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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==
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