Valuation of Financial Derivatives in Discrete-Time Models

The core subject of financial mathematics concerns the issue of pricing financial assets such as complex financial derivatives. The pricing technique is pervaded by the concept of arbitrage: mis-pricing will be spotted and exploited, resulting in a risk free return for any arbitrageur. A mispriced financial asset will expose the issuer to be exploited by the market as a money-pump. To prevent arbitrage, when pricing one turns to mathematics. The no-arbitrage pricing is thus formalized as a mathematical problem and it is possible to prove a mathematical pricing relationship for a financial derivative. In some specific cases it is even possible to calculate an explicit price. This thesis will consider the pricing technique of a widely used financial derivative - the option. Black-Scholes theory is, since its introduction in 1973, the main tool used for option pricing. The theory that derives the famous Black-Scholes formula involves a great amount of financial and mathematical theory, however often ignored by the user. This thesis tries to bring key concepts into light, hopefully leaving the reader (and writer) with a deeper understanding. Finance, in general, involves a great amount of uncertainty. To be able to express this uncertainty in a mathematical manner, one introduces probability theory. There will be a go-trough of basic probability theory needed to fully adopt the concept of an equivalent martingale measure which is the essential tool in arbitrage-free pricing. By introducing the time-discrete Cox-Ross-Rubinstein model and prove existence and uniqueness of an equivalent martingale measure, one is able to state the arbitrage-free price of a European call option. The model is then compared to the continues-time Black-Scholes model and in conclusion it is proved and showed that the asymptotic price of the CRR model is the same as the price calculated by the Black-Scholes formula

An Information-Based Neural Approach to Constraint Satisfaction

A novel artificial neural network approach to constraint satisfaction problems is presented. Based on information-theoretical considerations, it differs from a conventional mean-field approach in the form of the resulting free energy. The method, implemented as an annealing algorithm, is numerically explored on a testbed of K-SAT problems. The performance shows a dramatic improvement to that of a conventional mean-field approach, and is comparable to that of a state-of-the-art dedicated heuristic (Gsat+Walk). The real strength of the method, however, lies in its generality -- with minor modifications it is applicable to arbitrary types of discrete constraint satisfaction problems.Comment: 13 pages, 3 figures,(to appear in Neural Computation

The cosmic origin of fluorine and sulphur: Infrared spectroscopic studies of red giants

Disregarding the small primordial traces of the lightest elements, all metals have been formed in stellar processes, which means that the relative amount of metals in the Universe increases for every stellar generation. This build-up of elements is called chemical evolution and might be used both to constrain stellar models as well as understanding the formation and evolution of stellar populations. In this thesis I determine abundances of two of the least studied elements, fluorine and sulphur, in three different stellar populations in the Milky Way using infrared spectroscopy of giants. Regarding fluorine the chemical evolution is very unclear because the number of previous observations are small. The cosmic origin of fluorine could still be one or more of three different sources: asymptotic giant branch stars, core-collapse supernovae, or the winds of Wolf-Rayet stars. If the latter is confirmed by observations, fluorine would make a great proxy for the determining whether the initial mass function in the Bulge is different from the solar neighborhood, which has been suggested in several other types of works, but not all. If confirmed, that would tell us that the central parts of our Galaxy have evolved differently than the local Disk. In the thesis I find that all the fluorine in the solar neighborhood most likely was produced by asymptotic giant branch stars, but at the same time find possible indications of fluorine production by Wolf-Rayet stars in the Bulge, indeed suggesting an initial mass function of the Bulge that is skewed towards more massive stars as compared to the solar neighborhood. When it comes to sulphur, there have been several proposed trends for metal-poor stars. Interestingly some of these observations cannot be explained with classic models of Galactic evolution, thereby questioning some of our understanding of the formation and evolution of the Milky Way. In this thesis I find a Galactic evolution-trend of sulphur following the expected trend from established models and cannot confirm any of the more exotic trends

An information-based neural approach to generic constraint satisfaction

AbstractA novel artificial neural network heuristic (INN) for general constraint satisfaction problems is presented, extending a recently suggested method restricted to boolean variables. In contrast to conventional ANN methods, it employs a particular type of non-polynomial cost function, based on the information balance between variables and constraints in a mean-field setting. Implemented as an annealing algorithm, the method is numerically explored on a testbed of Graph Coloring problems. The performance is comparable to that of dedicated heuristics, and clearly superior to that of conventional mean-field annealing

A Modeling Study on How Cell Division Affects Properties of Epithelial Tissues Under Isotropic Growth

Cell proliferation affects both cellular geometry and topology in a growing tissue, and hence rules for cell division are key to understanding multicellular development. Epithelial cell layers have for long times been used to investigate how cell proliferation leads to tissue-scale properties, including organism-independent distributions of cell areas and number of neighbors. We use a cell-based two-dimensional tissue growth model including mechanics to investigate how different cell division rules result in different statistical properties of the cells at the tissue level. We focus on isotropic growth and division rules suggested for plant cells, and compare the models with data from the Arabidopsis shoot. We find that several division rules can lead to the correct distribution of number of neighbors, as seen in recent studies. In addition we find that when also geometrical properties are taken into account other constraints on the cell division rules result. We find that division rules acting in favor of equally sized and symmetrically shaped daughter cells can best describe the statistical tissue properties

It's about time: Analysing simplifying assumptions for modelling multi-step pathways in systems biology.

Thoughtful use of simplifying assumptions is crucial to make systems biology models tractable while still representative of the underlying biology. A useful simplification can elucidate the core dynamics of a system. A poorly chosen assumption can, however, either render a model too complicated for making conclusions or it can prevent an otherwise accurate model from describing experimentally observed dynamics. Here, we perform a computational investigation of sequential multi-step pathway models that contain fewer pathway steps than the system they are designed to emulate. We demonstrate when such models will fail to reproduce data and how detrimental truncation of a pathway leads to detectable signatures in model dynamics and its optimised parameters. An alternative assumption is suggested for simplifying such pathways. Rather than assuming a truncated number of pathway steps, we propose to use the assumption that the rates of information propagation along the pathway is homogeneous and, instead, letting the length of the pathway be a free parameter. We first focus on linear pathways that are sequential and have first-order kinetics, and we show how this assumption results in a three-parameter model that consistently outperforms its truncated rival and a delay differential equation alternative in recapitulating observed dynamics. We then show how the proposed assumption allows for similarly terse and effective models of non-linear pathways. Our results provide a foundation for well-informed decision making during model simplifications

Modeling auxin transport and plant development

The plant hormone auxin plays a critical role in plant development. Central to its function is its distribution in plant tissues, which is, in turn, largely shaped by intercellular polar transport processes. Auxin transport relies on diffusive uptake as well as carrier-mediated transport via influx and efflux carriers. Mathematical models have been used to both refine our theoretical understanding of these processes and to test new hypotheses regarding the localization of efflux carriers to understand auxin patterning at the tissue level. Here we review models for auxin transport and how they have been applied to patterning processes, including the elaboration of plant vasculature and primordium positioning. Second, we investigate the possible role of auxin influx carriers such as AUX1 in patterning auxin in the shoot meristem. We find that AUX1 and its relatives are likely to play a crucial role in maintaining high auxin levels in the meristem epidermis. We also show that auxin influx carriers may play an important role in stabilizing auxin distribution patterns generated by auxin-gradient type models for phyllotaxis

Shifting foundations: the mechanical cell wall and development.

The cell wall has long been acknowledged as an important physical mediator of growth in plants. Recent experimental and modelling work has brought the importance of cell wall mechanics into the forefront again. These data have challenged existing dogmas that relate cell wall structure to cell/organ growth, that uncouple elasticity from extensibility, and those which treat the cell wall as a passive and non-stressed material. Within this review we describe experiments and models which have changed the ways in which we view the mechanical cell wall, leading to new hypotheses and research avenues. It has become increasingly apparent that while we often wish to simplify our systems, we now require more complex multi-scale experiments and models in order to gain further insight into growth mechanics. We are currently experiencing an exciting and challenging shift in the foundations of our understanding of cell wall mechanics in growth and development.Work in the authors’ groups is funded by The Gatsby Charitable Foundation (GAT3396/PR4, SB; GAT3395/PR4, HJ), the Swedish Research Council (VR2013‐4632, HJ), the Knut and Alice Wallenberg Foundation via ShapeSystems (HJ), and the BBSRC (BB.L002884.1, SB).This is the final version of the article. It first appeared from Elsevier via https://doi.org/10.1016/j.pbi.2015.12.00

Modelling meristem development in plants

Meristems continually supply new cells for post-embryonic plant development and coordinate the initiation of new organs, such as leaves and flowers. Meristem function is regulated by a large and interconnected dynamic system that includes transcription networks, intercellular protein signalling, polarized transport of hormones and a constantly changing cellular topology. Mathematical modelling, in which the dynamics of a system are simulated using explicitly defined interactions, can serve as a powerful tool for examining the expected behaviour of such a system given our present knowledge and assumptions. Modelling can also help to investigate new hypotheses in silico both to validate ideas and to obtain inspiration for new experiments. Several recent studies have used new molecular data together with modelling and computational techniques to investigate meristem function