Application of Markov diffusion processes in economics and finance

                         Davydov, Dmitry; PhD

                         UNIVERSITY OF MICHIGAN, 2000

                         ECONOMICS, FINANCE (0508); POLITICAL SCIENCE, GENERAL (0615)

                         The dissertation is a collection of four papers. The papers utilize the common technique of modeling
                         political and financial variables as Markov diffusion processes. In the first chapter we build a model of a
                         political system, in which the ruling party chooses the time of election. We demonstrate that such a
                         system significantly prolongs the expected duration of the ruling party's stay in power. The time value of
                         the right crucially depends on the volatility of the public opinion. We show how to express the ruling
                         party's expected duration of stay in power as a solution to a free-boundary problem that we solve
                         numerically. In the second chapter we generalize the Black-Scholes-Merton option pricing model to a
                         wide class of Markov diffusion processes, including the constant elasticity of variance (CEV) process.
                         The CEV model exhibits an implied volatility smile that is a convex and monotonically decreasing function
                         of strike. We derive closed-form solutions for the prices of barrier and lookback options and demonstrate
                         that, in the presence of a CEV-based volatility smile, barrier and lookback prices and hedge ratios can
                         deviate dramatically from the values under a lognormal specification. The third chapter we analyze double
                         barrier step options with the payoff dependent on the occupation time outside the prespecified price
                         range during the life of the option. Occupation time-based contracts are easier to hedge than standard
                         barrier options and, therefore, smaller bid-ask spreads over the theoretical price are required. We obtain
                         the price and hedge ratios of occupation-time derivatives under assumption of lognormal process. The
                         solutions are represented in the form of single or double inverse Laplace transforms. In the fourth
                         chapter, we build a bond-pricing model with a positive probability of default. A firm may suffer a
                         random-size loss that occurs at the first jump of a Poisson process with random intensity. The default
                         happens only if the firm's equity, which is assumed to follow the lognormal process, is not large enough
                         to cover the loss. The analytical tractability is achieved through the approximation of the hazard rate. We
                         derive the analytical formulae for the price of the risky bonds and the spread.


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