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

Question

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 $\omega\in\lbrace 0,1\rbrace^2$. 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 $\min_{t\in\lbrace 0,1,2\rbrace}S_t^1(\omega)$?

Answer 1

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 $$ \begin{split} \frac{1}{(1+r)^2}\mathbb{E}^\mathbb{Q}[\gamma]&=\frac{400}{441}\left(\mathbb{E}^\mathbb{Q}[S_2^1]-\mathbb{E}^\mathbb{Q}\left[\min_{t\in\lbrace 0,1,2\rbrace}S_t^1\right]\right)\\ &= \frac{400}{441}(0.25\cdot(81+108+108+144)-0.25\cdot(81+100+100+90))\\ &=15.87 \end{split} $$

Could someone validate that ? Furthermore, it does not really answer the question on how to proceed, if no specific numbers are given. How should one approach this problem in general ?