Continuous-Time Evolutionary Stock and Bond
Markets with Time-Dependent Strategies�
Zhaojun Yang
School of Economics and Trade, Hunan University,
Changsha 410079 ( e-mail: zjyang@ )
Abstract
This paper develops a general continuous-time evolutionary �nance
model with time-dependent strategies. It is shown that the continuous
model, which is a limit of a general discrete model, is well-de�ned and
if there exists one completely diversi�ed strategy in the market, then
there is no sudden bankruptcy. After that a deterministic evolutionary
bond market is studied in detail. It is certi�ed that a bond market is
evolutionary stable, which is equal to arbitrage-free if and only if the
total returns de�ned in this paper across all the assets are the same, or
each bond is evaluated by an improper integral in which the integrand
is a discounted value of the dividend payo� with the discount rate
being market consumption parameter. Last an approach to compute
the benchmark interest rate is provided.
Keywords: Continuous Evolutionary Finance, Time-dependent Strat-
egy, Evolutionary Stable Bond Market, Bond Valuation, Benchmark Inter-
est Rate.
JEL subject classi�cation: D59, G11, G12
1 Introduction
The Darwinian principle "Survival of the Fittest" is well known to evolution-
ary biologists but rather less known for its applicability in �nancial market
�Research supported by China Social Science Fund (06BJL022), Economic Opening
and Trade Development Project in Philosophy and Social Science Innovation Foundation
of Hunan University 985 Engineering. I am grateful to Professor Klaus Reiner Schenk-
Hopp�e and Professor Jiaan Yan. This paper was initiated by Professor Schenk-Hopp�e
while I visited the University of Leeds in 2005 and the main work was done while I visited
Chinese Academy of Sciences in 2007. A substantial revision is made in this version and
all remaining errors are mine.
1
theory. However this major principle, as applied to �nancial markets, means
nothing else that those traders with the most successful trading strategies
will dominate the market at last, after an evolutionary process has taken
place. This evolutionary process can be understood as a process of adapta-
tion and imitation, rather than a process of inheritance in evolutionary biol-
ogy. From an evolutionary point, the market is completely determined by the
corresponding evolutionary stable trading strategy. Evolutionary �nancial
market models have been considered in Blume and Easley (1992), Evstig-
ineev, Hens and Schenk-Hopp�e (2006), Farmer and Lo (1999). The main
point in setting up an evolutionary �nancial market model is the speci�ca-
tion of an evolutionary dynamic which determines the market shares of the
relevant trading strategies in time. In general such a dynamic will depend on
the stochastic payo�s, dividends or prices of the underlying assets, as well as
the trading strategies, which are assumed to be adapted to the underlying
information. The models developed in Evstigineev, Hens and Schenk-Hopp�e
(2006) and Hens and Schenk-Hopp�e (2005) assume that stochastic dividends
or payo�s of the underlying assets are exogenously given, but that in contrast
to other models the asset prices are determined by the trading strategies and
a market clearing condition.
This paper employs a similar approach as in Evstigineev, Hens and
Schenk-Hopp�e (2006) and Hens and Schenk-Hopp�e (2005) but sets up a
model in continuous time rather than in discrete time. The choice of con-
tinuous time brings with it the usual technical problems which lie in the
analytical formulation of the model, in particular in a probabilistic frame-
work, but has the major bene�t. In particular, the methods from classical
analysis such as PDE and stochastic calculus become applicable and pro-
vide powerful tools for the solution of problems. It is therefore necessary
to extend discrete time evolutionary Evstigineev, Hens and Schenk-Hopp�e
(2006) �nance models to continuous time framework.
As a �rst step into this direction, Yang and Ewald (2008) considers a
model in which the trading strategies are assumed to be �x-mix, that is, the
relative budget fractions are constant in time. In this model Yang and Ewald
(2008) sets up a continuous time stochastic dynamic which describes the evo-
lution of market shares in a population of �nitely many trading strategies.
Yang and Ewald (2008) identi�es evolutionary stable investment strategies,
. those strategies that prevent entrants to the �nancial market from gain-
ing wealth in the long run. Under the assumption that the relative dividends
are �rst-order stationary and ergodic, Yang and Ewald (2008) derives an evo-
lutionary stable investment rule. Fix strategies are simple and also suitable
in some economic environments but are of course too restrictive.
Along this research line, Buchmann and Weber (2007) derives a con-
tinuous time approximation of the evolutionary market selection model of
Blume and Easley (1992). Conditions on the payo� structure of the assets
are identi�ed that guarantee convergence. It is shown that the continuous
2
time approximation equals the solution of an integral equation in a random
environment. For constant asset returns, the long-run asymptotic behavior
is discussed in detail.
However, Buchmann and Weber (2007) is substantially simpler than
what studied in this paper. First Buchmann and Weber (2007) considers a
market with only short-lived assets and thus capital gains are omitted, which
is much less interesting. Second, to guarantee convergence, Buchmann and
Weber (2007) imposes an unnatural condition on the payo� structure of the
assets and so, strictly speaking, its continuous-time model is not a limit of
the model of Blume and Easley (1992). Last, the assumption of constant
asset returns, which is much simpler than the one supposed by this paper, is
extremely restrictive and thus its strategies are �x or time-invariant as well.
Taking a step further into the continuous evolutionary �nance model,
this paper considers a model in which strategies are time-dependent rather
than time-invariant. A continuous time approximation of the evolutionary
market selection of a general discrete-time model is �rst derived, which is a
generalization of Evstigineev, Hens and Schenk-Hopp�e (2006). It is shown
that the continuous model is well-de�ned and only if one of the strategies
is completely diversi�ed - this condition almost imposes no restrictiveness,
there is no sudden bankruptcy in the market. Second, a bond market is
studied in which the dividend process paid by each asset is deterministic
and prices and wealth vary due to market interaction. It is certi�ed that
a bond market is evolutionary stable if and only if all the total returns of
bonds de�ned in this paper are equal to each other although the same total
return may change along the time. Any other market can be invaded just by
a portfolio that invests all wealth in an asset, which pays o� the largest total
return. When introduced on the market with arbitrarily small initial wealth,
this portfolio increases its market share at the incumbent's expense. By this
way the necessary and su�cient conditions are derived for the evolutionary
stable portfolio rule. It is shown that a bond market is evolutionary stable
if and only if each bond is evaluated by an improper integral in which the
integrand is a discounted value of the dividend payo� with the discount rate
being market consumption parameter. A formula is provided to evaluate the
discount rate or the market consumption rate based on the trading prices
in an relative e�ective bond market. By this way an approach to compute
benchmark interest rate is presented.
The structure of the article is as follows. Section 2 sets up a general
discrete-time market selection dynamic where the length of the trading time
interval is an arbitrary positive. Then a continuous-time evolutionary stock
market model is derived and discussed. Section 3 investigates an evolution-
ary bond market in which the dividend process is deterministic. Section 4
provides two formulas to price bonds and recover the benchmark interest
rate respectively. The main conclusions of this paper are summarized in
section 5.
3
2 Continuous-TimeMarket Selection Process with
Time-Dependent Strategies
Following Lucas (1978), this section introduces an in�nite horizon asset mar-
ket model. First a discrete market model is established with an arbitrary
trading interval and then a general continuous-time model is derived by
taking a limit of that the trading interval converges to zero.
Let (
;F ;P) be a probability space endowed with the �ltration (Ft)0�t�1,
where Ft is an information �ltration for market participants up to time t. All
the random variables and stochastic processes in this paper will be de�ned
on this base.
Consider an asset market with K � 1 long-lived assets and a single
perishable consumption good. The assets are indexed by k 2 �. Time is
discrete and denoted by t 2 � � fl�t j l = 0; 1; 2; � � � g, where �t is the
length of the time interval. In this market there are �nitely many players
(investors) and each player plays one strategy. All players or strategies
indexed by i = 1; � � � ; I; I � 2 compete with each other for the market
capital, and the amount of the wealth managed by strategy i is Ft�adapted
and denoted by wit; i = 1; � � � ; I. Each asset pays o� a dividend in cash or
just a single perishable consumption good as implied by Lucas (1978). As
seen in the following, whether the dividend is cash or good does not matter
only if it is supposed to be totally consumed. The amount of the dividend
paid by asset k at period t is Ft�adapted and denoted by Dkt . In this paper,
the following assumption is imposed.
Assumption 1. All strategies have a common consumption rate at each
period t, denoted by ct(�t), which is Ft�adapted satisfying 0 < ct (�t) <
1; t 2 �.
The assumption on common consumption rate is obviously necessary in
order to compare performances among strategies, since one of the aims of the
paper is to study what strategy will dominate the market at last. If, on the
contrary, the consumption rate is di�erent, then the strategy dominates the
market probably also because it consumes less. Moreover, the assumption
that 0 < ct (�t) < 1 says every strategy or player must consume at a strictly
positive consumption rate at each period but is not permitted to consume
all his wealth.
Denote by �it;k the fraction of the remaining wealth after consumption,
wit (1� ct (�t)), which strategy i assigns to the purchase of the asset k in
period t. Formally, a strategy is a stochastic process, which is Ft�adapted.
Further, the next assumption is made.
Assumption 2. (i) For every strategy, the fraction invested in each asset is
Ft�adapted and non-negative, . �it;k � 0 for almost every sample path and
all t; k; i. (ii) there is at least one completely diversi�ed strategy, . given
4
that t 2 � or t � 0 for the continuous time model, there is a j 2 f1; � � � ; Ig
with wjt > 0 such that 0 < �
j
t;k < 1 for almost every sample path and all k.
Assumption 2 means that short selling is prohibited (though which is
quite possibly unnecessary to get the main conclusions of this paper). Mean-
while Assumption 2 also makes sure that the prices of all assets are strictly
positive thanks to (1) below, actually which is just the reason why it is
assumed.
In this framework, all the assets are just K long-lived assets, which pay
o� dividends in cash or produce the same perishable consumption good, say
milk. The cash or perishable consumption good are consumed completely
and especially can not be used to reinvest. For this reason the total amount
of assets traded in the market keeps constant. On account of a stock split,
the following assumption is imposed without loss of generality:
Assumption 3. The supply of each asset is normalized to 1.
According to this assumption, the market-clearing price, denoted by �kt ,
is given by
�kt =
IX
i=1
�it;kw
i
t (1� ct (�t)): (1)
Denote the aggregate market wealth by Wt, . Wt =
PI
i=1w
i
t, then the
market-clearing price of consumption good or cash, denoted by �0t ;8 t 2 �,
is determined by
Wtct (�t) = �0tDt; (2)
where Dt is the aggregate dividend, . Dt =
PK
k=1D
k
t . In contrast to (2),
Evstigineev, Hens and Schenk-Hopp�e (2006) takes a normalized price for
cash, . �0t � 1. Following this, (2) is written as Wtct (�t) = Dt, which is
however a bit more arti�cial, although the di�erence is not important.
Let � � f1; � � � ;Kg. For a given strategy pro�le and the wealth wit; i =
1; � � � ; I, the percentage of all shares issued of asset k that strategy i invests
during period t is
�it;k =
�it;kw
i
tPI
j=1 �
j
t;kw
j
t
; t 2 �; k 2 �: (3)
It is evident that
KX
k=1
�it;k = 1;
IX
i=1
�it;k = 1: (4)
Since the change of a wealth results only from dividends and capital gains,
the wealth of strategy i at the beginning of period t + �t, . wit+1 is
determined by
wit+�t � (1� ct(�t))wit =
KX
k=1
�
�0t+�tD
k
t+�t + (�
k
t+�t � �kt )
�
�it;k; (5)
5
or according to (1) and (3)�(4), determined by
wit+�t =
KX
k=1
�
�0t+�tD
k
t+�t + �
k
t+�t
�
�it;k: (6)
While the total amount of assets traded in the market keeps constant, the
aggregate market wealth Wt does change with time because of trading ran-
domly. However, social welfare has evidently no relation to the level of the
market wealth Wt and a big Wt only means higher prices of the assets and
consumption good or cash. Actually, there exist in�nitely many solutions
to the wealth dynamics (6) since it is by (2) homogeneous for all wealth
variables, . wit; t 2 �; i = 1; � � � ; I. In particular, any solution multiplied
by an arbitrary non-zero number is still a solution. For this reason and
an economic consideration, the aggregate market wealth Wt is taken as a
num�eraire in the following.
Based on this num�eraire, the normalized prices for asset k and con-
sumption good are pkt � �kt
�
Wt and pk0 � �k0
�
Wt respectively for each pe-
riod t. Further, the normalized wealth or the market share of strategy i is
rit � wit
�
Wt. Accordingly, it follows directly from (1) and (3) that
pkt =
IX
i=1
�it;kr
i
t (1� ct(�t)); �it;k =
(1� ct(�t))�it;krit
pkt
and
IX
i=1
rit = 1: (7)
Obviously the market share process rit is the most important index that
tells how strategy i performs. For example, the market share process deter-
mines whether the strategy is dominated or dominating. Consequently, the
dynamics of market share process is studied below in detail.
Similar to Yang and Ewald (2008), by an elementary manipulation, for
all i; k; t it follows from (1)�(2) and (6) that
rit+�t =
KX
k=1
0@ct (�t) dkt+�t + (1� ct (�t)) IX
j=1
�jt+�t;kr
j
t+�t
1A�it;k; (8)
where dkt represents the relative dividend payment of asset k, . d
k
t �
Dkt
�
Dt. Equation (8) is called market selection equation.
Obviously rit � 0 thanks to (8) if ri0 = 0. In order to avoid this un-
interesting case, it is supposed throughout the paper that ri0 > 0 for each
i 2 f1; � � � ; Ig. Further, it is not di�cult to prove that (8) is well-de�ned.
Formally, the following theorem holds.
Theorem 1. Given a strategy pro�le and all the relative dividend payment
processes, fdkt gt2�; k 2 �, and provided that Assumption 1�2 hold, there
exists a path-wise unique market share process fritgt2� satisfying (8) for
each strategy i; i = 1; � � � ; I.
6
Proof. A sketch of a proof is given here and for more details refer to
Evstigineev, Hens and Schenk-Hopp�e (2006). In fact, this theorem can be
proved by solving (8) step by step forward. At each step, say period t+�t,
(8) is a system of I linear equations in I variables, . rit+�t; i 2 f1; � � � ; Ig
and the coe�cient matrix can be proved to be invertible by verifying that it
has a column dominant diagonal thanks to Assumption 1�2 and (4). Thus
the theorem is proved. �
A continuous-time evolutionary �nance model with time-dependent strate-
gies is established in the following by letting �t! 0 in (8). To achieve this
goal, more assumptions are needed and shown below.
Assumption 4. For all t 2 �; k 2 �; i 2 f1; � � � ; Ig and every sample path,
the following limits exist and the equalities hold:
lim
�t!0
dkt+�t = d
k
t ; lim
�t!0
�it+�t;k = �
i
t;k; lim
�t!0
�it+�t;k � �it;k
�t
� _�it;k (9)
Throughout this paper, "dot" notation is used for derivatives. This
assumption seems too restrictive but actually not. The �rst equality of
the assumption says the relative dividend process is continuous as well as
stochastic, which describes the economic phenomenon that the relative div-
idend changes gradually instead of suddenly. In addition, it is well-known
that a continuous-time portfolio in mathematical �nance is mostly left con-
tinuous with right hand limits. Consequently the portfolio is generally not
continuous, let alone being di�erentiable. On the contrary, asset prices in
this setup are endogenous instead of exogenous as in mathematical �nance
and they do change along with the portfolio. Accordingly, once a portfo-
lio is performed, the wealth managed by any players or strategies will be
changed in the same time through a complicated rebalancing process for
market-clearing. This rebalancing process will make the amount of wealth
managed by any players changed after trading, which is totally di�erent
from the self-�nancing trading assumed by mathematical �nance. As a re-
sult, every alteration of a portfolio will incur a strong resistance from the
market and the speed of alteration improbably gets too quick - that is to
say, the portfolio is altered slowly and thus the di�erential assumptions in
(9) are also acceptable.
Following the seemly strongly restrictive Assumption 4, the next assump-
tion is intuitively reasonable.
Assumption 5. The common consumption rate for each player or strategy
is a function of the time interval �t and the following limits exist for almost
every sample path and all t 2 �:
lim
�t!0
ct (�t) = 0 and lim
�t!0
ct (�t)
�t
� �ct: (10)
7
Assumption 5 says that the longer the time interval, the more the con-
sumption rate and there exists an instantaneous speed of the change of the
consumption rate, which is �ct for all t 2 �.
Now it is ready to derive the continuous-time evolutionary �nance model
with time-dependent strategies. In order to get a deep insight, the following
text turns to the case I = 2, which is particularly important from evolution-
ary game theory. Under this case, only two strategies compete with each
other for the market capital and the market shares satisfy that r1t + r
2
t = 1
for all t 2 �. Therefore the dynamics of only one market share, say r2t ; t 2 �
will be studied below in detail.
The following theorem is the main conclusion of this section, in which
(12) is called continuous-time market selection equation with time-dependent
strategies.
Theorem 2. Suppose a strategy pro�le f�it;kgt2�; i = 1; 2; k 2 �, is given
and Assumption 1�5 hold:
(i) For all t 2 � and i = 1; 2, 0 < rit+�t < 1 if and only if 0 < rit < 1;
(ii) For almost every sample path,
lim
�t!0
r1t+�t = r
1
t ; lim
�t!0
r2t+�t = r
2
t ; (11)
(iii) Further, for almost every sample path, the market share rit; i = 1; 2
is di�erentiable and r2t satis�es the following random di�erential equation
_r2t =
��ctr2t +
PK
k=1
�2t;kr
2
t ( _�1t;k(1�r2t )+_�2t;kr2t+�ctdkt )
�1t;k(1�r2t )+�2t;kr2tPK
k=1
�1t;k�
2
t;k
�1t;k(1�r2t )+�2t;kr2t
; (12)
with initial value r20, while _r
1
t = � _r2t :
Proof. It su�ces to verify the conclusions about the market share of
strategy 2. First, It follows from (8) and the case I = 2 that
r2t+�t =
ct(�t)
PK
k=1 d
k
t+�t�
2
t;k + (1� ct(�t))
PK
k=1 �
1
t+�t;k�
2
t;k
ct(�t) + (1� ct(�t))
PK
k=1
�
�1t+�t;k�
2
t;k + �
2
t+�t;k�
1
t;k
� : (13)
As a result, part (i) of this theorem is immediately proved thanks to the
assumptions of the theorem and
r1t + r
2
t = 1; t 2 �: (14)
From (4) and (7) it is concluded that
r1t
KX
k=1
�1t;k�
2
t;k + r
2
t
KX
k=1
�2t;k�
2
t;k =
KX
k=1
�
r1t �
1
t;k + r
2
t �
2
t;k
�
�2t;k = r
2
t : (15)
8
Consequently, one gets that
r2t+�t � r2t
=
ct(�t)
PK
k=1(d
k
t+�t�r2t )�2t;k+(1�ct(�t))
PK
k=1 �
2
t;k
P2
i=1 r
i
t(�
i
t+�t;k��it;k)
ct(�t)+(1�ct(�t))
PK
k=1(�1t+�t;k�2t;k+�2t+�t;k�1t;k)
:
(16)
Next, let the time interval shrink to zero, . �t! 0, then by (9)�(10) and
(14) the numerator converges to zero and its denominator of the right-hand
side of (16) converges to
KX
k=1
�1t;k�
2
t;k +
KX
k=1
�2t;k�
1
t;k =
KX
k=1
�1t;k�
2
t;k
�1t;k(1� r2t ) + �2t;kr2t
> 0: (17)
Accordingly, the second equality of (11) is derived. In the same way, one
infers from (13) that
_r2t = lim
�t!0
r2t+�t � r2t
�t
=
�ct
PK
k=1(d
k
t � r2t )�2t;k +
PK
k=1 �
2
t;k
P2
i=1 r
i
t
_�it;kPK
k=1
�1t;k�
2
t;k
�1t;k(1�r2t )+�2t;kr2t
:
Therefore, part (iii) of the theorem is shown after a simple substitution from
(7). �
Market selection equation (12) di�ers from Ito^ Stochastic di�erential
equation since its integral can be de�ned path by path as a Riemann inte-
gral. (12) demonstrates explicitly that the evolution of each market share
is determined by all strategies as well as the relative dividend and it is well
de�ned in the sense that its solution for arbitrary initial value 0 � r20 � 1
exists and is path-wise unique. With a view to proving this conclusion, a
general existence and uniqueness are shown in the following.
Let
h(t; !; x) �
��ct +
PK
k=1
�2t;k( _�1t;k(1�x)+ _�2t;kx+�ctdkt )
�1t;k(1�x)+�2t;kxPK
k=1
�1t;k�
2
t;k
�1t;k(1�x)+�2t;kx
;
and g(t; !; x) � xh(t; !; x) then (12) can be equivalently written as
_x = g(t; !; x) or _x = xh(t; !; x); (18)
in which initial value t0 = 0; x(t0) = r20 and r
2
t � x(t). Generally speaking,
there is a path-wise unique solution to (18) for an arbitrary initial value
(t0; x(t0)) 2 [0;1) � (�1;1) instead of only t0 = 0; x(t0) = r20 2 [0; 1].
In order to get this result, the next assumption instead of Assumption 2 is
su�cient.
Assumption 6. The consumption rate and the derivatives of the fraction
invested in each asset are almost surely path-wise continuous, �nite and Ft-
adapted. That is to say, 0 < �ct < 1; j _�it;kj < 1 and �ct; _�it;k are continuous
functions of t for all t � 0; k 2 �; i 2 f1; 2g and almost every sample path.
9
This assumption nearly puts no extra restriction from an economic view-
point on account of the comments following Assumption 4. Now, the exis-
tence and uniqueness of (18) are formally presented below.
Theorem 3. If Assumption 6 holds for a given strategy pro�le, then for an
arbitrary initial value (t0; x(t0)) 2 [0;1)� (�1;1) with �1t0;k(1� x(t0)) +
�2t0;kx(t0) 6= 0 for all k 2 �, there exists a path-wise unique local solution of
random di�erential equation (18).
Proof. According to the assumption of the theorem, there is a null set
N 2 F , such that for every ! 2 N c, one can �nd an open subset U satisfying
(t0; x(t0)) 2 U � [0;1)� (�1;1), on which function g(t; !; x) is Lipschitz
continuous in the third argument, uniformly with respect to the �rst (for
more details refer to the proof of Theorem 4 below). Hence, the theorem is
proved thanks to the well-known Picard-Lindelof theorem. �
Based on Theorem 3 and Assumption 2, the existence and uniqueness
of market selection equation (12) are discussed next and a more insightful
result is summed up in the following.
Theorem 4. For a given strategy pro�le, Assumption 2 and 6 hold:
(i) For an arbitrary initial value 0 < ri0 < 1; i = 1; 2, there exists a
path-wise unique solution of market selection equation (12), which satis�es
0 < rit < 1; i = 1; 2 for all t > 0;
(ii) For almost every sample path, if ri0 = 0, then r
i
t = 0 for i = 1; 2 and
t � 0; if ri0 = 1, then rit = 1 for i = 1; 2 and t � 0.
Proof. According to Theorem 3 part (ii) of the theorem is obvious
and thus, without loss of generality, only part (i) for i = 2 is shown in the
following.
(12) can be written as
_r2t = g(t; !; r
2
t ) or _r
2
t = r
2
t h(t; !; r
2
t ); (19)
with initial value 0 < r20 < 1. Thanks to Theorem 3 and part (ii) of the
theorem, the phase space of (19) is included in the interval [0; 1] for an
arbitrary initial value 0 < r20 < 1.
First, there exists a positive t1 > 0 such that there is a path-wise unique
solution of the initial value problem (19) for a given sample path up to t1
owing to Theorem 3 and in particular, 0 < r2t < 1 can be guaranteed for
0 � t � t1. Second, the interval [0; t1) is extended as wide as possible in the
following way.
For a given strategy pro�le and an arbitrary positive integer n � 2,
de�ne stopping time
Tn(r20; !) � inf
n
t � 0
���9k 2 �; . �1t;k(1� r2t ) + �2t;kr2t < 1n
or maxf�ct; j _�1t;kj; j _�2t;kjg > n
o
;
10
where the in�mum of the empty set is understood to be 1. Note that for
all x 2 [0; 1] and 0 � t < Tn(r20; !),
1
n
� minf�1t;k; �2t;kg � �1t;k(1� x) + �2t;kx � maxf�1t;k; �2t;kg � 1
and
1
n
� minf�1t;k; �2t;kg �
�1t;k�
2
t;k
�1t;k(1� x) + �2t;kx
� maxf�1t;k; �2t;kg � 1;
and thus, it follows from a length but elementary calculation that for all
x 2 [0; 1], 0 � t < Tn(r20; !) and ! 2
,
jf 0x(t; !; x)j < 6n4; and h(t; !; x) > �2n3:
For this reason, there exists a path-wise unique solution of (19) up to
Tn(r20; !) by means of extensibility of solutions. Moreover, according to
the second equality of (19), the solution satis�es
r2t = r
2
0 exp
�Z t
0
h(u; !; r2u)du
�
> r20 exp
�Z t
0
�2n3du
�
> 0 (20)
for 0 � t � Tn(r20; !). On the other hand, it is evident that r2t < 1 since
r1t + r
2
t = 1 and one can also show r
1
t > 0 in the same way.
Last, it can be veri�ed that limn!1 Tn(r20; !) =1; a:s:. In fact, if there
is a measurable set A 2 F with P(A) > 0, such that limn!1 Tn(r20; !) =
�(!) < 1 for all ! 2 A, then there exists k 2 � and i 2 f1; 2g, such that
�1�(!);k(1 � r2�(!)) + �2�(!);kr2�(!) = 0, or �c�(!) = 1, or j _�i�(!);kj = 1 for
all ! 2 A. The last two equalities contradict Assumption 6 and the �rst
equality leads to r2�(!) = 0, . w
2
�(!) = 0 and �
1
�(!);k = 0 or r
1
�(!) = 0,
. w1�(!) = 0 and �
2
�(!);k = 0, which contradicts Assumption 2. Hence the
proof is �nished. �
It follows directly from Theorem 4 that if one of the strategies is com-
pletely diversi�ed then there is no sudden death or bankruptcy in the econ-
omy even some strategies are not completely diversi�ed.
For example, suppose in a market there are only two assets: One pays 1
and another pays nothing all the time, . d1t = 1 and d
2
t = 0 respectively.
The consumption parameter is constant, . �ct � c > 0. Strategy 1 is
completely diversi�ed with �1t;1 = � and �
1
t;2 = 1 � �, 0 < � < 1 all the
time but strategy 2 is not with �2t;1 = 0 and �
2
t;2 = 1. Strategy 2 is clearly
very bad since it invests all the wealth in asset 2, which pays nothing all
the time. However, the bad strategy will not go bankrupt forever only if its
initial wealth is strictly positive, . 0 < r20 < 1. In fact, a simple calculation
leads to
r2t =
(1� �)r20 exp(�ct)
1� �+ �r20 (1� exp(�ct))
> 0; for all t � 0;
although limt!1 r2t = 0 and thus limt!1 r1t = 1, . strategy 1 dominates
the market at last.
11
3 Evolutionary Stable Bond Market
Market selection equation (12) is a rather general continuous-time evolution-
ary �nance model, in which it is di�cult to answer what is the "optimal"
strategy since the strategies are time-dependent in a stochastic environment.
In order to avoid a too complex problem suddenly while studying this general
model, Yang and Ewald (2008) deals with the case of constant proportions
strategies though in a stochastic world, which is substantially simpler than
the general model established in last section. For the same reason, neverthe-
less from another point of view, the following text focuses on a deterministic
bond market but keeps the strategies time-dependent.
By a deterministic bond market we mean nothing other than the mar-
ket discussed above in which the relative dividend process is a deterministic
function of time t. While fundamentals are �xed, prices and wealth vary
due to market interaction. This market appears too restrictive but in fact
includes a variety of marketable treasury securities issued by the United
States Department of the Treasury through the Bureau of the Public Debt:
Treasury bills, Treasury notes, Treasury bonds, and Treasury In
ation Pro-
tected Securities (TIPS). This market also includes Account Treasury bonds
issued by the China Department of Finance, which were �rst issued in 1997
but the amount increases with an extremely quick speed.
In a deterministic world, the dividend is decided in advance and thus for
a given strategy pro�le, (12) is an ordinary di�erential equation. Let
f(r2t ;�
1
t;k; �
2
t;k) �
��ctr2t +
PK
k=1
�2t;kr
2
t ( _�1t;k(1�r2t )+_�2t;kr2t+�ctdkt )
�1t;k(1�r2t )+�2t;kr2tPK
k=1
�1t;k�
2
t;k
�1t;k(1�r2t )+�2t;kr2t
:
The market selection equation (12) in a deterministic world is rewritten as
_r2t = f(r
2
t ;�
1
t;k; �
2
t;k) and _r
1
t = � _r2t : (21)
There are obviously only two �xed points in (21), . (1; 0) and (0; 1) re-
spectively according to part (ii) of Theorem 4.
Without loss of generality, the stability of �xed point (1,0) is discussed
below. Namely we investigate a bond market where there are two players:
One is an incumbent or the dominant market strategy, strategy 1 with an
initial market share r10 � 1 and another is a mutant, strategy 2 with an
initial market share r20 � 0. It is our aim to answer whether the incumbent
is evolutionary stable.
At �rst sight, the assumption that there are only such two strategies in
a market seems unrealistic but in fact not. For example, one can interpret
strategy 1 as the market portfolio and strategy 2 as an arbitrary strategy,
say one played by an individual or even a �nancial institution.
12
For this reason, whether the incumbent is evolutionary stable equals
to whether the market is e�ective. And if the incumbent is evolutionary
stable then a "fair price" of each asset can be derived by the following (24).
Accordingly, a study on a market even with only such two strategies can
still leads to a lot of interesting conclusions.
It is clear that the evolution of (21) is determined by the sign of function
f(r2t ;�
1
t;k; �
2
t;k). In particular, if the sign is positive the market share of strat-
egy 2 will increase but decrease on the contrary. Therefore, the following
de�nition is presented.
De�nition 1. (i) Strategy 1 is called evolutionary stable, if there exists a
positive " > 0 and for every strategy denoted by strategy 2 with an initial
market share 0 < r20 < ", it is impossible to �nd a time interval, say (a; b) �
(0;1) such that f(r2t ;�1t;k; �2t;k) > 0 for t 2 (a; b) while f(r2t ;�1t;k; �2t;k) � 0
for t 2 (a; b)c and t � 0; (ii) On the contrary, strategy 1 is called evolution-
ary unstable, if there exists a time interval, say (a; b) � (0;1) and a strategy
denoted by strategy 2 with an arbitrary initial market share 0 < r20 < 1, such
that f(r2t ;�
1
t;k; �
2
t;k) > 0 for t 2 (a; b) while f(r2t ;�1t;k; �2t;k) � 0 for t 2 (a; b)c
and t � 0.
It follows from this de�nition that an incumbent is evolutionary stable if
and only if there is no arbitrage opportunity in the market. Note that at the
beginning of the entry of a mutant, the market shares satisfy r1t � 1; r2t � 0
and thus the sign of f(r2t ;�
1
t;k; �
2
t;k) is determined by its linear approximation:
f(r2t ;�
1
t;k; �
2
t;k) �
��ct +
KX
k=1
�2t;k
�1t;k
�
�ctdkt + _�
1
t;k
�!
r2t : (22)
De�nition 2. For a given incumbent f�1t;k j k 2 �; t � 0g, the total return
of asset k is de�ned by
�t(k) �
�ctdkt + _�
1
t;k
�1t;k
; k 2 �: (23)
�t(k) is called the total return because it completely determine the in-
vestment value of the asset as seen in the following Theorem 5. In fact,
�t(k) is a sum of the capital gain and the dividend return. To make this
clear, recall (7) and (10) and note that r1t � 1; r2t � 0. Then one �nds
pkt = �
1
t;kr
1
t + �
2
t;kr
2
t � �1t;k; t � 0; k 2 �; (24)
which means the relative price of an asset approximately equals to the frac-
tion invested in this asset by the incumbent. Especially, if the market port-
folio is taken as the unique strategy in a market, . r1t = 1 all the time,
then the relative price is just the fraction invested by the market portfolio
according to (24).
13
It follows from (24) that _�1t;k=�
1
t;k � _pkt =pkt and �ctdkt =�1t;k � �ctdkt
�
pkt . Clearly
_pkt
�
pkt is a capital gain while �ctd
k
t
�
pkt represents a dividend return.
(23) says that the bigger the changing speed �ct of consumption rate,
the more dependent the total return on the relative dividend and the less
dependent on the capital gain. Hence, it is explained that in a society with
excess consumption, the dividend paid by an asset should be much more
appreciated.
Theorem 5. For a given incumbent, provided that the total return of a
market with the incumbent is not identical for all assets at some time t, this
incumbent is evolutionary unstable.
Proof. In fact, if a mutant invests all wealth in an asset with the largest
total return at �rst and then gradually revise the portfolio to equal to the
incumbent, then the mutant will gain the market. Namely f(r2t ;�
1
t;k; �
2
t;k) is
strictly positive in a small neighborhood of (1; 0) and keep non-negative all
the time. Hence this theorem is shown from De�nition 1. �
This theorem means that a market is arbitrage-free only if the total re-
turns of an incumbent de�ned by (23) are the same across all assets. To
take one step further, next we prove that if the total returns of an incum-
bent de�ned by (23) are the same across all assets, then the incumbent is
evolutionary stable, . the market is arbitrage-free.
Assumption 7. For a given incumbent, strategy 1, the total return de�ned
by (23) is identical for all assets all the time.
This assumption also appears to be too restrictive but as proved in The-
orem 5, if the assumption does not hold then there exists an arbitrage oppor-
tunity in the market, which is not interesting to study also in mathematical
�nance.
By virtue of Assumption 7, the value �t(k) is independent of k and it is
therefore denoted by �t in the following. Accordingly, one gets
�ctdkt + _�
1
t;k = �t�
1
t;k; k 2 �: (25)
Note that
KX
k=1
dkt = 1;
KX
k=1
�1t;k = 1;
KX
k=1
_�1t;k = 0: (26)
Aggregating (25) over assets, one �nds �t = �ct, namely
�ctdkt + _�
1
t;k = �ct�
1
t;k: (27)
Consequently, under Assumption 7, the linear approximation in (22) is al-
ways zero, which, however, does not directly mean strategy 1 is evolutionary
stable. Hence a more detailed analysis of the second and even higher order
14
approximation is necessary. For this purpose, a second-order approximation
of function f(r2t ;�
1
t;k; �
2
t;k) is shown below:
_r2t = f(r
2
t ;�
1
t;k; �
2
t;k) �
�
r2t
�2 KX
k=1
� �ct
�1t;k
�
�2t;k
�2 + _�2t;k + �ctdkt
�1t;k
�2t;k: (28)
Equivalently, for an arbitrary terminal time T , one has
1
r2T
� 1
r20
+
Z T
0
KX
k=1
�ct
�1t;k
�
�2t;k
�2 � _�2t;k + �ctdkt
�1t;k
�2t;k
!
dt: (29)
Therefore, if the target of a mutant is to maximize its market share at a
given terminal time T , . r2T then the optimal mutant f�2t;kg is a solution
of the following variation problem:
min
�2t;k; 0�t�T; k2�
R T
0
PK
k=1 �
2
t;k
�
�ct�2t;k
�1t;k
� _�
2
t;k+�ctd
k
t
�1t;k
�
dt
s:t:
( PK
k=1 �
2
t;k = 1;
0 � �2t;k � 1; 0 � t � T; k 2 �;
(30)
where strategy 2 is constrained to be di�erential according to Assumption
4. Regarding this problem there exists an "�optimal strategy shown below.
Theorem 6. Let �1T;k� be a minimum value of the set f�1T;k jk 2 �g. Then
for any given small " > 0, an "-optimal solution to (30) is given by�
�2t;k = �
1
t;k, 0 � t � T � �(");
�2T;k = 0; k 6= k�; �2T;k� = 1:
(31)
In addition, �2t;k; k 2 ��fk�g, which passes
�
T � �("); �1T��(");k
�
and (T; 0),
and �2t;k�, which passes
�
T � �("); �1T��(");k�
�
and (T; 1), are di�erentiable
at T � �(") and T , where �(") is a su�ciently small positive number. The
optimal objective function equals to 1=2�1=(2�1t;k�)+", which can be strictly
negative since " is a su�ciently small positive number.
Proof. It follows from (27) that
1
�1t;k
!0
t
=
�ctdkt � �ct�1t;k�
�1t;k
�2 ;
Consequently, applying the rule for integration by parts, (30) is equivalent
15
to
min
�2t;k; 0�t�T; k2�
PK
k=1
�
(�20;k)
2
2�10;k
� (�
2
T;k)
2
2�1T;k
�
+
R T
0
PK
k=1 �
2
t;k
��
�ct
�1t;k
+
�ctdkt��ct�1t;k
2(�1t;k)
2
�
�2t;k � �ctd
k
t
�1t;k
�
dt
s:t:
( PK
k=1 �
2
t;k = 1
0 � �2t;k � 1; 0 � t � T; k 2 �:
(32)
Obviously the optimal strategy is myopic. In addition, the objective function
is quadric and the constrained conditions are convex. Thus there exists a
unique globally optimal solution to (32). More concretely, three optimization
problems are solved below: The �rst one is
min
�20;k; k2�
PK
k=1
(�20;k)
2
2�10;k
s:t:
( PK
k=1 �
2
0;k = 1;
0 � �20;k � 1; k 2 �;
in which, there is a unique global solution - that is �20;k = �
1
0;k; k 2 �, and
the optimal value of the objective function is 1=2. The second one is
max
�2T;k; k2�
PK
k=1
(�2T;k)
2
2�1T;k
s:t:
( PK
k=1 �
2
T;k = 1;
0 � �2T;k � 1; k 2 �;
in which, there is a unique global solution - that is �2T;k� = 1; while �
2
T;k =
0; k 2 ��fk�g, and the optimal value of the objective function is (2�1t;k�)�1.
The last one is
min
�2t;k; 0�t�T; k2�
R T
0
PK
k=1 �
2
t;k
��
�ct
�1t;k
+
�ctdkt��ct�1t;k
2(�1t;k)
2
�
�2t;k � �ctd
k
t
�1t;k
�
dt
s:t:
( PK
k=1 �
2
t;k = 1
0 � �2t;k � 1; 0 � t � T; k 2 �;
(33)
in which there is a unique global solution, . �2t;k = �
1
t;k; 0 � t � T; k 2 �
and the optimal value of the objective function is zero. Accordingly, on
account of that the strategies must be di�erential, the theorem is proved. �
This theorem shows that the optimal mutant is almost the same as the
incumbent, except that, while approaching to the terminal time, the mutant
gradually but quickly invests all the wealth in only one asset, of which the
fraction of the wealth the incumbent assigns to the purchase is minimum at
the terminal time. It is also shown that the optimal objective function is
strictly negative. This implies from (29) that the market share of the mutant
16
is strictly increasing at the terminal time although it keeps unchanged prior
to the end. However, that does not represent the incumbent is evolutionary
unstable.
To make this point clear, one notes that, if the mutant plays the "-
optimal strategy given by (31), the total return on asset k� must be the
smallest among all assets at the terminal time in the new market portfolio
including the investment of the mutant after the entry. As a result, As-
sumption 7 does not hold anymore at time T and the only asset the mutant
purchases has the least value for investment since no rational investor will
buy it at the purchase price of the mutant according to Theorem 5. Mean-
while, if the portfolio of the mutant keeps invariant, the right-hand side of
(22) is strictly negative at least in a short period after the terminal time
and so the mutant's market share will strictly decrease.
For example, suppose an institutional investor buys an asset on a large
scale during a short period. Undoubtedly the asset price will shot up and
his market share will increase. However, this phenomenon will not last long
and the price will de�nitely fall. Afterwards he will probably lose what he
gained.
Accordingly, another condition is imposed on strategy 2 played by the
mutant. The aim is to make sure that the new extended market portfolio
after the entry satis�es Assumption 7 still at the terminal time T - that is to
say, the constrain �2T;k = �
1
T;k; k 2 � is added in the following. Therefore,
the variation problem (30) is changed into
min
�2t;k; 0�t�T; k2�
R T
0
PK
k=1 �
2
t;k
�
�ct�2t;k
�1t;k
� _�
2
t;k+�ctd
k
t
�1t;k
�
dt
s:t:
8><>:
PK
k=1 �
2
t;k = 1;
0 � �2t;k � 1; 0 � t � T; k 2 �;
�2T;k = �
1
T;k; k 2 �:
(34)
Based on Theorem 6, the following theorem is evident and hence the proof
is omitted.
Theorem 7. If Assumption 7 holds, then there is a unique solution of (34),
which is given by
�2t;k = �
1
t;k, 0 � t � T; k 2 �: (35)
The optimal objective function and f(r2t ;�
1
t;k; �
2
t;k) equal to zero for all t 2
[0; T ]. For an arbitrary strategy, which is di�erent from the strategy given
by (35) the objective function and f(r2t ;�
1
t;k; �
2
t;k) at some time t 2 [0; T ],
are strictly negative in a small neighborhood of (1; 0). Hence, the incumbent
satisfying Assumption 7 is evolutionary stable.
Notice that the terminal time T is arbitrarily selected and in particular if
let T !1, then one concludes (35) holds anytime. Therefore the following
corollary is direct from Theorem 5 and 7.
17
Corollary 1. An incumbent is evolutionary stable if and only if the portfolio
of the incumbent satis�es Assumption 7.
4 Explicit Expression for Bond Valuation and Bench-
mark Interest Rate
This section provides an expression for bond valuation if the the market
is arbitrage-free. Owing to De�nition 1, a market is arbitrage-free if and
only if the incumbent, . the market portfolio is evolutionary stable or
equivalently, if and only if Assumption 7 holds according to Corollary 1.
Thus it is supposed throughout this section that Assumption 7 holds.
Clearly, Assumption 7 leads to (27), which yields by integration
�1T;k = exp
�Z T
t
�cudu
��
�1t;k �
Z T
t
�cudku exp
�
�
Z u
t
�csds
�
du
�
; t � 0:
or equivalently
�1t;k = �
1
T;k exp
�
�
Z T
t
�cudu
�
+
Z T
t
�cudku exp
�
�
Z u
t
�csds
�
du; t � 0: (36)
Following that, the main result in this section are provided below.
Theorem 8. (i) A bond market is evolutionary stable if and only if each
bond (relative) price is given by
pkt =
Z 1
t
�cudku exp
�
�
Z u
t
�csds
�
du; t � 0; k 2 �; (37)
(ii) On the contrary, if the bond prices are collected already in an evolu-
tionary stable market, then the discount rate (consumption parameter) is
recovered by
�ct =
_pkt
pkt � dkt
; k 2 �; t � 0: (38)
Proof. In this bond market, the market portfolio can be considered as
the unique strategy, say strategy 1 with r1t � 1. Therefore, we conclude
from (24) that pkt = �
1
t;k for k 2 �. Let T !1 in (36), then (37) is derived
since
lim
T!1
�1T;k exp
�
�
Z T
t
�cudu
�
= 0:
Last, (38) follows directly from (27) and the equality pkt = �
1
t;k for all k 2 �.
�
Since pkt = �
1
t;k as shown in the above proof, part (i) of the theorem
accords with economic intuition. It says that in an evolutionary stable bond
18
market, one should divide wealth across assets according to the relative asset
prices, . the arbitrage-free prices, which equal to the discounted values
of the relative dividends with the discount rate just being the consumption
parameter �ct.
However, the conclusion of part (ii) appears more interesting. For exam-
ple, the benchmark interest rate is extremely important in �nance industry,
but how to �x it is open to my best knowledge. In order to approach this
problem, part (ii) of Theorem 8 suggests that in a relatively e�ective (.
evolutionary stable) bond market, the relative bond prices are collected �rst
and then the consumption parameter �ct are recovered by (38). This param-
eter should be a good candidate for the benchmark interest rate.
5 Conclusions
This paper develops a continuous-time evolutionary �nance model with
time-dependent strategies, from which an evolutionary stable bond market
is studied. First a general discrete-time evolutionary �nance is established
with an arbitrary trading time interval. A continuous-time model is de-
rived by letting the time interval converge to zero. It is shown that the
continuous-time model is well-de�ned and there is no sudden bankruptcy in
the general market only if all the asset prices keep positive.
Second, as a special case of the general model, a bond market with de-
terministic dividend payo�s is discussed in detail. This bond market is not
so arti�cial as one may consider at �rst sight since it includes a variety
of marketable treasury securities, say Treasury bills, Treasury notes, Trea-
sury bonds, and Treasury In
ation Protected Securities (TIPS) issued by
the United States Department of the Treasury. This market also includes
Account Treasury bonds issued by the China Department of Finance. It is
certi�ed that a bond market is evolutionary stable, which implies the market
is arbitrage-free, if and only if the total return of each asset in the market
is identical all the time.
Further, the arbitrage-free price of each bond is derived, which equals
an improper integral with the integrand being a discounted value of the
dividend payo�. The discount rate is identical for all bonds and equals the
market consumption parameter. To my best knowledge, there is not an
accepted approach to evaluate the discount rate or the benchmark interest
rate although this rate is vital for asset pricing. (38) at least provides a new
attempt to attack this problem. For example, if one gets bond prices in a
relatively e�ective or evolutionary stable bond market then the discount rate
or the benchmark interest rate can be recovered from (38).
In the end, although a general continuous-time evolutionary stock market
with time-dependent strategies is established in this paper, this paper after-
wards focuses on an evolutionary bond market, in which the fundamentals
19
are deterministic and both prices and wealth vary due to market interaction.
Clearly a profound study on continuous-time evolutionary stock market in
a stochastic world is more interesting and of course more challenging. This
work is left in future research.
References
Blume, L. - Easley, D. (1992): Evolution and Market Behavior. Journal of
Economic Theory, vol. 58, pp. 9-40.
Buchmann, B. - Weber, S. (2007): A continuous Time Approximation of
An Evolutionary Stock Market Model. International Journal of Theoretical
and Applied Finance, vol. 10, pp. 1229-1253.
Evstigineev, . - Hens, T. - Schenk-Hopp�e, . (2006): Evolutionary
Stable Stock Markets. Economic Theory, vol. 27, pp. 449-468.
Farmer, J. D. - Lo, W. (1999): Frontiers of Finance : Evolution and
E�cient Markets. Proceedings of the National Academy of Sciences, vol.
96, pp. 9991-9992.
Hens, T. - Schenk-Hopp�e, . (2005): Evolutionary Stability of Portfo-
lio Rules in Incomplete Markets. Journal of Mathematical Economics, vol.
41, pp. 43-66.
Lucas, R. (1978): Asset Prices in an Exchange Economy. Econometrica,
vol. 46, pp. 1429-1445.
Yang, . - Ewald, . (2008): Continuous Time Evolutionary Market
Dynamics: The Case of Fix-Mix Strategies. Investment Management and
Financial Innovation, vol. 5, pp. 34-42.
20
