Small‐cost asymptotics for long‐term growth rates in incomplete markets

Abstract

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.

A Primer on Portfolio Choice with Small Transaction Costs

“…To focus on the main ideas and computational issues, mathematical formalism is treated liberally throughout this survey. Rigorous verification theorems for the results presented here can be found in [82,78,71,2,74,1,38,17,18,68].…”

“…This requires that the effect of transaction costs is small even when compounded over a long horizon. To make this argument precise, one can directly consider the infinite-horizon problem as in[42,54,68] 19. We denote by ∆π(f ) the halfwidth of the stationary no-trade region obtained from the long-run portfolioπ(f ) via(4.15).…”

Portfolio choice with small temporary and transient price impact

“…2 Related work on other small transaction costs includes Shreve and Soner (1994); Whalley and Wilmott (1997); Korn (1998); Janeček and Shreve (2004); Soner and Touzi (2013); Bichuch (2014); Martin (2014); Altarovici, Muhle-Karbe, and Soner (2015); Kallsen and Li (2015); Possamaï, Soner, and Touzi (2015); Feodoria (2016); Cai, Rosenbaum, and Tankov (2017a,b); Kallsen and Muhle-Karbe (2017); Herdegen and Muhle-Karbe (2018); Melnyk and Seifried (2018). 3 The same statistic also plays a key role in optimal execution problems (Almgren & Chriss, 2001;Schied & Schöneborn, 2009) and models with asymmetric information (Muhle-Karbe & Webster, 2018).…”

“…Related work on other small transaction costs includes Shreve and Soner (); Whalley and Wilmott (); Korn (); Janeček and Shreve (); Soner and Touzi (); Bichuch (); Martin (); Altarovici, Muhle‐Karbe, and Soner (); Kallsen and Li (); Possamaï, Soner, and Touzi (); Feodoria (); Cai, Rosenbaum, and Tankov (,b); Kallsen and Muhle‐Karbe (); Herdegen and Muhle‐Karbe (); Melnyk and Seifried ().…”

Optimal portfolio selection under vanishing fixed transaction costs

“…Here, the authors derive solutions for the discounted consumption criterion for the linear utility, asymptotically for the exponential utility, and existence of optimal strategies, resp. Asymptotic results for vanishing fixed costs were recently obtained in [3], [12], [25], where in particular the last paper considers a generalization of our setting. These results are, however, different in nature to the results obtained in this article as the authors do not show convergence of the optimal strategies, but construct asymptotically optimal strategies, which are obviously suboptimal for all fixed positive costs.…”