent if S′0S″0.
Let us consider some characteristic properties of decomposable laws of two variables, which proceed from the following theorems.
THEOREM B.– Referring to definitions (2.10) and (2.13), an interest law is weakly decomposable if and only if, for each choice of subsequent times X<y<z, the initial accumulation factor from Y to Z is equal to the continuing accumulation factor from Y to Z of an accumulation started in X. In symbols: r(X;Y,Z)=r(Y;Y,Z)=m(Y,Z). Therefore, the decomposability implies independence of r(X;Y,Z) from the time of investment, and vice versa. There is an analogous condition in relation to the discount factors (2.17) and (2.20), for each choice of time X>Y>Z holds for a weakly decomposable discount law.
THEOREM C.– An interest law is weakly decomposable if and only if the instantaneous intensity δ(X,T), continuous by hypothesis, does not depend on the initial time X but only on the current time T. The analogous condition on the intensity θ(X,T) holds for a weakly decomposable discount law. Under the same condition necessary and sufficient on the instantaneous intensity of interest anddiscount, a strong decomposability of an exchange law specified by the factor identified by (2.5′) which satisfies (
