Poisson summation formula

Template:Prone to spam 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.

When buying a binary option the potential return it offers is certain and known before the purchase is made. Binary options can be bought on virtually any financial product and can be bought in both directions of trade either by buying a “Call”/“Up” option or a “Put”/“Down” option. This means that an investor can go long or short on any financial product simply by buying a binary option. Binary options are offered against a fixed expiry time which may be e.g. 60 seconds and up to 30 minutes, an hour ahead or to the close of the trading day.

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. Due to the difficulty in hedging binary options that are near expiry, they are much less traded than vanilla options, but can be approximated with vertical spreads.

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 websites called binary option trading platforms have been offering a simplified version of exchange-traded binary options. It is estimated that around 90 such platforms (including white label products) have been in operation as of January 2012, offering options on some 125 underlying assets.

The platforms offer standardized short-term binary options with a pre-determined profit/loss, that cannot be liquidated (buy or sell to close) before expiry.Template:Disputed-inline

The US Securities and Exchange Commission (SEC) and Commodity Futures Trading Commission (CFTC) have issued a joint warning to investors regarding binary options.

Business model

The platforms do not charge fees from their investors. Their profit comes from the difference between the options that expire in the money to options that expire out of the money. This difference can be found by the formula below. In this (for each base asset with same expiry characteristics), (W) is the in the money option payout in percentage terms (e.g. 1.7), (L) is the out of the money option payout in percentage terms (e.g. 0.15), $(V_1),(V_2)$ are the turnovers of transactions made for each outcome (e.g. $1,000), (S) is the platform's gain.

$S=-[V_1(W-1)+V_2(L-1)]$

In this example the platform's turnover is $2,000 and its profit is $150 or 7.5% on turnover. As the platform’s gain comes from the above formula, most platforms will be indifferent as to the outcome of a single trade. Note that if $V_1$ is not equal to $V_2$ then the platform will have to act as a market maker. This can cause the platform gain (S) to be more volatile than in the above formula. In order for a trader to make a long term profit he has to predict correctly roughly 54.5% of the time (depending on in-the-money and out-of-the money payouts).

Regulation and compliance

On non-regulated platforms, clients’ money is not necessarily kept in a trust account, as required by regulation, and transactions are not monitored by a third party to ensure fair play.

On May 3, 2012, the Cyprus Securities and Exchange Commission (CySEC) announced a policy change regarding the classification of binary options as financial instruments. The effect is that binary options platforms operating in Cyprus (where many of the platforms are based) will have to be CySEC regulated within six months of the date of the announcement. CySEC was the first EU MiFID-member regulator to treat binary options as financial instruments. In March 2013, Malta's Financial Services Authority announced that Binary options regulation would be transferred away from Malta's Lottery and Gaming Authority. On the 18th June 2013, Malta’s Financial Services Authority confirmed that in their view binary options fell under the scope of the Markets In Financial Instruments Directive (MiFID),2004/39/EC. With this announcement Malta became the second EU jurisdiction to regulate binary options as a financial instrument, providers will now have to gain a category 3 Investment Services licence and conform to MiFID's minimum capital requirements. Prior to this announcement it had been possible for firms to operate from the jurisdiction provided the firm had a valid Lottery and Gaming Authority licence.  

The Commodity Futures Trading Commission (CFTC) is a government agency that oversees the regulation of futures, options and swaps trading in the United States. On the 6th June 2013, Banc de Binary was charged by both the CTFC and SEC for alleged violations of US Financial Regulation. Both the CTFC and SEC working in conjunction with one another filed civil suits against the company, seeking disgorgement, plus financial penalties as well other preliminary and permanent injunctions against Banc De Binary. In civil suites filed in Nevada, the SEC and CTFC allege that Banc de Binary were offering off exchange traded options to US Customers and unlawfully solicited US customers to buy and sell options. The CTFC and SEC's suit also alleges that Banc de Binary did not limit its offerings to eligible contract participants contrary to use regulation.

Criticism

These platforms may be considered by some as gaming platforms rather than investment platforms because of their negative cumulative payout (they have an edge over the investor) and because they require little or no knowledge of the stock market to trade. According to Gordon Pape, writing in Forbes:

"[...] this sort of thing can quickly become addictive [...] no one, no matter how knowledgeable, can consistently predict what a stock or commodity will do within a short time frame."

Exchange-traded binary options

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.

In 2009, Nadex, a US based binary options provider launched binary options on a range of forex, commodities and stock indices markets.

Example of a binary options trade

A trader who thinks that the EUR/USD 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 price will close at or below 1.2500 at 3:00 p.m. can buy a put option or sell a call option contract.

At 2:00 p.m. the EUR/USD 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 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.

Black–Scholes valuation

In the Black–Scholes model, the price of the option can be found by the formulas below. In fact, the Black–Scholes formula for the price of a vanilla call option (or put option) can be interpreted by decomposition a call option into an asset-or-nothing call option minus a cash-or-nothing call option, and similarly for a put – the binary options are easier to analyze, and correspond to the two terms in the Black–Scholes formula.

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. $\mathrm{\Phi }$ denotes the cumulative distribution function of the normal distribution,

$\mathrm{\Phi }(x)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{x}{\int }}{e}^{-\frac{1}{2}{z}^2}dz.$

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,

$C=e^{-rT}\mathrm{\Phi }(d_2).$

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,

$P=e^{-rT}\mathrm{\Phi }(-d_2).$

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,

$C=S{e}^{-qT}\mathrm{\Phi }(d_1).$

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,

$P=S{e}^{-qT}\mathrm{\Phi }(-d_1).$

Foreign exchange

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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,

$C=e^{-r_{DOM}T}\mathrm{\Phi }(d_2)$

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,

$P=e^{-r_{DOM}T}\mathrm{\Phi }(-d_2)$

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,

$C=S{e}^{-r_{FOR}T}\mathrm{\Phi }(d_1)$

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,

$P=S{e}^{-r_{FOR}T}\mathrm{\Phi }(-d_1)$

Skew

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Also visit my web site ... hostgator1centcoupon.info 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:Template:Page needed,

$C=\underset{ϵ\to 0}{\lim }\frac{C_v(K-ϵ)-C_v(K)}{ϵ}$

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:

$C=-\frac{d{C}_v}{dK}$

When one takes volatility skew into account, $\sigma$ is a function of $K$:

$C=-\frac{d{C}_v(K,\sigma (K))}{dK}=-\frac{\partial C_v}{\partial K}-\frac{\partial C_v}{\partial \sigma }\frac{\partial \sigma }{\partial K}$

The first term is equal to the premium of the binary option ignoring skew:

$-\frac{\partial C_v}{\partial K}=-\frac{\partial (S\mathrm{\Phi }(d_1)-K{e}^{-rT}\mathrm{\Phi }(d_2))}{\partial K}=e^{-rT}\mathrm{\Phi }(d_2)=C_{noskew}$

$\frac{\partial C_v}{\partial \sigma }$ 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.

$C=C_{noskew}-Veg{a}_v*Skew$

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.

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.

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.

See also

• Options strategies
• Options spread
• Options arbitrage
• Synthetic position

External links

• CFTC investor alert

References

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