Most decision making processes involve risk associated with the uncertainty of a range of outcomes that may, or may not, occur at some time in the future. In recent years, many of the recent advances in risk theory have come from the field of financial mathematics. While the latter subject is now well developed we claim that there are still some important aspects of decision-making involving risk that are not covered by the existing theory. In particular, we believe that the following three questions require further investigations. 1. In the financial portfolio management context, if decisions need to be taken at multiple stages, is it possible to devise a time consistent risk minimization policy? 2. If the value of an asset of interest is strongly influenced by a climatic variable (such as temperature or rainfall) that is best modelled by techniques that are not normally used in financial modelling, how should financial derivatives on such an asset be priced? 3. If the undesirable risky phenomenon depends on both the upper and lower tails of a, possibly asymmetric, probability distribution (e.g., that of amount of rainfall), what is a suitable measure of risk in this context? This thesis supplies, at least partial, answers to each of these challenging questions. We hope that these results will motivate others to develop even better solutions. / First, for the purpose of defining a dynamic measure of risk, we introduce the time consistency concept that is inspired by the so-called principle of optimality of dynamic programming and demonstrate - via an example - that the conditional value-at-risk (CVaR) need not be time consistent in a multi-stage case. Then, we give the formulation of the target-percentile risk measure which is time consistent and hence more suitable in the multi-stage investment context. / Second, in order to hedge the risk related to the uncertainty of weather we study pricing of derivatives where the underlying asset is sensitive to the weather and the price is a function of a weather variable, for instance, temperature or rain-fall. We use time series to model the temperature and assume that the price of the underlying asset is a deterministic function of the temperature. We discuss both the continuous and discrete time models using the replicating portfolio approach. We obtain a partial differential equation that is similar to the Black-Scholes PDE in the continuous time case. We also provide a binomial approximation for the continuous time model and the corresponding PDE, and report on some numerical experiments. / Next, we analyze the risk encountered in many environmental problems that appear to exhibit special "two-sided" characteristics. For instance, in a given area and in a given period, farmers do not want to see too much or too little rainfall. We formulate and solve this problem with the help of a "two-sided loss function" that depends on the above ranges. Even in financial portfolio optimization a loss and a gain are "two sides of a coin", so it is desirable to deal with them in a manner that reflects an investor's relative concern. Consequently, we define Type I risk: "the loss is too big" and Type II risk: "the gain is too small". Ideally, we would want to minimize the two risks simultaneously. However, this may be impossible and hence we try to balance these two types of risk. / Thesis (PhDMathematics)--University of South Australia, 2006.
| Identifer | oai:union.ndltd.org:ADTP/267235 |
| Creators | Kang, Boda. |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | copyright under review |
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