Essays in capital markets

Abstract

This thesis consists of three essays in capital markets. In the first essay, given a European derivative security with an arbitrary payoff function and a corresponding set of underlying securities on which the derivative security is based, we solve the optimal-replication problem: find a self-financing dynamic portfolio strategy-involving only the underlying securities-that most closely approximates the payoff function at maturity. By applying stochastic dynamic programming to the minimization of a mean-squared- error loss function under Markov state-dynamics, we derive recursive expressions for the optimal-replication strategy that are readily implemented in practice. The approximation error or ... of the optimal-replication strategy is also given recursively and may be used to quantify the "degree" of market incompleteness. To investigate the practical significance of these c-arbitrage strategies, we consider several numerical examples including path-dependent options and options on assets with stochastic volatility and jumps. In the second essay we study the tracking error, resulting from the discrete-time application of continuous-time delta-hedging procedures for European options. We characterize the asymptotic distribution of the tracking error as the number of discrete time periods increases, and its joint distribution with other assets. We introduce the notion of temporal granularity of the continuous time stochastic model that enables us to characterize the degree to which discrete time approximations of continuous time models track the payoff of the option. We derive closed form expressions for the granularity for a put or call option on a stock that follows a geometric Brownian motion and a mean-reverting process. These expressions offer insight into the tracking error involved in applying continuous-time delta hedging in discrete time. We also introduce alternative measures of the tracking error and analyze their properties. The third essay presents a general equilibrium model of financial asset prices with irreversible real investment. The focus is on the effects of the irreversibility of real investment on financial asset prices. The model shows how this irreversibility leads to time variation in volatility and systematic risk of stock returns. Changes in these variables are driven by real economic activity, in particular, by firms' investment decisions. Thus, systematic risk of stock returns and their volatility are affected by economy-wide and industry-specific shocks. Firm-specific variables, particularly market-to-book ratios, are linked to real activity and contain information about the dynamic behavior of stock returns. The model of this paper also provides a framework for analyzing futures prices. A comparison between the economy with irreversible investment and an identical economy without the irreversibility shows that all of these results should be attributed to the irreversibility of real investment.