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Say we have the following binary option $B$ on asset $S$ with strike K and expiration time T, assume also that the following relation holds at time $0$:
$B > N*C(K,T)-N*C(K+1/N,T)$
Where $N$ is some natural number and $C(K,T)$ is the call option on asset $S$ with strike K and expiration time T
How is it possible to find an arbitrage strategy in that case , under the assumption that we can buy call options at any strike?
We assume that the inequality is given by \begin{align*} B > N C(K-1/N, T) - N C(K, T).\tag{1} \end{align*} The argument for the case with the inequality \begin{align*} B < N C(K, T) - N C(K+1/N, T) \end{align*} is similar. $$$$ For the binary option, \begin{align*} \pmb{1}_{\{S_T \ge K\}} = \begin{cases} 1, & \textrm{if } S_T \ge K,\\ 0, & \textrm{otherwise}, \end{cases} \end{align*} while for the portfolio with payoff \begin{align*} X_T &= N\bigg[S_T-\Big(K-\frac{1}{N}\Big)\bigg]^+ - N (S_T-K)^+\\ &= \begin{cases} 1, & \textrm{if } S_T \ge K,\\ N\bigg[S_T-\Big(K-\frac{1}{N}\Big)\bigg], & \textrm{if } K-\frac{1}{N} \le S_T \le K,\\ 0, & \textrm{otherwise}. \end{cases} \end{align*} Then, it is obvious that \begin{align*} \pmb{1}_{\{S_T \ge K\}} \le X_T,\tag{2} \end{align*} and, consequently, \begin{align*} B \le N C(K-1/N, T) -N C(K, T). \end{align*}
$$$$ However, if (1) holds, we can then short the binary option, short $N$ units call option with strike $K$, and long $N$ units call option with strike $K-1/N$. We have a net profit \begin{align*} B - \big[N C(K-1/N, T) - N C(K, T)\big]. \end{align*} Moreover, at maturity $T$, we have the portfolio payoff \begin{align*} X_T - \pmb{1}_{\{S_T \ge K\}} \ge 0, \end{align*} as per (2) above.
