Binary option is one of the simplest types of exotic options. The owner of a binary option receives a fixed payoff if the price of the underlying asset is below or above a certain point at the time of expiry (or before) and receives no payout at all in all other cases.
Despite the apparent simplicity of this derivative, a binary option is classified as an exotic option because its payout cannot be replicated exactly by the set of standard options. However, instead of treating a binary option as an exotic option and using the Monte Carlo method to calculate the fair value of the option, in practice many traders use option spreads (call and put spreads) when pricing binary options. In addition, the only acceptable way to manage and hedge the risk (i.e. Greeks) of a binary option is to use option spreads (such as call and put spreads). The vast majority of binary options are European. Therefore only European binary options are considered here.
Let us consider the following example. An investor buys a 3 month European binary option on a JP Morgan Chase (JPM) stock, which pays $10 if after 3 months the stock price exceeds $70, but if the stock price of JPM is below $70 on the expiration date, the investor will get nothing and lose the option premium (option purchase price). The figure below shows the payoff chart at the time of expiry for this binary JPM stock option.
A trader who sells a binary option on JPM can easily replicate the value of that binary option by purchasing multiple call spreads. The number of spreads required depends on the width of the call spread (difference between strikes). The wider the call spread, the smaller the number of spreads you will need. The risk profile of a wide call spread and a binary option differs significantly.
For example, a trader believes that he or she can successfully manage the risk (from selling a binary option) with a strike difference of $2.5 in the call spread. In other words, the trader will determine the value of the binary option on the basis of selling a 4 call spreads of $67.5-70, the leverage of which will be 4x (when paying $10 and the difference between strikes of $2.5). To see how a call spread imitates the behavior of a binary option, let's look at three different stock prices of JP Morgan Chase at the time of expiry.
1: First, let's assume that the price of JPM shares at the time of expiration is $75. According to the terms of the binary option, the investor (the buyer of the option) must receive $10. This is exactly what the trader's position dictates. In fact, the trader sold 4 call spreads of $67.5/70, and each call spread gives a payout of $2.5 if the stock price is at $75. As a result, the total payout to the investor is $10.
2: Second, if the stock price of JPM is $69 at expiration option, the investor (buyer of the binary option) should not receive any payout on the binary option. However, the 4-call spread of $67.5/70 involves a payout of $6 ($69 - $67.5 = $1.5 per call spread), resulting in the trader (the seller of the binary option, or 4-call spreads) losing $6 from selling the call spreads (i.e., the binary option based on 4x call spreads). This obviously shows that the trader has very conservatively (low) estimated the value of the binary option by choosing the strikes of $67.5/70. However, it is the width of the call spread that determines the gamma risk and the pin risk around the stock price of $70. If the stock trades at $70 at expiration, a binary option is worth nothing, but if the stock price rises to $70.01, the value of the option will jump immediately to $10. Obviously, it is extremely difficult for the trader (the binary option writer) to manage the risk of the delta and the gamma, since with a stock around $70, the delta can change dramatically from 0 to 100.
When hedging a binary option using call spreads, the trader still has to manage the large gamma pin risk (Read Binary Options - Pre-history). However, the fact that the trader initially estimates the value of a binary option based on the call spreads allows him to increase control over his position.
3: Third, if the stock price of JPM is at $66, neither the binary option nor the call spreads will have any payoff to the investor.
The choice of strikes for call spread
A natural question for a beginner would be “why did the trader choose not $70/72.5 call spread when pricing a binary option?” The answer is that the $70/72.5 spread achieves the exact opposite goal that the trader seeks to achieve, namely, to provide himself with a security margin.
In a way, the $70/72.5 spread makes the trader feel "richer" than he really is. For example, if the stock price reaches $71, the profit from a 4x call spread $70/72.5 is $4, while the binary option requires a $10 payout. In other words, buying the $70/72.5 call spread will not allow the trader to earn $10 at stock price $70, while the $67.5/70.0 call spread will generate $10 at stock price of $70.
Call spread as the foundation for successful binary options trading
Call spreads are used not only for binary options pricing, but also as an instrument that the trader actively trades to hedge the risks of the binary options portfolio. In other words, when a trader sells a binary option, one can say that he is selling the call spread in terms of risk profile. In this case, the trader can manage the risk (Greeks) from the binary option position in the same way as the call spread. The above shows that the call spread is a conservative proxy for a binary option. Therefore, when trading European binary options, a trader can compare the payoff of a binary option to the payoff of a call spread.
If these payoffs differ significantly some time before the expiration, there is an opportunity for a profitable trade in the market.
Replicating a binary option with a call spread
The smaller the difference between strikes of the call spread, the more call spreads you need to buy/sell to replicate the binary option. The narrower the call spread (the difference between strikes), the higher the risk and the more complex the risk profile is, as the gamma can change size and sign very quickly, which makes hedging difficult. In order to replicate a binary option more accurately, a trader needs to find a very narrow call spread. Thus, the smaller the difference between strikes of the call spread, the more the call spread resembles a binary option.