Multiple Lévy subordinator processes with applications in finance

There is a vast literature that has sought to model the dynamics of asset returns. The assumption typically made is that asset returns follow a normal distribution despite the preponderance of empirical evidence that rejects this distribution. There are several stylized facts about asset returns that should be recognized in modeling the dynamics of asset returns. Specifically, asset returns exhibit asymmetry and heavy tails. Modeling and analyzing the tail properties of asset returns are crucial for asset managers and risk managers. Consequently, the usefulness of the results of models that assume asset returns follow the normal law are questionable. To deal with non-normality, the method of subordination has been proposed in the literature to include business time and allow the variance of the normal distribution to change over time. The subordination process in finance, also called random time change is a technique employed to introduce additional parameters to the return model to reflect the heavy tail phenomena present in most asset returns and to generalize the classical asset pricing model. Subordinated Levy price processes and local volatility price processes are now the main tools in modern dynamic asset pricing theory. This thesis aims to study the theory of double Levy subordinator processes with application in finance. We focus on the multiple subordinated methods for modeling asset returns and the intuition behind the modeling process and its applications. The motivation for introducing one further layer of subordinator is twofold. The first is to capture the heavy tails of the stock return distribution that cannot be explained well by the existing Levy subordinated model. Some researchers worked with one-layer Levy subordinated model (normal inverse Gaussian) to explain the equity premium puzzle. We will show that the inability of the normal inverse Gaussian distribution to explain the equity premium puzzle is partly caused by the distribution tails that are not heavy enough to fit in rare events.
The second motivation for introducing one further layer of subordinator is to incorporate the views of investors in log-return and option pricing models. It has become clear, according to behavioral finance theory, the views of investors change the underlying asset process models. Investors view positive and negative returns on financial assets differently according to the disposition effect (i.e., the manner in which investors treat capital gains). Thus, to obtain more realistic asset prices, it is essential to incorporate the views of investors in log-return and option pricing models. To be consistent with dynamic asset pricing theory, the views of investors can be taken into account by introducing one further layer of subordinator, intrinsic time process, what we refer to as a behavioral subordinator. The process is subordinated to the Brownian motion process in the well-known log-normal model, resulting in a new log-price process.

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