Delta hedging in discrete time under stochastic interest rate

We propose a methodology based on the Laplace transform to compute the variance of the hedging error due to time discretization for financial derivatives when the interest rate is stochastic. Our approach can be applied to any affine model for asset prices and to a very general class of hedging strategies, including Delta hedging. We apply it in a two-dimensional market model, obtained by combining the models of Black-Scholes and Vasicek, where we compare a strategy that correctly takes into account the variability of interest rates to one that erroneously assumes that they are deterministic. We show that the differences between the two strategies can be very significant. The factors with stronger influence are the ratio between the standard deviations of the equity and that of the interest rate, and their correlation. The methodology is also applied to study the Delta hedging strategy for an interest rate option in the Cox-Ingersoll and Ross model, measuring the variance of the hedging error as a function of the frequency of the rebalancing dates. We compare the results obtained to those coming from a classical Monte Carlo simulation

Spreading and shortest paths in systems with sparse long-range connections

Spreading according to simple rules (e.g. of fire or diseases), and shortest-path distances are studied on d-dimensional systems with a small density p per site of long-range connections (``Small-World'' lattices). The volume V(t) covered by the spreading quantity on an infinite system is exactly calculated in all dimensions. We find that V(t) grows initially as t^d/d for t>t^*$, generalizing a previous result in one dimension. Using the properties of V(t), the average shortest-path distance \ell(r) can be calculated as a function of Euclidean distance r. It is found that \ell(r) = r for r<r_c=(2p \Gamma_d (d-1)!)^{-1/d} log(2p \Gamma_d L^d), and \ell(r) = r_c for r>r_c. The characteristic length r_c, which governs the behavior of shortest-path lengths, diverges with system size for all p>0. Therefore the mean separation s \sim p^{-1/d} between shortcut-ends is not a relevant internal length-scale for shortest-path lengths. We notice however that the globally averaged shortest-path length, divided by L, is a function of L/s only.Comment: 4 pages, 1 eps fig. Uses psfi

Entropy and Long range correlations in literary English

Recently long range correlations were detected in nucleotide sequences and in human writings by several authors. We undertake here a systematic investigation of two books, Moby Dick by H. Melville and Grimm's tales, with respect to the existence of long range correlations. The analysis is based on the calculation of entropy like quantities as the mutual information for pairs of letters and the entropy, the mean uncertainty, per letter. We further estimate the number of different subwords of a given length $n$. Filtering out the contributions due to the effects of the finite length of the texts, we find correlations ranging to a few hundred letters. Scaling laws for the mutual information (decay with a power law), for the entropy per letter (decay with the inverse square root of $n$) and for the word numbers (stretched exponential growth with $n$ and with a power law of the text length) were found.Comment: 8 page

Statistics of finite-time Lyapunov exponents in the Ulam map

The statistical properties of finite-time Lyapunov exponents at the Ulam point of the logistic map are investigated. The exact analytical expression for the autocorrelation function of one-step Lyapunov exponents is obtained, allowing the calculation of the variance of exponents computed over time intervals of length $n$. The variance anomalously decays as $1/n^2$. The probability density of finite-time exponents noticeably deviates from the Gaussian shape, decaying with exponential tails and presenting $2^{n-1}$ spikes that narrow and accumulate close to the mean value with increasing $n$. The asymptotic expression for this probability distribution function is derived. It provides an adequate smooth approximation to describe numerical histograms built for not too small $n$, where the finiteness of bin size trimmes the sharp peaks.Comment: 6 pages, 4 figures, to appear in Phys. Rev.

Implicit incentives for fund managers with partial information

We study the optimal asset allocation problem for a fund manager whose compensation depends on the performance of her portfolio with respect to a benchmark. The objective of the manager is to maximise the expected utility of her final wealth. The manager observes the prices but not the values of the market price of risk that drives the expected returns. Estimates of the market price of risk get more precise as more observations are available. We formulate the problem as an optimization under partial information. The particular structure of the incentives makes the objective function not concave. Therefore, we solve the problem by combining the martingale method and a concavification procedure and we obtain the optimal wealth and the investment strategy. A numerical example shows the effect of learning on the optimal strategy

Self-calibrating highly sensitive dynamic capacitance sensor: Towards rapid sensing and counting of particles in laminar flow systems

In this report we propose a sensor architecture and a corresponding read-out technique on silicon for the detection of dynamic capacitance change. This approach can be applied to rapid particle counting and single particle sensing in a fluidic system. The sensing principle is based on capacitance variation of an interdigitated electrode (IDE) structure embedded in an oscillator circuit. The capacitance scaling of the IDE results in frequency modulation of the oscillator. A demodulator architecture is employed to provide a read-out of the frequency modulation caused by the capacitance change. A self-calibrating technique is employed at the read-out amplifier stage. The capacitance variation of the IDE due to particle flow causing frequency modulation and the corresponding demodulator read-out has been analytically modelled. Experimental verification of the established model and the functionality of the sensor chip were shown using a modulating capacitor independent of fluidic integration. The initial results show that the sensor is capable of detecting frequency changes of the order of 100 parts per million (PPM), which translates to a shift of 1.43 MHz at 14.3 GHz operating frequency. It is also shown that a capacitance change every 3 μs can be accurately detected

On the relationship between directed percolation and the synchronization transition in spatially extended systems

We study the nature of the synchronization transition in spatially extended systems by discussing a simple stochastic model. An analytic argument is put forward showing that, in the limit of discontinuous processes, the transition belongs to the directed percolation (DP) universality class. The analysis is complemented by a detailed investigation of the dependence of the first passage time for the amplitude of the difference field on the adopted threshold. We find the existence of a critical threshold separating the regime controlled by linear mechanisms from that controlled by collective phenomena. As a result of this analysis we conclude that the synchronization transition belongs to the DP class also in continuous models. The conclusions are supported by numerical checks on coupled map lattices too

Genome Biol.

Background Whole genome sequencing of marine cyanobacteria has revealed an unprecedented degree of genomic variation and streamlining. With a size of 1.66 megabase-pairs, Prochlorococcus sp. MED4 has the most compact of these genomes and it is enigmatic how the few identified regulatory proteins efficiently sustain the lifestyle of an ecologically successful marine microorganism. Small non-coding RNAs (ncRNAs) control a plethora of processes in eukaryotes as well as in bacteria; however, systematic searches for ncRNAs are still lacking for most eubacterial phyla outside the enterobacteria. Results Based on a computational prediction we show the presence of several ncRNAs (cyanobacterial functional RNA or Yfr) in several different cyanobacteria of the Prochlorococcus-Synechococcus lineage. Some ncRNA genes are present only in two or three of the four strains investigated, whereas the RNAs Yfr2 through Yfr5 are structurally highly related and are encoded by a rapidly evolving gene family as their genes exist in different copy numbers and at different sites in the four investigated genomes. One ncRNA, Yfr7, is present in at least seven other cyanobacteria. In addition, control elements for several ribosomal operons were predicted as well as riboswitches for thiamine pyrophosphate and cobalamin. Conclusion This is the first genome-wide and systematic screen for ncRNAs in cyanobacteria. Several ncRNAs were both computationally predicted and their presence was biochemically verified. These RNAs may have regulatory functions and each shows a distinct phylogenetic distribution. Our approach can be applied to any group of microorganisms for which more than one total genome sequence is available for comparative analysis

Small-world networks: Evidence for a crossover picture

Watts and Strogatz [Nature 393, 440 (1998)] have recently introduced a model for disordered networks and reported that, even for very small values of the disorder $p$ in the links, the network behaves as a small-world. Here, we test the hypothesis that the appearance of small-world behavior is not a phase-transition but a crossover phenomenon which depends both on the network size $n$ and on the degree of disorder $p$. We propose that the average distance $\ell$ between any two vertices of the network is a scaling function of $n / n^*$. The crossover size $n^*$ above which the network behaves as a small-world is shown to scale as $n^*(p \ll 1) \sim p^{-\tau}$ with $\tau \approx 2/3$.Comment: 5 pages, 5 postscript figures (1 in color), Latex/Revtex/multicols/epsf. Accepted for publication in Physical Review Letter