Twelve Dollars for the Finish, Seven for the Average, and the Root Three Between Them
An option settling on the mean of a share's closes is strictly cheaper than one settling on the closing price, and the reason is convex order rather than any pricing model: for a martingale share every intermediate price is a forecast of the last one, so the average is dominated at every strike, for calls and for puts. Quantitatively the time average of a Brownian path carries variance T/3 against T, a swing ratio of 1/sqrt(3) = 0.57735, which turns 11.9235 into 6.9013 at a 30 percent volatility. A finite grid of 252 fixings sits at 0.33532 rather than 1/3, which accounts for most of the gap to the 6.918 measured by simulation on the true arithmetic average.