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I am I quite new to the topic and at the moment I am self studying the CRR model. My notations are: $N$ number of periods, $\delta T$ length of one period, $S_0$ stock price at time $t=0$, $f_0$ price of the call option at time $t=0$, $r$ risk free interest rate (I assume continuously compounding) and $\widetilde{p}$ risk neutral probability.

I want to find a closed formula for the "fair" price of the call option at time $t=i<N$ after $j$ up movements in the stock price. I already derived the following formula via backward induction for the "fair" price of a call option at time $t=i$ and implemented it in R and I get correct results.
\begin{equation} f_{i,j}=e^{-r \delta T}\left(\widetilde{p}f_{i+1,j+1}+(1+\widetilde{p})f_{i+1,j} \right), \quad j=0,1, \dots,i \end{equation} $j$ denotes the number of up movements in the stock price. I wonder if I can represent this formula in a closed form. For example for $t=0$ I know the formula \begin{equation} f_0=e^{-rN\delta T}\sum_{j=0}^N {N \choose j}\widetilde{p}^j(1-\widetilde{p})^{N-j}\max\left\{S_0u^jd^{N-j};0\right\} \end{equation} which can be rewritten as \begin{equation} f_0=S_0\sum_{j=k}^N {N \choose j} \hat{p}^j(1-\hat{p})^{N-j}-Ke^{-r \cdot N \cdot \delta T}\sum_{j=k}^N {N \choose j} \widetilde{p}^j(1-\widetilde{p})^{N-j} \end{equation} where $k$ is the smallest integer such that $S_0u^kd^{N-k}\geq K$ and $\hat{p}=e^{-r \delta T} \widetilde{p} u$.

Thanks in advance !

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    $\begingroup$ This looks off-topic here. I think it would better fit on Quantitative Finance Stack Exchange. $\endgroup$ – Richard Hardy Mar 1 at 14:56
  • $\begingroup$ Sorry, I did not know that such a thing exists. Thanks ! $\endgroup$ – Lars Mar 1 at 15:45

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