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 Schenk-Hoppé (University of Leeds)

In the paper by Dempster, Evstigneev and Taksar (Annals of Finance, 2006), it has been shown that the von Neumann-Gale growth model provides a convenient and natural framework for the analysis of questions of asset pricing and hedging under transaction costs. The talk will review recent results pertaining to a different area of applications of the model in Finance. It will demonstrate how methods and concepts developed in the context of von Neumann-Gale dynamical systems can be used to build a complete and self-consistent theory of optimal financial growth under transaction costs.

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.

✄ ✂ ✄ ✁ � � ✄ ✁ ✁ Von Neumann - Gale dynamical systems : the deterministic case . Given: R n ; a closed convex cone A 1,2,... , a set-valued mapping for each t G t a , A , A , a a G t a ☎✝✆ satisfying the following condition: the graph of the mapping G t , a , b ✞ A : b Z t A G t a is a closed convex cone . Most of the classical deterministic theory has been developed for R ✟ n . A This will be assumed throughout the talk.

✁ ✄ ✄ ✄ ✁ ✁ Multivalued dynamical system defined by G t . A path ( trajectory ) b 0 , b 1 , b 2 ,... : G t b t � 1 , t 1,2,... b t or, equivalently, b t � 1 , b t Z t . In economics contexts, states of the system 1 ,..., b t 0 are typically interpreted as b t b t n commodity vectors . The process of economic growth is regarded as an evolution of b t in time. Elements Z t are feasible input - output pairs , or a , b technological processes (for the time period t ✂ 1, t ). The sets Z t are termed technology sets . In financial applications, which will be considered in detail later, vectors b t represent portfolios of assets , and the sets Z t describe self - financing ( solvency ) constraints .

✁ ✁ ✁ ✁ ✄ Example : von Neumann ( 1937 ) model . The cone Z t is polyhedral : there is a finite set of basic technological processes � 1 ✁ , b � 1 ✁ ,..., a � m ✁ , b � m a and m � j ✁ , b � j ✁ , d j a a , b a , b Z t ✂ 1 j where d 1 0,..., d m 0 are intensities of operating the technological processes � 1 ✁ , b � 1 ✁ ,..., a � m ✁ , b ✁ . a � m Gale ( 1956 ): general, not necessarily polyhedral, cones.

✁ ✟ ✁ � ✄ � ✂ ✁ ✁ ✄ ✄ � ✁ ✄ ✁ ✁ ✁ ✂ ✁ ✄ ✁ ✁ ✄ ✂ � � � � � � � ✁ Stochastic von Neumann – Gale dynamical systems Given : � , F , P probability space; ... ... ... F F 0 F 1 F t F filtration. � 1 1,2,... , let For each t ✁ , a ✁ , a R ✟ n G t be a set-valued mapping assigning to each and ✁ , a ✟ n a set G t ✟ n so that R R a ✁ , the graph (i) for each ✁ , a : a , b : b Z t G t ✁ , of the mapping G t is a closed convex cone ( transition cone ); ✁ , a ✁ , a is G t (ii) the set-valued mapping ✟ n -measurable. ✞ B R F t Random ( multivalued ) dynamical system : Paths ( trajectories ) , y 1 ,... y 0 ✁ , y t � 1 (a.s.) , t 1,2,..., y t G t or, equivalently, , y t y t � 1 Z t (a.s.) and � , F t , P , R n . y t L

✁ ✁ ✄ ✁ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ Dynamic securities market model . � , F , P , F t . There are n assets traded on the Given 0,1,... . A (contingent) portfolio of market at dates t assets held by an investor at date t is a vector 1 ,..., y t n y t y t where y t 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 y t is F t -measurable. In the applications we will deal with (capital growth theory), the classical model excludes short selling, and so the vectors y t are supposed to be non-negative. A trading strategy is a sequence of contingent portfolios y 0 , y 1 , y 2 ,... We will focus on the analysis of self-financing strategies. They are defined as follows. In the model, ✟ n (defining the R ✞ R ✟ n we are given sets Z t self - financing constraints ), and a strategy y 0 , y 1 , y 2 ,... is called self - financing if , y t y t � 1 Z t (a.s.) for all t . is a closed convex cone It is assumed that Z t ✁ . This assumption depending F t -measurably on means that the model takes into account proportional transaction costs. The cones Z t define a stochastic von Neumann-Gale model. Trading strategies are paths in this model.

✄ ✁ � � ✂ ✁ ✁ ✁ ✁ ✟ ✟ � ✄ ✂ ✁ ✁ ✁ ✟ ✁ ✁ � ✁ ✄ ✄ ✁ ✁ ✁ ✁ ✁ ✟ Examples No transaction costs . Let S t 1 ,..., S t n S t be the vector of asset prices at time t (F t -measurable). Define n n S t i ✟ n : a i . R ✞ R : a , b ✟ n b i Z t i S t � 1 ✂ 1 ✂ 1 i i A portfolio a can be rebalanced to a portfolio b (without transaction costs) if and only if a , b Z t . Proportional transaction costs : single currency . Let ✟ n for which R ✞ R be the set of those a , b ✟ n Z t n n S t i S t i 1 1 b i a i a i ✂ b i ✟ , t , i t , i i i S t � 1 S t � 1 ✁✄✂ ✂ ☎✂ ✂ 1 ✂ 1 i i where : max a ,0 . a The transaction cost rates for buying and selling are 0 and 1 0 , given by the numbers t , i t , i ✆✝✂ respectively. A portfolio a can be rebalanced to a portfolio b (with transaction costs) if and only if a , b Z t . The above inequality expresses the fact that purchases of assets are made only at the expense of sales of other assets.

✁ � ✁ � ✆ � ✁ ✁ � ✁ � ✆ Proportional transaction costs : several currencies . There are n currencies. A matrix ij ij 0 and 1 ii with t t t is given, specifying the exchange rates of the 1,2,..., n (including transaction costs). currencies i 0 of currencies can be exchanged to a A portfolio a portfolio b at date t if and only if there exists a ji ( exchange matrix ) such that nonnegative matrix d t n n ji , 0 ij ij . a i b i d t d t t ✂ 1 ✂ 1 j j ij ( i Here, d t j ) stands for the amount of currency j ii of currency i exchanged into currency i . The amount d t is left unexchanged. The second inequality says that, at time t , the i th position of the portfolio cannot be greater ij obtained as a result of the n ij d t than the sum t ✂ 1 j exchange. A version of the models considered by Kabanov, Stricker and others.

✁ � ✁ � ✁ � ✁ � ✁ � ✁ ✁ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ � Stationary models . This is an important class of models in which the random cone-valued process Z t 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 � , F , P ( time shift ) such that probability space (a) the filtration ... ... ... is F F 0 F 1 F t � 1 invariant with respect to the time shift � 1 F t ✟ 1 , T F t (b) for each t , . Z t T Z t ✟ 1 The last condition means that the probabilistic structure of the transition cones is invariant with respect to the time shift (stationarity of the cone-valued process Z t ). Examples . In the above examples, the transition cones form stationary processes if the vector-valued processes of asset returns 1 n S t ,..., S t , R t 1 n S t � 1 S t � 1 or the matrix-valued processes ij n M t i , j ✂ 1 t of the currencies’ exchange rates are stationary.

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✄ ✁ ✄ � ✂ ✁ � ✂ ✁ � 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 our 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). b 1 ,..., b n , put | b | | b 1 | ... | b n | . If For a vector b 0 , then | b | b 1 ... 0 represents a b b n , and if b portfolio, then | b | is the value of this portfolio—the total amount of money invested in all its assets. Definition . Let y 0 , y 1 ,... be an investment strategy. It is called asymptotically optimal if for any other investment � , y 1 � ,... there exists a supermartingale strategy y 0 t such that � | | y t t , t 0,1,... (a.s.) | y t | Note that the above property remains valid if | | is replaced by any (possibly random) function , where c | a | C | a | , where 0 a c C are non-random constants.