# Pricing look-back option

I have the monthly price data of a stock starting from December 2020 and I am considering a EU style look-back option issued in December 2020. The payoff at maturity of the look-back option is given by max(Smin-K, 0) where Smin is the minimum monthly stock price during the life of the option (the fist and last stock price are included in the minimum) and K denotes the strike price. The average stock return is 10% and the stock volatility is 30%. This look-back call option matures next month and the strike price is 30. The continuously compounded risk free rate is 2.5% per annum. I should compute the current value of this option with an one-period binomial option pricing model. What I am not getting is what is exactly meant by Smin. Is it the minimum monthly stock price during the life of the option (thus the minimum of two historical prices that I have (December 2020 and January 2021)? Is this unrealistic given that in December I cannot know the January price and thus I should estimate it with a binomial tree? (To solve this problem I am using security state prices computing up as ((R-D)/R(U-D) and down as ((U-R)/R(U-D) where R=1+risk free rate compounded for the period (δt=1/12). Thank you.