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CAPITAL GROWTH THEORY UNDER TRANSACTION COSTS : AN APPROACH BASED ON - - PDF document
CAPITAL GROWTH THEORY UNDER TRANSACTION COSTS : AN APPROACH BASED ON - - PDF document
CAPITAL GROWTH THEORY UNDER TRANSACTION COSTS : AN APPROACH BASED ON THE VON NEUMANN - GALE MODEL Michael Taksar (University of Missouri) Joint work with Wael Bahsoun (University of Manchester) Igor Evstigneev (University of Manchester) Klaus
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The term ”von Neumann-Gale model” refers to a special class of multivalued (set-valued) dynamical
- systems. The classical theory (von Neumann 1937,
Gale 1956, and others) deals with deterministic models and aims basically at economic applications. First attempts to build stochastic generalizations of the von Neumann-Gale model were undertaken in the pioneering work of Dynkin, Radner and their research groups in the 1970s. The initial attack on the problem left many questions
- unanswered. Significant progress was achieved only
recently. The progress was motivated and the new methods were suggested by the applications of the model in Finance.
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Von Neumann-Gale dynamical systems: the deterministic case. Given: a closed convex cone A
- Rn;
for each t
✁1,2,..., a set-valued mapping a
✂Gt a , a
✄A,
☎✝✆Gt a
- A,
satisfying the following condition: the graph of the mapping Gt , Zt
✁a,b
✄A
✞ A : b ✄Gt a is a closed convex cone. Most of the classical deterministic theory has been developed for A
✁R
✟n.This will be assumed throughout the talk.
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Multivalued dynamical system defined by Gt . A path (trajectory) b0,b1,b2,... : bt
✄Gt bt 1 , t
✁1,2,...
- r, equivalently,
bt 1,bt
✄Zt. In economics contexts, states of the system bt
✁bt
1,...,bt n
✁0 are typically interpreted as commodity vectors. The process of economic growth is regarded as an evolution of bt in time. Elements a,b
✄Zt are feasible input-output pairs, or technological processes (for the time period t
✂ 1,t).The sets Zt are termed technology sets. In financial applications, which will be considered in detail later, vectors bt represent portfolios of assets, and the sets Zt describe self-financing (solvency) constraints.
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Example: von Neumann (1937) model. The cone Zt is polyhedral: there is a finite set of basic technological processes a
1✁ ,b 1✁ ,..., a m✁ ,b m ✁and a,b
✄Zt a,b
✁j
✂ 1m
dj a
j ✁ ,b j ✁ ,where d1
✁0,...,dm
✁are intensities of operating the technological processes a
1✁ ,b 1✁ ,..., a m✁ ,b m ✁ .Gale (1956): general, not necessarily polyhedral, cones.
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Stochastic von Neumann–Gale dynamical systems Given:
,F,P probability space;...
- F
- F0
- F1
- ...
- Ft
- ...
- F filtration.
For each t
✁1,2,..., let
✁ ,a ✂Gt
✁ ,a- R
be a set-valued mapping assigning to each
✁ ✄- and
a
✄R
✟n a set Gt ✁ ,a- R
(i) for each
✁ , the graphZt
✁:
✁a,b : b
✄Gt
✁ ,a- f the mapping Gt
is a closed convex cone (transition cone); (ii) the set-valued mapping
✁ ,a ✂Gt
✁ ,a isFt
✞ B R ✟n -measurable.Random (multivalued) dynamical system: Paths (trajectories) y0
✁,y1
✁,... yt
✁ ✄Gt
✁ ,yt 1 ✁(a.s.), t
✁1,2,...,
- r, equivalently,
yt 1
✁,yt
✁ ✄Zt
✁(a.s.) and yt
✄L
✟ ✂ ,Ft,P,Rn . SLIDE 8
Dynamic securities market model. Given
,F,P ,Ft . There are n assets traded on the market at dates t
✁0,1,.... A (contingent) portfolio of assets held by an investor at date t is a vector yt
✁ ✁yt
1
✁,...,yt
n
✁where yt
i
✁is the amount of money invested in asset i (the value of asset i in the portfolio in terms of the current market prices). It is supposed that yt
✁is Ft-measurable. In the applications we will deal with (capital growth theory), the classical model excludes short selling, and so the vectors yt are supposed to be non-negative. A trading strategy is a sequence of contingent portfolios y0,y1,y2,... We will focus on the analysis of self-financing
- strategies. They are defined as follows. In the model,
we are given sets Zt
✁- R
self-financing constraints), and a strategy y0,y1,y2,... is called self-financing if yt 1
✁,yt
✁ ✄Zt
✁(a.s.) for all t. It is assumed that Zt
✁is a closed convex cone depending Ft-measurably on
✁ . This assumptionmeans that the model takes into account proportional transaction costs. The cones Zt
✁define a stochastic von Neumann-Gale model. Trading strategies are paths in this model.
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Examples No transaction costs. Let St
✁ ✁St
1
✁,...,St
n
✁be the vector of asset prices at time t (Ft-measurable). Define Zt
✁:
✁a,b
✄R
✟n ✞ R ✟n :i
✂ 1n
bi
- i
n
St
i
✁St 1
i
✁ai . A portfolio a can be rebalanced to a portfolio b (without transaction costs) if and only if a,b
✄Zt
✁. Proportional transaction costs: single currency. Let Zt
✁be the set of those a,b
✄R
✟n ✞ R ✟n for whichi
✂ 1n
1
✁✄✂t,i
✟bi
✂St
i
St 1
i
ai
✟- i
n
1
✂☎✂t,i
- St
i
St 1
i
ai
✂ bi ✟ ,where a
✟:
✁max a,0 . The transaction cost rates for buying and selling are given by the numbers
✂t,i
✟ ✁ ✁0 and 1
✆✝✂t,i
- ✁
0,
- respectively. A portfolio a can be rebalanced to a
portfolio b (with transaction costs) if and only if a,b
✄Zt
✁. The above inequality expresses the fact that purchases of assets are made only at the expense
- f sales of other assets.
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Proportional transaction costs: several currencies. There are n currencies. A matrix
- t
ij
✁with
- t
ij
✆0 and
- t
ii
✁1 is given, specifying the exchange rates of the currencies i
✁1,2,...,n (including transaction costs). A portfolio a
✁0 of currencies can be exchanged to a portfolio b at date t if and only if there exists a nonnegative matrix dt
ji (exchange matrix) such that
ai
j
✂ 1n
dt
ji, 0
bi
j
✂ 1n
- t
ij
✁dt
ij.
Here, dt
ij (i
✆j) stands for the amount of currency j exchanged into currency i. The amount dt
ii of currency i
is left unexchanged. The second inequality says that, at time t, the ith position of the portfolio cannot be greater than the sum
j
✂ 1n
- t
ijdt ij obtained as a result of the
exchange. A version of the models considered by Kabanov, Stricker and others.
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Stationary models. This is an important class of models in which the random cone-valued process Zt
✁is stationary. Formally, in a stationary model we are given in addition to the above data a measure preserving one-to-one transformation T of the probability space
,F,P (time shift) such that(a) the filtration ...
- F
- F0
- F1
- ...
- Ft
- ... is
invariant with respect to the time shift T
1Ft ✁Ft
✟ 1,(b) for each t, Zt T
✁ ✁Zt
✟ 1 ✁. The last condition means that the probabilistic structure
- f the transition cones is invariant with respect to the
time shift (stationarity of the cone-valued process Zt).
- Examples. In the above examples, the transition cones
form stationary processes if the vector-valued processes of asset returns Rt
✁ ✁St
1
✁St 1
1
✁,..., St
n
✁St 1
n
✁,
- r the matrix-valued processes
Mt
✁ ✁- t
ij
✁i,j
✂ 1n
- f the currencies’ exchange rates are stationary.
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CAPITAL GROWTH THEORY How to invest in order to maximize the asymptotic growth rate of wealth? Pioneers: Shannon, Kelly, Breiman (1950s and 1960s). Most general results without transaction costs: Algoet and Cover (Ann. Probability, 1988). A central goal of
- ur work is to extend the results of Algoet and Cover to
the case of proportional transaction costs. How to define asymptotic optimality? In the definition below, we follow essentially Algoet and Cover (1988). For a vector b
✁b1,...,bn , put |b|
✁|b1|
✁...
✁|bn|. If b
✁0, then |b|
✁b1
✁...
✁bn, and if b
✁0 represents a portfolio, then |b| is the value of this portfolio—the total amount of money invested in all its assets.
- Definition. Let y0,y1,... be an investment strategy. It is
called asymptotically optimal if for any other investment strategy y0
,y1 ,... there exists a supermartingale ✁t such
that |yt
||yt|
- ✁
t, t
✁0,1,... (a.s.) Note that the above property remains valid if | | is replaced by any (possibly random) function
✂, where c|a|
- ✂
a
- C|a|, where 0
c
✄C are non-random constants.
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Implications of asymptotic optimality. The strength
- f the above definition, which might seem not
immediately intuitive, is illustrated by the following implications of asymptotic optimality. As long as |yt
||yt|
- ✁
t, t
✁0,1,... (a.s.), where
✁t is a supermartingale, the following properties
hold. (a) With probability one sup
t
|yt
||yt|
✄✁, i.e. for no strategy wealth can grow asymptotically faster than for y0,y1,... (a.s.). (b) The strategy y0,y1,... a.s. maximizes the exponential growth rate of wealth lim supt
✂ ✂1 t ln|yt|. (c) We have sup
t E |yt
||yt|
✄✄and sup
t Eln |yt
||yt|
✄✄ . SLIDE 14
This work aims at obtaining results on asymptotic
- ptimality in models with transaction costs.
The main results: existence of asymptotically optimal strategies in general (non-stationary) models; existence of asymptotically optimal strategies of a special structure (so-called balanced strategies) in stationary models. From now on we will use the terms ”trading strategy” in the dynamic securities market model and ”path” in the associated stochastic von Neumann-Gale model interchangeably. Relevant studies.
- M. I. Taksar, M. J. Klass, and D. Assaf. Diffusion model
for optimal portfolio selection in the presence of brokerage fees. Math. Oper. Res., 13(2), 1988.
- A. Kalai and A. Blum. Universal portfolios with and
without transaction costs. In Proc. 10th COLT, 1997.
- G. Iyengar and T. M. Cover. Growth optimal investment
in horse race markets with costs. IEEE Trans. Info. Theory, 46, 2000.
- M. Akian, A. Sulem, and M. I. Taksar. Dynamic
- ptimization of long term growth rate for a mixed
portfolio with transactions costs. Math. Fin., 11, 2001.
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The idea of our approach. Our main tool for analysing the questions of asymptotic optimality is the concept of a rapid path in the stochastic von Neumann-Gale model. Rapid paths. A path x0,x1,... (finite or infinite) is called rapid if there is a sequence of random vectors p0,p1,... , pt
✄L
✟1 ,Ft,P,Rn , such thatptxt
✁1 (a.s.) (1) and any of the four equivalent conditions holds: for all x,y
✄Zt (a.s.), Eln pty pt 1x
- Eln
ptxt pt 1xt 1
✁0, (2) E pty pt 1x
- E
ptxt pt 1xt 1
✁1, (3) Epty
- Ept 1x,
(4) E pty|Ft 1
- pt 1x.
(5) In the financial applications, the notion of a rapid path is a counterpart of the notion of a numeraire portfolio (Long 1990). Note that for any path y0,y1,... , the sequence p0y0,p1y1,p2y2,... is a supermartingale. The proofs of the main results regarding asymptotic
- ptimality are based on the following fact. Under quite
general assumptions, an infinite rapid path is asymptotically optimal.
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Existence results for rapid paths (implying the existence results for asymptotically optimal paths) are based on the following assumptions:
- Assumptions. The transition cones Zt
satisfy: (A1) The set Zt
✁contains with every a,b all a
,b- such that a
- ✁
a and 0
- b
- b.
(A2) There exists a constant M such that |b|
- M|a| for
all a,b
✄Zt
✁. (A3) There exists a constant
- ✆
0 such that e,
e ✄Zt
✁, where e
✁1,1,...,1 . (A4) There exists an integer l
✁1 such that for every t
✁0 and i
✁1,...,n there is a path yt,i,...,yt
✟ l,i satisfyingyt,i
✁ei,...,yt
✟ l,i ✁ e,where ei
✁0,0,...,1,...,0 (all the coordinates are 0 except the ith which is 1).
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Existence results for non-stationary models. Finite rapid paths: Evstigneev and Flåm (1998) –
- btained by the maximization of logarithmic functionals:
Eln|xN|
- max
- ver all paths
x0,...,xN with fixed x0 and N. Infinite rapid paths: Bahsoun, Evstigneev and Taksar (2007) – obtained by passing to the limit from finite rapid paths with the help of the Fatou lemma in several dimensions (a conditional version of it).
- Lemma. Let G be a sub-
,
✁ ✄ , N ✁1,2,..., are random vectors with values in R
✟n and the conditional expectations E wN ✁|G are finite and converge a.s. to a random vector z
✁, then exists a sequence of integer-valued random variables 1
✄N1
✁ ✄N2
✁ ✄... and a random vector w
✁such that limwNk
- ✂
w
✁(a.s.) and E w
✁|G
- z
(a.s.). This result is a conditional version of the multidimensional Fatou lemma (Schmeidler 1970).
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Existence results for rapid paths in stationary models. Recall that in stationary models the random cones Zt
✁form a stationary process: Zt
✁ ✁Z Tt
✁. In such models an important role is played by the notion of a balanced path. Balanced paths are paths of the form yt
✁ ✂1
✂2...✂ ty
- t,
where
✂t
✁ ✁ ✂Tt 1
✁, y
- t
y Tt
✁. The scalar function
✂ ✁ ✆0 is supposed to be F1-measurable and bounded. The vector function y
✁ ✁0 is F0-measurable and satisfies |y
✁|
✁1. A balanced path grows with stationary proportions (given by the stationary vector process y
- t
) and at a stationary rate, growth rates at times t
✁1,2,... being
✂1,
✂2,....
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Von Neumann path: balanced path maximizing Elog
✂t
among all balanced paths yt
✁ ✂1
✂2...✂ ty
- t.
By virtue of stationarity, the functional Elog
✂t does not
depend on t. To find a von Neumann path, we have to solve the following variational problem maximize Eln
✂- ver all
L
✟ ✂ ,F1,P,R1 , y ✄L
✟ ✂ ,F0,P,Rnsatisfying y
✁,
✂ ✁y T
✁ ✄Z1
✁(a.s.), |y
✁|
✁1 (a.s.).
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The main result on rapid paths for stationary models.
- Theorem. Under assumptions (A1) - (A4), a von