Binary option
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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 it 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
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 and r is the risk free rate. N denotes the cumulative distribution function of the normal distribution and K denotes the strike price.
Cash-or-nothing call
- [ P = Qe^N(d_2) \,]
- [ d_1 = \frac + (r-q+v^/2)T}} \,]
- [ d_2 = d_1-v\sqrt \,]
Cash-or-nothing put
- [ P = Qe^N(-d_2) \,]
- [ d_1 = \frac + (r-q+v^/2)T}} \,]
- [ d_2 = d_1-v\sqrt \,]
Asset-or-nothing call
- [ P = Se^N(d_1) \,]
- [ d_1 = \frac + (r-q+v^/2)T}} \,]
- [ d_2 = d_1-v\sqrt \,]
Asset-or-nothing put
- [ P = Se^N(-d_1) \,]
- [ d_1 = \frac + (r-q+v^/2)T}} \,]
- [ d_2 = d_1-v\sqrt \,]
See also
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