This dissertation consist of three contributions to financial and insurance mathematics. The first part considers numerical methods for dynamic portfolio optimisation in the expected utility model. The aim is to compare the risk-neutral computational approach (RNCA) also known as the martingale approach to stochastic dynamic programming (SDP) in a discrete-time setting. The main idea of the RNCA is to use the completeness and the arbitrage free properties of the market to compute the optimal consumption rules and then determine the trading strategy that finance this optimal consumption. In contrast, SDP solves for the optimal consumption and investment rules simultaneously using backward recursion and the principle of optimality. The setting that we consider is a discrete time and state space lattice. We provide some new theoretical results relating to the Hyperbolic Absolute Risk Aversion class of utility functions as well as propose a straightforward implementation of RNCA in binomial and trinomial lattices. Moreover, instead of discretizing the Hamilton-Jacobi-Bellman equation with possibly more than one state variable, we use symbolic algorithms to implement stochastic dynamic programming. This new approach provides a simpler numerical procedure for computing optimal consumption-investment policies. A comparison of the RNCA with SDP demonstrates the superiority of the RNCA in terms of computation. The second part considers the pricing of contingent claims using an approach developed and applied in applied in insurance. This approach utilize probability distortion functions as the dual of the utility functions used in financial theory. The main idea of the dual theory is to distort the subjective probabilities rather than outcomes to express the investor????????s risk aversion. In the first part, the RNCA for asset allocation uses the same principle as risk-neutral valuation for derivative pricing. The idea of the second part of this research is to show that the risk-neutral valuation can be recovered from the probability distortion function approach, thereby establishing consistency between the insurance and the financial approaches. We prove that pricing contingent claims under the real world probability measure using an appropriate distortion operator produces arbitrage-free prices when the underlying asset prices are log-normal. We investigate cases when the insurance-based approach fails to produce arbitrage-free prices and determine the appropriate distortion operator under more general assumptions than those used in Black-Scholes option pricing. In the third part we introduce dynamic portfolio optimisation with risk measures based on probability distortion function and provide a formal treatment of this class of risk measures. We employ the RNCA to study the consumption-investment problem in discrete time with preferences consistent with Yaari????????s dual (non-expected utility) theory of choice. As an application, we first consider risk measures based on the Proportional Hazard Transform that treats the upside and downside of the risk differently and secondly a risk measure based on the standard Normal cumulative distribution function. When the objective is to maximise a dual utility of wealth, and the underlying security returns are normal, the efficient frontier is found to be the same as in the mean-variance portfolio problem for an equivalent risk tolerance. When the objective is to maximise a dual utility of consumption, then ????????plunging???????????? behaviour occurs ( investing everything is the risky asset). Other properties of the optimal consumption-investment policies in the dual theory are also investigated and discussed.
| Identifer | oai:union.ndltd.org:ADTP/187786 |
| Date | January 2001 |
| Creators | Hamada, Mahmoud, Actuarial Studies, Australian School of Business, UNSW |
| Publisher | Awarded by:University of New South Wales. School of Actuarial Studies |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | Copyright Mahmoud Hamada, http://unsworks.unsw.edu.au/copyright |
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