Stable processes: theory and applications in finance - Kateregga, Michael (Division of Actuarial Science, 2018)
This thesis is a study on stable distributions and some of their applications in understanding financial markets. Three broad problems are explored: First, we study a parameter and density estimation problem for stable distributions using commodity market data. We investigate and compare the accuracy of the quantile, logarithmic, maximum likelihood (ML) and empirical characteristic function (ECF) methods. It turns out that the ECF is the most recommendable method, challenging literature that instead suggests the ML. Secondly, we develop an affine theory for subordinated random processes and apply the results to pricing commodity futures in markets where the spot price includes jumps. The jumps are introduced by subordinating Brownian motion in the spot model by an α-stable process, α ε (0; 1] which leads to a new pricing approach for models with latent variables. The third problem is the pricing of general derivatives and risk management based on Malliavin calculus. We derive a Bismut-Elworthy-Li (BEL) representation formula for computing financial Greeks under the framework of subordinated Brownian motion by an inverse α-stable process with α ε (0; 1]. This subordination by an inverse α-stable process allows zero returns in the model rendering it fit for illiquid emerging markets. In addition, we demonstrate that the model is best suited for pricing derivatives with irregular payoff functions compared to the traditional Euler methods.