Mathematical methods for valuation and risk assessment of investment projects and real options

In this thesis, we study the problems of risk measurement, valuation and hedging of financial positions in incomplete markets when an insufficient number of assets are available for investment (real options). We work closely with three measures of risk: Worst-Case Scenario (WCS) (the supremum of expected values over a set of given probability measures), Value-at-Risk (VaR) and Average Value-at-Risk (AVaR), and analyse the problem of hedging derivative securities depending on a non-traded asset, defined in terms of the risk measures via their acceptance sets. The hedging problem associated to VaR is the problem of minimising the expected shortfall. For WCS, the hedging problem turns out to be a robust version of minimising the expected shortfall; and as AVaR can be seen as a particular case of WCS, its hedging problem is also related to the minimisation of expected shortfall. Under some sufficient conditions, we solve explicitly the minimal expected shortfall problem in a discrete-time setting of two assets driven by correlated binomial models. In the continuous-time case, we analyse the problem of measuring risk by WCS, VaR and AVaR on positions modelled as Markov diffusion processes and develop some results on transformations of Markov processes to apply to the risk measurement of derivative securities. In all cases, we characterise the risk of a position as the solution of a partial differential equation of second order with boundary conditions. In relation to the valuation and hedging of derivative securities, and in the search for explicit solutions, we analyse a variant of the robust version of the expected shortfall hedging problem. Instead of taking the loss function $l(x) = [x]^+$ we work with the strictly increasing, strictly convex function $L_{\epsilon}(x) = \epsilon \log \left( \frac{1+exp\{−x/\epsilon\} }{ exp\{−x/\epsilon\} } \right)$. Clearly $lim_{\epsilon \rightarrow 0} L_{\epsilon}(x) = l(x)$. The reformulation to the problem for L_{\epsilon}(x) also allow us to use directly the dual theory under robust preferences recently developed in [82]. Due to the fact that the function $L_{\epsilon}(x)$ is not separable in its variables, we are not able to solve explicitly, but instead, we use a power series approximation in the dual variables. It turns out that the approximated solution corresponds to the robust version of a utility maximisation problem with exponential preferences $(U(x) = −\frac{1}{\gamma}e^{-\gamma x})$ for a preferenes parameter $\gamma = 1/\epsilon$. For the approximated problem, we analyse the cases with and without random endowment, and obtain an expression for the utility indifference bid price of a derivative security which depends only on the non-traded asset

Planning drinking water for airplanes

The management of the Dutch national airline company KLM intends to bring a sufficient amount of water on board of all flights to fulfill customer’s demand. On the other hand, the surplus of water after a flight should be kept to a minimum to reduce fuel costs. The service to passengers is measured with a service level. The objective of this research is to develop models, which can be used to minimize the amount of water on board of flights such that a predefined service level is met. The difficulty that has to be overcome is the fact that most of the available data of water consumption on flights are rounded off to the nearest eighth of the water tank. For wide-body aircrafts this rounding may correspond to about two hundred litres of water. Part of the problem was also to define a good service level. The use of a service level as a model parameter would give KLM a better control of the water surplus. The available data have been analyzed to examine which aspects we had to take into consideration. Next, a general framework has been developed in which the service level has been defined as a Quality of Service for each flight: The probability that a sufficient amount of water is available on a given flight leg. Three approaches will be proposed to find a probability distribution function for the total water consumption on a flight. The first approach tries to fit a distribution for the water consumption based on the available data, without any assumptions on the underlying shape of the distribution. The second approach assumes normality for the total water consumption on a flight and the third approach uses a binomial distribution. All methods are validated and numerically illustrated. We recommend KLM to use the second approach, where the first approach can be used to determine an upper bound on the water level

The management of acute venous thromboembolism in clinical practice - study rationale and protocol of the European PREFER in VTE Registry

Background: Venous thromboembolism (VTE) is a major health problem, with over one million events every year in Europe. However, there is a paucity of data on the current management in real life, including factors influencing treatment pathways, patient satisfaction, quality of life (QoL), and utilization of health care resources and the corresponding costs. The PREFER in VTE registry has been designed to address this and to understand medical care and needs as well as potential gaps for improvement. Methods/design: The PREFER in VTE registry was a prospective, observational, multicenter study conducted in seven European countries including Austria, France Germany, Italy, Spain, Switzerland, and the UK to assess the characteristics and the management of patients with VTE, the use of health care resources, and to provide data to estimate the costs for 12 months treatment following a first-time and/or recurrent VTE diagnosed in hospitals or specialized or primary care centers. In addition, existing anticoagulant treatment patterns, patient pathways, clinical outcomes, treatment satisfaction, and health related QoL were documented. The centers were chosen to reflect the care environment in which patients with VTE are managed in each of the participating countries. Patients were eligible to be enrolled into the registry if they were at least 18 years old, had a symptomatic, objectively confirmed first time or recurrent acute VTE defined as either distal or proximal deep vein thrombosis, pulmonary embolism or both. After the baseline visit at the time of the acute VTE event, further follow-up documentations occurred at 1, 3, 6 and 12 months. Follow-up data was collected by either routinely scheduled visits or by telephone calls. Results: Overall, 381 centers participated, which enrolled 3,545 patients during an observational period of 1 year. Conclusion: The PREFER in VTE registry will provide valuable insights into the characteristics of patients with VTE and their acute and mid-term management, as well as into drug utilization and the use of health care resources in acute first-time and/or recurrent VTE across Europe in clinical practice. Trial registration: Registered in DRKS register, ID number: DRKS0000479