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Special Semester on Stochastics with Emphasis on Finance
Linz, September 2008 - December 2008
Presentation: Thou Shalt Still Buy and Hold

Workshop: Optimization and Optimal Control

Time: Thu, October 23, 2008, 09:30-10:20

Speaker: Xunyu Zhou

Abstract

An investor holding a stock needs to decide when to sell it over a given investment horizon. It is tempting to think that she should sell at the maximum price over the entire horizon, which is however impossible to achieve. A close yet realistic goal is to sell the stock at a time when
the expected relative error between the selling price and the aforementioned maximum price is minimized. This problem is investigated thoroughly for a Black–Scholes market. The optimal selling time turns out to be of a bang-bang nature, i.e., one should either sell at the terminal time or sell immediately, depending on how good the stock is (which is
defined precisely). Moreover, under the optimal selling policy the resulting expected relative error diminishes to zero when the stock is sufficiently good. Furthermore, the optimality of the bang-bang policy is largely preserved if one applies a convex "utility function" to the relative error. The convexity, in turn, has interesting connection with behavioural finance. All these results justify the widely accepted financial wisdom that one should buy and hold a stock – if it is good, that is.

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