Optimal derivatives: portfolios, payoffs and preferences

Abstract

This article presents an extension to the growth optimal derivative that can accommodate risk preferences differing from those of logarithmic utility. Analysis of the optimal derivative provides interesting insights into the behaviour of power investors. We show that power investors under the real-world probability can be viewed as logarithmic investors under the myopic probability of Guasoni and Robertson [(2012). “Portfolios and Risk Premia for the Long Run.” Annals of Applied Probability, 22 (1), 239–284]. Furthermore, this intuition provides criteria for establishing whether fractional Kelly betting is optimal for power investors. Finally, the Black–Scholes model is used to demonstrate how the optimal derivative can be implemented and we show that our approach is consistent with classical techniques.

Keywords:

• portfolio selection
• growth optimal portfolios
• derivative securities
• power utility
• myopic probability

Disclosure statement

No potential conflict of interest was reported by the authors.