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I am interested in the following problem.

We have a Multi-Step Binomial Model with discrete time $T=1,\dots,n$. We also assume that the stock $S_t$ is a martingale and there is a risk-free bond with $r=1$.

How to evaluate the price of the European option $$X_n=\max_{1\le i \le n}S_i-\min_{1\le j \le n}S_j$$ $$=\max_{1\le i < j \le n}|S_i-S_j|?$$

Here $X_n$ represents the biggest change of the stock price during the given time period. Is this option well known or maybe can it be rewritten as a combination of well known payoffs? How should one proceed in order to find it's price?

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This is the sum of look back call and lookback put with floating strike. You can then price it using the formulas in wikipedia:

https://en.wikipedia.org/wiki/Lookback_option

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