Editing Black–Scholes model (section)

==Extensions of the model==
The above model can be extended for variable (but deterministic) rates and volatilities. The model may also be used to value European options on instruments paying dividends. In this case, closed-form solutions are available if the dividend is a known proportion of the stock price. [[Option style|American options]] and options on stocks paying a known cash dividend (in the short term, more realistic than a proportional dividend) are more difficult to value, and a choice of solution techniques is available (for example [[Lattice model (finance)|lattices]] and [[Finite difference methods for option pricing|grids]]).

===Instruments paying continuous yield dividends===
For options on indices, it is reasonable to make the simplifying assumption that dividends are paid continuously, and that the dividend amount is proportional to the level of the index.

The dividend payment paid over the time period <math>[t, t + dt]</math> is then modelled as:
:<math>qS_t\,dt</math>

for some constant <math>q</math> (the [[dividend yield]]).

Under this formulation the arbitrage-free price implied by the Black–Scholes model can be shown to be:
:<math>C(S_t, t) = e^{-r(T - t)}[FN(d_1) - KN(d_2)]\,</math>

and
:<math>P(S_t, t) = e^{-r(T - t)}[KN(-d_2) - FN(-d_1)]\,</math>

where now
:<math>F = S_t e^{(r - q)(T - t)}\,</math>

is the modified forward price that occurs in the terms <math>d_1, d_2</math>:
:<math>d_1 = \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r - q + \frac{1}{2}\sigma^2\right)(T - t)\right]</math>

and
:<math>d_2 = d_1 - \sigma\sqrt{T - t} = \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r - q - \frac{1}{2}\sigma^2\right)(T - t)\right]</math>.<ref name="finance.bi.no, 2017">{{cite web|title=Extending the Black Scholes formula|url=http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node9.html|website=finance.bi.no|access-date=July 21, 2017|date=October 22, 2003}}</ref>

===Instruments paying discrete proportional dividends===
It is also possible to extend the Black–Scholes framework to options on instruments paying discrete proportional dividends. This is useful when the option is struck on a single stock.

A typical model is to assume that a proportion <math>\delta</math> of the stock price is paid out at pre-determined times <math>t_1, t_2, \ldots, t_n </math>. The price of the stock is then modelled as:
:<math>S_t = S_0(1 - \delta)^{n(t)}e^{ut + \sigma W_t}</math>

where <math>n(t)</math> is the number of dividends that have been paid by time <math>t</math>.

The price of a call option on such a stock is again:
:<math>C(S_0, T) = e^{-rT}[FN(d_1) - KN(d_2)]\,</math>

where now
:<math>F = S_{0}(1 - \delta)^{n(T)}e^{rT}\,</math>

is the forward price for the dividend paying stock.

===American options===
The problem of finding the price of an [[American option]] is related to the [[optimal stopping]] problem of finding the time to execute the option. Since the American option can be exercised at any time before the expiration date, the Black–Scholes equation becomes a variational inequality of the form:
:<math>\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV \leq 0</math><ref>{{cite web|author = André Jaun |url=http://www.lifelong-learners.com/opt/com/SYL/s6node6.php|title=The Black–Scholes equation for American options|access-date=May 5, 2012}}</ref>
together with <math>V(S, t) \geq H(S)</math> where <math>H(S)</math> denotes the payoff at stock price <math>S</math> and the terminal condition: <math>V(S, T) = H(S)</math>.

In general this inequality does not have a closed form solution, though an American call with no dividends is equal to a European call and the Roll–Geske–Whaley method provides a solution for an American call with one dividend;<ref name="Ødegaard">{{cite web|author=Bernt Ødegaard |year=2003|url=http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node9.html#SECTION00920000000000000000|title=Extending the Black Scholes formula|access-date=May 5, 2012}}</ref><ref name="Chance2">{{cite web|author=Don Chance|year=2008 |url=http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN98-01.pdf|title= Closed-Form American Call Option Pricing: Roll-Geske-Whaley|access-date=May 16, 2012}}</ref> see also [[Black's approximation]].

Barone-Adesi and Whaley<ref>{{cite journal|author= Giovanni Barone-Adesi|author2= Robert E Whaley|name-list-style= amp|title=Efficient analytic approximation of American option values|journal=Journal of Finance|volume=42 | issue = 2|date=June 1987|pages=301–20|url=https://ideas.repec.org/a/bla/jfinan/v42y1987i2p301-20.html|doi=10.2307/2328254|jstor= 2328254}}</ref> is a further approximation formula. Here, the stochastic differential equation (which is valid for the value of any derivative) is split into two components: the European option value and the early exercise premium. With some assumptions, a [[quadratic equation]] that approximates the solution for the latter is then obtained. This solution involves [[root-finding algorithms|finding the critical value]], <math>s*</math>, such that one is indifferent between early exercise and holding to maturity.<ref name="Ødegaard2">{{cite web|author=Bernt Ødegaard |year=2003|url=http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node13.html|title=A quadratic approximation to American prices due to Barone-Adesi and Whaley|access-date=June 25, 2012}}</ref><ref name="Chance3">{{cite web|author=Don Chance|year=2008 |url=http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN98-02.pdf|title= Approximation Of American Option Values: Barone-Adesi-Whaley|access-date=June 25, 2012}}</ref>

Bjerksund and Stensland<ref>Petter Bjerksund and Gunnar Stensland, 2002.  [http://brage.bibsys.no/nhh/bitstream/URN:NBN:no-bibsys_brage_22301/1/bjerksund%20petter%200902.pdf Closed Form Valuation of American Options]</ref> provide an approximation based on an exercise strategy corresponding to a trigger price. Here, if the underlying asset price is greater than or equal to the trigger price it is optimal to exercise, and the value must equal <math>S - X</math>, otherwise the option "boils down to: (i) a European [[Barrier option#Types|up-and-out]] call option… and (ii) a rebate that is received at the knock-out date if the option is knocked out prior to the maturity date". The formula is readily modified for the valuation of a put option, using [[put–call parity]]. This approximation is computationally inexpensive and the method is fast, with evidence indicating that the approximation may be more accurate in pricing long dated options than Barone-Adesi and Whaley.<ref>[http://www.global-derivatives.com/index.php?option=com_content&task=view&id=14 American options]</ref>

==== Perpetual put ====
Despite the lack of a general analytical solution for American put options, it is possible to derive such a formula for the case of a perpetual option – meaning that the option never expires (i.e., <math>T\rightarrow \infty</math>).<ref>{{Cite book|title=Heard on the Street: Quantitative Questions from Wall Street Job Interviews|last=Crack|first=Timothy Falcon|publisher=Timothy Crack|year=2015|isbn=978-0-9941182-5-7|edition=16th|pages=159–162}}</ref> In this case, the time decay of the option is equal to zero, which leads to the Black–Scholes PDE becoming an ODE:<math display="block">{1\over{2}}\sigma^{2}S^{2}{d^{2}V\over{dS^{2}}} + (r-q)S{dV\over{dS}} - rV = 0</math>Let <math>S_{-}</math> denote the lower exercise boundary, below which it is optimal to exercise the option. The boundary conditions are:<math display="block">V(S_{-}) = K-S_{-}, \quad {dV\over{dS}}(S_{-}) = -1, \quad V(S) \leq K</math>The solutions to the ODE are a linear combination of any two linearly independent solutions:<math display="block">V(S) = A_{1}S^{\lambda_{1}} + A_{2}S^{\lambda_{2}}</math>For <math>S_{-} \leq S</math>, substitution of this solution into the ODE for <math>i = {1,2}</math> yields:<math display="block">\left[ {1\over{2}}\sigma^{2}\lambda_{i}(\lambda_{i}-1) + (r-q)\lambda_{i} - r \right]S^{\lambda_{i}} = 0</math>Rearranging the terms gives:<math display="block">{1\over{2}}\sigma^{2}\lambda_{i}^{2} + \left(r-q - {1\over{2}} \sigma^{2}\right)\lambda_{i} - r = 0</math>Using the [[quadratic formula]], the solutions for <math>\lambda_{i}</math> are:<math display="block">\begin{aligned}
\lambda_{1} &= {-\left(r-q-{1\over{2}}\sigma^{2} \right ) + \sqrt{\left(r-q-{1\over{2}}\sigma^{2} \right )^{2} + 2\sigma^{2}r}\over{\sigma^{2}}} \\
\lambda_{2} &= {-\left(r-q-{1\over{2}}\sigma^{2} \right ) - \sqrt{\left(r-q-{1\over{2}}\sigma^{2} \right )^{2} + 2\sigma^{2}r}\over{\sigma^{2}}}
\end{aligned}</math>In order to have a finite solution for the perpetual put, since the boundary conditions imply upper and lower finite bounds on the value of the put, it is necessary to set <math>A_{1} = 0</math>, leading to the solution <math>V(S) = A_{2}S^{\lambda_{2}}</math>. From the first boundary condition, it is known that:<math display="block">V(S_{-}) = A_{2}(S_{-})^{\lambda_{2}} = K-S_{-} \implies A_{2} = {K-S_{-}\over{(S_{-})^{\lambda_{2}}}}</math>Therefore, the value of the perpetual put becomes:<math display="block">V(S) = (K-S_{-})\left( {S\over{S_{-}}} \right)^{\lambda_{2}}</math>The second boundary condition yields the location of the lower exercise boundary:<math display="block">{dV\over{dS}}(S_{-}) = \lambda_{2}{K-S_{-}\over{S_{-}}} = -1 \implies S_{-} = {\lambda_{2}K\over{\lambda_{2}-1}}</math>To conclude, for <math display="inline">S \geq S_{-} = {\lambda_{2}K\over{\lambda_{2}-1}}</math>, the perpetual American put option is worth:<math display="block">V(S) = {K\over{1-\lambda_{2}}} \left( {\lambda_{2}-1\over{\lambda_{2}}}\right)^{\lambda_{2}} \left( {S\over{K}} \right)^{\lambda_{2}}</math>

===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.