> We can see that a higher mean and a higher standard deviation result in higher option value.
This is a pretty unconvincing argument, because increasing the standard deviation increases the expected value of a lognormal distribution, even without any weird option payoffs being involved.
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.
This is a pretty unconvincing argument, because increasing the standard deviation increases the expected value of a lognormal distribution, even without any weird option payoffs being involved.