Semi-Markov Switching Lévy Processes and their Applications in Finance
Date
2020-07-23
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Abstract
We, as humans, learn from our mistakes. Ultimately, we progress and grow, both as an individual and as a society. After the catastrophic recession period due to the sub-prime mortgage crisis in 2007, researchers and mathematicians began to look for answers about the massive damages done by the crisis. These events suggested that there must be some other factors that had hidden and needed to be accounted for in the previous prevailing models of our financial markets. As many different explanations arose, the main focus was to find a way to account for these more significant crises with some economically interpretable assumptions. The existence of different states or regimes in our financial market is one of the most acceptable ideas so far, as we tend to notice that the market has been switching between cycles of booms and recessions. This idea has triggered many studies on regime-switching models, but mostly with Markov regimes, as they are mathematically simple and relatively easy to solve. However, some of the assumptions made behind the Markov switching models have been questionable and unrealistic. For instance, Markov regimes imply that each one of them can switch to any other state no matter how long we have stayed in the current state. This is deviating from what we would expect, as we usually observe that the likelihood of recovery has some correlation with the recession’s age; the longer we stay in a recession, the harder for the market to recover. Thus, the primary purpose of this thesis is to find a better model with a better explanation. The introduction of semi-Markov markets intuitively has a time-varying propensity of regime changes using the conditional intensity matrix. In general, the semi-Markov switching models should be more in line with our financial market and generate better results when simulating financial derivatives prices. In this thesis, we will start by introducing some related definitions and theorems first. We will develop a semi-martingale representation for both the discrete-time semi-Markov chains and continuous-time semi-Markov processes, with some examples and applications. Then, we will construct the theoretical framework of a stochastic model under a semi-Markov regime-switching process driven by Lévy processes. The first step is to derive its Itô’s formula, as we need it to find the semi-closed form formulas for the characteristic function of log prices. Then, we will be developing the risk-neutral measure specific to our semi-Markov switching models. As some of us may already know, the Lévy driven regime-switching markets are incomplete, which means that there is more than one risk-neutral measure when pricing financial derivatives. When pricing a European-style option, since we already have the semi-closed form of the characteristic function for log asset prices, it allows us to use a Fourier transform method, first derived by Carr and Madan, namely the Fractional Fast Fourier Transform (FRFT) algorithm to obtain the estimated option prices. When comparing with Markov switching models, estimations and simulations show that the semi-Markov model performs better. It also offers more insight into the dynamics of market regimes, providing us with a better explanation of where the financial market is headed to next.
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Zhang, Y. I. (2020). Semi-Markov Switching Lévy Processes and their Applications in Finance (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.