Applied Diffusion Processes from Engineering to Finance E-Book

Jacques Janssen

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 373

Veröffentlichungsjahr: 2013

Beschreibung

The aim of this book is to promote interaction between engineering, finance and insurance, as these three domains have many models and methods of solution in common for solving real-life problems. The authors point out the strict inter-relations that exist among the diffusion models used in engineering, finance and insurance. In each of the three fields, the basic diffusion models are presented and their strong similarities are discussed. Analytical, numerical and Monte Carlo simulation methods are explained with a view to applying them to obtain the solutions to the different problems presented in the book. Advanced topics such as nonlinear problems, Lévy processes and semi-Markov models in interactions with the diffusion models are discussed, as well as possible future interactions among engineering, finance and insurance. Contents 1. Diffusion Phenomena and Models. 2. Probabilistic Models of Diffusion Processes. 3. Solving Partial Differential Equations of Second Order. 4. Problems in Finance. 5. Basic PDE in Finance. 6. Exotic and American Options Pricing Theory. 7. Hitting Times for Diffusion Processes and Stochastic Models in Insurance. 8. Numerical Methods. 9. Advanced Topics in Engineering: Nonlinear Models. 10. Lévy Processes. 11. Advanced Topics in Insurance: Copula Models and VaR Techniques. 12. Advanced Topics in Finance: Semi-Markov Models. 13. Monte Carlo Semi-Markov Simulation Methods.

Leseprobe

Contents

Introduction

Chapter 1: Diffusion Phenomena and Models

1.1. General presentation of diffusion process

1.2. General balance equations

1.3. Heat conduction equation

1.4. Initial and boundary conditions

Chapter 2: Probabilistic Models of Diffusion Processes

2.1. Stochastic differentiation

2.2. Itô’s formula

2.3. Stochastic differential equations (SDE)

2.4. Itô and diffusion processes

2.5. Some particular cases of diffusion processes

2.6. Multidimensional diffusion processes

2.7. The Stroock–Varadhan martingale characterization of diffusions (Karlin and Taylor [KAR 81])

2.8. The Feynman–Kac formula (Platen and Heath [PLA 06])

Chapter 3: Solving Partial Differential Equations of Second Order

3.1. Basic definitions on PDE of second order

3.2. Solving the heat equation

3.3. Solution by the method of Laplace transform

3.4. Green’s functions

Chapter 4: Problems in Finance

4.1. Basic stochastic models for stock prices

4.2. The bond investments

4.3. Dynamic deterministic continuous time model for instantaneous interest rate

4.4. Stochastic continuous time dynamic model for instantaneous interest rate

4.5. Multidimensional Black and Scholes model

Chapter 5: Basic PDE in Finance

5.1. Introduction to option theory

5.2. Pricing the plain vanilla call with the Black–Scholes–Samuelson model

5.3. Pricing no plain vanilla calls with the Black-Scholes-Samuelson model

5.4. Zero-coupon pricing under the assumption of no arbitrage

Chapter 6: Exotic and American Options Pricing Theory

6.1. Introduction

6.2. The Garman–Kohlhagen formula

6.3. Binary or digital options

6.4. “Asset or nothing” options

6.5. Numerical examples

6.6. Path-dependent options

6.7. Multi-asset options

6.8. American options

Chapter 7: Hitting Times for Diffusion Processes and Stochastic Models in Insurance

7.1. Hitting or first passage times for some diffusion processes

7.2. Merton’s model for default risk

7.3. Risk diffusion models for insurance

Chapter 8: Numerical Methods

8.1. Introduction

8.2. Discretization and numerical differentiation

8.3. Finite difference methods

Chapter 9: Advanced Topics in Engineering: Nonlinear Models

9.1. Nonlinear model in heat conduction

9.2. Integral method applied to diffusive problems

9.3. Integral method applied to nonlinear problems

9.4. Use of transformations in nonlinear problems

Chapter 10: Lévy Processes

10.1. Motivation

10.2. Notion of characteristic functions

10.3. Lévy processes

10.4. Lévy–Khintchine formula

10.5. Examples of Lévy processes

10.6. Variance gamma (VG) process

10.7. The Brownian–Poisson model with jumps

10.8. Risk neutral measures for Lévy models in finance

10.9. Conclusion

Chapter 11: Advanced Topics in Insurance: Copula Models and VaR Techniques

11.1. Introduction

11.2. Sklar theorem (1959)

11.3. Particular cases and Fréchet bounds

11.4. Dependence

11.5. Applications in finance: pricing of the bivariate digital put option [CHE 04]

11.6. VaR application in insurance

Chapter 12: Advanced Topics in Finance: Semi-Markov Models

12.1. Introduction

12.2. Homogeneous semi-Markov process

12.3. Semi-Markov option model

12.4. Semi-Markov VaR models

12.5. Conclusion

Chapter 13: Monte Carlo Semi-Markov Simulation Methods

13.1. Presentation of our simulation model

13.2. The semi-Markov Monte Carlo model in a homogeneous environment

13.3. A credit risk example

13.4. Semi-Markov Monte Carlo with initial recurrence backward time in homogeneous case

13.5. The SMMC applied to claim reserving problem

13.6. An example of claim reserving calculation

Conclusion

Bibliography

Index

First published 2013 in Great Britain and the United States 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:

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© ISTE Ltd 2013

The rights of Jacques Janssen, Oronzio Manca, Raimondo Manca 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 Control Number: 2012955110

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-249-7

Introduction

The aim of this book is to facilitate interaction among engineering, finance and the insurance sectors as there are a lot of common models and solution methods for solving real-life problems in these three fields.

In the 19th Century, many problems in physics, for example heat diffusion, were theoretically solved using partial differential equations (PDEs). This led to new problems in mathematical analysis and later in function analysis; in particular, concerning the existence and unicity for the solutions of such PDE equations giving initial conditions of a Cauchy type that is the knowledge of the unknown function on a regular curve of the adequate Euclidean space.

Unfortunately, such PDEs have, in general, no explicit solution and so the problem of numerical treatment was posed. Although mathematicians could, indeed, formulate algorithms to give a good numerical approximation of the solution, it was nevertheless difficult to use such algorithms in practice, and it is only in the late 20th Century that this became possible with, of course, the building of more and more powerful computers together with elaborate software for numerical analysis.

In the 1970s, stochastic finance came into existence with the work of Black, Scholes and Merton, just after the fundamental results of Markowitz and Sharpe.

The main result is the celebrated Black and Scholes formula giving the value of a European call option with a closed formula. It can only be obtained by the authors with a laborious analytical transformation of their PDE arriving at the resolution of a well-known equation in physics called the diffusion equation.

Without this result, it is probable that the Black and Scholes formula would not exist.

So with partial differential equations as the vehicle, the interaction among engineering, physics and finance plays a fundamental role, and this book will show that this role is of major importance to get new results in finance and that, moreover, it could also be applied in the other spheres.

In Chapters 1–3, basic diffusion phenomena and models, probabilistic models of diffusion processes and the related PDEs including the heat equation are presented together with some fundamental results in stochastic calculus such as Itô’s and Feynman–Kac’s formulas.

Chapter 4 presents fundamental problems in finance concerning the stochastic evolution of stock prices and interest rates, while Chapter 5 shows how PDEs are necessary to price financial products, such as options and zero-coupon bonds, and how the interaction with PDE in physics works. It also shows that some important methods in finance, such as the use of the risk-neutral measure with Girsanov’s theorem, are nothing other than the use of Green’s function that is presented in Chapter 3.

Chapter 6 thoroughly analyzes stochastic finance with the consideration of more sophisticated derivative products than the plain vanilla European options (exotic options, American options, etc.). The pricing of such financial products is done with the resolution of PDEs with the methods of physics and engineering as presented in Chapter 3.

Chapter 7 presents some applications in stochastic insurance using hitting times for diffusion processes such as the Merton model for credit risk and asset liability models (ALM) to model the risk of banks and insurance companies.

Chapter 8 first describes the finite difference method and its application for solving numerically the PDEs of Black and Scholes as presented in the preceding chapters.

The next four chapters (Chapters 9–12) discuss some recent advanced topics such as non-linear problems, Lévy processes, and the copula approach and semi-Markov models in interaction with diffusion models.

This is, in particular, important for the evolution from Gaussian to non-Gaussian stochastic finance in future years as, indeed, recent crises imply considering the case of non-efficient and incomplete markets. In particular, this extension can be done with jumps models, generalizing the Merton model for option pricing. We can also use an economic-financial environmental process using semi-Markov theory. These last processes are also useful for pricing American options with a discrete-time model.

The last chapter (Chapter 13) presents some simulation results as it is a fact that for some real situations, there do not exist simple closed formulas and so simulation is the only possibility.

Finally, the Conclusion deals with actual and future interactions among engineering, finance and insurance as a fructuous source of developments for new models that are more adapted to approach the complexity of our three basic fields, thereby showing the great originality of this book.

This book is intended for a large audience of professional, research and academic disciplines, including engineers, mathematicians, physicists, actuaries and finance researchers with a good knowledge of probability theory.

Chapter 1

Diffusion Phenomena and Models

The aim of this chapter is to obtain the differential diffusion equation from the macroscopic point of view starting from a microscopic point of view. The approach is heuristic and a rigorous analysis is found in the current literature as also suggested in the following sections. The equation is obtained with reference to the mass diffusion phenomena and also by analogy to heat conduction. Then the analysis is carried out with reference to this last physical aspect. The parabolic and elliptic equations are presented and the initial and boundary conditions are also given.

In doing so, we can see in the following chapters why stochastic finance uses the results of diffusion theory.

1.1. General presentation of diffusion process

In general, a diffusion phenomenon is a process in which some physical properties are transported at molecular or atomic level from one part of the space to another part. The process is the result of random migration of small particles inside the physical space. It determines the motion of matter as well as energy. From a general point of view, the diffusion concept or phenomenon is also related to the random movement of small particles, and a very simple example is given by an observer on a skyscraper watching a crowded square: people move in all directions randomly but uniformly. Another example is a red wine drop in a glass filled with water. After some time the water becomes uniformly light pink in color. This suggests that the wine overruns the water, the molecules of wine are everywhere and the wine is said to have diffused into the water. This mass transport is due to the molecular agitation with the result that zones with a high concentration of wine determine a net molecular mass movement in all directions toward zones with lower wine concentration. In fact, an individual molecule of wine moves randomly and in a dilute solution each molecule of wine acts independently of the other molecules and undergoes collisions with the water molecules. The motion of a single molecule of wine can be depicted by the term of a “random walk” as shown in Figure 1.1. The picture of random molecular motions should adapt with the fact that a transfer of molecules from the region of higher concentration to the region of lower concentration is observed. If two thin zones are considered with equal volumes, one with a higher concentration and the other with a lower concentration, there is a dynamic exchange. A net transfer of molecules from the higher concentration to the lower concentration is obtained according to the second law of thermodynamics. Some other examples and descriptions are found in several books on this topic [BAK 08, CRA 75, CUS 09, GHE 88].

Figure 1.1.Path of Brownian motion

The molecular transfer determines a mass diffusion and, consequently, a diffusion of the other physical properties, such as the energy or more precisely an energy flux in conduction mode, is present. It needs to describe mathematically the molecular random transfer and to obtain a macroscopic description by means of a continuous model [BAK 08, BER 93, GHE 88, MAZ 09, WEI 94]. In the following, the term particle will be substitute for molecule. To characterize diffusive spreading, it is convenient to consider points on a line with an arbitrary origin, as indicated in Figure 1.2.

Figure 1.2.Particle exchange: a) particle number Ni(t) at location (site) i – 1, i and i + 1 at time t; b) the continuous distribution n and derived flux φ

[1.1]

and the net number of particles from site i – 1 to site i is given by:

[1.2]

The difference between the two net numbers of particles determines the time rate of change and allows us to evaluate all possible transitions to and from the i-th site:

[1.3]

Equation [1.3] expresses the rate of change in terms of a difference in the number of particles and of the nearest neighbor distribution around site i.

The main interest of most engineering problems is not addressed in the molecular behavior of a substance but how the substance acts as a continuum medium. The following step is to find an analog of equation [1.3] in the continuum, treating the problem as a molecular motion along a single axis coordinate x. It is assumed for simplicity that the sites are equidistant and Δx is the jump or lattice distance. The i-th site has the coordinate xi equal to iΔx. If the number of particles at each point along the x-axis, at time t, is known, it is possible to evaluate the net number of particles that will move across the unit area in the unit time from point xi to point xi + Δx, the flux of particles φx(x, t). A continuous distribution n(x, t) that satisfies the relation is defined as:

[1.4]

[1.5]

with A the area normal to the x coordinate and Δt the considered time interval. It should be underlined that equation [1.5] is a general expression of the generic flux along the assigned direction x of the net exchange of particles. The flux could be a net exchange of molecules or a current or heat. By simple manipulations it is obtained:

[1.6]

where Г and η depend on the particles exchanged, the mass or the current or the heat diffusion. Further, Г is a characteristic coefficient linked to the material whereas a difference of η determines a flux along the considered direction. In the limit Δx → 0, the partial variation of η is obtained along the x coordinate and a generic relation is obtained between the flux and the component of gradient along the x direction:

[1.7]

Equation [1.7] indicates that the flux along the assigned direction is related to the component of gradient along the assigned direction. Г is the diffusivity or diffusion coefficient. Moreover, equation [1.2] in the continuum domain and for the unit area allows us to carry out the following relation:

[1.8]

In equation [1.8], it is indicated that the rate of the η variation inside a spatial neighbor depends on the variation of φx along the x-axis. It is obtained by equations [1.7] and [1.8]:

[1.9]

When the diffusion coefficient is independent of η(x, t), equation [1.9] becomes:

[1.10]

Equations [1.9] and [1.10] are the diffusion equations in a one-dimensional space variable x. It allows us to evaluate the distribution of the η(x, t) with respect to the time and the spatial variable x. Equations [1.9] and [1.10] can be extended to a three-dimensional time-dependent problem and the diffusion equation takes the form:

[1.11]

where P is the point in the three-dimensional space and the flux component along the generic direction, xk, is given by:

[1.12]

and in vector form:

[1.13]

In equation [1.13], the flux is a vector and due to a gradient of a property η it represents a constitutive relation. Examples of constitutive relations are Fick’s law for mass diffusion and Fourier’s law for heat conduction. The first is a linear relationship between the mass flux and the concentration gradient [CRA 75, CUS 09, GHE 88, MIK 84]. The second determines a link between the heat diffusion and the temperature gradient [MIK 84, ÖZI 93, YEN 08, WAN 08].

The next step is to obtain the diffusion equation starting from a description in terms of continuum and to employ a global balance of an extensive quantity which is additive in the sense that its value for a set of subsystems is an algebraic sum of its values for each subsystem.

1.2. General balance equations

The balance of a generic extensive quantity or extensive property related to a physical, economic, financial or insurance phenomena should be given on an assigned system which is, from a thermodynamics point of view, an assigned portion of space or matter or, from a more general point of view, a set of points defined by independent variables. In financial as well as in economic problems, the independent variables are the financial or the economic ones, respectively.

A logical form of a balance of an extensive quantity is applied on a system, V, as depicted in Figure 1.3, defined by means of a surface or a border, A, which contains the system in an assigned time interval Δt. It is given by:

[1.14]

If the extensive quantity does not present any source term, positive (production) or negative (destruction), the equation [1.14] is a conservation equation of the extensive quantity or property and is given by:

Figure 1.3.Assigned system to apply the balance of an extensive quantity

[1.15]

The various terms of this equation [1.14] are evaluated starting from the flux and the normal to the external surface and considering the variation per unit of time (Δt→0), i.e. rate of extensive quantity through the bounding surfaces, as:

[1.16]

where is the flux vector of the extensive quantity and n is the outward-drawn normal unit vector to the surface element dA. The minus sign assures that if the net rate is entering, the scalar product is negative but the total quantity should be increased and the opposite is obtained if the net rate is exiting. Moreover, applying the divergence theorem to convert the surface integral to a volume integral, we obtain:

[1.17]

The rate of extensive quantity generation in the considered system is evaluated as:

[1.18]

where g is the generation per unit volume and time. The rate of extensive quantity storage in the system is evaluated as:

[1.19]

[1.20]

where χ is the specific value of the extensive quantity per unit volume and the derivative D/Dt is the substantial derivative or total derivative of the substance of the extensive quantity contained in the volume V. Applying the Reynolds transport theorem [ARP 84] and considering a velocity field v, we obtain:

[1.21]

The second term in the left-hand side of equation [1.21] is the convective flux through the surface A which enters and exits together with the macroscopic motion of the substance. The balance equation in logical form, equation [1.14], can be written in its integral form as:

[1.22]

Considering equations [1.18] and [1.21] and the Gauss divergence theorem [1.17], we obtain:

[1.23]

then

[1.24]

Equation [1.24] is derived on an arbitrary volume V and is valid for each V, and, consequently, the local balance equation is given by:

[1.25]

The two terms χ(P,t)v(P,t) and φ(P, t) are the convective flux or flow and the diffusive flux or flux of the extensive property χ. They are associated with the macroscopic mass flow rate and the microscopic diffusion of the considered property, respectively.

If the volume V is fixed in the space (a solid or a fluid in rest), we obtain [MIK 84, ÖZI 93]:

[1.26]

In this case, through the surface A there is only the molecular flux or diffusion which enters and exits together with the molecular motion of the substance. The substitution of equations [1.16], [1.18] and [1.26] into equation [1.14] provides the following relation:

[1.27]

Equation [1.27] is evaluated on an arbitrary volume V and is valid for each V, which is:

[1.28]

and, consequently, the following is obtained:

[1.29]

Considering equation [1.13], we obtain:

[1.30]

1.3. Heat conduction equation

Equation [1.30] is a general form of a diffusion equation in a differential form and it is a partial differential equation (PDE). In a physical space, it is a threedimensional PDE. Equation [1.30] describes the physical diffusions phenomena such as heat conduction and mass diffusion. In the following, equation [1.30] is considered with reference to heat conduction. In this case, the flux vector, φ, is the heat flux vector, q, which represents the heat flow per unit time, per unit area of an isothermal surface in the direction of decreasing temperature, T, and the physical dimensions of the heat flux are expressed in W/m2 and the temperature in K or °C. Equation [1.13] is written as [MIK 84, ÖZI 93, YEN 08, WAN 08]:

[1.31]

[1.32]

Equation [1.32] is the general form of the heat conduction equation for an isotropic medium. When the thermal conductivity is assumed to be independent of position and temperature, equation [1.32] simplifies to:

[1.33]

[1.34]

Equations [1.32] and [1.34] in a physical domain, the Euclidian geometrical space, can be expressed in different geometrical coordinate systems, such as a Cartesian or a rectangular coordinate system, a cylindrical coordinate system or a spherical coordinate system.

For a one-dimensional geometrical heat conduction problem, along the x-axis, equation [1.32] is similar to equation [1.9] and becomes:

[1.35]

and for k which is independent of temperature or space and time, equation [1.35] simplifies to:

[1.36]

If the term g(x, t) is equal to 0, a one-dimensional Fourier equation is obtained:

[1.37]

Equation [1.36] becomes the Poisson equation if the problem is independent of time, a steady state condition, and is given by:

[1.38]

[1.39]

The Laplace equation in a three-dimensional problem is given by:

[1.40]

[1.41]

1.4. Initial and boundary conditions

The initial condition is written, with reference to Figure 1.4, as the temperature distribution in the assigned domain V at time equal to 0:

[1.42]

In general, three different types of boundary conditions can be considered in heat conduction problems [DUF 06, MIK 84, ÖZI 93, YEN 08, WAN 08]:

1) Boundary condition of the first kind or a Dirichlet condition when the temperature distribution is assigned to the boundary surface, as indicated in Figure 1.5:

[1.43]

where the surface temperature f(P, t) is a function of position and time. The particular case

[1.44]

is the homogeneous boundary condition of the first kind or homogeneous Dirichlet condition.

Figure 1.5.Assigned temperature distribution at the boundary surface

2) Boundary condition of the second kind or a Neumann condition when the heat flux is assigned to the boundary surface, as depicted in Figure 1.6:

[1.45]

where the derivative is along the outward drawn normal to the surface and f(P, t) is the assigned heat flux. The particular case:

[1.46]

3) Boundary condition of the third kind or a Robin condition when the surface heat transfer is assigned to the boundary surface, as shown in Figure 1.7:

[1.47]

where the derivative is again along the outward drawn normal to the surface and f(P, t) is an assigned function which represents a heat flux at boundary surface. The particular case:

[1.48]

is the homogeneous boundary condition of the third kind or homogeneous Robin condition.

Figure 1.6.Assigned heat flux distribution at the boundary surface

If two different materials having two different thermal conductivity are in contact, such as shown in Figure 1.8 and there is no relative motion and the contact is perfect between the materials, then the heat flux and the temperature at two surfaces at the interface are equal and are defined as:

[1.49a]

[1.49b]

Figure 1.7.Surface heat transfer assigned at the boundary surface

There are other types of boundary conditions, such as radiative heat transfer, change of phase or the moving interface of two media, the interface of two solids in relative motion or a nonlinear convective heat transfer. These can be written following the procedure given to formulate the above boundary conditions.

Figure 1.8.Two different materials in contact along the surface S

EXAMPLE 1.1.–

Consider a plane, depicted in Figure 1.9, with an assigned thickness, L, subject to surface heat transfer on the two external surfaces by means of the heat transfer surface coefficients, h1 and h2. The two external temperatures are T∞1 and T∞2. The initial distribution of temperature inside the plate is uniform and equal to Ti. Write the heat conduction equation, the initial and boundary conditions in the hypothesis that the solid is homogeneous and isotropic and the thermal conductivity is k.

[1.50a]

the initial and boundary conditions are obtained:

[1.50b]

[1.50c]

[1.50d]

Figure 1.9.Geometric configuration and temperature values at external surface and initial temperature distribution of example 1.1

The conductive heat transfer problem [1.50], heat conduction equation [1.50a] together with initial [1.50b] and boundary [1.50c] and [1.50d] are a well-posed problem and should have one solution that satisfies equation [1.50a] and the conditions [1.50b]–[1.50d]. If the h1 and h2 values are very high with respect to the k value (hi >> k) equations [1.50c] and [1.50d] become:

[1.50e]

[1.50f]

and two boundary conditions of the first kind are assigned, i.e. the temperature values on the two external surfaces of the plate are known. Moreover, if the h1 is equal to 0, we obtain:

[1.50g]

Chapter 2

Probabilistic Models of Diffusion Processes

This chapter presents the basic results concerning Itô’s calculus – also called stochastic calculus, one of the main tools used in stochastic finance, and also the most important notions and results concerning diffusion processes intensively used in finance and insurance.

2.1. Stochastic differentiation

2.1.1. Definition

On the probability space (Ω,ℑ,(ℑ, ≥ 0),), let us consider an adapted standard Brownian motion and two adapted processes and , which are sufficiently smooth.