Essays on Asset Pricing Models with Jump Processes

This dissertation contains four autonomous academic papers on asset pricing models with jump processes, including the studies of equilibrium asset pricing model, option pricing model, and empirical test. The common thread between them is the application of jump processes that links them in asset price modeling. The first three papers study Lévy process and its inhomogeneous extensions, while the last one studies contagious Hawkes processes. The first essay proposes a novel equilibrium asset pricing model under the semiparametric jump diffusion framework, including drift, volatility and jump intensity in a general time-varying form. The corresponding pricing kernel provides insights on option pricing, and equity premium puzzle [Mehra & Prescott (1985)]. The analytical solutions of equity premium and European call option are given as well. The second essay introduces a new econometric method/procedure to disentangle the three time-varying components of drift, volatility and jump in asset prices. By combining Hodrick-Prescott filter and particle filters, I decomposed the three timevarying components in the S&P500 index, and observed the clustering of volatility and jumps, though the clustering effects are more pronounced when the time-varying drift is negative. Empirical results support the proposed time-varying jump diffusion asset pricing model in Chapter 2. The third essay studies the (un)importance of small jumps in option pricing models. The option pricing literature argues that the behavior of small jumps in a Geometric Lévy model is of paramount importance [Carr et al. (2002)]. This is evidently true for very short time horizons and very deep in- and out-of-the-money options. In this paper, we took the complementary view and asked what values of time to maturity and option moneyness in a Geometric Lévy model lead to option prices, which are practically indistinguishable from the price of plain vanilla options in the BlackScholes model. In other words, in what situation that the Lévy model in question can be replaced with a Brownian motion with minimal pricing error. We produced explicit tight bounds in the case of a Poisson jump process, and related heuristic bounds for arbitrary Lévy process with exponentially decaying jump intensity. The fourth essay models and tests contagious jumps in bull/bear market regimes, in which I developed the regime switching Hawkes processes to model the contagious asset jumps in the international stock market. This new model allows serial and regional contagion in international asset prices, in which contagious impact can be flexible to accommodate different market conditions