Editing Black–Scholes model (section)

===Binary options===
By solving the Black–Scholes differential equation with the [[Heaviside function]] as a boundary condition, one ends up with the pricing of options that pay one unit above some predefined strike price and nothing 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 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.

====Cash-or-nothing call====
This pays out one unit of cash if the spot is above the strike at maturity. Its value is given by:
:<math> C =e^{-r (T-t)}N(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 is given by:
:<math> P = e^{-r (T-t)}N(-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 is given by:
:<math> C = Se^{-q (T-t)}N(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 is given by:
:<math> P = Se^{-q (T-t)}N(-d_1),</math>

====Foreign Exchange (FX) ====
{{Further|Foreign exchange derivative}}

Denoting by ''S'' the FOR/DOM exchange rate (i.e., 1 unit of foreign currency is worth S units of domestic currency) one 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 by taking <math>r_{f}</math>, the foreign interest rate, <math>r_{d}</math>, the domestic interest rate, and the rest as above, the following results can be obtained:

In the case of a digital call (this is a call FOR/put DOM) paying out one unit of the domestic currency gotten as present value:
:<math> C = e^{-r_{d} T}N(d_2) \,</math>
In the case of a digital put (this is a put FOR/call DOM) paying out one unit of the domestic currency gotten as present value:
:<math> P = e^{-r_{d}T}N(-d_2) \,</math>
In the case of a digital call (this is a call FOR/put DOM) paying out one unit of the foreign currency gotten as present value:
:<math> C = Se^{-r_{f} T}N(d_1) \,</math>
In the case of a digital put (this is a put FOR/call DOM) paying out one unit of the foreign currency gotten as present value:
:<math> P = Se^{-r_{f}T}N(-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>Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. Journal of business, 621-651.</ref><ref>Gatheral, J. (2006). The volatility surface: a practitioner's guide (Vol. 357). John Wiley & Sons.</ref>
:<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 N(d_1) - Ke^{-r(T-t)} N(d_2))}{\partial K} = e^{-r (T-t)} N(d_2) = C_\text{no skew}</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". If the skew is typically negative, the value of a binary call will be higher when taking skew into account.
:<math> C = C_\text{no skew} - \text{Vega}_v \cdot \text{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.