Numerical methods for option pricing.

This thesis aims to introduce some fundamental concepts underlying option valuation theory including implementation of computational tools. In many cases analytical solution for option pricing does not exist, thus the following numerical methods are used: binomial trees, Monte Carlo simulations and finite difference methods. First, an algorithm based on Hull and Wilmott is written for every method. Then these algorithms are improved in different ways. For the binomial tree both speed and memory usage is significantly improved by using only one vector instead of a whole price storing matrix. Computational time in Monte Carlo simulations is reduced by implementing a parallel algorithm (in C) which is capable of improving speed by a factor which equals the number of processors used. Furthermore, MatLab code for Monte Carlo was made faster by vectorizing simulation process. Finally, obtained option values are compared to those obtained with popular finite difference methods, and it is discussed which of the algorithms is more appropriate for which purpose.

Hybrid wind-diesel power plant valuation and optimization using real option theory and dynamic programming

El sistema de energía eólica-diesel híbrido tiene un gran potencial en la prestación de suministro de energía a comunidades remotas. En comparación con los sistemas tradicionales de diesel, las plantas de energía híbridas ofrecen grandes ventajas tales como el suministro de capacidad de energía extra para "microgrids", reducción de los contaminantes y emisiones de gases de efecto invernadero, y la cobertura del riesgo de aumento inesperado del precio del combustible. El principal objetivo de la presente tesis es proporcionar nuevos conocimientos para la evaluación y optimización de los sistemas de energía híbrido eólico-diesel considerando las incertidumbres. Dado que la energía eólica es una variable estocástica, ésta no puede ser controlada ni predecirse con exactitud. La naturaleza incierta del viento como fuente de energía produce serios problemas tanto para la operación como para la evaluación del valor del sistema de energía eólica-diesel híbrido. Por un lado, la regulación de la potencia inyectada desde las turbinas de viento es una difícil tarea cuando opera el sistema híbrido. Por otro lado, el bene.cio económico de un sistema eólico-diesel híbrido se logra directamente a través de la energía entregada a la red de alimentación de la energía eólica. Consecuentemente, la incertidumbre de los recursos eólicos incrementa la dificultad de estimar los beneficios globales en la etapa de planificación. La principal preocupación del modelo tradicional determinista es no tener en cuenta la incertidumbre futura a la hora de tomar la decisión de operación. Con lo cual, no se prevé las acciones operativas flexibles en respuesta a los escenarios futuros. El análisis del rendimiento y simulación por ordenador en el Proyecto Eólico San Cristóbal demuestra que la incertidumbre sobre la energía eólica, las estrategias de control, almacenamiento de energía, y la curva de potencia de aerogeneradores tienen un impacto significativo sobre el rendimiento del sistema. En la presente tesis, se analiza la relación entre la teoría de valoración de opciones y el proceso de toma de decisiones. La opción real se desarrolla con un modelo y se presenta a través de ejemplos prácticos para evaluar el valor de los sistemas de energía eólica-diesel híbridos. Los resultados muestran que las opciones operacionales pueden aportar un valor adicional para el sistema de energía híbrida, cuando esta flexibilidad operativa se utiliza correctamente. Este marco se puede aplicar en la optimización de la operación a corto plazo teniendo en cuenta la naturaleza dependiente de la trayectoria de la política óptima de despacho, dadas las plausibles futuras realizaciones de la producción de energía eólica. En comparación con los métodos de valoración y optimización existentes, el resultado del caso de estudio numérico muestra que la política de operación resultante del modelo de optimización propuesto presenta una notable actuación en la reducción del con- sumo total de combustible del sistema eólico-diesel. Con el .n de tomar decisiones óptimas, los operadores de plantas de energía y los gestores de éstas no deben centrarse sólo en el resultado directo de cada acción operativa, tampoco deberían tomar decisiones deterministas. La forma correcta es gestionar dinámicamente el sistema de energía teniendo en cuenta el valor futuro condicionado en cada opción frente a la incertidumbre. ABSTRACT Hybrid wind-diesel power systems have a great potential in providing energy supply to remote communities. Compared with the traditional diesel systems, hybrid power plants are providing many advantages such as providing extra energy capacity to the micro-grid, reducing pollution and greenhouse-gas emissions, and hedging the risk of unexpected fuel price increases. This dissertation aims at providing novel insights for assessing and optimizing hybrid wind-diesel power systems considering the related uncertainties. Since wind power can neither be controlled nor accurately predicted, the energy harvested from a wind turbine may be considered a stochastic variable. This uncertain nature of wind energy source results in serious problems for both the operation and value assessment of the hybrid wind-diesel power system. On the one hand, regulating the uncertain power injected from wind turbines is a difficult task when operating the hybrid system. On the other hand, the economic profit of a hybrid wind-diesel system is achieved directly through the energy delivered to the power grid from the wind energy. Therefore, the uncertainty of wind resources has increased the difficulty in estimating the total benefits in the planning stage. The main concern of the traditional deterministic model is that it does not consider the future uncertainty when making the dispatch decision. Thus, it does not provide flexible operational actions in response to the uncertain future scenarios. Performance analysis and computer simulation on the San Cristobal Wind Project demonstrate that the wind power uncertainty, control strategies, energy storage, and the wind turbine power curve have a significant impact on the performance of the system. In this dissertation, the relationship between option pricing theory and decision making process is discussed. A real option model is developed and presented through practical examples for assessing the value of hybrid wind-diesel power systems. Results show that operational options can provide additional value to the hybrid power system when this operational flexibility is correctly utilized. This framework can be applied in optimizing short term dispatch decisions considering the path-dependent nature of the optimal dispatch policy, given the plausible future realizations of the wind power production. Comparing with the existing valuation and optimization methods, result from numerical example shows that the dispatch policy resulting from the proposed optimization model exhibits a remarkable performance in minimizing the total fuel consumption of the wind-diesel system. In order to make optimal decisions, power plant operators and managers should not just focus on the direct outcome of each operational action; neither should they make deterministic decisions. The correct way is to dynamically manage the power system by taking into consideration the conditional future value in each option in response to the uncertainty.

Optimal and Long-Term Dynamic Transport Policy Design: Seeking Maximum Social Welfare through a Pricing Scheme.

This article presents an alternative approach to the decision-making process in transport strategy design. The study explores the possibility of integrating forecasting, assessment and optimization procedures in support of a decision-making process designed to reach the best achievable scenario through mobility policies. Long-term evaluation, as required by a dynamic system such as a city, is provided by a strategic Land-Use and Transport Interaction (LUTI) model. The social welfare achieved by implementing mobility LUTI model policies is measured through a cost-benefit analysis and maximized through an optimization process throughout the evaluation period. The method is tested by optimizing a pricing policy scheme in Madrid on a cordon toll in a context requiring system efficiency, social equity and environmental quality. The optimized scheme yields an appreciable increase in social surplus through a relatively low rate compared to other similar pricing toll schemes. The results highlight the different considerations regarding mobility impacts on the case study area, as well as the major contributors to social welfare surplus. This leads the authors to reconsider the cost-analysis approach, as defined in the study, as the best option for formulating sustainability measures.

Optimal and Long-Term Dynamic Transport Policy Design: Seeking Maximum Social Welfare through a Pricing Scheme

This article presents an alternative approach to the decision-making process in transport strategy design. The study explores the possibility of integrating forecasting, assessment and optimization procedures in support of a decision-making process designed to reach the best achievable scenario through mobility policies. Long-term evaluation, as required by a dynamic system such as a city, is provided by a strategic Land-Use and Transport Interaction (LUTI) model. The social welfare achieved by implementing mobility LUTI model policies is measured through a cost-benefit analysis and maximized through an optimization process throughout the evaluation period. The method is tested by optimizing a pricing policy scheme in Madrid on a cordon toll in a context requiring system efficiency, social equity and environmental quality. The optimized scheme yields an appreciable increase in social surplus through a relatively low rate compared to other similar pricing toll schemes. The results highlight the different considerations regarding mobility impacts on the case study area, as well as the major contributors to social welfare surplus. This leads the authors to reconsider the cost-analysis approach, as defined in the study, as the best option for formulating sustainability measures.

Valuation of a new entrant in an oligopolistic market, including its option to abandon: A real-life case

The purpose of this paper is to increase current empirical evidence on the relevance of real options for explaining firm investment decisions in oligopolistic markets. We study an actual investment case in the Spanish mobile telephony industry, the entrant in the market of a new operator, Yoigo. We analyze the option to abandon in order to show the relevance of the possibility of selling the company in an oligopolistic market where competitors are not allowed free entrance. The NPV (net present value) of the new entrant is calculated as a starting point. Then, based on the general approach proposed by Copeland and Antikarov (2001), a binomial tree is used to model managerial flexibility in discrete time periods, and value the option to abandon. The strike price of the option is calculated based on incremental EBITDA margins due to selling customers or merging with a competitor.

Next Generation of Energy Residential Gateways for Demand Response and Dynamic Pricing

Predictions about electric energy needs, based on current electric energy models, forecast that the global energy consumption on Earth for 2050 will double present rates. Using distributed procedures for control and integration, the expected needs can be halved. Therefore implementation of Smart Grids is necessary. Interaction between final consumers and utilities is a key factor of future Smart Grids. This interaction is aimed to reach efficient and responsible energy consumption. Energy Residential Gateways (ERG) are new in-building devices that will govern the communication between user and utility and will control electric loads. Utilities will offer new services empowering residential customers to lower their electric bill. Some of these services are Smart Metering, Demand Response and Dynamic Pricing. This paper presents a practical development of an ERG for residential buildings.

Improving the analysis of road pricing acceptability surveys by using hybrid models

Several international studies have analyzed the acceptability of road pricing schemes by means of an attitude survey in combination with the results of a stated choice experiment using both a descriptive analysis and a discrete-choice model with binary choice (?accept? or ?not accept? the toll). However, the use of hybrid discrete choice models constitutes an innovative alternative for integrating subjective attitudes and perceptions deriving from the survey of attitudes with the more objective variables from the stated choice experiment. This paper analyzes the results of applying these models to measure the acceptability of interurban road pricing among different groups of stakeholders (road freight and passenger operators, highway concessionaires, and associations of private car users) with qualitatively significant opinions on road pricing measures. Our results show that hybrid models are better suited to explaining the acceptability of a road pricing scheme by different groups of stakeholders than a separate analysis of the survey of attitudes and a discrete-choice model applied on a stated choice experiment. A particular finding was that the strong psycho-social latent variable of the perception of fairness explains the rejection or acceptance of a toll scheme by road stakeholders.

Equity pricing in the mining sector: evidence from NYSE and LSE

The study examines the Capital Asset Pricing Model (CAPM) for the mining sector using weekly stock returns from 27 companies traded on the New York Stock Exchange (NYSE) or on the London Stock Exchange (LSE) for the period of December 2008 to December 2010. The results support the use of the CAPM for the allocation of risk to companies. Most companies involved in precious metals (particularly gold), which have a beta value less than unity (Table 1), have been actuated as shelter values during the financial crisis. Values of R2 do not shown very explanatory power of fitted models (R2 < 70 %). Estimated coefficients beta are not sufficient to determine the expected returns on securities but the results of the tests conducted on sample data for the period analysed do not appear to clearly reject the CAPM

Assessment of irrigators’ preferences for different water supply risk management tools: option contract and insurance.

Irrigators face the risk of not having enough water to meet their crops’ demand. There are different mechanisms to cope with this risk, including water markets (option contracts) or insurance. A farmer will purchase them when the expected utility change derived from the tool is positive. This paper presents a theoretical assessment of the farmer’s expected utility under two different option contracts, a drought insurance and a combination of an option contract and the insurance. We analyze the conditions that determine farmer’s reference for one instrument or the other and perform a numerical application that is relevant for a Spanish region.

Integration of congestion pricing and intertemporal preference rate in social welfare function

Assessing social benefits in transport policy implementation has been studied by many researchers using theoretical or empirical measures. However, few of them measure social benefit using different discount rates including the inter-temporal preferences rate of users, the private investment discount rate and the inter-temporal preferences rate of the government. In general, the social discount rate used is the same for all social actors. Therefore, this paper aims to assess a new method by integrating different types of discount rate belonging to different social actors in order to measure the real benefits of each actor in the short, medium and long term. A dynamic simulation is provided by a strategic Land-Use and Transport Interaction (LUTI) model. The method is tested by optimizing a cordon toll scheme in Madrid considering socio- economic efficiency and environmental criteria. Based on the modified social welfare function (WF), the effects on the measure of social benefits are estimated and compared with the classical WF results as well. The results of this research could be a key issue to understanding the relationship between transport system policies and social actors' benefits distribution in a metropolitan context. The results show that the use of more suitable discount rates for each social actor had an effect on the selection and definition of optimal strategy of congestion pricing. The usefulness of the measure of congestion toll declines more quickly overtime.

Integration of congestion pricing and intertemporal preferences rate integration in social welfare function

Many researchers have used theoretical or empirical measures to assess social benefits in transport policy implementation. However, few have measured social benefits by using discount rates, including the intertemporal preference rate of users, the private investment discount rate, and the intertemporal preference rate of the government. In general, the social discount rate used is the same for all social actors. This paper aims to assess a new method by integrating different types of discount rates belonging to different social actors to measure the real benefits of each actor in the short term, medium term, and long term. A dynamic simulation is provided by a strategic land use and transport interaction model. The method was tested by optimizing a cordon toll scheme in Madrid, Spain. Socioeconomic efficiency and environmental criteria were considered. On the basis of the modified social welfare function, the effects on the measure of social benefits were estimated and compared with the classical welfare function measures. The results show that the use of more suitable discount rates for each social actor had an effect on the selection and definition of optimal strategy of congestion pricing. The usefulness of the measure of congestion toll declines more quickly over time. This result could be the key to understanding the relationship between transport system policies and the distribution of social actors? benefits in a metropolitan context.

Water option contracts for reducing water supply risks: an application to the Tagus-Segura Transfer

El agua es un recurso cada vez más escaso y valioso. Por ello, los recursos hídricos disponibles deben asignarse de una forma eficiente entre los diferentes usos. El cambio climático aumentará la frecuencia y severidad de los eventos extremos, y podría incrementar la demanda de agua de los cultivos. El empleo de mecanismos flexibles de asignación de agua puede ser imprescindible para hacer frente a este aumento en la variabilidad del balance hídrico y para asegurar que los riesgos de suministro, y no solo los recursos, son compartidos de manera eficiente entre los usuarios. Los mercados de agua permiten la reasignación de los recursos hídricos, favoreciendo su transferencia desde los usos de menor a los de mayor valor. Diferentes tipos de mercados de agua se han establecido en diferentes partes del mundo, ayudando a los participantes a afrontar los problemas de escasez de agua en esas zonas. En España, los intercambios de agua están permitidos desde 1999, aunque la participación de los usuarios en el mercado ha sido limitada. Hay varios aspectos de los mercados de agua en España que deben mejorarse. Esta tesis, además de proponer una serie de cambios en el marco regulatorio, propone la introducción de contratos de opción de agua como una posible mejora. La principal ventaja de este tipo de contratos es la estabilidad legal e institucional que éstos proporcionan tanto a compradores como vendedores. Para apoyar esta propuesta, se han llevado a cabo diferentes análisis que muestran el potencial de los contratos de opción como herramienta de reducción del riesgo asociado a una oferta de agua inestable. La Cuenca del Segura (Sureste de España), la Cuenca del Tajo y el Acueducto Tajo- Segura han sido seleccionados como casos de estudio. Tres análisis distintos aplicados a dicha región se presentan en esta tesis: a) una evaluación de los contratos de opción como mecanismo para reducir los riesgos de disponibilidad de agua sufridos por los regantes en la Cuenca del Segura; b) un marco teórico para analizar las preferencias de los regantes por diferentes mecanismos de gestión del riesgo de disponibilidad de agua, su disposición a pagar por ellos y los precios aproximados de estos instrumentos (seguro de sequía y contratos de opción de agua); y c) una evaluación del papel de los contratos de opción en las decisiones de aprovisionamiento de agua de una comunidad de regantes ante una oferta de agua incierta. Los resultados muestran el potencial de reducción del riesgo de los contratos de opción para regantes en España, pero pueden ser extrapolados a otros sectores o regiones. Las principales conclusiones de esta tesis son: a) la agricultura será uno de los sectores más afectados por el cambio climático. Si los precios del agua aumentan, la rentabilidad de los cultivos puede caer hasta niveles negativos, lo que podría dar lugar al abandono de cultivos de regadío en algunas zonas de España. Las políticas de cambio climático y de agua deben estar estrechamente coordinadas para asegurar un uso de agua eficiente y la rentabilidad de la agricultura; b) aunque los mercados de agua han ayudado a algunos usuarios a afrontar problemas de disponibilidad del recurso en momentos de escasez, hay varios aspectos que deben mejorarse; c) es necesario desarrollar mercados de agua más flexibles y estables para garantizar una asignación eficiente de los recursos entre los usuarios de agua; d) los resultados muestran los beneficios derivados del establecimiento de un contrato de opción entre usuarios de agua del Tajo y del Segura para reducir el riesgo de disponibilidad de agua en la cuenca receptora; e) la disposición a pagar de los regantes por un contrato de opción de agua o un seguro de sequía hidrológica, que representa el valor que tienen estos mecanismos para aquellos usuarios de agua que se enfrentan a riesgos relacionados con la disponibilidad del recurso, es consistente con los resultados obtenidos en estudios previos y superior al precio de mercado de estos instrumentos, lo que favorece la viabilidad de estos mecanismos de gestión del riesgo ; y f) los contratos de opción podrían ayudar a optimizar las decisiones de aprovisionamiento de agua bajo incertidumbre, proporcionando más estabilidad y flexibilidad que los mercados temporales de agua. ABSTRACT Water is becoming increasingly scarce and valuable. Thus, existing water resources need to be efficiently allocated among users. Climate change is expected to increase the frequency and severity of extreme events, and it may also increase irrigated crops' water demand. The implementation of flexible allocation mechanisms could be essential to cope with this increased variability of the water balance and ensure that supply risks, and not only water resources, are also efficiently shared and managed. Water markets allow for the reallocation of water resources from low to high value uses. Different water trading mechanisms have been created in different parts of the world and have helped users to alleviate water scarcity problems in those areas. In Spain, water trading is allowed since 1999, although market activity has been limited. There are several issues in the Spanish water market that should be improved. This thesis, besides proposing several changes in the legislative framework, proposes the introduction of water option contracts as a potential improvement. The main advantage for both buyer and seller derived from an option contract is the institutional and legal stability it provides. To support this proposal, different analyses have been carried out that show the potential of option contracts as a risk reduction tool to manage water supply instability. The Segura Basin (Southeast Spain), the Tagus Basin and the Tagus-Segura inter-basin Transfer have been selected as the case study. Three different analyses applied to this region are presented in this thesis: a) an evaluation of option contracts as a mechanisms to reduce water supply availability risks in the Segura Basin; b) a theoretical framework for analyzing farmer’s preferences for different water supply risk management tools and farmers’ willingness to pay for them, together with the assessment of the prices of these mechanisms (drought insurance and water option contracts); and c) an evaluation of the role of option contracts in water procurement decisions under uncertainty. Results show the risk-reduction potential of option contracts for the agricultural sector in Spain, but these results can be extrapolated to other sectors or regions. The main conclusions of the thesis are: a) agriculture would be one of the most affected sectors by climate change. With higher water tariffs, crop’s profitability can drop to negative levels, which may result in the abandoning of the crop in many areas. Climate change and water policies must be closely coordinated to ensure efficient water use and crops’ profitability; b) although Spanish water markets have alleviated water availability problems for some users during water scarcity periods, there are several issues that should be improved; c) more flexible and stable water market mechanisms are needed to allocate water resources and water supply risks among competing users; d) results show the benefits derived from the establishment of an inter-basin option contract between water users in the Tagus and the Segura basins for reducing water supply availability risks in the recipient area; e) irrigators’ willingness to pay for option contracts or drought insurance, that represent the value that this kind of trading mechanisms has for water users facing water supply reliability problems, are consistent with results obtained in previous works and higher than the prices of this risk management tools, which shows the feasibility of these mechanisms; and f) option contracts would help to optimize water procurement decisions under uncertainty, providing more flexibility and stability than the spot market.

Active Learning and Dynamic Pricing Policies

In this paper, we address the problem of dynamic pricing to optimize the revenue coming from the sales of a limited inventory in a finite time-horizon. A priori, the demand is assumed to be unknown. The seller must learn on the fly. We first deal with the simplest case, involving only one class of product for sale. Furthermore the general situation is considered with a finite number of product classes for sale. In particular, a case in point is the sale of tickets for events related to culture and leisure; in this case, typically the tickets are sold months before the event, thus, uncertainty over actual demand levels is a very a common occurrence. We propose a heuristic strategy of adaptive dynamic pricing, based on experience gained from the past, taking into account, for each time period, the available inventory, the time remaining to reach the horizon, and the profit made in previous periods. In the computational simulations performed, the demand is updated dynamically based on the prices being offered, as well as on the remaining time and inventory. The simulations show a significant profit over the fixed-price strategy, confirming the practical usefulness of the proposed strategy. We develop a tool allowing us to test different dynamic pricing strategies designed to fit market conditions and seller s objectives, which will facilitate data analysis and decision-making in the face of the problem of dynamic pricing.

How varies optimal welfare pricing with income distribution? The case of the untolled alternative.

In some countries, such as Spain, it is very common that in the same corridor there are two roads with the same origin and destination but with some differences. The most important contrast is that one is a toll highway which offers a better quality than the parallel road in exchange of a price. The users decide if the price of the toll is worth to pay for the advantages offered. This problem is known as the untolled alternative and it has been largely studied in the academic literature, particularly related to economic efficiency and the optimal welfare toll. However, there is a gap in the literature academic regarding how it affects income distribution to the optimal toll. The main objective of the paper is to fill this gap. In this paper a theoretical model in order to obtain the optimal welfare price in a toll highway that competes for capturing the traffic with a conventional road is developed. This model is done for non-usual users who decide over the expectation of free flow conditions. This model is finally applied to the variables we want to focus on: average value of travel time (VTT) which is strongly related with income, dispersion of this VTT and traffic levels, from free flow to congestion. Derived from the results, we conclude that the higher the average VTT the higher the optimal price, the higher the dispersion of this VTT the lower the optimal price and finally, the more the traffic the higher the optimal toll.

Fuel Cells: A Real Option for Unmanned Aerial Vehicles Propulsion

The possibility of implementing fuel cell technology in Unmanned Aerial Vehicle (UAV) propulsion systems is considered. Potential advantages of the Proton Exchange Membrane or Polymer Electrolyte Membrane (PEMFC) and Direct Methanol Fuel Cells (DMFC), their fuels (hydrogen and methanol), and their storage systems are revised from technical and environmental standpoints. Some operating commercial applications are described. Main constraints for these kinds of fuel cells are analyzed in order to elucidate the viability of future developments. Since the low power density is the main problem of fuel cells, hybridization with electric batteries, necessary in most cases, is also explored.