We consider the problem of rational decision making in the presence of nonlinear constraints. By using tools borrowed from spin glass and random matrix theory, we focus on the portfolio optimisation problem. We show that the number of "optimal" solutions is generically exponentially large: rationality is thus de facto of limited use. In addition, this problem is related to spin glasses with Lévy-like (long-ranged) couplings, for which we show that the ground state is not exponentially degenerate.

We study the scaling behavior in currency exchange rates. Our results suggest that they satisfy scaling with an exponent close to 0.5, but that it differs qualitatively from that of a simple random walk. Indeed price variations cannot be considered as independent variables and subtle correlations are present. Furthermore, we introduce a novel statistical analysis for economic data which makes the physical properties of a signal more evident and eliminates the systematic effects of time periodicity.

We reconsider Eigen's quasispecies model for competing self-reproductive macromolecules in populations characterized by a single-peaked fitness landscape. The use of ideas and tools borrowed from polymer theory and statistical mechanics allows us to exactly solve the model for generic DNA lengths d. The mathematical shape of the quasispecies confined around the master sequence is perturbatively found in powers of 1/d at large d. We rigorously prove the existence of the error-threshold phenomena and study the quasispecies formation in the general context of critical phase transitions in physics. No sharp transitions exist at any finite d, and at d→ϱ the transition is of first order. The typical rms amplitude of a quasispecies around the master sequence is found to diverge algebraically with exponent Ќ ϭ1 at the transition to the delocalized phase in the limit d→ϱ.

This paper introduces a general framework for market models, named Market Model Approach, through the concept of admissible sets of forward swap rates spanning a given tenor structure. We relate this concept to results in graph theory by showing that a set is admissible if and only if the associated graph is a tree. This connection enables us to enumerate all admissible models for a given tenor structure. Three main classes are identified within this framework and correspond to the co-terminal, co-initial, and cosliding model. We prove that the LIBOR market model is the only admissible model of a co-sliding type. By focusing on the co-terminal model in a lognormal setting, we develop and compare several approximating analytical formulae for caplets, while swaptions can be priced by a simple Black-type formula. A novel calibration technique is introduced to allow simultaneous calibration to caplet and swaption prices. Empirical calibration of the co-terminal model is shown to be faster, more robust, and more efficient than the same procedure applied to the LIBOR market model. We then argue that the co-terminal approach is the simplest and most convenient market model for pricing and hedging a large variety of exotic interest-rate derivatives.

A simple stochastic model of investment based on communication theory is introduced and analyzed in detail. We solve it exactly in a simple case and we use a weak disorder expansion to deal with the small fluctuations of the capital between two consecutive trading periods. Some possible generalizations are also discussed. ͓S1063-651X͑96͒52011-8͔PACS number͑s͒: 02.50. Le, 05.20.Ϫy Let us consider an investor, whose aim is to increase a given capital Z by investing it in several stocks i (iϭ1, . . . ,M) with a given strategy. The total amount of capital is the sum Zϭ͚ iϭ1 M Z(i)ϩZ (0) where Z(i) is the fraction of the capital invested in the stock i and Z(0) is the capital left at the bank. The capital Z(i) is multiplied, at each trading period, by a factor a, a being a stochastic variable whose distribution is, in general, unknown. For our simple model we assume uncorrelated distributions; however, in reality, it is well known that such data can be correlated in time. We believe the following analysis can be readily generalized to the correlated case as well.The investment strategy is the following: at each ''time'' n (nϭ1, . . . ,N), one takes a fraction i ͓0,1͔ of the capital Z(i) and transfers it to the cash bank; at the same time, a fraction /M of the bank capital is taken from the bank itself and then added to Z(i). We can then write our system aswhere, as above defined, we suppose that a i are independent random variables with common distribution given by (a).The system is completely specified by a (M ϩ1)ϫ(M ϩ1) random transfer matrix A n since we can write Z ជ nϩ1 ϭA n Z ជ n . Note that the system is normalized: if a i ϭ1 for all i, i.e., if there is no gain or loss at each time step, then the total capital is conserved: Z nϩ1 ϭZ n . The first equation implies that after a trading period the amount deposited at the bank is the sum of two contributions: the part which has not been transferred and the part coming from Z(i). On the other hand, for each stock i, we have that Z(i) is incremented or diminished depending on the value of the stochastic variable a i . Note that the choice of ͑1͒ is not the only one possible: one may define Z nϩ1 (i)ϭ/M Z n (0)ϩa i (1Ϫ i )Z n (i), e.g., we may decide not to bet the fraction of the capital transferred from the bank. This choice, however, has the disadvantage of rendering calculations much harder and leaving the qualitative behavior unchanged. The goal is to optimize the capital gain by varying the set ͕͖ϭ͕, 1 , . . . , M ͖ ͓that is to have the largest Zϭ ͚ iϭ0 M Z(i) after N trading periods, in the limit N→ϱ͔.How is it possible to decide the best strategy once we know (a), and with fixed M ? We find that a special case of this problem can be mapped to a classical one in communication theory. Indeed in the 1950s Kelly considered this type optimization, albeit using information theory ͓1͔. His model is based on the concept of the so-called rate of transmission in a communication channel, as introduced by Shannon ͓2͔. Suppose one considers a channel and uses it to tra...