Increased volatility of the underlying always increases option prices. Moreover, a higher standard deviation decreases the compounded returns of a log normal d

Increased volatility of the underlying always increases option prices. Moreover, a higher standard deviation decreases the compounded returns of a log normal distribution, not decreases it. This is called volatility drag, and is closesly related to the AM-GM inequality: the geometric and arithmetic means of a series are only equal if all terms are identical. If this is not true, an arithmetic average will always be greater.