Inspired by volatility stabilized market models introduced by Robert Fernholz and Ioannis Karatzas [2], we introduce a class of processes which are polynomial in the sense of [1] to model both, asset prices (or market capitalizations of companies in an equity market) and their corresponding market weights. More precisely, we characterize the class of polynomial diff usion models for the asset price process whose market weights process is again a polynomial diff usion process on the unit simplex. Explicit parameter conditions assuring the existence of relative arbitrages with respect to the market portfolio are given and the connection to non-attainment of the boundary is discussed. We also consider extensions to models with jumps and the computation of optimal relative arbitrage strategies.
References
[1] C. Cuchiero, M. Keller-Ressel, and J. Teichmann. Polynomial processes and their applications to mathematical finance. Finance and Stochastics, 16(4):711-740, 2012.
[2] R. Fernholz and I. Karatzas. Relative arbitrage in volatility-stabilized markets. Annals of Finance, 1(2):149-177, 2005.
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