Binary option

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).

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.

American vs European

Binary options are usually European-style - for a call, the price of the underlying must be above the strike at the expiration date. American exist also, but these automatically exercise whenever the price "touches" the strike price, yielding very different behaviour. See a comparison of binary options to standard (or American) options.

Exchange-traded binary options

Binary option 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.

In 2007, the Options Clearing Corporation proposed a rule change to allow binary options, and the Securities and Exchange Commission approved listing cash-or-nothing binary options in 2008. 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. 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. 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.

CBOE offers binary options on the S&P 500 (SPX) and the CBOE Volatility Index (VIX). The tickers for these are BSZ and BVZ, 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.

Black-Scholes Valuation

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 $\sigma$ 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,

$d_{1}={\frac {\ln {\frac {S}{K}}+(r-q+\sigma ^{2}/2)T}{\sigma {\sqrt {T}}}},\,d_{2}=d_{1}-\sigma {\sqrt {T}}.\,$

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

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,

$P=Se^{-r_{FOR}T}\Phi (-d_{1})\,$

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: ,

$C=\lim _{\epsilon \to 0}{\frac {C_{v}(K-\epsilon )-C_{v}(K)}{\epsilon }}$

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.

$C=C_{noskew}-Vega_{v}*Skew$

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, 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.

A portfolio of binary options can also be used to synthetically recreate (or valuate) any other option (analogous to integration).

External links

• How Binary Options Work at Financial-edu.com
• Binary Options Strategies at BinaryOptionWIki.com