# 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 (or a precise predefined quantity or units 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]

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, often referred to as an expiry date, the stock is trading at above $100,$1,000 is received. If the stock is trading below $100, no 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.

Though binary options sometimes trade on regulated exchanges, they are generally unregulated, trading on the internet, and prone to fraud.[1] 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.[2]

## 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.[3]

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.[4]

In March 2013, Malta's Financial Services Authority announced that binary options regulation would be transferred away from Malta's Lottery and Gaming Authority.[5] On 18 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.[6] 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.[7] 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.[8]

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 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.[9]

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.[10][11]

In the UK, the Department for Culture, Media and Sport have written to the Gambling Commission suggesting binary options are to be reclassified from "fixed odds bets" to "financial instruments".[12] This would mean regulation falls under the remit of the Financial Conduct Authority. This would be a significant step in the regulation of binary options as FCA regulation would carry much more weight both with European members and UK consumers.

## 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]

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 in order 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 decomposing 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.

## References

1. ^ a b Binary Option Definition Investopedia. Retrieved 2013-06-30.
2. ^ "Investor Alert Binary Options and Fraud" (PDF). Securities Exchange Commission. Retrieved 20 November 2013.
3. ^ "warning against unauthorised websites offering binary options trading". AMF. Retrieved 14 January 2012.
4. ^ "regarding the supervision of Binary Options" (PDF). CySEC. 3 May 2012. Retrieved 4 June 2012.
5. ^ Gaming or Trading? That is the Question – MFSA To Regulate Binary Options as a Financial Product
6. ^ "Malta The Next Cyprus? MFSA Announces Regulatory Framework For Binary Options". Finance Magnates. July 18, 2013.
7. ^ "Warning" (PDF). Retrieved 27 March 2014.
8. ^ "Warning" (PDF). Retrieved 27 March 2014.
9. ^ "The projects of the CySEC regulator in terms of binary options in 2014". Retrieved 27 March 2014.
10. ^
11. ^ "Binary Options and Fraud". Retrieved 23 May 2014.
12. ^ "Binary Options brokers to be regulated by the FCA". Retrieved 14 May 2015.
13. ^ Pape, Gordon (27 July 2010). "Don’t Gamble On Binary Options". Forbes.com. Archived from the original on 2013-06-21. Retrieved 15 April 2011.
14. ^ Securities and Exchange Commission, Release No. 34-56471; File No. SR-OCC-2007-08, September 19, 2007. “Self-Regulatory Organizations; The Options Clearing Corporation; Notice of Filing of a Proposed Rule Change Relating to Binary Options”.
15. ^ Frankel, Doris (9 June 2008). "CBOE to list binary options on S&P 500, VIX". Reuters.
16. ^ "System and methods for trading binary options on an exchange", World Intellectual Property Organization filing. Wipo.int. Retrieved on 2013-01-12.
17. ^ a b BINARY OPTIONS ON SPXSM AND VIX®. cboe.com
18. ^ SPX Binary Contract Specifications. Cboe.com (2012-04-16). Retrieved on 2013-01-12.
19. ^ VIX Binary Contract Specifications. Cboe.com (2012-04-16). Retrieved on 2013-01-12.
20. ^ Nadex 2009 Press Release. Retrieved September 20th, 2011. (PDF) . Retrieved on 2013-01-12.
21. ^ Hull, John C. (2005). Options, Futures and Other Derivatives. Prentice Hall. ISBN 0-13-149908-4.
22. ^ Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. Journal of business, 621-651.
23. ^ Gatheral, J. (2006). The volatility surface: a practitioner's guide (Vol. 357). John Wiley & Sons.