What's the price of a lookback call option in the arbitrage-free CRR-model?

If we consider the CRR-model in two periods, i.e. T=2. Let $S^1$ be the risky asset with $S_0^1=100$ and $S^0$ the bond with $S_0^0=1$ . Furthermore, we assume the model is arbitrage-free with $y_b=-0.1<r=0.05<y_g=0.2$ . Therefore, an unique equivalent martingale measure $\mathbb{Q}$ exists with $\mathbb{Q}(\lbrace\omega\rbrace) =0.5^{Z_1(\lbrace\omega\rbrace)+Z_2(\lbrace\omega\rbrace)}\cdot 0.5^{2-Z_1(\lbrace\omega\rbrace)-Z_2(\lbrace\omega\rbrace)}$ , where $\omega\in\lbrace 0,1\rbrace^2$ .