Mathematical Finance

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Mathematical Finance

Deterministic and Stochastic Models

Jacques Janssen

Raimondo Manca

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Mathematical Finance

Deterministic and Stochastic Models

Jacques Janssen

Raimondo Manca

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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.

27-37 St George’s Road 111 River Street

London SW19 4EU Hoboken, NJ 07030

UK USA

www.iste.co.uk www.wiley.com

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 332.01'51922--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

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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

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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

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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.

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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.

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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

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.

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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

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.

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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 6(2)_{the Cartesian}

orthogonal reference system OXS (with abscissas X and ordinate S). 6(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);

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– 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 6(2)_{, with reference system OXS, by the point P}₀

_{

_[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

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:

– 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).

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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.

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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:

(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 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

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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 P}1 and P3, all indifferent to P0. If S0<0, the continuous

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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

I = B – A 0. (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

D = A – B 0 (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}

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.

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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/t}2_{, force ml/t}2_{, 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 m1_t-1_);

2) rate, defined as amount over amount (thus a “pure number”, with dimension

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3) intensity, defined as amount over the product of amount multiplied by time (then with dimension m0_t-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”.

amount (m1_t0_{) flow}_(m1_t-1₎

rate (m0_t0_{) intensity}_(m0_t-1₎

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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

S' = fc (S, T; T') (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

S" = fa (S', T'; T") (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.

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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';

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– 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

S* = fa [{ fc (S,T;T')},T';T ] (2.3)

If (S,T,T') is S* = S, the f_a neutralizes the effect of fc, acting as the inverse

function, and the following investment or anticipation operation is called the

corresponding operation; in this case the laws expressed by fc and fa are said to be

conjugated.

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

S' = f (S,T;T') (2.4)

where if T T' then f = fc, whereas if T >T' then f = fa .

1 Lending and discounting operations are the same thing because in both cases there is an exchange of a lower amount in a previous time for a greater amount in a future time. The only difference is that in the first case the lower and previous amount is fixed, whereas in the second case the greater and future amount is fixed.

2 We will not use “equivalent” – even if it is used in practice – in the cases that we will consider later where § gives rise to an equivalence relation (see footnote 6 of Chapter 1).

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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 Tƍ-T the amount in Tƍ 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 f (S,T;T’ ) = f_c(S,T;T’ ), if T < T’ S , if T = T’ f_a(S,T;T’ ), if T > T’ ® ° ¯ ° (2.5)

If the symmetric law holds in the considered set P, then

S = fa[{ fc (S,T,T')},T',T ], (S,T,T'), T<T' 5 (2.6)

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

S = f [{ f (S,T,T')},T',T], (S,T,T') (2.6')

3 Let us recall that a binary relation * between elements a, b, ... of a set satisfies the reflexive law if: a* a, a .

4 Let us recall that a binary relation * between elements a, b, ... of a set satisfies the symmetric law if: a* b b* a, a,b .

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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

S' = S. m(T,T'), T T' (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

S" = S'. a(T',T"), T' T" (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

m(T,T) = a(T,T) = 1, T (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

m(T,T') . a (T',T) = 1, TT' (2.8)

6 The symmetric case – far from being realistic in the contracts with companies and banks, due to the different conditions and onerousness of the lending market (which leads to costs for the companies) compared to the investment market (which leads to profits for the companies) – can be applied to the contracts between persons or linked companies and, from a theoretical point of view, makes it possible to deal with the two systems in a similar manner.

7 The property of proportional amounts is normally used in theoretical schemes, but should only be used with smaller amounts. The financial profits for the unit of invested capital can change according to the value of the capital and the contractual strength of the investors.

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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 z(X,Y ) = m(X,Y ) , if X < Y 1 , if X = Y a(X,Y ) , if X > Y ® ° ¯ ° (2.5')

(2.5ƍ) being a particular case of (2.5).

To summarize, given an indifference relation §, the corresponding exchange law expressed by the factor z(X,Y), such that (X,S1) § (Y,S2), is equivalent to S2 = S1 z(X,Y). The exchange factor z(X,Y) is a function defined for each couple

(X,Y) of exchange times, which “brings” the values from X to Y forward (=

accumulation) if X<Y and backward (= discounting) if X>Y. We will now assume that

z(X,Y) > 0, X,Y) (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.

In geometric terms, let us consider the Cartesian plane OXY with the points

G

{

(X,Y) with the aforementioned meaning8_{. The exchange factor is then the point}

function z(G). Because of (2.5'), z(G)=1 if G is on the bisector of the coordinate axes. Furthermore, if G is over the bisector (i.e. if X<Y), then z(G) = m(X,Y) > 1; otherwise, if G is under the bisector (i.e. if X>Y), z(G) = a(X,Y)<1 and more precisely because of (2.5"): 0<a(X,Y)<19_.

8 Note that the functions m(X,Y) and a(X,Y) are defined in the disjoint half-planes X<Y and

X>Y, i.e. over and under the bisector of coordinate axes. It can be useful to extend their

definition on the bisector Y=X, recalling (2.7) and putting m(X,X) = a(X,X) = 1.

9 (2.5') brings to a general formulation of exchange value of two variables, which does not imply the symmetry of financial relations. It follows that the law z(X,Y) can be used to schematize not just the time variability of the cost and profit parameters, but also their difference in investment and discount operations which are of interest to any company. For example, if a company obtains liquid assets through anticipation of future credits and uses

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Recalling the considerations of Chapter 1 (especially criteria d) for positive amounts, given that z(X,Y) is the exchange value of unitary amount), in the hypothesis of positive returns for the money the contour curves z(X,Y) = const. are graphs of strictly increasing functions Y=\(X)10_.

If relation § expressed by z(X,Y) satisfies the symmetric property, as a particular case of (2.6') the below condition follows:

z(X,Y).z(Y,X) = 1; (X,Y) (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.

It is obvious that if the indifference relation is symmetric, it is enough to be able to define z(X,Y) in one of the two half-planes to obtain the value of z in the second half-plane using the following rule: the values of z for points which are symmetric

with respect to the bisector are reciprocal. In this case z(X,Y) = 1/z(Y,X), (X,Y) then the couples of contour curves of accumulation factor z = k >1 and discount factor z = 1/k <1 are functions which are mutually inverse.

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

iaf:= m(X,Y) (2.10)

them in financial operations, and if the parameters a and m used in such an operation and summarized in z are not reciprocal, a non-zero spread is created.

10 In fact if we assume z(X,Y) to be continuous and partially differentiable everywhere, it follows that: w z

w X< 0, w z

w Y > 0 X,Y). Therefore, the contour curves z(X,Y) = const. are continuous and strictly increasing; they are graphs of functions Y = \(X) invertible. In fact, for a theorem on implicit function, it follows that: \'(X) = - w z

w X/ w z

w Y , where in the aforementioned hypothesis \(X) is continuous and \'(X) > 0.

11 In this section we will denote with roman capital letters the temporal variables meaning

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(:= 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

iir:= m(X,Y) – 1 (2.11)

– the initial interest (per period) intensity, expressed by

iii {m(X,Y) – 1}/(Y-X) = {m(X,X+t) – 1}/ t (2.12) where t = Y-X > 0.

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

caf r(X;Y,Z) = m(X,Z)/m(X,Y) = m(X,Y+u)/m(X,Y) (2.13)

– the continuing interest (per period) rate from Y to Z (= interest for unitary

accumulated amount in Y passing from Y to Z=Y+u, u>0, for the investment started in X) is expressed by

cir caf - 1 = {m(X,Z) - m(X,Y)}/m(X,Y) (2.14) = {m(X,X + u) – m(X,Y)}/m (X,Y)

– the continuing interest (per period) intensity from Y to Z = Y+u, u>0 is expressed by ( ; , ) 1 ( , ) - ( , ) ( , ) - ( , ) cii:= = ( - ) ( , ) ( , ) r X Y Z m X Z m X Y m X Y u m X Y Z Y Z Y m X Y u m X Y _(2.15)

Mathematical Finance

Mathematical Finance

Deterministic and Stochastic Models

Jacques Janssen

Raimondo Manca

Mathematical Finance

Deterministic and Stochastic Models

Jacques Janssen

Raimondo Manca

Table of Contents

Introductory Elements

to Financial Mathematics

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Theory of Financial Laws

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