or:

$\begin{array}{cc}{c}_{0}& ={S}_{0}\left[{\displaystyle \sum}_{j=a}^{T}\frac{T!}{j!(T-j)!}{q}^{j}{(1-q)}^{T-j}\frac{{u}^{j}{d}^{T-j}}{{(1+r)}^{T}}\right]-\frac{X}{{(1+r)}^{T}}\left[{\displaystyle \sum}_{j=a}^{T}\frac{T!}{j!(T-j)!}{q}^{j}{(1-q)}^{T-j}\right]\end{array}$

or, in shorthand form:^{12}

${c}_{0}={S}_{0}B\left[T,q\prime \right]-X{(1+r)}^{-T}B[T,q]$

where *q′*=*qu*/(1+*r*) and 1−*q′*=*d*(1−*q*)/(*1*+*r*). The values *q′*, *q*, and *T* are the parameters for the two binomial distributions. Three points are worth further discussion regarding this simplified binomial model:

1. First, as *T* approaches infinity, the binomial distribution will approach the normal distribution, ...

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