POSSIBILITIES OF BROWNIAN MOTION APPLICATION FOR OPTION PRICING

Ponašanje cijena dionica u veoma kratkom vremenu osnovica je vrednovanja uvjetovanih tražbina, posebno onih koje se javljaju kao formalizirane opcije, ali i u parametrima koji defi niraju dugoročno ponašanje u dionicama skrivenih opcija. Promjenama cijena dionica mijenja se rizičnost uvjetovanih tražbina, pa za vrednovanje većine njih nije moguće primijeniti koncept ekonomske vrijednosti. Jedna je od mogućnosti vrednovanja opcija oslanjanje na stohastičko ponašanje cijena dionica ili neke druge vezane imovine, pa je važno odrediti stohastički proces koji opisuje to ponašanje. Stohastički proces u kontinuiranom vremenu, poznat pod nazivom geometrijsko Brownovo gibanje, odnosno Wienerov proces sastavni je dio mnogih modela vrednovanja opcija. Najpoznatiji model vrednovanja opcija jest Black–Scholes model, a polazi od pretpostavke da cijene dionica slijede geometrijsko Brownovo gibanje. Bez obzira na kritike i postojanje brojnih alternativnih modela vrednovanja opcija, B/S model najkorišteniji je model i zajedno s različitim prilagodbama može poslužiti za vrednovanje različitih opcija i poslovnih procesa koji sadrže određene opcijske elemente.The behaviour of stock prices in a very short period of time is the base for valuing contingent claims, especially those that appear as formalized options, but also in the parameters that define long-term behaviour in stock’s hidden options. Changes in stock prices alter the risk of contingent claims, and because of that, for valuing the most of them it is not possible to apply the economic value concept. One of the possibilities for pricing the options is to rely upon the stochastic behaviour of stock prices or other underlying asset. It is important to determine the stochastic process that describes this behaviour. The continuous time stochastic process is known as geometric Brownian motion or Wiener process and it is a building block for all kinds of models for options pricing. The best-known model for option pricing is the Black-Scholes model and it assumes that stock prices follow geometric Brownian motion. Despite critics and the existence of alternative models for option pricing, the Black-Scholes model is widely used and with appropriate adjustments it can be used for pricing of different options and business processes with embedded options

A Holistic Approach to Sustainability Analysis of Industrial Networks

The aim of this thesis is to support the evaluation of sustainable development strategies for industrial networks in the context of industrial ecology (IE). Industrial networks are a group of units which carry out, or contribute to, industrial activity, and are connected by material and energy flows, but also capital and information exchanges. The components of an industrial network encompass resource extraction, processing and refining, forming and assembly, use, disposal, as well as recycling and reprocessing. The motivation behind this research is the realisation that much of the current environmental system analysis focus within IE lacks a structured approach to considering: • system environment • dynamic nature of the system and its environment • economic and social impacts • the effect of uncertainty on analysis outcomes. It is argued in this thesis that current environmental analysis approaches used in IE can be improved in their capacity to capture the complexity of industrial systems, with the objective of promoting sustainable development. While IE emphasises the benefit of a systems approach to identifying environmental strategies in industry, analysis tools have to date not engaged extensively with important aspects such as the influence of system environment and dynamics on the viability of an environmental strategy, or with the economic or social impacts of industrial system development, which are equally important for sustainable development. Nor is the assessment of the effect of uncertainty on analysis outcomes an integral part of environmental analysis tools in IE. This is particularly significant when, in fact, the degree of uncertainty in assumptions and data used increases with the scope, and therefore the abstraction, of the system under consideration. IE will have to engage with the network and contextual complexities to a greater degree if it is to evolve from a concept to the application of its principles in practice. The main contribution of this thesis is therefore the development of a structured approach to analysing industrial networks for the purpose of identifying strategies to encourage sustainable development, while accounting for the complexity of the underlying system as well as the problem context. This analysis is intended to allow the identification of preferred network development pathways and to test the effectiveness of sustainable development strategies. A top-down, prescriptive approach is adopted for this purpose. This approach is chosen as the industrial network analysis is intended to identify how a network should develop, rather than focusing on how it could develop. Industrial networks are systems which are complex in both their structure and behaviour. This thesis also delivers a characterisation of these networks, which serves two purposes – quantifying key elements of structure and behaviour; and using this information to build a foundation for subsequent industrial network analysis. The value of such an approach can be seen in the following example. With a detailed understanding of individual network characteristics, both separately and collectively, it is possible to determine the source of issues, the means available to address them, any barriers that might exist, and the consequences of implementing any strategic interventions. The analysis approach proposed in this thesis is based on multi-criteria decisions analysis (MCDA), which, as a process, combines initial problem structuring and subsequent quantitative analysis stages. The tools employed within MCDA have been employed variously around considerations of sustainable development. Their value in this thesis is their integration within a rigorous analytical framework. Rigorous problem structuring is attractive as it helps elucidate the complexities of the system and its environment and is, by definition, designed to deal with multiple environmental social and economic criteria that would have to be considered to promote sustainable development. For the quantitative analysis, the industrial network analysis draws from existing analysis tools in IE, but predominately from other systems research disciplines, such as process systems engineering (PSE) and supply chain management (SCM). These fields, due to their maturity and practical focus, have invested a lot of research into system design and strategic planning, capturing system dynamics and uncertainty to ensure, within selected system constraints, that a proposed system or changes to a system are viable, and that the system is capable of achieving the stated objectives. Both PSE and SCM rely heavily on optimisation for system design and planning, and achieve good results with it as an analytical tool. The similarity between industrial networks and process systems / supply chains, suggests that an optimisation platform, specifically multi-objective dynamic optimisation, could be employed fruitfully for the analysis of industrial networks. This is the approach taken in this thesis. It is consistent with the “top down” approach advocated previously, which is deemed preferable for the identification and implementation analysis of strategic interventions. This enables the determination of a structure (design) that is “best” able to operate under future conditions (planning) with respect to the chosen sustainable development objectives. However, an analysis is only ever as good as its underlying data and assumptions. The complexity and scope of the industrial network and the challenge of articulating sustainable development target(s) give rise to significant uncertainties. For this reason a framework is developed within this thesis that integrates uncertainty analysis into the overall approach, to obtain insight into the robustness of the analysis results. Quantifying all the uncertainties in an industrial network model can be a daunting task for a modeller, and a decision-maker can be confused by modelling results. Means are therefore suggested to reduce the set of uncertainties that have to be engaged with, by identifying those which impact critically on model outcomes. However, even if uncertainty cannot be reduced, and the implementation of any strategy retains a degree of risk, the uncertainty analysis has the benefit that it forces an analyst to engage in more detail with the network in question, and to be more critical of the underlying assumptions. The analysis approach is applied to two case studies in this thesis: one deals with waste avoidance in an existing wood-products network in a large urban metropolis; the other with the potential for renewable energy generation in a developing economy. Together, these case studies provide a rich tableau within which to demonstrate the full features of the industrial network analysis. These case studies highlight how the context within which the relevant industrial network functions influences greatly the evolution of the network over time; how uncertainty is managed; and what strategies are preferred in each case in order to enhance the contribution of each network to sustainable development. This thesis makes an intellectual contribution in the following areas: • the characterisation of industrial networks to highlight sources of environmental issues, role the characteristics (could) play in the identification of (preferred) sustainable development strategies, and the need to explicitly consider these in a systems analysis. • the synthesis, adaptation and application of existing tools to fulfil the need for analysis tools in IE that can handle both contextual and system complexity, and address the above mentioned issues of lacking consideration of o system environment o dynamic nature of the system and its environment o economic and social impacts o the effect of uncertainty on analysis outcomes. • the development and demonstration of an industrial network analysis approach that o is flexible enough to model any industrial network at the inter-firm level, regardless of form and configuration of materials and products circulated, and depending on the existing network and the proposed strategies. o is able to encompass a wide range of environmental strategies, either individually or in combination depending on what best suits the situation, rather than focusing on any strategy in particular. o ensures long term viability of strategies, rather than short term solutions delivering incremental improvement. • the development of a comprehensive approach to capturing and assessing the effect of uncertainty on solution robustness for industrial network analysis, including the screening to determine the most important parameters, considering valuation and technical uncertainties, including future uncertainty. The industrial network analysis approach presented in this thesis looks more to how a network should develop (according to a set of sustainable development objectives), rather than how it may in actual fact develop. Consequently, the influence of agent interests and behaviour is not considered explicitly. This may be construed as a limitation of the industrial analysis approach. However, it is argued that the “top down” modelling approach favoured here is useful at a policy-making level. Here, for example, government instrumentalities, trade organisations and industry groupings, non-government organisations and community-based organisations are likely to be interested more in the performance of the network as a whole, rather than (necessarily) following the behaviour of individual agents within the network. Future work could well entertain the prospect of a mixed approach, in which the top-down approach of this thesis is complemented by a “bottom-up”, agent-based analysis. In this manner, it would be possible to give an indication of how attainable the identified industrial network development pathways are. Furthermore, the use of government incentives can be explored to assess if network development could approach the preferred development pathway which is identified using the methodology and results articulated in this thesis

A Holistic Approach to Sustainability Analysis of Industrial Networks

The aim of this thesis is to support the evaluation of sustainable development strategies for industrial networks in the context of industrial ecology (IE). Industrial networks are a group of units which carry out, or contribute to, industrial activity, and are connected by material and energy flows, but also capital and information exchanges. The components of an industrial network encompass resource extraction, processing and refining, forming and assembly, use, disposal, as well as recycling and reprocessing. The motivation behind this research is the realisation that much of the current environmental system analysis focus within IE lacks a structured approach to considering: • system environment • dynamic nature of the system and its environment • economic and social impacts • the effect of uncertainty on analysis outcomes. It is argued in this thesis that current environmental analysis approaches used in IE can be improved in their capacity to capture the complexity of industrial systems, with the objective of promoting sustainable development. While IE emphasises the benefit of a systems approach to identifying environmental strategies in industry, analysis tools have to date not engaged extensively with important aspects such as the influence of system environment and dynamics on the viability of an environmental strategy, or with the economic or social impacts of industrial system development, which are equally important for sustainable development. Nor is the assessment of the effect of uncertainty on analysis outcomes an integral part of environmental analysis tools in IE. This is particularly significant when, in fact, the degree of uncertainty in assumptions and data used increases with the scope, and therefore the abstraction, of the system under consideration. IE will have to engage with the network and contextual complexities to a greater degree if it is to evolve from a concept to the application of its principles in practice. The main contribution of this thesis is therefore the development of a structured approach to analysing industrial networks for the purpose of identifying strategies to encourage sustainable development, while accounting for the complexity of the underlying system as well as the problem context. This analysis is intended to allow the identification of preferred network development pathways and to test the effectiveness of sustainable development strategies. A top-down, prescriptive approach is adopted for this purpose. This approach is chosen as the industrial network analysis is intended to identify how a network should develop, rather than focusing on how it could develop. Industrial networks are systems which are complex in both their structure and behaviour. This thesis also delivers a characterisation of these networks, which serves two purposes – quantifying key elements of structure and behaviour; and using this information to build a foundation for subsequent industrial network analysis. The value of such an approach can be seen in the following example. With a detailed understanding of individual network characteristics, both separately and collectively, it is possible to determine the source of issues, the means available to address them, any barriers that might exist, and the consequences of implementing any strategic interventions. The analysis approach proposed in this thesis is based on multi-criteria decisions analysis (MCDA), which, as a process, combines initial problem structuring and subsequent quantitative analysis stages. The tools employed within MCDA have been employed variously around considerations of sustainable development. Their value in this thesis is their integration within a rigorous analytical framework. Rigorous problem structuring is attractive as it helps elucidate the complexities of the system and its environment and is, by definition, designed to deal with multiple environmental social and economic criteria that would have to be considered to promote sustainable development. For the quantitative analysis, the industrial network analysis draws from existing analysis tools in IE, but predominately from other systems research disciplines, such as process systems engineering (PSE) and supply chain management (SCM). These fields, due to their maturity and practical focus, have invested a lot of research into system design and strategic planning, capturing system dynamics and uncertainty to ensure, within selected system constraints, that a proposed system or changes to a system are viable, and that the system is capable of achieving the stated objectives. Both PSE and SCM rely heavily on optimisation for system design and planning, and achieve good results with it as an analytical tool. The similarity between industrial networks and process systems / supply chains, suggests that an optimisation platform, specifically multi-objective dynamic optimisation, could be employed fruitfully for the analysis of industrial networks. This is the approach taken in this thesis. It is consistent with the “top down” approach advocated previously, which is deemed preferable for the identification and implementation analysis of strategic interventions. This enables the determination of a structure (design) that is “best” able to operate under future conditions (planning) with respect to the chosen sustainable development objectives. However, an analysis is only ever as good as its underlying data and assumptions. The complexity and scope of the industrial network and the challenge of articulating sustainable development target(s) give rise to significant uncertainties. For this reason a framework is developed within this thesis that integrates uncertainty analysis into the overall approach, to obtain insight into the robustness of the analysis results. Quantifying all the uncertainties in an industrial network model can be a daunting task for a modeller, and a decision-maker can be confused by modelling results. Means are therefore suggested to reduce the set of uncertainties that have to be engaged with, by identifying those which impact critically on model outcomes. However, even if uncertainty cannot be reduced, and the implementation of any strategy retains a degree of risk, the uncertainty analysis has the benefit that it forces an analyst to engage in more detail with the network in question, and to be more critical of the underlying assumptions. The analysis approach is applied to two case studies in this thesis: one deals with waste avoidance in an existing wood-products network in a large urban metropolis; the other with the potential for renewable energy generation in a developing economy. Together, these case studies provide a rich tableau within which to demonstrate the full features of the industrial network analysis. These case studies highlight how the context within which the relevant industrial network functions influences greatly the evolution of the network over time; how uncertainty is managed; and what strategies are preferred in each case in order to enhance the contribution of each network to sustainable development. This thesis makes an intellectual contribution in the following areas: • the characterisation of industrial networks to highlight sources of environmental issues, role the characteristics (could) play in the identification of (preferred) sustainable development strategies, and the need to explicitly consider these in a systems analysis. • the synthesis, adaptation and application of existing tools to fulfil the need for analysis tools in IE that can handle both contextual and system complexity, and address the above mentioned issues of lacking consideration of o system environment o dynamic nature of the system and its environment o economic and social impacts o the effect of uncertainty on analysis outcomes. • the development and demonstration of an industrial network analysis approach that o is flexible enough to model any industrial network at the inter-firm level, regardless of form and configuration of materials and products circulated, and depending on the existing network and the proposed strategies. o is able to encompass a wide range of environmental strategies, either individually or in combination depending on what best suits the situation, rather than focusing on any strategy in particular. o ensures long term viability of strategies, rather than short term solutions delivering incremental improvement. • the development of a comprehensive approach to capturing and assessing the effect of uncertainty on solution robustness for industrial network analysis, including the screening to determine the most important parameters, considering valuation and technical uncertainties, including future uncertainty. The industrial network analysis approach presented in this thesis looks more to how a network should develop (according to a set of sustainable development objectives), rather than how it may in actual fact develop. Consequently, the influence of agent interests and behaviour is not considered explicitly. This may be construed as a limitation of the industrial analysis approach. However, it is argued that the “top down” modelling approach favoured here is useful at a policy-making level. Here, for example, government instrumentalities, trade organisations and industry groupings, non-government organisations and community-based organisations are likely to be interested more in the performance of the network as a whole, rather than (necessarily) following the behaviour of individual agents within the network. Future work could well entertain the prospect of a mixed approach, in which the top-down approach of this thesis is complemented by a “bottom-up”, agent-based analysis. In this manner, it would be possible to give an indication of how attainable the identified industrial network development pathways are. Furthermore, the use of government incentives can be explored to assess if network development could approach the preferred development pathway which is identified using the methodology and results articulated in this thesis

Reduced order modelling of bone resorption and formation.

The bone remodelling process, performed by the Bone Multicellular Unit (BMU) is a key multi-hierarchically regulated process, which provides and supports various functionality of bone tissue. It is also plays a critical role in bone disorders, as well as bone tissue healing following damage. Improved modelling of bone turnover processes could play a significant role in helping to understand the underlying cause of bone disorders and thus develop more effective treatment methods. Moreover, despite extensive research in the field of bone tissue engineering, bonescaffold development is still very empirical. The development of improved methods of modelling the bone remodelling process should help to develop new implant designs which encourage rapid osteointegration. There are a number of limitations with respect to previous research in the field of mathematical modelling of the bone remodelling process, including the absence of an osteocyte loop of regulation. It is within this context that this research presented in this thesis utilises a range of modelling methods to develop a framework for bone remodelling which can be used to improve treatment methods for bone disorders. The study concentrated on dynamic and steady state variables that in perspective can be used as constraints for optimisation problem considering bone remodelling or tissue remodelling with the help of the grafts/scaffolds.The cellular and combined allosteric-regulation approaches to modelling of bone turnover, based on the osteocyte loop of regulation, have been studied. Both approaches have been studied different within wide range of rate parameters. The approach to the model validation has been considered, including a statistical approach and parameter reduction approach. From a validation perspective the cellular class of modes is preferable since it has fewer parameters to validate. The optimal control framework for regulation of remodelling has been studied. Future work in to improve the models and their application to bone scaffold design applications have been considered. The study illustrates the complexity of formalisation of the metabolic processes and the relations between hierarchical subsystems in hard tissue where a relatively small number of cells are active. Different types/modes of behaviour have been found in the study: relaxational, periodical and chaotic modes. All of these types of behaviour can be found, in bone tissue. However, a chaotic or periodic modes are ones of the hardest to verify although a number of periodical phenomena have been observed empirically in bone and skeletal development. Implementation of the allosteric loop into cellular model damps other types of behaviour/modes. In this sense it improves the robustness, predictability and control of the system. The developed models represent a first step in a hierarchical model of bone tissue (system versus local effects). The limited autonomy of any organ or tissue implies differentiation on a regulatory level as well as physiological functions and metabolic differences. Implementation into the cellular phenomenological model of allosteric-like loop of regulation has been performed. The results show that the robustness of regulation can be inherited from the phenomenological model. An attempt to correlate the main bone disorders with different modes of behaviour has been undertaken using Paget’s disorder in bone, osteoporosis and some more general skeleton disorders which lead to periodical changes in bone mass, reported by some authors. However, additional studies are needed to make this hypothesis significant. The study has revealed a few interesting techniques. When studying a multidimensional phenomenon, as a bone tissue is, the visualisation and data reduction is important for analysis and interpretation of results. In the study two novel technical methods have been proposed. The first is the graphical matrix method to visualise/project the multidimensional phase space of variables into diagonal matrix of regular combination of two-dimensional graphs. This significantly simplifies the analysis and, in principle, makes it possible to visualise the phase space higher than three-dimensional. The second important technical development is the application of the Monte-Carlo method in combination with the regression method to study the character and stability of the equilibrium points of a dynamic system. The advantage of this method is that it enables the most influential parameters that affect the character and stability of the equilibrium point to be identified from a large number of the rate parameters/constants of the dynamic system. This makes the interpretation of parameters and conceptual verification of the model much easier