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Small‐cost asymptotics for long‐term growth rates in incomplete markets
Author(s) -
Melnyk Yaroslav,
Seifried Frank Thomas
Publication year - 2018
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12152
Subject(s) - dynamic programming , term (time) , transaction cost , economics , stochastic volatility , path dependent , econometrics , markov process , volatility (finance) , markov chain , order (exchange) , path (computing) , mathematics , mathematical optimization , mathematical economics , computer science , statistics , microeconomics , finance , physics , quantum mechanics , programming language
This paper provides a rigorous asymptotic analysis of long‐term growth rates under both proportional and Morton–Pliska transaction costs. We consider a general incomplete financial market with an unspanned Markov factor process that includes the Heston stochastic volatility model and the Kim–Omberg stochastic excess return model as special cases. Using a dynamic programming approach, we determine the leading‐order expansions of long‐term growth rates and explicitly construct strategies that are optimal at the leading order. We further analyze the asymptotic performance of Morton–Pliska strategies in settings with proportional transaction costs. We find that the performance of the optimal Morton–Pliska strategy is the same as that of the optimal one with costs increased by a factor of 2 . Finally, we demonstrate that our strategies are in fact pathwise optimal, in the sense that they maximize the long‐run growth rate path by path.