This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

A fully nonlinear problem arising in financial modelling

Maria do Rosário Grossinho1* and Eva Morais12

Author Affiliations

1 ISEG, CEMAPRE - Technical University of Lisbon, Rua do Quelhas 6, Lisboa, 1200-781, Portugal

2 Department of Mathematics, University of Trás-os-Montes e Alto Douro, Apartado 1013, Vila Real, 5001-801, Portugal

For all author emails, please log on.

Boundary Value Problems 2013, 2013:146  doi:10.1186/1687-2770-2013-146


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/146


Received:29 December 2012
Accepted:24 May 2013
Published:13 June 2013

© 2013 Grossinho and Morais; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We state existence and localisation results for a fully nonlinear boundary value problem using the upper and lower solutions method. With this study we aim to contribute to a better understanding of some analytical features of a problem arising in financial modelling related to the introduction of transaction costs in the classical Black-Scholes model. Our result concerns stationary solutions which become interesting in finance when the time does not play a relevant role such as, for instance, in perpetual options.

1 Introduction

In 1973, Fisher Black and Myron Scholes suggested a model that became fundamental for the valuation of financial derivatives in a complete frictionless market. Along with the no-arbitrage possibilities, the classical Black-Scholes model assumes that in order to replicate exactly the returns of a certain derivative, the hedging portfolio is continuously adjusted by transactioning the underlying asset of the derivative. This fact can only happen if no transaction costs exist when buying or selling financial assets. Otherwise, a continuous adjustment would imply that those costs, such as taxes or fees, would become infinitely large.

Hence the introduction of transaction costs in the model is a problem that has been motivating the work of several authors and has led to the study of new models that generalise the classical Black-Scholes model.

In this paper, we aim to give a contribution to better understanding of some analytical features of the problem. We are concerned with the existence and localisation results for the nonlinear second-order Dirichlet boundary problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M2">View MathML</a> and p, q are positive constants (p and q can be assumed nonnegative, but the assumption is neither interesting from the mathematical viewpoint nor reasonable for applications to finance). We also consider that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M3">View MathML</a>. This assumption turns out to be quite natural in some financial settings, for instance, if we are dealing with call options.

This problem is related to the study of stationary solutions of a nonlinear parabolic equation that models the valuation of a call option in presence of transaction costs. These stationary solutions give the option value V as a function of the stock price, which can be interesting when dealing with a model where the time does not play a relevant role such as, for instance, in perpetual options.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M4">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M5">View MathML</a>, the equation of (1) is of the following type:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M6">View MathML</a>

However, we will consider it in the form presented in (1) in order to keep some similarity to the financial setting problem, as we will see later.

The arguments used rely on the upper and lower solutions method (see [1,2]).

The results we present were motivated by the work of Amster et al. (see [3]). There, the authors studied an analogous problem but without obtaining a localisation result. Then they proceeded their study using a limit of a nonincreasing (nondecreasing) sequence of upper (respectively lower) solutions. Those results were proved under some Lipschitz condition, which has some financial implications. We do not assume such a condition and, moreover, we obtain localisation information on the solution.

In this paper, we state some existence results for the problem (1), following a previous study contained in [4], and improving the localisation result. Relating to the financial model, this implies more accuracy in the information of the option regarding the behaviour of the stock value, in a model where transaction costs are considered. In Section 2, we consider an auxiliary problem. Using the results proved there, we state in Section 3 an existence and localisation theorem for (1). In Section 4, we make some comments on the financial problem that suggested our analytical study. With this section we aim to illustrate the interplay between finance and mathematics which, in fact, is very challenging and often stimulates the use of innovative mathematical and computational techniques.

2 Auxiliary result

Observe that the equation of (1) is from the algebraic point of view a second-order equation in the variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M7">View MathML</a>. So, solving it algebraically in order of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M8">View MathML</a>, we obtain the equivalent form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M9">View MathML</a>

which leads us to consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M10">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M11">View MathML</a>

This fact suggests the study of the auxiliary problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M12">View MathML</a>

(2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M13">View MathML</a>

The main argument to prove Theorem 2 relies on the method of upper and lower solutions.

We recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M14">View MathML</a> is a lower solution of (2) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M15">View MathML</a>

(3)

Similarly, an upper solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M16">View MathML</a> of (2) is defined by reversing the inequalities in (3). A solution of (2) is a function u which is simultaneously a lower and an upper solution. A function f is said to satisfy the Nagumo condition on some given subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M17">View MathML</a> if there exists a positive continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M19">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M20">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M21">View MathML</a>

(4)

Lemma 1Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M22">View MathML</a>

(5)

Then:

1. The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M23">View MathML</a>is a lower solution of the problem (2).

2. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M24">View MathML</a>is small enough, then the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M25">View MathML</a>

is a lower solution of the problem (2).

3. The function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M26">View MathML</a>

is an upper solution of the problem (2).

Proof 1. If we plug <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M23">View MathML</a> in the first member of the equation in (2), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M28">View MathML</a>

and by (5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M29">View MathML</a>

2. Consider now the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M30">View MathML</a>

Observe first that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M31">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M32">View MathML</a>

On the other hand, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M33">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M24">View MathML</a> .

So, if we plug the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M35">View MathML</a> in the first member of the equation in (2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M36">View MathML</a>

which, for k small enough, is non-negative.

3. Consider now the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M37">View MathML</a>

Then, in an analogous way, we plug <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M38">View MathML</a> in the first member of the equation in (2). We observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M39">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M40">View MathML</a>

So, the three assertions of the lemma are proved. □

Remark 1 The value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M41">View MathML</a> is a suitable upper bound for the possible values that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M42">View MathML</a> can take in the above assertion 2. In fact, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M4">View MathML</a> and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M44">View MathML</a>, easy computations show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M45">View MathML</a>

Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M46">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M47">View MathML</a>

which is used in the last step of the proof of assertion 2.

Now we state an existence and localisation result for the auxiliary problem (2).

Theorem 1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M44">View MathML</a>. Then:

1. The problem (2) has a solutionVsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M49">View MathML</a>

2. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M24">View MathML</a>is small enough, the problem (2) has a solutionVsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M51">View MathML</a>

3. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M52">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M53">View MathML</a>is a solution of the problem (2).

Proof By the previous lemma, we know already that there are lower and upper solutions for the problem (2). It is also clear that they are well ordered, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M54">View MathML</a>

So, if the function H satisfies the Nagumo condition, the thesis will follow by [1] or [2].

In order to prove assertions 1 and 2, we consider the sets, respectively,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M55">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M56">View MathML</a>

Function H satisfies the Nagumo condition in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M57">View MathML</a> and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M58">View MathML</a>. In fact, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M59">View MathML</a>

Thus, for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M61">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M62">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M63">View MathML</a>

So, by what was said above, the first two assertions of the thesis hold.

The third assertion follows directly from obvious computations. □

Proposition 1Consider the problem (2) and the solutionVgiven by Theorem 1. ThenVis convex and satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M64">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M65">View MathML</a>.

Proof The convexity of the solution V follows easily by the equation of the problem (2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M66">View MathML</a>

Noting that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M67">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M68">View MathML</a>

But, using Theorem 1, assertion 1, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M69">View MathML</a>. In fact, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M70">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M71">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M72">View MathML</a>. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M73">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M74">View MathML</a>.

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M75">View MathML</a>

 □

3 Existence and localisation result

We return to the original problem (1) and state an existence and uniqueness result. We also provide information on the localisation of the solution.

Theorem 2Consider the nonlinear Dirichlet boundary value problem (1). The following assertions hold:

1. The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M76">View MathML</a>is a solution of the problem (1) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M77">View MathML</a>.

2. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M78">View MathML</a>, then the problem (1) has a convex solutionVsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M79">View MathML</a>

(6)

3. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M24">View MathML</a>is small enough, then the problem (1) has a convex solutionVsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M82">View MathML</a>

(7)

4. Moreover, Vis the unique convex solution of (1) in any of the above cases.

Proof 1. It is easily verified that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M76">View MathML</a> satisfies the equation of (1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M84">View MathML</a>, and that the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M85">View MathML</a> results if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M77">View MathML</a>.

2. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M87">View MathML</a> denote a convex solution of the auxiliary problem (2) given by Theorem 1. Then, using (2) and Proposition 1, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M88">View MathML</a>

therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M87">View MathML</a> is a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M90">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M92">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M87">View MathML</a> is a solution of the Dirichlet boundary value problem (1). Statement (6) is obtained from Theorem 1.

3. The proof is the same as for the previous case.

4. The uniqueness result follows immediately from Theorem 2.1 of [3]. □

Remark 2 We remark that Theorem 2.1 of [3] guarantees the existence and uniqueness of a solution but does not give information on localisation, due to the method used. However, using that uniqueness result, from the above theorem, assertions 2 and 3, we can derive the following corollary.

Corollary 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M78">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M95">View MathML</a>is small enough, then the nonlinear Dirichlet boundary value problem (1) has a convex solutionVsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M96">View MathML</a>

(8)

Figure 1 illustrates the localisation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M4">View MathML</a> of the convex solution for the problem (1) whose existence is given by assertion 2 of Theorem 2.

thumbnailFigure 1. Localisation of the convex solution for the problem (1) stated by Theorem 2, assertion 2.

Assertion 3 of Theorem 2 improves the localisation result contained in assertion 2 as long as k is small enough. In fact, more accuracy is obtained near c.

Figure 2 is an illustration of the new localisation of the convex solution in this case.

thumbnailFigure 2. Localisation of the convex solution for the problem (1) stated by Corollary 1.

4 Application to a generalised Black-Scholes model

For better understanding how that above equation appears from financial option pricing, we present in this section some of the financial and mathematical features that lie behind the model. Namely, we will refer to the introduction of transaction costs in the classical Black-Scholes (BS) model. The classical BS model concerns the price of a call or a put option on an underlying asset when the price of this asset itself is modelled as a geometric Brownian motion. We consider the BS equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M98">View MathML</a>

with the final condition, which represents the pay-off function,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M99">View MathML</a>

where V is the option price, S is the stock price at time t, T is the maturity time, r is the interest short rate and σ is the volatility of the stock returns.

When approaching the question of introducing transaction costs in the classical Black-Scholes model, one of the first main references is the work presented by Leland in [5], that suggests a market with proportional transaction costs. That is, considering ν the number of shares (it is positive if the agent buys or negative if the agent sells) and S the price of the asset at time t, the costs of the transaction of ν shares at time t are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M100">View MathML</a>

where the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M101">View MathML</a> depends on the parts involved in the transaction.

Leland [5] suggested a new strategy using transaction costs proportional to the monetary value of any buy or sale of the asset. Leland’s replication strategy consisted in using the common Black-Scholes formulae in periodical revisions of the portfolio but with an appropriately enlarged volatility. This is a model widely accepted in the financial industry. However, there are some mathematical problems with this approach, as referred by Kabanov and Safarian [6]: the terminal value of the replicating portfolio does not converge to the terminal payoff of the derivative if the transaction costs do not depend on the number of revisions (tending to infinity), limiting discrepancy that can be calculated explicitly.

In this work we use a slightly different structure for the transaction costs that was presented by Amster et al.[3]. We assume that the individual cost of the transaction of each share diminishes as the number of shares transactioned increases, which is represented by considering the cost as the percentage given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M102">View MathML</a>

where ν is the number of shares traded (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M103">View MathML</a> in a buy or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M104">View MathML</a> in a sale) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M105">View MathML</a> are constants depending on the individual investor.

We consider the hedging portfolio consisting in an option of value V (short position) and Δ shares of the underlying asset of price S (long position), and consider a strategy where the portfolio is reviewed every δt, with δt a finite, fixed time step.

The inclusion of transaction costs in the classical Black-Scoles model leads us to the following equation: (see [4])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M106">View MathML</a>

where μ is the drift coefficient and σ is the volatility of the underlying asset.

Considering a small enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M107">View MathML</a>

(9)

the following nonlinear equation is obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M108">View MathML</a>

(10)

The equation obtained (10) is clearly an extension of the classical Black-Scholes equation. The introduction of our particular model for the transaction costs in the option pricing market led us to a partial differential equation that contains the Black-Scholes terms with an additional nonlinear term modelling the presence of transaction costs. We also use an adjusted volatility in the model <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M109">View MathML</a>, not the real volatility σ.

When concerned with the existence and localisation of stationary solutions of the above equation, in order to consider models where the influence of time does not play an important role, we are led to the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M110">View MathML</a>

Dividing by the positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M111">View MathML</a> and writing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M113">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/146/mathml/M114">View MathML</a>

This equation is precisely the equation of the problem (1) studied in the previous section. Contributions to better understanding of the analytical features of the model can be useful for future developments of the financial model itself. In fact, research is full of many examples that illustrate the fruitful interplay between finance and mathematics with reciprocal advantages.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors collaborated in all the steps concerning the research, discussion and achievements presented in the final manuscript.

Acknowledgements

This research was supported by the Fundação para a Ciência e Tecnologia through the project PEst-OE/EGE/UI0491/2011 and by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE - Novel Methods in Computational Finance. We thank the anonymous referees for their useful comments.

References

  1. Mawhin, J: Points fixes, points critiques et problèmes aux limites, Presses University Montreal, Montreal (1985)

  2. De Coster, C, Habets, P: Two-Point Boundary Value Problems Lower and Upper Solutions, Elsevier, Amsterdam (2006)

  3. Amster, P, Averbuj, CG, Mariani, MC, Rial, D: A Black-Scholes option pricing model with transaction costs. J. Math. Anal. Appl.. 303, 688–695 (2005). Publisher Full Text OpenURL

  4. Grossinho, MR, Morais, E: A note on a stationary problem for a Black-Scholes equation with transaction costs. Int. J. Pure Appl. Math.. 51, 579–587 (2009). PubMed Abstract OpenURL

  5. Leland, HE: Option pricing and replication with transaction costs. J. Finance. 40, 1283–1301 (1985). Publisher Full Text OpenURL

  6. Kabanov, Y, Safarian, M: Markets with Transaction Costs - Mathematical Theory, Springer, Berlin (2009)