# Binary option

In finance, a binary option is a type of option where the payoff is either some fixed amount of some asset or nothing at all. 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. Thus, the options are binary in nature because there are only two possible outcomes. 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). Binary options are usually European-style options.

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$1000. Then, if at the future maturity date, the stock is trading at or above $100,$1000 is received. If its stock is trading below $100, nothing is received. In the popular Black-Scholes model, the value of a digital option can be expressed in terms of the cumulative normal distribution function. ## Non exchange-traded binary options Binary option 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 web-sites called binary option trading platforms have been offering a simplified version of exchange-traded binary options. It is estimated that around 30 such platforms (including white label products) have been in operation as of January 2011, offering options on some 70 underlying assets. ## Exchange-traded binary options In 2007, the Options Clearing Corporation proposed a rule change to allow binary options,[1] and the Securities and Exchange Commission approved listing cash-or-nothing binary options in 2008.[2] 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.[3] Amex calls binary options "Fixed Return Options"; 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.[4] The American Stock Exchange and Donato A. Montanaro submitted a patent application for exchange-listed binary options using a volume-weighted settlement index in 2005.[5] CBOE offers binary options on the S&P 500 (SPX) and the CBOE Volatility Index (VIX).[6] The tickers for these are BSZ[7] and BVZ,[8] respectively. 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.[6][9]

In 2009 Nadex, the North American Derivatives Exchange, launched and now offers a suite of binary options vehicles.[10]. Nadex binary options are available on a range Stock Index Futures, Spot Forex, Commodity Futures, and Economic Events.[11]

## Example of a Binary Options Trade

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

At 2:00 p.m. the EUR/USD spot 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 = $1000. If the EUR/USD strike 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:$1000 - $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$1000. 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.[12] 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 σ is the volatility. Φ 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

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 rFOR , the foreign interest rate, rDOM , 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.

To take volatility skew into account, a more sophisticated analysis based on call spreads can be used.

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 Cv is a vanilla European call:[page needed],[13][14]

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

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, σ 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 = CnoskewVegav * 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 US Presidential election can be interpreted as an estimate of 65% likelihood of him winning.

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.

## References

1. ^ 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”.
2. ^ Frankel, Doris (June 9, 2008). "CBOE to list binary options on S&P 500, VIX". Reuters.
3. ^ http://www.amex.com/options/prodInf/OptPiFROs.jsp
4. ^ http://www.amex.com/options/prodInf/fros.settlementindex.pdf
5. ^ "System and methods for trading binary options on an exchange", World Intellectual Property Organization filing.
6. ^ a b http://www.cboe.com/micro/binaries/BinariesQRG.pdf
7. ^ SPX Binary Contract Specifications
8. ^ VIX Binary Contract Specifications
9. ^ http://www.amex.com/options/prodInf/fros.specifications.pdf
10. ^ http://www.nadex.com/content/files/pressrelease-01.pdf Nadex 2009 Press Release. Retrieved September 20th, 2011
11. ^ Cannon Trading Company, Inc. What Are Binary Options? Retrieved September 20th, 2011
12. ^ Hull, John C. (2005). Options, Futures and Other Derivatives. Prentice Hall. ISBN 0131499084.
13. ^ Taleb, Nassim Nicholas (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. Wiley Finance. ISBN 0471152803.
14. ^ Lehman Brothers, "Listed Binary Options", July 2008, http://www.cboe.com/Institutional/pdf/ListedBinaryOptions.pdf

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