Derivative Pricing and Portfolio Choice under Flexible Non-Linear Return Dynamics

Many empirical results have shown the nonlinearities in different asset return dynamics. My dissertation contributes to the literature by exploring the effects of nonlinear dynamics of returns on portfolio choice and option pricing models. The first essay in Chapter 1 studies the dynamic consumption and asset allocation problem in the presence of regimes in asset returns. Previous solutions either require numerical methods or define the investor's preference over moments of the terminal wealth distribution. I introduce a new way to model the investor's belief process, which is an approximation of the one obtainable under Bayes' updating rule, and provide a closed-form solution. Optimal asset allocations are found to vary considerably across states. For a continuous rebalancing investor, the optimal allocation to the risky asset is always an increasing function of the investment horizon, and the information about the current state is used more aggressively. Under buy-and-hold settings, the investor holds less of the risky asset in the bull state and more of the risky asset in the bear state as the investment horizon expands. The underlying regime has small, yet still significant, impacts on consumption-wealth ratios. Substantial welfare costs confirm the economic importance of accounting for the presence of regimes in asset returns. In Chapter 2, I use the SNP approach to estimate the dynamics of the conditional state price density (SPD) implied by option prices. SNP is a method ii to estimate the conditional density of a nonlinear stationary process based on a Hermite expansion. The SNP method embeds a large class of models, including VAR, ARCH, and nonlinear nonparametric processes. Without pre-imposing restrictions on conditional moments, the SNP approach provides a consistent estimation of the law of motion when the option prices follow a nonlinear process with conditional heterogeneities. When assuming the log of the underlying asset price has an SNP conditional SPD, the martingale restriction on the expected return is derived and a closed-form solution is obtained for European options. Finally, in an empirical application, I extract the SPD implied by the S&P 500 option prices and compare it with corresponding Black-Scholes SPD.

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