# Binary Option

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In 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.

Binary options contracts have long been available 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 90 such platforms (including white label products) have been in operation as of January 2012, offering options on some 125 underlying assets.

In 2007, 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.[5]

CBOE offers binary options on the 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.[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.

Binary option trading is now an international industry. It is most widely recognized in the United States, but is increasing gaining popularity in the Middle East, Western Europe, Australia and Asia.[11]

## Example of a binary options trade

A trader who thinks that the EUR/USD 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 $sigma$ 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,

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

### 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 $r_{FOR}$, 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.

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 $C_v$ is a vanilla European call:[[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, $sigma$ 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:

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

$frac{partial C_v}{partial sigma}$ 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 US Presidential election can be interpreted as an estimate of 65% likelihood of him winning.

In financial markets, 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.