Portfolio selection formulated in terms of Semimartingales and Quadratic Variation

Every since Harry Markowitz published his remarkable piece on portfolio diversification in the 50s which then evolved into Modern Portfolio Theory (MPT), the trade-off between return which is commonly measured by expected return, and risk which is commonly measured by expected standard deviation, has been at the heart of investors’ decision making process. Over time the simplicity of this approach has proven to be powerful enough to outweigh its long list of theoretical shortcomings listed in the paper and its popularity with both academics and practitioners has remained intack. The aim of this paper is to present an alternative way of measuring risk when the underling investment instrument is modeled as a semimartingale process. This alternative measurement called Quadratic Variation we argue offer better insight into the riskiness of a stochastic process when the assumptions like normal distribution is no longer satisfied. Also we think given the advance computing power that is accessible to professional investors nowadays, estimating quadratic variation can be simple enough that it offers an appealing alternative to standard deviation in practical field as well. In order to better illustrate the difference between Quadratic variation and Standard deviation in Portfolio optimization study, we formed an investment portfolio with two instruments: Nokia equity which represents equity, and German Bunds Futures which represent fixed income market. We then performed two mean-variance optimization exercises, one using standard deviation as the risk measurement and the other one using the estimate of quadratic variation as the risk measurement. On the result of risk estimation per se, we also used different ways of estimating quadratic variation taken into account the issue of market microstructure. The result showed that using quadratic variation the optimal portfolio contains roughly 10% of holding in Nokia stock while in the traditional mean-variance framework the corresponding figure would have be around 13%