Mathematical Finance. Deterministic and Stochastic Models. Jacques Janssen Raimondo Manca Ernesto Volpe di Prignano

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1 Mathematical Finance Deterministic and Stochastic Models Jacques Janssen Raimondo Manca Ernesto Volpe di Prignano

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3 Mathematical Finance

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5 Mathematical Finance Deterministic and Stochastic Models Jacques Janssen Raimondo Manca Ernesto Volpe di Prignano

6 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 Ltd John Wiley & Sons, Inc St George s Road 111 River Street London SW19 4EU Hoboken, NJ UK USA ISTE Ltd, 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 Library of Congress Cataloging-in-Publication Data Janssen, Jacques, Mathematical finance : deterministic and stochastic models / Jacques Janssen, Raimondo Manca, Ernesto Volpe. p. cm. Includes bibliographical references and index. ISBN Finance--Mathematical models. 2. Stochastic processes. 3. Investments--Mathematics. I. Manca, Raimondo. II. Volpe, Ernesto. III. Title. HG106.J ' dc British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: Printed and bound in Great Britain by CPI Antony Rowe Ltd, Chippenham, Wiltshire.

7 Table of Contents Preface...xvii Part I. Deterministic Models... 1 Chapter 1. Introductory Elements to Financial Mathematics The object of traditional financial mathematics Financial supplies. Preference and indifference relations The subjective aspect of preferences Objective aspects of financial laws. The equivalence principle The dimensional viewpoint of financial quantities Chapter 2. Theory of Financial Laws Indifference relations and exchange laws for simple financial operations Two variable laws and exchange factors Derived quantities in the accumulation and discount laws Accumulation Discounting Decomposable financial lawas Weak and strong decomposability properties: equivalence relations Equivalence classes: characteristic properties of decomposable laws Uniform financial laws: mean evaluations Theory of uniform exchange laws An outline of associative averages Average duration and average maturity Average index of return: average rate Uniform decomposable financial laws: exponential regime... 39

8 vi Mathematical Finance Chapter 3. Uniform Regimes in Financial Practice Preliminary comments Equivalent rates and intensities The regime of simple delayed interest (SDI) The regime of rational discount (RD) The regime of simple discount (SD) The regime of simple advance interest (SAI) Comments on the SDI, RD, SD and SAI uniform regimes Exchange factors (EF) Corrective operations Initial averaged intensities and instantaneous intensity Average length in the linear law and their conjugates Average rates in linear law and their conjugated laws The compound interest regime Conversion of interests The regime of discretely compound interest (DCI) The regime of continuously compound interest (CCI) The regime of continuously comound discount (CCD) Complements and exercises on compound regimes Comparison of laws of different regimes Chapter 4. Financial Operations and their Evaluation: Decisional Criteria Calculation of capital values: fairness Retrospective and prospective reserve Usufruct and bare ownership in discrete and continuous cases Methods and models for financial decisions and choices Internal rate as return index Outline on GDCF and internal financial law Classifications and propert of financial projects Decisional criteria for financial projects Choice criteria for mutually exclusive financial projects Mixed projects: the TRM method Dicisional criteria on mixed projects Appendix: outline on numberical methods for the solution of equations General aspects The linear interpolation method Dichotomic method (or for successive divisions) Secants and tangents method Classical interation method...143

9 Table of Contents vii Chapter 5. Annuities-Certain and their Value at Fixed Rate General aspects Evaluation of constant installment annuities in the compound regime Temporary annual annuity Annual perpetuity Fractional and pluriannual annuities Inequalities between annuity values with different frequency: correction factors Evaluation of constant installment annuities according to linear laws The direct problem Use of correction factors Inverse problem Evaluation of varying installment annuities in the compound regime General case Specific cases: annual annuities in arithmetic progression Specific cases: fractional and pluriannual annuities in arithmetic progression Specific cases: annual annuity in geometric progression Specific cases: fractional and pluriannual annuity in geometric progression Evaluation of varying installment annuities according to linear laws General case Specific cases: annuities in arithmetic progression Specific cases: annuities in geometric progression Chapter 6. Loan Amortization and Funding Methods General features of loan amortization General loan amortization at fixed rate Gradual amortizatin with varying installments Particular case: delayed constant installment amortization Particular case: amortization with constant principal repayments Particular case: amortization with advance interests Particular case: American amortization Amortization in the continuous scheme Life amortization Periodic advance payments Periodic payments with delayed principal amounts Continuous payment flow Periodic funcing at fixed rate Delayed payments Advance payments Continuours payments...251

10 viii Mathematical Finance 6.5. Amortizations with adjustment of rates and values Amortizations with adjustable rate Amortizations with adjustment of the outstanding loan balance Valuation of reserves in unshared loans General aspects Makeham s formula Usufructs and bare ownership valuation for some amortization forms Leasing operation Ordinary leasing The monetary adjustment in leasing Amortizations of loans shared in securities An introduction on the securities Amortization from the viewpoint of the debtor Amortization from the point of view of the bondholder Drawing probabiity and mean life Adjustable rate bonds, indexed bonds and convertible bonds Rule variations in bond loans Valuation in shared loans Introduction Valuation of bonds with given maturity Valuation of drawing bonds Bond loan with varying rate or values adjusted in time Chapter 7. Exchanges and Prices on the Financial Market A reinterpretation of the financial quantities in a market and price logic: the perfect market The perfect market Bonds Spot contracts, price and rates. Yield rate Forward contracts, prices and rates The implicit structure of prices, rates and intensities Term structures Structures with discrete payments Structures with fractional periods Structures with flows in continuum Chapter 8. Annuities, Amortizations and Funding in the Case of Term Structures Capital value of annuities in the case of term structures Amortizations in the case of term structures Amortization with varying installments...337

11 Table of Contents ix Amortization with constant installments Amortization with constant principal repayments Life amortization Updating of valuations during amortization Funding in term structure environments Valuations referred to shared loans in term structure environments Financial flows by the issuer s and investors point of view Valuations of price and yield Chapter 9. Time and Variability Indicators, Classical Immunization Main time indicators Maturity and time to maturity Arithmetic mean maturity Average maturity Mean financial time length or duration Variability and dispersion indicators nd order duration Relative variation Elasticity Convexity and volatility convexity Approximated estimations of price fluctuation Rate risk and classical immunization An introductin to financial risk Preliminaries to classic immunization The optimal time of realization The meaning of classical immunization Single liability cover Multiple liability cover Part II. Stochastic Models Chapter 10. Basic Probabilistic Tools for Finance The sample space Probability space Random variables Expectation and independence Main distribution probabilities The binominal distribution The Poisson distribution The normal (or Laplace Gauss) distribution The log-normal distribution The negative exponential distribution...432

12 x Mathematical Finance The multidimensional normal distribution Conditioning Stochastic processes Martingales Brownian motion Chapter 11. Markov Chains Definitions State classification Occupation times Absorption probabilities Asymptotic behavior Examples A management problem in an insurance company A case study in social insurance Chapter 12. Semi-Markov Processes Positive (J-X) processes Semi-Markov and extended semi-markov chains Primary properties Examples Markov renewal processes, semi-markov and associated counting processes Particular cases of MRP Renewal processes and Markov chains MRP of zero order Continuous Markov processes Markov renewal functions The Markov renewal equation Asymptotic behavior of an MRP Asymptotic behavior of SMP Irreducible case Non-irreducible case Non-homogenous Markov and semi-markov processes General definitions Chapter 13. Stochastic or Itô Calculus Problem of stochastic integration Stochastic integration of simple predictable processes and semi-martingales General definition of the stochastic integral...523

13 Table of Contents xi Itô s formula Quadratic variation of a semi-martingale Itô s formula Stochastic integral with standard Brownian motion as integrator process Case of predictable simple processes Extension to general integrand processes Stochastic differentiation Definition Examples Back to Itô s formula Stochastic differential of a product Itô s formula with time dependence Interpretation of Itô s formula Other extensions of Itô s formula Stochastic differential equations Existence and unicity general thorem Solution of stochastic differntial equations Diffusion processes Chapter 14. Option Theory Introduction The Cox, Ross, Rubinstein (CRR) or binomial model One-period model Multi-period model The Black-Scholes formula as the limit of the binomial model The lognormality of the underlying asset The Black-Scholes formula The Black-Scholes continuous time model The model The Solution of the Black-Scholes-Samuelson model Pricing the call with the Black-Scholes-Samuelson model Exercises on option pricing The Greek parameters Introduction Values of the Greek parameters Excercises The impact of dividend repartition Estimation of the volatility Historic method Implicit volatility method Black-Scholes on the market...587

14 xii Mathematical Finance Empirical studies Smile effect Exotic options Introduction Garman-Kohlhagen formula Greek parameters Theoretical models Binary or digital options Asset or nothing options The barrier options Lockback options Asiatic (or average) options Rainbow options The formula of Barone-Adesi and Whaley (1987): formula for American options Chapter 15. Markov and Semi-Markov Option Models The Janssen-Manca model The Markov extension of the one-period CRR model The multi-period discrete Markov chain model The multi-period discrete Markov chain limit model The extension of the Black-Scholes pricing formula with Markov environment: the Janssen-Manca formula The extension of the Black-Scholes pricing formula with a semi-markov environment: the Janssen-Manca-Volpe formula Introduction The Janssen-Manca-Çinlar model Call option pricing Stationary option pricing formula Markov and semi-markov option pricing models with arbitrage possibility Introduction The homogenous Markov model for the underlying asset Particular cases Numerical example for the Markov model The continuous time homogenous semi-markov model for the underlying asset Numerical example for the semi-markov model Conclusion...640

15 Table of Contents xiii Chapter 16. Interest Rate Stochastic Models Application to the Bond Pricing Problem The bond investments Introduction Yield curve Yield to maturity for a financial investment and for a bond Dynamic deterministic continuous time model for instantaneous interest rate Instantaneous interest rate Particular cases Yield curve associated with instantaneous interest rate Example of theoretical models Stochastic continuous time dynamic model for instantaneous interest rate The OUV stochastic model The CIR model (1985) The HJM model (1992) Zero-coupon pricing under the assumption of no arbitrage Stochastic dynamics of zero-coupons Application of the no arbitrage principle and risk premium Partial differential equatin for the structure of zero coupons Values of zero coupons without arbitrage opportunity for particular cases Values of a call on zero-coupon Option on bond with coupons A numerical example Appendix (solution of the OUV equation) Chapter 17. Portfolio Theory Quantitative portfolio management Notion of efficiency Exercises Markowitz theory for two assets Case of one risky asset and one non-risky asset Chapter 18. Value at Risk (VaR) Methods and Simulation VaR of one asset Introduction Definition of VaR for one asset Case of the normal distribution...705

16 xiv Mathematical Finance Example II: an internal model in case of the lognormal distribution Trajectory simulation Coherence and VaR extensions Risk measures General form of the VaR VaR extensions: TVaR and conditional VaR VaR of an asset portfolio VaR methodology General methods for VaR calculation VaR implementation VaR for a bond portfolio VaR for one plain vanilla option VaR and Monte Carlo simulation methods Introduction Case of one risk factor Case of several risk factors Monte Carlo simulation scheme for the VaR calculation of an asset portfolio Chapter 19. Credit Risk or Default Risk Introduction The Merton model Evaluation model of a risky debt Interpretation of Merton s result Spreads The Longstaff and Schwartz model (1995) Construction of a rating with Merton s model for the firm Rating construction Time dynamic evolution of a rating Discrete time semi-markov processes Purpose DTSMP definition Semi-Markov credit risk models NHSMP with backward conditioning time Examples Homogenous SMP application Non-homogenous downward example Non-homogenous downward backward example...784

17 Table of Contents xv Chapter 20. Markov and Semi-Markov Reward Processes and Stochastic Annuities Reward processes Homogenous and non-homogenous DTMRWP Homogenous and non-homogenous DTSMRWP The immediate cases The due cases MRWP and stochastic annuities Stochastic annuities Motorcar insurance application DTSMRWP and generalized stochastic annuities (GSA) Generalized stochastic annuities (GSA) GSA examples References Index...839

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19 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

20 xviii Mathematical Finance 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

21 Preface xix 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.

22 xx Mathematical Finance 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.

23 Part I Deterministic Models

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25 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 values 1. 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 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.

26 4 Mathematical Finance this context decision theory in uncertain conditions, which contains actuarial mathematics and more generally the theory of random financial operations 2. 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 Financial supplies. Preference and indifference relations The subjective aspect of preferences Let us call financial supply a dated amount, that is, a prefixed amount to place at a given payment deadline. A supply can be formally represented as an ordinate couple (X,S) where S = monetary amount (transferred or accounted from one subject to another) and X = time of payment. 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 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.

27 Introductory Elements to Financial Mathematics 5 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 (2) the Cartesian orthogonal reference system OXS (with abscissas X and ordinate S). (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,S 1 ) and (X,S 2 ) 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 Z 3, 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 (X 1,S 1 ) compared to (X 2,S 2 ) if he considers the first one more advantageous than the second; we then have (X 1,S 1 ) (X 2,S 2 ); 3 It is known that the symbol has the meaning for all.

28 6 Mathematical Finance the decision maker expresses a judgment of weak preference, indicated with, of the supply (X 1,S 1 ) compared to (X 2,S 2 ), if he does not consider the second one more advantageous than the first; we then have (X 1,S 1 ) (X 2,S 2 ) 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 (X 0,S 0 ), 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 (X 0,S 0 ) on the plane (2), with reference system OXS, by the point P 0 [X 0,S 0 ]. Then, considering the four quadrants adjacent to P 0, based only on criteria c) and d), it turns out that: 1) Comparing S 0 >0 to supplies with a positive amount, identified by the points P i (i=1,,4) (see Figure 1.1), being incomes, it is convenient to anticipate their collection. Therefore, all points P 2 [X 2,S 2 ] in the 2 nd quadrant (NW) are preferred to P 0 because they have income S 2 greater than S 0 and are available at time X 2 previous to time X 0 ; whereas P 0 is preferred to all points P 4 [X 4,S 4 ] in the 4 th quadrant (SE) because they have income S 4 smaller than S 0 and are available at time X 4 later than X 0 ; it is not possible to conclude anything about the preference between P 0 and points P 1 [X 1,S 1 ] in the 1 st quadrant (NE) or points P 3 [X 3,S 3 ] in the 3 rd quadrant (SW). Figure 1.1. Preferences with positive amounts 4 The judgment of weak preference is equivalent to the merging of strong preference of (X 1,S 1 ) with respect to (X 2,S 2 ) and of (X 2,S 2 ) with respect to (X 1,S 1 ). In other words: weak preference = strong preference or indifference; indifference = no strong preference of one supply with respect to another. 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).

29 Introductory Elements to Financial Mathematics 7 2) Comparing S 0 <0 to supplies with a negative amount, identified by the points P i (i=1,,4) (see Figure 1.2), being outcomes, it is convenient to postpone their time of payment. Therefore all points P 1 [X 1,S 1 ] in the 1 st quadrant (NE) are preferred to P 0 because they have outcome S smaller than S 0 and are payable at time X later than X 0 ; whereas P 0 is preferred to all points P 3 [X 3,S 3 ] in the 3 rd quadrant (SW) because they have outcome S 3 greater than S 0 and are payable at time X 3, which is later than X 0. Nothing can be concluded on the preference between P 0 and all points P 2 [X 2,S 2 ] of the 2 nd quadrant (NW) or all points P 4 [X 4,S 4 ] of the 4 th 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 P 0, 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 P 0 (we have dominance of P' on P 0 when the operator prefers P to P 0 ) and other supplies P" dominated by P 0 (when he prefers P 0 to P"), but the comparability with P 0 is incomplete because there are infinite supplies P'" not comparable with P 0 based on criteria c) and d). To make the comparability of P 0 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 P 0, indifferent on P 0 or dominated by P 0 is the aim of the following discussion. Figure 1.2. Preferences with negative amounts

30 8 Mathematical Finance 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 P 0 and in indifference relation with P 0 ; 2) the second phase, according to the transitivity of preferences, is to select the advantageous and disadvantageous preferences with respect to P 0, 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: (X,A) (Y,B) (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 P 0 [X 0,S 0 ] the point representing the supply for which the indifference is searched. Then: if S 0 >0 (see Figure 1.3), with X=X 0, Y=X 1 >X 0, the rightward movement from P0 to A 1 [X 1,S 0 ] 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 S 1 >S 0 which gives the compensation, where P 0 and P 1 [X 1,S 1 ], obtained from A 1 moving upwards, and represents indifferent supply (or, in brief, P 1 and P 0 are indifferent points). Instead, if Y=X 3 <X 0, the leftwards movement from P 0 to A 3 [X 3,S 0 ] is advantageous for the income anticipation; therefore, in order to have indifference, there needs to be a decrease in the income from S 0 to S 3, obtained through a survey with downward movement of the indifference point P 3 [X 3,S 3 ] with S 3 <S 0 ;