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Binary option

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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.

Contents

Non exchange-traded binary options

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 web-sites called binary option trading platforms have been

offering a simplified version of exchange-traded binary options. It is estimated that around

50 such platforms (including white label products) have been in operation as of January

2011, offering options on some 70 underlying assets.

[edit] 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.

[edit] 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.

[edit] Black-Scholes Valuation

In the Black-Scholes model, the price of the option can be found by the formulas below.[11]

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}. \,

[edit] 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). \,

[edit] 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). \,

[edit] 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). \,

[edit] 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). \,

[edit] 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) \,

[edit] 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:[page needed],[12][13]

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

[edit] 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.[14]

[edit] 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.

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

including conventional options.[14]

[edit] Structured Binary Options Strategies

It may come as a surprise to many interested in the options space that put options were not

introduced on the CBOE until 1977, nine years after call options were. The binary options

market at present is in the same 'no-mans-land' where there is a vibrant FX binary options

market with sophisticated binary options strategies, while at the other extreme there are a

plethora of platforms offering one-hour bets dressing themselves up as 'investments'.

But the binary options market too has its range of straddles, strangles, call spreads,

butterflies, condors etc.. which as yet have not been explored by the mainstream

exchanges. Tunnels, aka rangebets, aka corridors are reasonably well-known and are

priced in the manner of a conventional call spread although the tunnel is primarily a

volatility trade. Others such as the Duke of York, Tug of War, Accumulators provide a rich

seam of varied instruments providing distinct and unique P&L profiles.

As indicated above, binary options are generally perceived as European-style options that

cannot be exercised before expiry. The American-style binary options are out there but are

usually referred to as one-touch options. A comprehensive list of binary options strategies

would include European and American binary options, 'knock-in' binary options, 'knock-out'

binary options and two-asset binary options.

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