Financial Applications of the Conditional Expectation

Abstract

This dissertation examines different financial applications of some conditional expectation estimators. In the first application, we provide some theoretical motivations behind the use of the moving average rule as one of the most popular trading tools among practitioners. In particular, we examine the conditional probability of the price increments and we study how this probability changes over time. In the second application, we present different approaches to evaluate the presence of the arbitrage opportunities in the option market. In particular, we investigate empirically the well-known put-call parity no-arbitrage relation and the state price density. We first measure the violation of the put-call parity as the difference in implied volatilities between call and put options. Furthermore, we propose alternative approaches to estimate the state price density under the classical hypothesis of the Black and Scholes model. In the third application, we investigate the implications for portfolio theory of using conditional expectation estimators. First, we focus on the approximation of the conditional expectation within large-scale portfolio selection problems. In this context, we propose a new consistent multivariate kernel estimator to approximate the conditional expectation. We show how the new estimator can be used for the return approximation of large-scale portfolio problems. Moreover, the proposed estimator optimizes the bandwidth selection of kernel type estimators, solving the classical selection problem. Second, we propose new performance measures based on the conditional expectation that takes into account the heavy tails of the return distributions. Third, we deal with the portfolio selection problem from the point of view of different non-satiable investors, namely risk-averse and risk-seeking investors. In particular, using a well-known ordering classification, we first identify different definitions of returns based on the investors’ preferences. The new definitions of returns are based on the conditional expected value between the random wealth assessed at different times. Finally, for each problem, we propose an empirical application of several admissible portfolio optimization problems using the US stock market.