Variance Bounds for Functions of Unimodal Random Variable

Abstract

For unimodal random variable S with fixed moments $\mathbb{E}{S}^i=m_i(i=1,2,\dots ,n)$ and mode m, a formula on variance of any functions of S is provided along the line of Khintchine transform. Upper bounds are derived on the variance of European call options and Gap options. The techniques are based on symmetrization, domination by quadratic functions of two variables and change of measures.

Keywords:

• Khintchine transform
• Moment problems
• Options
• Unimodal distribution
• Variance

Mathematics Subject Classification:

• 60C05
• 65K10
• 49M29

Additional information

Funding

This work is supported by the NSF grant DMS 1106938.