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 or digital 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.

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[edit] Closed-form solutions for binary options

In the Black-Scholes model, the price of the option can be found by the formulas below. 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 v is the volatility. 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,

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

[edit] 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}N(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}N(-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}N(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}N(-d_1) \,

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