Bounding a Binary Option with a Call Spread to Identify Arbitrage
Summary
This note shows how a binary option payoff can be bounded by a scaled call spread with nearby strikes. The spread buys calls at a lower strike and sells calls at the binary option’s strike. At expiration, its payoff is one or more in the interval between the strikes and zero below it, so it covers the binary payoff that pays when the underlying finishes at or above the higher strike.
If the binary option’s price exceeds the call spread’s price, the stated construction sells the binary and sells the higher-strike calls while buying the lower-strike calls. The investor receives a positive initial difference and has a nonnegative maturity payoff under the payoff comparison. The argument relies on the assumed strike spacing, matching expiration and underlying, and the ability to trade the options at the stated prices; it does not address transaction costs, liquidity, or settlement details. The document’s initial inequality and the answer’s adjusted strike convention differ, so the exact strike indexing should be checked when applying the construction.
Key ideas
- A suitably scaled call spread can dominate a binary option’s expiration payoff.
- The call spread uses a lower strike and the binary option’s threshold strike.
- If the binary is overpriced relative to the spread, the payoff dominance supports a short-binary, long-spread arbitrage.
- The construction depends on consistent strikes and expirations, as well as frictionless execution assumptions.
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Full text
# Arbitrage opportunity in discrete time
# Arbitrage opportunity in discrete time
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?
## Answer by Gordon (score 0, accepted)
https://quant.stackexchange.com/a/25381
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.