Binary option: Difference between revisions

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{{Short description|Financial exotic option with an all-or-nothing payoff}}

{{Prone to spam|date=January 2014}}

A '''binary option''' is a [[gamblingfinance|financial]] scheme[[exotic option]] in which the payoff is either some fixed monetary amount or nothing at all.<ref name="Breeden, D. T. 1978">Breeden, D. T., & Litzenberger, R. H. (1978). "Prices of state-contingent claims implicit in option prices". ''Journal of Business'', 621–651.</ref><ref name="Gatheral, J. 2006">Gatheral, J. (2006). ''The volatility surface: a practitioner's guide'' (Vol. 357). John Wiley & Sons.</ref> The two main types of binary options are the cash-or-nothing binary option and the asset-or-nothing binary option. The former pays some fixed amount of cash if the option expires [[in-the-money]] while the latter pays the value of the underlying security. They are also called '''all-or-nothing options''', '''digital options''' (more common in forex/interest rate markets), and '''fixed return options''' ('''FROs''') (on the [[NYSE American]]).<ref name="Ipedia">[http://www.investopedia.com/terms/b/binary-option.asp Binary Option Definition] Investopedia. Retrieved 2013-06-30.</ref>

While binary options may be used in theoretical asset pricing, they are prone to [[fraud]] in their applications and hence banned by regulators in many jurisdictions as a form of [[gambling]].<ref name="Globes 12-15-16" /> Many binary option outlets have been exposed as fraudulent.<ref>{{Cite web |url=https://www.fbi.gov/news/stories/binary-options-fraud |title=Binary Options Fraud |website=Federal Bureau of Investigation |language=en-us |access-date=2017-05-30}}</ref> The U.S. [[Federal Bureau of Investigation|FBI]] is investigating binary option scams throughout the world, and the Israeli police have tied the industry to criminal syndicates.<ref name="FBI">{{cite news |last1=Weinglass |first1=Simona |authorlink=Simona Weinglass|title=FBI says it's investigating binary options fraud worldwide, invites victims to come forward |url=http://www.timesofisrael.com/fbi-says-its-investigating-binary-options-fraud-worldwide-invites-victims-to-come-forward/ |access-date=February 15, 2017 |work =[[The Times of Israel]] |date=February 15, 2017}}</ref><ref name="ToIidesofM"/><ref name=":1" /> The [[European Securities and Markets Authority]] (ESMA) has banned retail binary options trading.<ref name="esma.europa.eu">{{Cite web |url=https://www.esma.europa.eu/press-news/esma-news/esma-agrees-prohibit-binary-options-and-restrict-cfds-protect-retail-investors |title=ESMA agrees to prohibit binary options and restrict CFDs to protect retail investors |website=www.esma.europa.eu|access-date=2019-03-21}}</ref> [[Australian Securities & Investments Commission]] (ASIC) considers binary options as a "high-risk" and "unpredictable" investment option, <ref>{{Cite news |url=http://www.ibtimes.com.au/binary-options-trading-australia-how-safe-it-1568192 |title=Binary Options Trading In Australia: How Safe Is It? |date=2018-05-14 |work=International Business Times AU |access-date=2018-05-22 |language=en}}</ref> and finally also banned binary options sale to retail investors in 2021.<ref name="asic.gov.au">{{cite web |url=https://asic.gov.au/about-asic/news-centre/find-a-media-release/2021-releases/21-064mr-asic-bans-the-sale-of-binary-options-to-retail-clients|title=21-064MR ASIC bans the sale of binary options to retail clients |website=[[Australian Securities & Investments Commission]]|access-date=8 March 2022}}</ref>

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Binary options "are based on a simple 'yes' or 'no' proposition: Will an underlying asset be above a certain price at a certain time?"<ref name="investopedia-guide">{{cite web |last1=Mitchell |first1=Cory |title=A Guide To Trading Binary Options In The U.S. |url=https://www.investopedia.com/articles/active-trading/061114/guide-trading-binary-options-us.asp |website=Investopedia |access-date=4 May 2018 |date=11 June 2014}}</ref> Traders place wagers as to whether that will or will not happen. If a customer believes the price of an underlying asset will be above a certain price at a set time, the trader buys the binary option, but if he or she believes it will be below that price, they sell the option. In the U.S. exchanges, the price of a binary is always under $100.<ref name="investopedia-guide"/>

[[Investopedia]] described the binary options trading process in the U.S. thus:

<blockquote>[A] binary may be trading at $42.50 (bid) and $44.50 (offer) at 1 p.m. If you buy the binary option right then you will pay $44.50, if you decide to sell right then you'll sell at $42.50.

Let's assume you decide to buy at $44.50. If at 1:30 p.m. the price of gold is above $1,250, your option expires and it becomes worth $100. You make a profit of $100 – $44.50 = $55.50 (less fees). This is called being "in the money".

But if the price of gold is below $1,250 at 1:30 p.m., the option expires at $0. Therefore you lose the $44.50 invested. This is called being "out of the money".

The bid and offer fluctuate until the option expires. You can close your position at any time before expiry to lock in a profit or a reduce a loss (compared to letting it expire out of the money).<ref name="investopedia-guide"/></blockquote>

In the U.S., every binary option settles at $100 or $0, $100 if the bet is correct, 0 if it is not.<ref name="investopedia-guide"/>

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The U.S. [[Commodity Futures Trading Commission]] warns that "some binary options Internet-based trading platforms may overstate the average return on investment by advertising a higher average return on investment than a customer should expect given the payout structure."<ref name="CFTC investor warning">{{cite web |url=https://www.cftc.gov/ConsumerProtection/FraudAwarenessPrevention/CFTCFraudAdvisories/fraudadv_binaryoptions.html |title=CFTC Fraud Advisories |website=www.cftc.gov |publisher=U.S. Commodity Futures Trading Commission |access-date= 4 May 2018}}</ref>

==Black–Scholes valuation==

In the [[Black–Scholes model]], the price of the option can be found by the formulas below.<ref>{{Cite book |last=Hull |first=John C. |year=2005 |title=Options, Futures and Other Derivatives |publisher=[[Prentice Hall]] |isbn=0-13-149908-4}}</ref> In fact, the [[Black–Scholes formula]] for the price of a vanilla [[call option]] (or [[put option]]) can be [[Black–Scholes#Interpretation|interpreted]] by decomposing 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 <math> \sigma </math> is the [[volatility (finance)|volatility]]. <math>\Phi</math> denotes the [[cumulative distribution function]] of the [[normal distribution]],

:<math> \Phi(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x e^{-z^2/2} dz. </math>

and,

:<math> d_1 = \frac{\ln\frac{S}{K} + (r-q+\sigma^{2}/2)T}{\sigma\sqrt{T}}. </math>

: <math> d_2 = d_1-\sigma\sqrt{T}. </math>

===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

:<math> C = e^{-rT}\Phi(d_2). \,</math>

===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

:<math> P = e^{-rT}\Phi(-d_2). \,</math>

===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

:<math> C = Se^{-qT}\Phi(d_1). \,</math>

===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:

:<math> P = Se^{-qT}\Phi(-d_1). \,</math>

===American style===

[[File:American binary put.png|thumb|American binary put with K = 100, r = 0.04, σ = 0.2, T = 1]]

An [[American option]] gives the holder the right to exercise at any point up to and including the expiry time <math>T</math>. That is, denoting by <math>K</math> the strike price, if <math>K\geq S</math> (resp. <math>K\leq S</math>), the corresponding American binary put (resp. call) is worth exactly one unit. Let

:<math> a=\frac{1}{\sigma}\ln(K/S)\text{, }\xi=\frac{r-q}{\sigma}-\frac{\sigma}{2}\text{, and }b=\sqrt{\xi^{2}+2r}. \,</math>

The price of a cash-or-nothing American binary put (resp. call) with strike <math>K<S</math> (resp. <math>K>S</math>) and time-to-expiry <math>T</math> is:

:<math> \frac{1}{2}e^{a\left(\xi-b\right)}\left\{ 1+\operatorname{sgn}(a)\operatorname{erf}\left(\frac{bT-a}{\sqrt{2T}}\right)+e^{2ab}\left[1-\operatorname{sgn}(a)\operatorname{erf}\left(\frac{bT+a}{\sqrt{2T}}\right)\right]\right\} \,</math>

where <math>\operatorname{erf}</math> denotes the [[error function]] and <math>\operatorname{sgn}</math> denotes the [[sign function]]. The above follows immediately from expressions for the Laplace transform of the distribution of the conditional first passage time of Brownian motion to a particular level.<ref>[http://parsiad.ca/post/closed-form-expressions-for-perpetual-and-finite-maturity-american-binary-options Closed-form expressions for perpetual and finite-maturity American binary options]{{Dead link|date=June 2019 |bot=InternetArchiveBot |fix-attempted=yes }}. parsiad.ca (2015-03-01). Retrieved on 2016-07-18.</ref>

===Foreign exchange===

{{Further|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 <math>r_{\mathrm{FOR}}</math>, the foreign interest rate, <math>r_{DOM}</math>, 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,

:<math> C = e^{-r_{\mathrm{DOM}} T}\Phi(d_2) \,</math>

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,

:<math> P = e^{-r_{\mathrm{DOM}}T}\Phi(-d_2) \,</math>

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,

:<math> C = Se^{-r_{\mathrm{FOR}} T}\Phi(d_1) \,</math>

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,

:<math> P = Se^{-r_{\mathrm{FOR}}T}\Phi(-d_1) \,</math>

===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. The Black–Scholes model relies on symmetry of distribution and ignores the [[skewness]] of the distribution of the asset. Market makers adjust for such skewness by, instead of using a single standard deviation for the underlying asset <math>\sigma</math> across all strikes, incorporating a variable one <math>\sigma(K)</math> where volatility depends on strike price, thus incorporating the [[volatility skew]] into account. The skew matters because it affects the binary considerably more than the regular options.

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 infinitesimally tight spread, where <math>C_v</math> is a vanilla European call:<ref name="Breeden, D. T. 1978"/><ref name="Gatheral, J. 2006"/>

:<math> C = \lim_{\epsilon \to 0} \frac{C_v(K-\epsilon) - C_v(K)}{\epsilon} </math>

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:

:<math> C = -\frac{dC_v}{dK} </math>

When one takes volatility skew into account, <math>\sigma</math> is a function of <math>K</math>:

:<math> 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}</math>

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

:<math> -\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_{\mathrm{noskew}}</math>

<math>\frac{\partial C_v}{\partial \sigma}</math> is the [[Greeks (finance)|Vega]] of the vanilla call; <math>\frac{\partial \sigma}{\partial K}</math> 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.

:<math> C = C_{\mathrm{noskew}} - \mathrm{Vega}_v * \mathrm{Skew}</math>

===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.

==Regulation and fraud==