# RESEARCH

Al Ramz Quantitative Research focuses on mathematical methods to deal with financial markets for equities, fixed income and their derivatives. Current activity focuses on traditional sciences, like probability theory, the use of integral operators and Boltzmann equations, applications of complex-variable functions to time-series, and the theory of generalized functions. It also engages, by time to time, in blended approaches with machine learning and AI.

We present a systematic approach to maximize the Sharpe ratio of a portfolio composed of several assets and/or strategies. This is also a common problem in multi-strategy hedge funds. We begin with the case of two assets, or strategies, and derive an exact analytical solution. Then, we draw a path to the more complex case of several assets or strategies, and we outline the approach based on numerical methods, along with the connection to quadratic forms. Some mathematical preliminaries are also presented in order ensure the reader is familiar with some theory of generalized functions, their integrals, and their applications to probability.

Over the last decades, much attention has been devoted to the Black-Scholes formula for option pricing. Variations of it have been proposed, and two approaches have been attempted. One approach uses probability density on the positive real axis of the price of the underlying asset, while the other uses probability densities on the full real axis of the logarithms of the price of the underlying asset. While the first approach is more intuitive, as it leads to integrals over ranges of prices, the second is mathematically more suitable to be used with Gaussian distributions of probability, as it extends over the entire real axis, and we choose to adopt this latter approach here. It is, however, seen in most publications about the subject that the transition from prices to their logarithms is done without much care, when in fact it leads to some inflationary phenomena that need to be renormalized. This is the scope of this article, whose theoretical derivation is done set-by-step. At the same time, it then becomes evident how renormalization leads to both a different pricing formula, and different Greek parameters, thus leading to potential strategies to outprice counterparts on the market. This is covered in the second part of the article. We finally include the Mathematica notebooks used in this derivation, along with the resulting pricing functions and greeks, for practical use in applications.