Three essays on continuous time finance
In first essay, I present a general equilibrium model of financial asset pricing where market can be incomplete. The focus is on the effects of economics fundamentals on capital asset pricing model. Using martingale approach, I am able to solve models of asset pricing, optimal consumption, production, leisure and portfolio choice in which heterogeneous agents maximize their expected utility functions which are defined over not only consumption but also leisure. Solving these models, I derive a CAPM expressing the returns in terms of covariance of return with economics fundamentals. Such CAPM captures the way investors perceive the relationship between risk and return. We also show that why consumption along is not enough to determine the asset return. My model is very attractive in two aspects: first the martingale approach allows the interaction between assets and real economy; secondly martingale approach allows me to solve the models where the representative agent method seems helpless.
In second essay, I study contingent claim valuation in an equilibrium framework and its Monte Carlo implementation. I examine a world in which asset prices follow a diffusion process, and market with stocks and bond is dynamically incomplete, therefore at least some contingent claims are non-redundant assets. I study the issue when the asset market can be completed by these contingent claims and derive equations that determine equilibrium price kernel where the security prices and state variable processes follow Markov process, and thus I show how to implement the computation of options by Monte Carlo Simulation. My approach allows for flexible specification for underlying asset prices and is applicable to pricing a variety of options (such as volatility swap) which is discretized path-dependent when underlying assets have stochastic volatility and jumps.
In third essay, I analyze the problem facing a specialist in NYSE who has an obligation to provide the maximum liquidity and who wishes to hedge that obligation when setting bid-ask spread is constrained. We develop a dynamic framework to model the liability of the specialist and its hedging activities using stocks and derivatives. We develop a solvable model in which we can characterize the optimal hedging strategy in terms of the specialist's risk aversion and volatility of stock. Our model provides an explicit formula for hedging strategy when the return on assets which are used to hedge the risk follow geometric Brownian motions; also we derive an explicit formula for hedging strategy using redundant derivatives. Most importantly, we show that hedging is not always possible and characterize the hedging policies in terms of correlation, interest rate and risk aversion.