CRR Binomial Call Pricing at Intermediate Tree Nodes

Summary

The document asks how to express the value of a European call at an intermediate node of a Cox–Ross–Rubinstein binomial tree. It defines the node by the time step and the number of upward moves, and describes a backward-induction recurrence that discounts the risk-neutral expected value of the next-step option payoffs.

The author also gives a time-zero call valuation as a binomial sum and rewrites it as two tail-probability sums, one weighted by a transformed probability and the other by the risk-neutral probability. The strike threshold is the smallest number of up moves that makes the terminal stock price reach the strike. The document presents the question and the starting formulas but does not provide an answer or derive a closed form for intermediate nodes. Its formulas therefore serve as context rather than a completed result, and depend on the stated binomial model assumptions and notation.

Key ideas

• A CRR tree node is identified by the time step and the number of upward stock moves.
• Backward induction values the option by discounting its risk-neutral expected value at the next step.
• At maturity, the call payoff depends on whether the terminal stock price exceeds the strike.
• The time-zero value can be written as binomial tail sums using a threshold number of upward moves.
• The document poses, but does not resolve, the request for an intermediate-node closed form.

Tags

# Cox-Ross-Rubinstein Model Closed Formula for Call Option


# Cox-Ross-Rubinstein Model Closed Formula for Call Option












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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