Editing Black–Scholes model (section)

=={{anchor|BSFormula}}Black–Scholes formula==
[[File:European Call Surface.png|thumbnail|A European call valued using the Black–Scholes pricing equation for varying asset price <math>S</math> and time-to-expiry <math>T</math>. In this particular example, the strike price is set to 1.]]

The Black–Scholes formula calculates the price of [[European option|European]] [[Put option|put]] and [[call option]]s. This price is [[Consistency|consistent]] with the Black–Scholes equation. This follows since the formula can be obtained [[Equation solving#Differential equations|by solving]] the equation for the corresponding terminal and [[Boundary Conditions|boundary conditions]]:

:<math>\begin{align}
& C(0, t) = 0\text{ for all }t \\
& C(S, t) \rightarrow S - K \text{ as }S \rightarrow \infty \\
& C(S, T) = \max\{S - K, 0\}
\end{align}</math>

The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:
:<math>\begin{align}
  C(S_t, t) &= N(d_+)S_t - N(d_-)Ke^{-r(T - t)} \\
     d_+ &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\
     d_- &= d_+ - \sigma\sqrt{T - t} \\
\end{align}</math>

The price of a corresponding put option based on [[put–call parity]] with [[discount factor]] <math>e^{-r(T-t)}</math> is:
:<math>\begin{align}
  P(S_t, t) &= Ke^{-r(T - t)} - S_t + C(S_t, t) \\
          &= N(-d_-) Ke^{-r(T - t)} - N(-d_+) S_t
\end{align}\,</math>

===Alternative formulation===
Introducing auxiliary variables allows for the formula to be simplified and reformulated in a form that can be more convenient (this is a special case of the [[Black model|Black '76 formula]]):
:<math>\begin{align}
  C(F, \tau) &= D \left[ N(d_+) F - N(d_-) K \right] \\
       d_+ &=
         \frac{1}{\sigma\sqrt{\tau}}\left[\ln\left(\frac{F}{K}\right) + \frac{1}{2}\sigma^2\tau\right] \\
       d_- &= d_+ - \sigma\sqrt{\tau}
\end{align}</math>

where:

<math>D = e^{-r\tau}</math> is the discount factor

<math>F = e^{r\tau} S = \frac{S}{D}</math> is the [[forward price]] of the underlying asset, and <math>S = DF</math>

Given put–call parity, which is expressed in these terms as:
:<math>C - P = D(F - K) = S - D K</math>

the price of a put option is:
:<math>P(F, \tau) = D \left[ N(-d_-) K - N(-d_+) F \right]</math>

===Interpretation===
It is possible to have intuitive interpretations of the Black–Scholes formula, with the main subtlety being the interpretation of <math>d_\pm</math> and why there are two different terms.<ref name="Nielsen"/>

The formula can be interpreted by first decomposing a call option into the difference of two [[binary option]]s: an [[asset-or-nothing call]] minus a [[cash-or-nothing call]] (long an asset-or-nothing call, short a cash-or-nothing call). A call option exchanges cash for an asset at expiry, while an asset-or-nothing call just yields the asset (with no cash in exchange) and a cash-or-nothing call just yields cash (with no asset in exchange). The Black–Scholes formula is a difference of two terms, and these two terms are equal to the values of the binary call options. These binary options are less frequently traded than vanilla call options, but are easier to analyze.

Thus the formula:
:<math>C = D \left[ N(d_+) F - N(d_-) K \right]</math>

breaks up as:
:<math>C = D N(d_+) F - D N(d_-) K,</math>

where <math>D N(d_+) F</math> is the present value of an asset-or-nothing call and <math>D N(d_-) K</math> is the present value of a cash-or-nothing call. The ''D'' factor is for discounting, because the expiration date is in future, and removing it changes ''present'' value to ''future'' value (value at expiry). Thus <math>N(d_+) ~ F</math> is the future value of an asset-or-nothing call and <math>N(d_-) ~ K</math> is the future value of a cash-or-nothing call. In risk-neutral terms, these are the [[expected value]] of the asset and the expected value of the cash in the risk-neutral measure.

A naive, and slightly incorrect, interpretation of these terms is that <math>N(d_+) F</math> is the probability of the option expiring in the money <math>N(d_+)</math>, multiplied by the value of the underlying at expiry ''F,'' while <math>N(d_-) K</math> is the probability of the option expiring in the money <math>N(d_-),</math> multiplied by the value of the cash at expiry ''K.'' This interpretation is incorrect because either both binaries expire in the money or both expire out of the money (either cash is exchanged for the asset or it is not), but the probabilities <math>N(d_+)</math> and <math>N(d_-)</math> are not equal. In fact, <math>d_\pm</math> can be interpreted as measures of [[moneyness]] (in standard deviations) and <math>N(d_\pm)</math> as probabilities of expiring ITM (''percent moneyness''), in the respective [[numéraire]], as discussed below. Simply put, the interpretation of the cash option, <math>N(d_-) K</math>, is correct, as the value of the cash is independent of movements of the underlying asset, and thus can be interpreted as a simple product of "probability times value", while the <math>N(d_+) F</math> is more complicated, as the probability of expiring in the money and the value of the asset at expiry are not independent.<ref name="Nielsen"/> More precisely, the value of the asset at expiry is variable in terms of cash, but is constant in terms of the asset itself (a fixed quantity of the asset), and thus these quantities are independent if one changes numéraire to the asset rather than cash.

If one uses spot ''S'' instead of forward ''F,'' in <math>d_\pm</math> instead of the <math display="inline">\frac{1}{2}\sigma^2</math> term there is <math display="inline">\left(r \pm \frac{1}{2}\sigma^2\right)\tau,</math> which can be interpreted as a drift factor (in the risk-neutral measure for appropriate numéraire). The use of ''d''<sub>−</sub> for moneyness rather than the standardized moneyness <math display="inline">m = \frac{1}{\sigma\sqrt{\tau}}\ln\left(\frac{F}{K}\right)</math>{{snd}} in other words, the reason for the <math display="inline">\frac{1}{2}\sigma^2</math> factor{{snd}} is due to the difference between the median and mean of the [[log-normal distribution]]; it is the same factor as in [[Itō's lemma#Geometric Brownian motion|Itō's lemma applied to geometric Brownian motion]]. In addition, another way to see that the naive interpretation is incorrect is that replacing <math>N(d_+)</math> by <math>N(d_-)</math> in the formula yields a negative value for out-of-the-money call options.<ref name="Nielsen"/>{{rp|6}}

In detail, the terms <math>N(d_+), N(d_-)</math> are the ''probabilities of the option expiring in-the-money'' under the equivalent exponential [[Martingale (probability theory)|martingale]] probability measure (numéraire=stock) and the equivalent martingale probability measure (numéraire=risk free asset), respectively.<ref name="Nielsen"/> The risk neutral probability density for the stock price <math>S_T \in (0, \infty)</math> is
:<math>p(S, T) = \frac{N^\prime [d_-(S_T)]}{S_T \sigma\sqrt{T}}</math>

where <math>d_- = d_-(K)</math> is defined as above.

Specifically, <math>N(d_-)</math> is the probability that the call will be exercised provided one assumes that the asset drift is the risk-free rate. <math>N(d_+)</math>, however, does not lend itself to a simple probability interpretation. <math>SN(d_+)</math> is correctly interpreted as the present value, using the risk-free interest rate, of the expected asset price at expiration, [[Conditional probability|given that]] the asset price at expiration is above the exercise price.<ref name="Chance 99-02">{{cite CiteSeerX |author=Don Chance |date=June 3, 2011 |title=Derivation and Interpretation of the Black–Scholes Model |citeseerx=10.1.1.363.2491 }}</ref> For related discussion{{snd}} and graphical representation{{snd}} see [[Datar–Mathews method for real option valuation#Transformation to the Black–Scholes Option|Datar–Mathews method for real option valuation]].

The equivalent martingale probability measure is also called the [[Financial mathematics#Derivatives pricing: the Q world|risk-neutral probability measure]]. Note that both of these are ''probabilities'' in a [[measure (mathematics)|measure theoretic]] sense, and neither of these is the true probability of expiring in-the-money under the [[financial mathematics#Risk and portfolio management: the P world|real probability measure]]. To calculate the probability under the real ("physical") probability measure, additional information is required—the drift term in the physical measure, or equivalently, the [[market price of risk]].

====Derivations====
{{See also|Martingale pricing}}
A standard derivation for solving the Black–Scholes PDE is given in the article [[Black–Scholes equation]].

The [[Feynman–Kac formula]] says that the solution to this type of PDE, when discounted appropriately, is actually a [[martingale (probability theory)|martingale]]. Thus the option price is the expected value of the discounted payoff of the option. Computing the option price via this expectation is the [[risk neutrality]] approach and can be done without knowledge of PDEs.<ref name="Nielsen">{{cite web |first= Lars Tyge |last= Nielsen | year=1993 | url= http://www.ltnielsen.com/wp-content/uploads/Understanding.pdf | title = Understanding ''N''(''d''<sub>1</sub>) and ''N''(''d''<sub>2</sub>): Risk-Adjusted Probabilities in the Black–Scholes Model |website=LT Nielsen}}</ref> Note the [[expected value|expectation]] of the option payoff is not done under the real world [[probability measure]], but an artificial [[risk-neutral measure]], which differs from the real world measure. For the underlying logic see section [[Rational pricing#Risk neutral valuation|"risk neutral valuation"]] under [[Rational pricing]] as well as section [[Mathematical finance#Derivatives pricing: the Q world|"Derivatives pricing: the Q world]]" under [[Mathematical finance]]; for details, once again, see [[John C. Hull (economist)|Hull]].<ref name="Hull">{{Cite book|last=Hull |first=John C. |year=2008| edition=7th |title=Options, Futures and Other Derivatives |publisher=[[Prentice Hall]] |isbn=978-0-13-505283-9}}</ref>{{rp|307–309}}