# 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 $$y_b=-0.1. Therefore, an unique equivalent martingale measure $$\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$$. Furthermore, the lookback call option is given by $$\gamma (\omega)=S_2^1(\omega)-\min_{t\in\lbrace 0,1,2\rbrace}S_t^1(\omega)$$. I see that the price of the option depends on these numbers a lot, but I want a general approach. In some literature I have found out that the hedging price ( or arbitrage-free price) is given by $$\mathbb{E}^\mathbb{Q}\left[\frac{f(S_2^1)}{(1+r)^2}\right]$$, but how exactly do I determine the function $$f$$. If it was a call option with a fixed strike price $$K$$, i.e. $$\gamma' (\omega)=(S_2^1(\omega)-K)^+$$ I assume the $$f$$ would be given by $$f(x)=\max(x-K,0)$$. But how do I model the dependency of $$\gamma$$ on $$\min_{t\in\lbrace 0,1,2\rbrace}S_t^1(\omega)$$?

I thought about it and think that in this case it does not really matter how $$f$$ looks exactly, because $$f(S_2^1(\omega ))=\gamma (\omega)$$ should be true. Then, with the formular for the equivalent risk-neutral measure $$\mathbb{Q}$$ given above we can compute the probabilities $$Q((0,0))=Q((1,0))=Q((0,1))=Q((1,1))=0.25$$. Furthermore, in the CRR-model it holds, that $$S_t^1(\omega )=S_0^1(1+y_g)^{D_t(\omega)}(1+y_b)^{t-D_t(\omega)}$$. So for the numbers above we get $$\begin{split} \frac{1}{(1+r)^2}\mathbb{E}^\mathbb{Q}[\gamma]&=\frac{400}{441}\left(\mathbb{E}^\mathbb{Q}[S_2^1]-\mathbb{E}^\mathbb{Q}\left[\min_{t\in\lbrace 0,1,2\rbrace}S_t^1\right]\right)\\ &= \frac{400}{441}(0.25\cdot(81+108+108+144)-0.25\cdot(81+100+100+90))\\ &=15.87 \end{split}$$