Banco de Mexico

Documentos de Investigacion

Banco de Mexico

Working Papers

N◦ 2008-03

An Analytical Approach to Merton’s Rational OptionPricing Theory

Rocıo Elizondo Pablo PadillaBanco de Mexico IIMAS, UNAM

March 2008

La serie de Documentos de Investigacion del Banco de Mexico divulga resultados preliminares detrabajos de investigacion economica realizados en el Banco de Mexico con la finalidad de propiciarel intercambio y debate de ideas. El contenido de los Documentos de Investigacion, ası como lasconclusiones que de ellos se derivan, son responsabilidad exclusiva de los autores y no reflejannecesariamente las del Banco de Mexico.

The Working Papers series of Banco de Mexico disseminates preliminary results of economicresearch conducted at Banco de Mexico in order to promote the exchange and debate of ideas. Theviews and conclusions presented in the Working Papers are exclusively the responsibility of theauthors and do not necessarily reflect those of Banco de Mexico.

Documento de Investigacion Working Paper2008-03 2008-03

An Analytical Approach to Merton’s Rational Option PricingTheory*

Rocıo Elizondo† Pablo Padilla‡Banco de Mexico IIMAS, UNAM

AbstractIn the early 70s Merton developed a theory based on economic arguments to study

the properties of option and warrant prices. The main tool in his proofs was the portfoliodominance principle. In the context where the price of a contingent claim satisfies a partialdifferential equation we provide analytical proofs of Merton’s rational option pricing theory.We use several versions of the maximum principle as well as the sliding and the moving planesmethods to prove our results. Our approach enables us to extend the theory to nonlinearmodels.Keywords: Merton, rational theory, option pricing, Black-Scholes, maximum principle.JEL Classification: C02, G10, G11.

ResumenEn los anos 70, Merton desarrollo una teorıa basada en argumentos economicos y fi-

nancieros para estudiar las propiedades del precio de una opcion. La herramienta principalen sus demostraciones fue el principio de dominancia de portafolios. En el contexto dondeel precio de una reclamacion contingente satisface una ecuacion diferencial parcial, nosotrosdamos demostraciones analıticas de la teorıa racional de opciones de Merton para el caso deuna opcion. Para demostrar estos resultados, utilizamos diferentes versiones del principio delmaximo, ası como los metodos de planos moviles y de deslizamiento. Nuestra aproximacionfacilita extender la teorıa a modelos no lineales.Palabras Clave: Merton, teorıa racional, estimacion de opciones, Black-Scholes, principiodel maximo.

*We thank Carlos Capistran, Jesus Castaneda, Daniel Chiquiar, Lai Kwong and referees of Analysisand Applications Journal for very helpful comments and suggestions, and the Consejo Nacional de Cienciay Tecnologıa (CONACYT) and Direccion General de Estudios de Posgrado (DGEP, UNAM) for financialsupport. Preprint of an article submitted for consideration in Analysis and Applications 2007 World ScientificPublishing Company, www.worldscinet.com/aa/aa.shtml.).

† Direccion General de Investigacion Economica. Email: melizondo@banxico.org.mx.‡ Departamento de Matematicas y Mecanica. Email: pablo@mym.iimas.unam.mx.

1 Introduction

The rational theory of option pricing, RTOP, is an attempt to derive proper-ties of option prices based on assumptions su�ciently weak to gain universalsupport [14]. To the extent that this attempt is successful, the resultingtheorems become necessary conditions to be satis�ed by any rational optionpricing theory. The assumptions are both of economic and �nancial nature.

In this article, we develop a similar analysis for the rational option pri-cing theory of Merton [14], but we use partial di�erential equations tech-niques. In particular, the maximum principle ([8], [18] and [5]), movingplanes and sliding methods [2] are used to obtain monotonicity propertiesof option prices as well as to establish comparison principles. The existingproofs, in contrast, relied on economic and �nancial arguments (dominanceof portfolios).

Our results are not only of theoretical relevance, by providing analyticalproofs to the theory, but are important in practice, e.g., when implementingnumerical schemes (see [10]). Besides, they allow extending Merton's theoryto some nonlinear models. These have become important also in practice ([6]and [16]). In particular, nonlinear models arise when pricing some bonds andin generalizing standard approaches, i.e., the Black-Scholes-Merton model.In these cases, representation formulas or Green's function techniques arenot available anymore. Nevertheless, it would be important to be able toestablish the same or similar qualitative results also in these cases.

In what follows, we will recall the essential features of Merton's RTOPas presented in [14].

In order to establish the restrictions on prices, we need the notion ofdominance: security (portfolio) A dominates security (portfolio) B if, onsome known date in the future, the return on A will exceed the return on Bfor some possible states, and will be at least as large as on B in all possiblestates.

We will also assume the following:

a) In perfect markets with no transactions costs and the ability to borrowand short-sell without restriction, the existence of a dominated securitywould be equivalent to the existence of an arbitrage opportunity.

b) If one assumes something like "symmetric market rationality" [13] andfurther that investors prefer more wealth to less, then any investorwilling to purchase security B would prefer to purchase A.

1

c) The rate at which one is granted a loan is named lending rate and thereturn of the lender is the saving return.

We consider four kinds of �nancial instruments: an "American" or "Eu-ropean" call or put option.

We denote by

S: the underlying asset.

E: exercise price.

T : exercise time.

r: risk free interest rate, i.e., we assume lending and saving rates are equal.

�: the volatility of underlying asset.

The basic principle considered by Merton is a necessary condition for aRTOP, namely, that the option be priced such that it is neither a dominantnor a dominated security.

The main results that Merton presents are the following:

1) Early exercise for an "American"-type put might be optimal.

2) The value of an "American"-type option is larger than or equal to thevalue of a "European"-type option, in other words, the "American"-type option dominates the "European".

3) If we consider two "European" or two"American"-type call options thatdi�er only in the expiration time then the option with the largestexpiration time has a larger value than the other.

4) If we consider two "European" or two "American"-type call options thatdi�er only in the exercise price then the option with the largest exerciseprice has a smaller value than the other.

5) It is not optimal to exercise an "American" call non-dividend-payingasset before expiring. In this case, the value of an "American"-typewarrant is the same as its "European" counterpart.

This paper is structured as follows: in section two, we recall some stan-dard tools related to the maximum principle. In section three we prove

2

Merton's results using the maximum principle, the Fokker-Planck equation,the sliding and the moving planes methods instead of economic and �nancialarguments. In section four we present the conclusions and some issues forfuture research.

2 Preliminaries

For the sake of completeness, in this section we state the precise versionsof the maximum principle that we will use in proving our results. Most ofthem are standard and we refer to [8] for details.

2.1 Maximum Principles for Parabolic Operators

2.1.1 Weak Maximum Principle

We assume that L is an operator of the form

Lu = ��ni;j=1aijuxixj +�ni=1biuxi + cu; (1)

where coe�cients aij , bi, and c are continuous. We will always assumethe uniform parabolicity (see [8]) condition and also that aij = aji (i; j =1; : : : ; n). We assume U to be an open, bounded subset of Rn and, setUT = (0; T ] � U for some �xed time T > 0. Recall also that the parabolicboundary of UT is �T = UT � UT .

Theorem 1 (Weak maximum principle for c � 0)Assume u 2 C21 (UT ) \ C(UT ) and c � 0 in UT .1

i) If

ut + Lu � 0 in UT ;

thenmaxUT

u = max�T

u:

1C21 means that the �rst derivative with respect to the time variable, t, is continuousand the second derivatives with respect to the space variables are also continuous.

3

ii) Likewise, if

ut + Lu � 0 in UT ;

thenminUT

u = min�T

u:

Theorem 2 (Weak maximum principle for c � 0)Assume u 2 C21 (UT ) \ C(UT ) and c � 0 in UT .

i) If

ut + Lu � 0 in UT ;

thenmaxUT

u � max�T

u+:

ii) If

ut + Lu � 0 in UT ;

thenminUT

u � min�T

u�:

Remark 1 In particular, if ut + Lu = 0 in UT , then

maxUT

j u j= max�T

j u j :

The following comparison lemma is immediate from the maximum prin-ciples already stated.

Lemma 1 We consider u1 and u2, two solutions of the equation

ut + Lu = 0 in UT ;

and satisfying the condition

u1 � u2 in �T ;

then

u1 � u2 in UT :

The corresponding results can be easily adapted for backward equations.

4

3 Main Results for European and American Calls

The Black-Scholes partial di�erential equation for a European call is (see[20])

@CEu@t

+1

2�2S2

@2CEu@S2

+ rS@CEu@S

� rCEu = 0; (2)

where CEu denotes the price of the European call, with payo� CEu(T; S) =max(S � E; 0), and boundary conditionsCEu(t; 0) = 0 and CEu(t; S)! S if S !1.

Similarly, the corresponding partial di�erential inequality for an Ameri-can call is

@CA@t

+1

2�2S2

@2CA@S2

+ rS@CA@S

� rCA � 0; (3)

where CA denotes the price of the American call, with payo� CA(t; S) =max(S � E; 0), boundary conditions CA(t; 0) = 0, and CA(t; S) ! S ifS !1, and CA(t; S) � max(S � E; 0) at the free boundary.

Analogously, we will denote with PEu and PA the corresponding pricesfor a European and an American put option.

Remark 2 For the price of a European or an American call option, V (t; S)!S when S ! 1, which is the standard boundary condition. For simpli-city and without loss of generality, we will consider the boundary condi-tion V (t; ~S) = ~S, for an ~S �xed and su�ciently large. We could applythe maximum principle in unbounded domains, but imposing further tech-nical assumptions. This is also necessary in order to accurately implementnumerical methods.

A �rst trivial statement, consequence of the maximum principle, is that

CEu(T; S) � 0 and CA(t; S) � 0, which ensures the positivity of prices(see [7]).

Recall that if we consider the Black-Scholes equation for a European callon a dividend-paying asset, we have

5

@CED@t

+1

2�2S2

@2CED@S2

+ (r �D0)S@CED@S

� rCED = 0; (4)

with payo� CED(T; S) = max(S � E; 0).For an American call on a dividend-paying asset, equality in equation

(4) becomes an inequality. In this case, we have the following conditionCAD(t; S) � max(S � E; 0) at the free boundary.

Lemma 2 An American call in absence of arbitrage opportunities must sa-tisfy

CA(t; S) � max(S � E; 0): (5)

Proof

This inequality has always been proved using economic and �nancialarguments as follows:

The put-call parity is given byCEu = PEu + S � PV (E), where PV stands for present value. Then

CEu = PEu + S � V P (E) + E � E= (S � E) + fE � V P (E)g+ PEu� S � E: (since the terms left out are positive)

Now we want to prove the same lemma using the maximum principle. Forthis example we give a detailed proof. Later, we will use similar argumentsand, thus, omit some of the details.

We want to show that CA(t; S) � max(S � E; 0), and we already knowthat CA(t; S) � 0. Thus, we only have to verify that CA(t; S) � S � E.

We consider the Black-Scholes inequality for an American call (3).By standard regularity arguments [9], CA(t; S) 2 C21 (Ut) \ C( �Ut) and

�t = �Ut � Ut.We make the next change of variable � = T � t. Then (3) is transformed

into

6

@CA@�

� LBS(CA)

=@CA@�

� 12�2S2

@2CA@S2

� rS @CA@S

+ rCA � 0; (6)

or,

@(�CA)@�

� LBS(�CA)

=@(�CA)@�

� 12�2S2

@2(�CA)@S2

� rS @(�CA)@S

+ r(�CA) � 0; (7)

since r � 0, the inequality satis�es the conditions to apply the weak maxi-mum principle with c = r � 0 (equation (2)). Then from equation (7), wewant to verify, applying the maximum principle, that

(S � E)� CA � 0:

Let u = (S � E)� CA, then we have to show that

u� � LBS(u) � 0:

Substituting the value of u, we get

(S � E)� + (�CA)� � LBS(S � E)� LBS(�CA) � �rS + rS � rE� �rE � 0;

since r � 0, and E � 0.If we apply the maximum principle, we have:

maxU�

((S � E)� CA) � max��((S � E)� CA)+ = 0;

given that we have in the boundary the following conditions:

i) If S > E

) CA = S�E ) (S�E)�CA = 0 ) max��((S�E)�CA)+ = 0:

7

ii) If S < E

) CA = 0 ) (S � E)� CA = S � E � 0 ) max��(S � E)+ = 0:

iii) If S = 0

) CA = 0 ) (S � E)� CA = �E � 0 ) max��(�E)+ = 0:

iv) If S is large

) CA � S ) (S � E)� CA � �E � 0 ) max��(�E)+ = 0:

The last analysis reveals that

max�U�((S � E)� CA) � 0

) (S � E)� CA � 0 in �U� ; so CA � (S � E):

This completes our proof. 2This �nancial interpretation leads to an important result: an American

call on a non-dividend-paying asset will never be exercised prior to expira-tion, hence, it has the same value as a European call.

If we consider now an American call on a dividend-paying asset, we alsorecover and extend some important results.

Let us consider the Black-Scholes inequality for an American call on adividend-paying asset

@CAD@t

+@CAD@S2

+ (r �D0)S@CAD@S

� rCAD � 0: (8)

Similarly to the last proof, we have the change of variable � = T � t, andtaking u = (S � E)� CAD we get

u� � LBSD(u) � �rS + SD0 + rS � rE � SD0 � rE:

In this case, we cannot apply the maximum principle because we do notknow the sign of SD0� rE. For this reason, the inequality CAD � S �E isnot always true, i.e., it tells us that the early exercise could occur.

8

The main idea of using the maximum principle is that we can say accu-rately when the early exercise is possible.

If SD0 < rE implies that u� +LBSD(u) � 0 and if we use the maximumprinciple, we have CAD � S�E. The last proof shows that the early exercisedoes not occur.

For the sake of consistency, we will verify that the early exercise for anAmerican put, PA, cannot be ruled out with our methodology. Proceedingby contradiction, we try to show that PA � (E � S) � 0.

Let u = PA � (E � S), and � = T � t

u� � LBS(u) � �rS � rE + rS � �rE � 0:

If we apply the maximum principle, we obtain the inequality

max�U�(PA � (E � S)) � max

��(PA � (E � S))+:

We verifying the boundary

i)

If S > E ) max��(S � E)+ = S � E:

ii)

If S < E ) max��(PA � (E � S))+ = 0:

iii)

If S = 0 ) max��(S)+ = S:

iv)

If S !1 ) max��(S)+ = S:

Since the boundary term does not have a de�nite sign, we do not obtainthe inequality PA � max(E � S; 0). This reveals that an American putoption can, in principle, be exercised any time before expiring.

We would like to prove Lemma 2 in a di�erent way. To do that, we provethe following theorem.

9

Theorem 3 If the underlying asset does not give dividends, the value of aEuropean call is the same as that of an American call, i.e., the early exercisefor an American call does not occur.

Before providing proof, some facts need to be presented. We need torefer to [7].

We have already proved this result, by showing that the American callsatis�es the condition CA � max(S�E), but now we want to establish thistheorem using other tools that will also be useful in other contexts. It iswell known that for a European option V , the solution to the Black-Scholesequation can be expressed using the Feynman-Kac formula [17]:

V (t; S) = E

�g�Xt;ST

�exp

��Z T

tr(�;Xt;S

� d�)

��; (9)

where V can be a put or a call value, g is the payo� and for � 2 [0; t] theprocess Xt;S

� is de�ned by considering Xt;S� � S and where for � 2 [t; T ]

the process Xt;S� is de�ned to be the solution of the stochastic di�erential

equation

dXt;S� = r

��;Xt;S

�

�Xt;S� dt+ �

��;Xt;S

�

�dB� and Xt;S

� = S: (10)

The expectation in (9) is considered with respect to the risk neutralmeasure.

This is the fundamental theorem of asset pricing, which states that thearbitrage free price can be obtained as an expected value with respect tothe risk neutral measure.

In turn, this can be written as an expected value, not with respect tothe risk neutral measure, but with respect to the physical measure, with anappropriate discount factor (the so-called Samuelson formula [3]). Moreover,the density of the physical measure satis�es a Fokker-Planck equation, whichcan be written once the dynamics of S is given by a SDE (see [15]).

In fact, an analogous formulation can be given for an American call (formore details see [7] and [19]). Recall that the equation for a European callis (2).

According to Samuelson's formula the price of a European call at time tand exercise time T is

CEu(t; S; T ) = e�r(T�t)Et;Sp

�g

�S(T�t)

Ep(S(T�t))S0e

r(T�t)��

; (11)

10

where g(�) is the payo�; Ep(St) = e�(T�t)S0, � is the return interest rate andEp is the expected value at time t, with respect to the physical measure.

If the price of an American call is written in terms of a European call,we get the following formula

CA(t; S; T ) = Et;Sp (CA(�; S; t)=�) =

Z t

0�(�; S)CEu(�; S; t)d�; (12)

with CEu(t; S; T ) as before a European call with exercise time T and timebefore to expire t, and �(t; S) is the probability that early exercise occurs attime t, when the underlying asset is S2 and where we have used the fact thata European and an American call non-dividend-paying stock are identical.

We know that �(t; S) satis�es the following Fokker-Planck equation(FP)

@�

@t=1

2

@2��2S2�

�@S2

� @ (�S�)@S

( see also [7] ); (13)

with boundary conditions: �(t; 0) = 0 8t 2 [0; T ], �(S; t) ! 0 as S ! 1,and the initial condition �(0; S) = �(S � S0) with S0 the initial underlyingasset.

Proof of Theorem 3

Using this approach we can easily see that CA(t; S) � CEu(t; S).

CA(t; S) =

Z t

0�(�; S)CEu(�; S)d�

�Z t

0�(�; S)CEu(t; S)d� ( by Theorem 4; see below )

= CEu(t; S)

Z t

0�(�; S)d� (since CEu(t; S) does not depend on �);

� CEu(t; S): ( since � is a probability distribution

function and integrates to one in [0; T ])

The opposite inequality, CA � CEu, is a general fact (see [14]) and followseasily from the maximum principle (see [7]). The proof is analogous to thatof Lemma 2.

2In geometric terms, �(t; S) is the probability that the process exists in the domainthrough the free boundary.

11

We can give an alternative proof by applying the maximum principle to�.

We consider the Fokker-Planck equation (13) and, according to the ma-ximum principle [Theorem 1], we have

a)

maxUt

(��) � max�Ut

(��)+ = 0;

or

�� � 0 ) � � 0:

The result of this inequality is that � � 0, because � is a probabilitydensity function.

b) On the other hand, it also satis�es

minUt

(��) � min�Ut(��)�:

Therefore

1) if � = 1

minUt

(��) � �1 ) �maxUt

(�) � �1 ) maxUt

(�) � 1:

2) if � = 0

minUt

(��) � 0 ) �maxUt

(�) � 0 ) maxUt

(�) � 0:

This means that in UT the function is identically zero, except perhapsat the top when t = T .

We have proved before that CA(t; S)� (S�E) � 0, i.e., the boundary inthe right hand side is not reached because the early exercise does not occur(Lemma 2).

Since the only portion of the boundary where � can be di�erent fromzero is at ft = Tg and

RUT� = 1, we see that this measure is a Dirac delta

concentrated in ft = Tg.

12

In this way, we get the price of an American call:

CA(t; S) =

Z t

0�(�; S)CEu(�; S; t)d�

=

Z t

0�(� � t; S)CEu(�; S; t)d�

= CEu(t; S):

We can say that the value of an American call is the same as that of aEuropean call, since the early exercise does not occur.2

3.1 Others Results about American Call Options

We mention other results about American and European calls. Their proofsrely not only on the maximum principle, but on the so-called sliding andmoving planes methods [2].

We consider the heat equation

@u

@t=@2u

@x2+ b

@u

@x+ cu;

in the rectangle [0; T ]� [0; L], with initial and boundary conditionsu(0; x) = u0(x) convex and u(t; 0) = u(t; L) = 0, 0 � t � T , respectively.

Theorem 4 Let u be a classical solution, then u is monotone in t, i.e.,

u(t; x) � u(t0; x) if t � t0 8x 2 [0; L]

Proof

By contradiction. Assume that u(t; x) < u(t0; x) with t � t0 and let

t� = supft j u(t; x) � u(t0; x); x 2 [0; L]g:

If t� > 0 then

u(t; x) � u(t0; x) 8x 2 [0; L] and 0 � t0�:

Notice that this is true since the initial condition is convex as it can beeasily veri�ed [7].

13

De�ne

u� = u(t+ �; x) for t 2 [t� � �; t� + �];

and � su�ciently small so that the measure of [t�� �; t�+ �]� [0; L] is smalland the maximum principle holds [5]. � is considered to be also small.

Then u� (t� � �; x) � u(t� � �; x), by the de�nition of t�.From the maximum principle it follows that the previous inequality is

valid in [t� � �; t� + �]� [0; L].This in turn implies that u(t; x) � u(t0; x) for t0� + �, and this is a

contradiction.2As a particular case, we have the next lemma.

Lemma 3 Given two call options on the same stock S, with the same ex-piring time and with the same exercise price E, but evaluated at di�erenttimes before expiring (real times), t2 > t1, they satisfy

C(t2; S; E) � C(t1; S; E): (14)

Notice that by Theorem 3, C can be either American or European.

Proof

Let C(t2; S; E) and C(t1; S; E) be two options with t2 > t1.We putS = Eex, ti = T � �i

12�2

and C(t; S;E) = E�(x; �i), for i = 1; 2.

According to this change of variables equation (2) is transformed in thefollowing heat equation and here the �i's correspond to the time of expiration

@�

@�i=@2�

@x2+ (k1 � 1)

@�

@x� k1�; k1 =

r12�

2

and the �nal condition is transformed in an initial condition of the followingform

�(0; x) = max(ex � 1; 0);

which is convex and its boundary condition is

�(t; 0) = �(t; L) = 0; 0 � t � T:

14

In the previous theorem we consider t = �1, t0 = �2, u(t; x) = �(�1; x)

and u(t0; x) = �(�2; x).2

Remark 3 Using the maximum principle, it is easy to show that for twogiven call options on the same stock S, with the same exercise price E, butwith di�erent expiring times, T2 > T1, they satisfy, C(T2; S) � C(T1; S).

Lemma 4 Given two identical calls options, American or European, withthe exception that one has an exercise price larger than the other, i.e., E2 >E1, they must satisfy the following inequality

C(t; S;E2) � C(t; S;E1): (15)

Proof

We are going to prove the lemma only for a European call. The case ofan American call is similar.

Let two European call options be CEu(t; S;E1) and CEu(t; S;E2), res-pectively. They satisfy equation (2) with boundary conditions:

� CEu(t; 0; Ei) = 0, for i = 1; 2

� CEu(t; S;Ei)! S if S !1, for i = 1; 2, and

� CEu(T; S;Ei) = max(S � Ei; 0), for i = 1; 2

We are going to use Lemma 1. We make a change of variables so that(2) becomes a parabolic equation

@u

@�=@2u

@x2i�1 < xi <1 y � > 0;

with xi = log(S=Ei) for i = 1; 2 and initial condition

u(0; xi) = max(e12(k1+1)xi � e(k1�1)xi ; 0):

If we apply Lemma 1, we have to prove that for x2 < x1,

u(0; x1) > u(0; x2):

First, we consider two cases

15

1) if S � Ei ) SEi� 1 ) ln( SEi ) � 0.

2) if S � Ei ) SEi� 1 ) ln( SEi ) � 0.

Without loss of generality, we only consider the �rst case (S � Ei),because for the second case, we have max(S � Ei; 0) = 0.

Let E1 < E2, S � 0 and S > Ei for i = 1; 2, then

x1 � x2:

Since k1 =2r�2> 0

e12(k1+1)x1 � e

12(k1+1)x2 ; (16)

and using direct calculus, we can prove

�e12(k1�1)x1 � �e

12(k1�1)x2 : (17)

From inequalities (16) and (17), we get

e12(k1+1)x1 � e

12(k1�1)x1 � e

12(k1+1)x2 � e

12(k1�1)x2 :

We have

max(e12(k1+1)x1 � e

12(k1�1)x1 ; 0) � max(e

12(k1+1)x2 � e

12(k1�1)x2 ; 0);

therefore u(0; x1) � u(0; x2) for x2 < x1.

Then, we can apply Lemma 1 and have

u(t; x1) � u(t; x1):

Thus, we have proved that

CEu(t; S;E2) � CEu(t; S;E1) if E1 < E2:2

Finally, according to Merton, now we want to establish that a Europeanoption is equivalent to a long position in the common stock levered by alimited-liability discount loan, where the borrower promises to pay E dollarsat the end of � periods, but in the event of default is only liable to the extent

16

of the value of the common stock at that time. If the present value of sucha loan is a decreasing function of the interest rate, then, for a given stockprice, the option price will be an increasing function of the interest rate.

Let P (�) be the price of a riskless (in terms of default) discounted loan(or bond), which pays one dollar � years from now. If it is assumed thatcurrent and future interest rates are positive, then

1 = P (0) > P (t1) > P (t2) > : : : > P (tn); for 0 < t1 < t2 < : : : < tn:

Theorem 5 If the exercise price of a European call option is E, the under-lying asset does not pay dividends and we build the common stock over theoption life , then

CEu(t; S) � max(0; S � EP (t)): (18)

Proof

The proof is analogous to that of Lemma 2.

4 Conclusions

As we mentioned in the introduction, and from the techniques used in theproofs, it is clear that basically all results can be extended to semilinearmodels and to fully nonlinear equations (see [6] and [16]).

A relevant direction of future research is to consider models that allowfor small arbitrage opportunities, in which case our approach would enableus to consider equations of the form (2) with variable interest rates. Inthis context, an arbitrage opportunity would be generated if r(t) becomesnegative for a short period.

However, the tools we employed would still be applicable provided thesetime "windows" are not very large. Along the same lines, using analyticaltechniques provides us with a more quantitative version of Merton's RTOP(see equation (8)).

References

[1] H.Ahn and P. Wilmott, On trading american options, OCIAM, OxfordUniversity, OFRC Working Papers Series, Number 1999mf02.

17

[2] H. Berestycki H. and L. Nirenberg, On the method of moving planesand the sliding method, Bol. Soc. Brasil. de Mat. 22 (1991), 1-37.

[3] M. Bladt, T. H. Rydberg, An actuarial approach to option pricing un-der the physical measure and without market assumptions, Insurance:Mathematics and Economics. 22(1) (1998), 65-73.

[4] T. Bj�ork, Arbitrage Ttheory in Continuos Time (Oxford UniversityPress, 1998).

[5] X. Cabr�e, On the Alexandro�-Bakelman-Pucci estimate and reversedH�older inequality for solutions or elliptic and parabolic equations, Com-mun. Pure Appl. Math. 48 (1995) 539-570.

[6] d-�ne Christ Churh College, Nonlineal Black Scholes Modelling: FDMvs FEM, Msc in Mathematical Finance Thesis, University of Oxford(2004).

[7] R. Elizondo, Implementaci�on de factores macroecon�omicos a algunosmodelos de productos derivados, PhD in Sciences Mathematics Thesis,IIMAS, UNAM (2008).

[8] L. C. Evans, Partial Di�erential Equations (American MathematicalSociety, 1998).

[9] D. Gilbarg and N. Trudinger, Elliptic Partial Di�erential Equations ofSecond Order, 2nd edn. (Springer-Verlag, 1983).

[10] H. Han and X. Wu, A fast numerical method for the Black-Scholesequation of American options, SIAM. J. Numer. Anal. 41(6) (2003)2081-2095.

[11] R. A. Jarrow, Preferences, continuity and the arbitrage pricing theory,The Review of Financial Studies 1(2) (1988) 159-172.

[12] F. Llenera-Garc�es, Una nota sobre valoraci�on de opciones Americanasy arbitraje, Investigaciones Econ�omicas 24(I) (2000) 207-218.

[13] H. M. Merton and F. Modigliani, Dividend Policy and Market Balua-tion: A Reply, The Journal of Business. 36(1) (1963) 116-119.

[14] R. C. Merton, Continuos-Time Finance, 4th. edn. (Blackwell, 1995).

18

[15] B. �ksndal, Stochastic Di�erential Equations: An introduction withAppplications (Springer-Verlag, Berlin, 1985).

[16] M. Sirbu and S. E. Shreve, A two person game for pricing convertiblebonds, SIAM J. Control Optim. 45(4) (2006) 1508-1539.

[17] J. M. Steele, Stochastic Calculus and Financial Applications (Springer-Verlag, 2001).

[18] A. Tijonov and A. Samarsky, Ecuaciones de la F��sica Matem�atica, (Edi-torial MIR, Moscu, 1983).

[19] N. Uys, Optimal stopping problems and American option, MSc disser-tation, University of Witwatersrand (2005).

[20] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathema-tical Models and Computations, (Oxford Financial Press, 1993).

19