# Binary option

In finance, a binary option is a type of option in which the payoff can take only two possible outcomes, either some fixed monetary amount of some asset or nothing at all (in contrast to ordinary financial options that typically have a continuous spectrum of payoff). The two main types of binary options are the cash-or-nothing binary option and the asset-or-nothing binary option. The cash-or-nothing binary option pays some fixed amount of cash if the option expires in-the-money while the asset-or-nothing pays the value of the underlying security. They are also called all-or-nothing options, digital options (more common in forex/interest rate markets), and fixed return options (FROs) (on the American Stock Exchange).[1]

When buying a binary option the potential return it offers is certain and known before the purchase is made. Binary options can be bought on virtually any financial product and can be bought in both directions of trade either by buying a “Call”/“Up” option or a “Put”/“Down” option. Binary options are offered against a fixed expiry time.[2]

For example, a purchase is made of a binary cash-or-nothing call option on XYZ Corp's stock struck at $100 with a binary payoff of$1,000. Then, if at the future maturity date, the stock is trading at above $100,$1,000 is received. If the stock is trading below $100, the money is received. And if the stock is trading at$100, the money is returned to the purchaser.

The value of a digital option can be expressed in terms of the probability of exceeding a certain value, that is, the cumulative distribution function, which in the Black-Scholes equation is the Gaussian. Due to the difficulty for market-makers to hedge binary options that are near the strike price around expiry, these are much less liquid than vanilla options. Dealers often replicate them using vertical spreads, which provides a rough, inexact hedge.

Binary options contracts have long been available over-the-counter (OTC), i.e., sold directly by the issuer to the buyer. They were generally considered "exotic" instruments and there was no liquid market for trading these instruments between their issuance and expiration. They were often seen embedded in more complex option contracts.

Since mid-2008 binary options websites called binary option trading platforms have been offering a simplified version of exchange-traded binary options. It is estimated that around 90 such platforms (including white-label products) have been in operation as of January 2012, offering options on some 200 underlying assets.

The platforms offer standardized short-term binary options with a pre-determined profit/loss, that cannot be liquidated (buy or sell to close) before expiry, unless the platform or broker allow such liquidation.[disputed ]

The U.S. Securities and Exchange Commission (SEC) and Commodity Futures Trading Commission (CFTC) have issued a joint warning to American investors regarding unregulated binary options.[3]

The platforms do not charge fees from their investors. Their profit comes from the difference between the options that expire in the money to options that expire out of the money. This difference can be found by the formula below. In this (for each base asset with same expiry characteristics), (W) is the in the money option payout in percentage terms (e.g., 1.7), (L) is the out of the money option payout in percentage terms (e.g., 0.15), $(V_{1}),(V_{2})$ are the turnovers of transactions made for each outcome (e.g., $1,000), (S) is the platform's gain. $S = -[V_{1}(W-1)+V_{2}(L-1)]$ In this example the platform's turnover is$2,000 and its profit is $150 or 7.5% on turnover. As the platform’s gain comes from the above formula, most platforms will be indifferent as to the outcome of a single trade. Note that if $V_{1}$ is not equal to $V_{2}$ then the platform will have to act as a market maker. This can cause the platform gain (S) to be more volatile than in the above formula. In order for a trader to make a long-term profit he has to predict correctly roughly 54.5% of the time (depending on in-the-money and out-of-the money payouts). ## Regulation and compliance On non-regulated platforms, client money is not necessarily kept in a trust account, as required by government financial regulation, and transactions are not monitored by third parties in order to ensure fair play.[4] On May 3, 2012, the Cyprus Securities and Exchange Commission (CySEC) announced a policy change regarding the classification of binary options as financial instruments. The effect is that binary options platforms operating in Cyprus, where many of the platforms are based, will have to be CySEC regulated within six months of the date of the announcement. CySEC was the first EU MiFID-member regulator to treat binary options as financial instruments.[5] On 7 January 2013, Banc De Binary became the first licensed binary option firm recognized as an investment firm by CySEC. In March 2013, Malta's Financial Services Authority announced that binary options regulation would be transferred away from Malta's Lottery and Gaming Authority.[6] On18 June 2013, Malta’s Financial Services Authority confirmed that in their view binary options fell under the scope of the Markets in Financial Instruments Directive (MiFID) 2004/39/EC. With this announcement Malta became the second EU jurisdiction to regulate binary options as a financial instrument, providers will now have to gain a category 3 Investment Services licence and conform to MiFID's minimum capital requirements.[7] Prior to this announcement it had been possible for firms to operate from the jurisdiction provided the firm had a valid Lottery and Gaming Authority licence. In 2013, CySEC prevailed over the disreputable binary options brokers and communicated intensively with traders in order to prevent the risks of using unregulated financial services. On September 19, 2013, Cyprus Securities and Exchange Commission (CySEC) sent out a press release warning investors against binary options broker TraderXP, CySEC stated that TraderXP is not and has never been licensed by CySEC.[8] On October 18, 2013, CySEC released an investor warning about binary options broker NRGbinary and its parent company NRG Capital (CY) Ltd., stating that NRGbinary is not and has never been licensed by CySEC.[9] The Cypriot regulator also temporarily suspended the license of the Cedar Finance on December 19, 2013. The decision was taken by Cyprus Securities and Exchange Commission(CySEC) because the potential violations referenced appear to seriously endanger the interests of the company’s (Cedar Finance | www.cedarfinance.com) customers and the proper functioning of capital markets, as described in the official issued press release. CySEC also issued a warning against binary option broker PlanetOption at the end of the year and another warning against binary option broker LBinary on January 10, 2014, pointing out that it is not regulated by the Commission and the Commission has not received any notification by any of its counterparts in other European countries to the effect of this firm being a regulated provider. As far as penalties are concerned, the Cyprus regulator imposed a penalty of €15,000 against ZoomTrader. OptionBravo and ChargeXP were also financially penalized. CySEC also indicated that it has voted to reject the ShortOption license application.[10] The U.S. Commodity Futures Trading Commission (CFTC) oversees the regulation of futures, options and swaps trading in the United States. On June 6, 2013, the CFTC and the U.S. Securities and Exchange Commission jointly issued an Investor Alert to warn about fraudulent promotional schemes involving binary options and binary options trading platforms. At the same time they charged Banc De Binary Ltd., a Cyprus-based company, with illegally selling binary options to U.S. investors.[11][12] ## Criticism These platforms may be considered by some as gaming platforms rather than investment platforms because of their negative cumulative payout (they have an edge over the investor) and because they require little or no knowledge of the stock market to trade. According to Gordon Pape, writing in Forbes, "this sort of thing can quickly become addictive...no one, no matter how knowledgeable, can consistently predict what a stock or commodity will do within a short time frame".[13] ## Exchange-traded binary options In 2007, the Options Clearing Corporation proposed a rule change to allow binary options,[14] and the Securities and Exchange Commission approved listing cash-or-nothing binary options in 2008.[15] In May 2008, the American Stock Exchange (Amex) launched exchange-traded European cash-or-nothing binary options, and the Chicago Board Options Exchange (CBOE) followed in June 2008. The standardization of binary options allows them to be exchange-traded with continuous quotations. Amex offers binary options on some ETFs and a few highly liquid equities such as Citigroup and Google. Amex calls binary options "Fixed Return Options" (FROs); calls are named "Finish High" and puts are named "Finish Low". To reduce the threat of market manipulation of single stocks, Amex FROs use a "settlement index" defined as a volume-weighted average of trades on the expiration day. Amex and Donato A. Montanaro submitted a patent application for exchange-listed binary options using a volume-weighted settlement index in 2005.[16] CBOE offers binary options on the S&P 500 (SPX) and the CBOE Volatility Index (VIX).[17] The tickers for these are BSZ[18] and BVZ, respectively.[19] CBOE only offers calls, as binary put options are trivial to create synthetically from binary call options. BSZ strikes are at 5-point intervals and BVZ strikes are at 1-point intervals. The actual underlying to BSZ and BVZ are based on the opening prices of index basket members. Both Amex and CBOE listed options have values between$0 and $1, with a multiplier of 100, and tick size of$0.01, and are cash settled.[17]

In 2009, Nadex, a U.S.-based binary options provider launched binary options on a range of forex, commodities and stock indices markets.[20]

## Example of a binary options trade

A trader who thinks that the EUR/USD price will close at or above 1.2500 at 3:00 p.m. can buy a call option on that outcome. A trader who thinks that the EUR/USD price will close at or below 1.2500 at 3:00 p.m. can buy a put option or sell a call option contract.

At 2:00 p.m. the EUR/USD price is 1.2490. The trader believes this will increase, so he buys 10 call options for EUR/USD at or above 1.2500 at 3:00 p.m. at a cost of $40 each. The risk involved in this trade is known. The trader’s gross profit/loss follows the "all or nothing" principle. He can lose all the money he invested, which in this case is$40 x 10 = $400, or make a gross profit of$100 x 10 = $1,000. If the EUR/USD price will close at or above 1.2500 at 3:00 p.m. the trader's net profit will be the payoff at expiry minus the cost of the option:$1,000 – $400 =$600.

The trader can also choose to liquidate (buy or sell to close) his position prior to expiration, at which point the option value is not guaranteed to be $100. The larger the gap between the spot price and the strike price, the value of the option decreases, as the option is less likely to expire in the money. In this example, at 3:00 p.m. the spot has risen to 1.2505. The option has expired in the money and the gross payoff is$1,000. The trader's net profit is \$600.

## Black–Scholes valuation

In the Black–Scholes model, the price of the option can be found by the formulas below.[21] In fact, the Black–Scholes formula for the price of a vanilla call option (or put option) can be interpreted by decomposition a call option into an asset-or-nothing call option minus a cash-or-nothing call option, and similarly for a put – the binary options are easier to analyze, and correspond to the two terms in the Black–Scholes formula.

In these, S is the initial stock price, K denotes the strike price, T is the time to maturity, q is the dividend rate, r is the risk-free interest rate and $\sigma$ is the volatility. $\Phi$ denotes the cumulative distribution function of the normal distribution,

$\Phi(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x e^{-\frac{1}{2} z^2} dz.$

and,

$d_1 = \frac{\ln\frac{S}{K} + (r-q+\sigma^{2}/2)T}{\sigma\sqrt{T}},\,d_2 = d_1-\sigma\sqrt{T}. \,$

### Cash-or-nothing call

This pays out one unit of cash if the spot is above the strike at maturity. Its value now is given by,

$C = e^{-rT}\Phi(d_2). \,$

### Cash-or-nothing put

This pays out one unit of cash if the spot is below the strike at maturity. Its value now is given by,

$P = e^{-rT}\Phi(-d_2). \,$

### Asset-or-nothing call

This pays out one unit of asset if the spot is above the strike at maturity. Its value now is given by,

$C = Se^{-qT}\Phi(d_1). \,$

### Asset-or-nothing put

This pays out one unit of asset if the spot is below the strike at maturity. Its value now is given by,

$P = Se^{-qT}\Phi(-d_1). \,$

### Foreign exchange

Further information: Foreign exchange derivative

If we denote by S the FOR/DOM exchange rate (i.e., 1 unit of foreign currency is worth S units of domestic currency) we can observe that paying out 1 unit of the domestic currency if the spot at maturity is above or below the strike is exactly like a cash-or nothing call and put respectively. Similarly, paying out 1 unit of the foreign currency if the spot at maturity is above or below the strike is exactly like an asset-or nothing call and put respectively. Hence if we now take $r_{FOR}$, the foreign interest rate, $r_{DOM}$, the domestic interest rate, and the rest as above, we get the following results.

In case of a digital call (this is a call FOR/put DOM) paying out one unit of the domestic currency we get as present value,

$C = e^{-r_{DOM} T}\Phi(d_2) \,$

In case of a digital put (this is a put FOR/call DOM) paying out one unit of the domestic currency we get as present value,

$P = e^{-r_{DOM}T}\Phi(-d_2) \,$

While in case of a digital call (this is a call FOR/put DOM) paying out one unit of the foreign currency we get as present value,

$C = Se^{-r_{FOR} T}\Phi(d_1) \,$

and in case of a digital put (this is a put FOR/call DOM) paying out one unit of the foreign currency we get as present value,

$P = Se^{-r_{FOR}T}\Phi(-d_1) \,$

### Skew

In the standard Black–Scholes model, one can interpret the premium of the binary option in the risk-neutral world as the expected value = probability of being in-the-money * unit, discounted to the present value. The Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset. Market makers adjust for such skewness by, instead of using a single standard deviation for the underlying asset $\sigma$ across all strikes, incorporating a variable one $\sigma(K)$ where volatility depends on strike price, thus incorporating the volatility skew into account. The skew matters because it affects the binary considerably more than the regular options.

A binary call option is, at long expirations, similar to a tight call spread using two vanilla options. One can model the value of a binary cash-or-nothing option, C, at strike K, as an infinitessimally tight spread, where $C_v$ is a vanilla European call:[22][23]

$C = \lim_{\epsilon \to 0} \frac{C_v(K-\epsilon) - C_v(K)}{\epsilon}$

Thus, the value of a binary call is the negative of the derivative of the price of a vanilla call with respect to strike price:

$C = -\frac{dC_v}{dK}$

When one takes volatility skew into account, $\sigma$ is a function of $K$:

$C = -\frac{dC_v(K,\sigma(K))}{dK} = -\frac{\partial C_v}{\partial K} - \frac{\partial C_v}{\partial \sigma} \frac{\partial \sigma}{\partial K}$

The first term is equal to the premium of the binary option ignoring skew:

$-\frac{\partial C_v}{\partial K} = -\frac{\partial (S\Phi(d_1) - Ke^{-rT}\Phi(d_2))}{\partial K} = e^{-rT}\Phi(d_2) = C_{noskew}$

$\frac{\partial C_v}{\partial \sigma}$ is the Vega of the vanilla call; $\frac{\partial \sigma}{\partial K}$ is sometimes called the "skew slope" or just "skew". Skew is typically negative, so the value of a binary call is higher when taking skew into account.

$C = C_{noskew} - Vega_v * Skew$

## Relationship to vanilla options' Greeks

Since a binary call is a mathematical derivative of a vanilla call with respect to strike, the price of a binary call has the same shape as the delta of a vanilla call, and the delta of a binary call has the same shape as the gamma of a vanilla call.

## Interpretation of prices

In a prediction market, binary options are used to find out a population's best estimate of an event occurring – for example, a price of 0.65 on a binary option triggered by the Democratic candidate winning the next U.S. presidential election can be interpreted as an estimate of 65% likelihood of victory.

In financial markets, expected returns on a stock or other instrument are already priced into the stock. However, a binary options market provides other information. Just as the regular options market reveals the market's estimate of variance (volatility), i.e., the second moment, a binary options market reveals the market's estimate of skew, i.e., the third moment.

In theory, a portfolio of binary options can also be used to synthetically recreate (or valuate) any other option (analogous to integration), although in practical terms this is not possible due to the lack of depth of the market for these relatively thinly traded securities.

In theory a portfolio of options can synthetically recreate any other financial instrument, including conventional options.