Approximating the Growth Optimal Portfolio and its applications in quantitative finance

The Benchmark Approach (BA) represents an alternative framework to quantitative finance that relies on the existence of a Growth Optimal Portfolio (GOP) to be used as Numeraire for financial modeling. When employed as a benchmark, the GOP makes any other non-negative portfolio either trendless or mean-decreasing. This property is known as supermartingale property, and it allows to exclude ex-ante some basic forms of arbitrage in the financial markets. Moreover, the GOP is constructed to maximize the long-term growth rate of the investment, and it delivers the best long-term performance when compared to any other strictly positive portfolio. These results are particularly interesting to apply the BA framework in the context of portfolio optimization and valuation of contingent claims. By introducing the Diversification Theorem, Platen and Heath (2006) show that Diversified Portfolios (DPs) converge to the Numeraire when composed by a sufficiently large number of constituents. In the present research, we build on this result and exploit naive-diversification as a tool to construct valid proxies of the GOP. More specifically, we follow the methodology proposed by Platen and Rendek (2020) to construct a Hierarchically Weighted Index (HWI), a particular class of equally-weighted strategies that proved to be very robust to approximate the GOP. We evaluate the performance of different specifications of the HWI against the traditional Equally Weighted Index (EWI) and the MSCI-World Index. As a final result, we prove that the HWI approximates well the GOP, by showing robust statistical evidence that the supermartingale property of benchmarked returns cannot be easily rejected when our preferred HWI specification is used as a benchmark.