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.
This problem is very strange, because the Smin can only be taken across a single path. For example, if you had say 100 in Dec 2020 and 105 in Jan 2021, the Smin will be 100. However, if you had 95 in Jan 2021, the Smin will be 95. Therefore, the Smin depends on the path taken by the stock simulated in the binomial tree.
What I understand by the problem is that the you can price the lookback by taking the expectation (using the risk-neutral probabilities) of the terminal prices of the option (determined by its Smin and strike at maturity) over the two paths, which is then discounted as the riskless rate to arrive at the price.
Please let me know if this unclear.