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binary option: Definition from Answers.com

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 or digital options.

For example, suppose I buy 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, I receive $1000. If its stock is trading below $100, I receive nothing.

In the popular Black-Scholes model, the value of a digital option can be expressed in terms of the cumulative normal distribution function.

Closed-form solutions to binary options

In the Black-Scholes model, the price of the option can be found by the formulas below, where Q is the cash payoff, S is the initial stock price, T is the time to maturity, q is the dividend rate, v is the volatility, r is the risk-free interest rate, N denotes the cumulative distribution function of the normal distribution,

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

and K denotes the strike price. In the formulas below,

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

Cash-or-nothing call

$C = Qe^{-rT}N(d_2) \,$

In case of Cash-or-nothing call on currency, i.e. if the underlier is an exchange ratio, then we must distinguish between the following two cases.

For Q defined in home (valuation) currency:

$C = Qe^{-r_{home} T}N(d_1) \,$

For Q defined in foreign currency:

$C = Qe^{-r_{foreign} T}N(d_2)/S_0 \,$

where S₀ is the exchange ratio (foreign divided by home) in valuation date.

Cash-or-nothing put

$P = Qe^{-rT}N(-d_2) \,$

In case of Cash-or-nothing put on currency, i.e. if the underlier is an exchange ratio, then we must distinguish between the following two cases.

For Q defined in home (valuation) currency:

$C = Qe^{-r_{home}T}N(-d_1) \,$

For Q defined in foreign currency:

$C = Qe^{-r_{foreign}T}N(-d_2)/S_0 \,$

where S₀ is the exchange ratio (foreign divided by home) in valuation date.

Asset-or-nothing call

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

Asset-or-nothing put

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

External Links

Online Binary Option Calculators, sitmo.com
How Binary Options Work at Financial-edu.com

Derivatives market
Derivative (finance)
Options	Terms: Strike price · Expiration · Open interest · Pin risk Vanilla options: Option styles · Call · Put · Warrants · Fixed income · Employee stock option · FX Exotic options: Asian · Lookback · Barrier · Binary · Swaption · Mountain range Options strategies: Covered call · Naked put · Collar · Straddle · Strangle · Butterfly Options spreads: Bull spread · Bear spread · Calendar spread · Vertical spread · Debit spread · Credit spread Valuation of options: Moneyness · Option time value · Put-call parity · Black-Scholes · Black · Binomial
Swaps	Interest rate swap · Total return swap · Equity swap · Credit default swap · Forex swap · Currency swap · Constant maturity swap · Basis swap · Volatility swap · Variance swap
Other derivatives	Credit derivative · Equity derivative · Interest rate derivative · Inflation derivatives

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