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| {{use mdy dates|date=August 2012}}
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| The '''Black–Scholes''' {{IPAc-en|ˌ|b|l|æ|k|_|ˈ|ʃ|oʊ|l|z}}<ref>{{cite web|title=Scholes|url=http://www.merriam-webster.com/dictionary/scholes|accessdate=March 26, 2012}}</ref> or '''Black–Scholes–Merton''' model is a [[mathematical model]] of a [[financial market]] containing certain [[Derivative (finance)|derivative]] investment instruments. From the model, one can deduce the '''Black–Scholes formula''', which gives a theoretical estimate of the price of [[option style|European-style]] [[option (finance)|options]]. The formula led to a boom in options trading and legitimised scientifically the activities of the [[Chicago Board Options Exchange]] and other options markets around the world.<ref name="mackenzie">{{cite book |title= An Engine, Not a Camera: How Financial Models Shape Markets|last= MacKenzie|first= Donald|authorlink=Donald Angus MacKenzie |year= 2006|publisher= MIT Press|location= Cambridge, MA|isbn=0-262-13460-8 }}</ref> lt is widely used, although often with adjustments and corrections, by options market participants.<ref name="bodie-kane-marcus">{{cite book |title= Investments|last= Bodie|first= Zvi|authorlink=Zvi Bodie |coauthors= Alex Kane, Alan J. Marcus|year= 2008|publisher= McGraw-Hill/Irwin|location= New York|edition=7th|isbn=978-0-07-326967-2 }}</ref>{{rp|751}} Many empirical tests have shown that the Black–Scholes price is "fairly close" to the observed prices, although there are well-known discrepancies such as the "[[Volatility smile|option smile]]".<ref name="bodie-kane-marcus"/>{{rp|770–771}}
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| The Black–Scholes was first published by [[Fischer Black]] and [[Myron Scholes]] in their 1973 paper, "The Pricing of Options and Corporate Liabilities", published in the ''[[Journal of Political Economy]]''. They derived a stochastic [[partial differential equation]], now called the [[Black–Scholes equation]], which estimates the price of the option over time. The key idea behind the model is to [[hedge (finance)|hedge]] the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk. This type of hedge is called [[delta hedging]] and is the basis of more complicated hedging strategies such as those engaged in by [[investment bank]]s and [[hedge fund]]s.
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| [[Robert C. Merton]] was the first to publish a paper expanding the mathematical understanding of the options pricing model, and coined the term "Black–Scholes [[options pricing]] model". Merton and Scholes received the 1997 [[Nobel Prize in Economics]] (The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) for their work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.<ref>{{cite web|url=http://nobelprize.org/nobel_prizes/economics/laureates/1997/press.html|title=Nobel Prize Foundation, 1997 Press release|accessdate=March 26, 2012|date=October 14, 1997}}</ref>
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| The model's assumptions have been relaxed and generalized in a variety of directions, leading to a plethora of models which are currently used in derivative pricing and [[risk management]]. It is the insights of the model, as exemplified in the [[Black-Scholes formula]], that are frequently used by market participants, as distinguished from the actual prices. These insights include [[no-arbitrage bounds]] and [[risk-neutral measure|risk-neutral pricing]]. The [[Black-Scholes equation]], a partial differential equation that governs the price of the option, is also important as it enables pricing when an explicit formula is not possible.
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| The Black–Scholes formula has only one parameter that cannot be observed in the market: the average future volatility of the underlying asset. Since the formula is increasing in this parameter, it can be inverted to produce a "[[volatility surface]]" that is then used to calibrate other models, e.g. for [[Derivative (finance)#OTC and exchange-traded|OTC derivatives]].
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| ==The Black-Scholes world==
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| The Black–Scholes model assumes that the market consists of at least one risky asset, usually called the stock, and one riskless asset, usually called the money market, cash, or bond.
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| Now we make assumptions on the assets (which explain their names):
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| * (riskless rate) The rate of return on the riskless asset is constant and thus called the [[risk-free interest rate]].
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| * (random walk) The instantaneous log returns of the stock price is an infinitesimal [[random walk]] with drift; more precisely, it is a [[geometric Brownian motion]], and we will assume its drift and volatility is constant (if they are time-varying, we can deduce a suitably modified Black–Scholes formula quite simply, as long as the volatility is not random).
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| * The stock does not pay a [[dividend]].<ref name="div_yield" group="Notes">Although the original model assumed no dividends, trivial extensions to the model can accommodate a continuous dividend yield factor.</ref>
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| Assumptions on the market:
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| * There is no [[arbitrage]] opportunity (i.e., there is no way to make a riskless profit).
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| * It is possible to borrow and lend any amount, even fractional, of cash at the riskless rate.
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| * It is possible to buy and sell any amount, even fractional, of the stock (this includes [[short selling]]).
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| * The above transactions do not incur any fees or costs (i.e., [[frictionless market]]).
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| With these assumptions holding, suppose there is a derivative security also trading in this market. We specify that this security will have a certain payoff at a specified date in the future, depending on the value(s) taken by the stock up to that date. It is a surprising fact that the derivative's price is completely determined at the current time, even though we do not know what path the stock price will take in the future. For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a [[Hedge (finance)|hedged position]], consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock".<ref>{{cite journal|author=Black, Fischer|author2=Scholes, Myron|title=The Pricing of Options and Corporate Liabilities|journal=Journal of Political Economy|volume=81|issue=3|pages=637–654}}</ref> Their dynamic hedging strategy led to a partial differential equation which governed the price of the option. Its solution is given by the Black–Scholes formula.
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| Several of these assumptions of the original model have been removed in subsequent extensions of the model. Modern versions account for dynamic interest rates (Merton, 1976){{Citation needed|date=November 2010}}, [[transaction cost]]s and taxes (Ingersoll, 1976){{Citation needed|date=November 2010}}, and dividend payout.<ref name="merton 1973">{{cite journal|last=Merton|first=Robert|title=Theory of Rational Option Pricing|journal=Bell Journal of Economics and Management Science|volume=4|issue=1|pages=141–183|doi=10.2307/3003143}}</ref>
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| ==Notation==
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| Let | |
| :<math>S</math>, be the price of the stock, which will sometimes be a random variable and other times a constant (context should make this clear).
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| :<math>V(S, t)</math>, the price of a derivative as a function of time and stock price.
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| :<math>C(S, t)</math> the price of a European call option and <math>P(S, t)</math> the price of a European put option.
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| :<math>K</math>, the [[strike price]] of the option.
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| :<math>r</math>, the annualized [[risk-free interest rate]], [[Continuous compounding|continuously compounded]] (the [[force of interest]]).
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| :<math>\mu</math>, the [[drift rate]] of <math>S</math>, annualized.
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| :<math>\sigma</math>, the volatility of the stock's returns; this is the square root of the [[quadratic variation]] of the stock's log price process.
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| :<math>t</math>, a time in years; we generally use: now=0, expiry=T.
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| :<math>\Pi</math>, the value of a [[Portfolio (finance)|portfolio]].
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| Finally we will use <math>N(x)</math> which denotes the [[standard normal]] [[cumulative distribution function]],
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| :<math>N(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-\frac{1}{2}z^2}\, dz</math>
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| <math>N'(x)</math> which denotes the standard normal [[probability density function]],
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| :<math>N'(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2} </math>
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| ==The Black–Scholes equation==
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| {{main|Black–Scholes equation}}
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| [[Image:Stockpricesimulation.jpg|thumb|right|Simulated geometric Brownian motions with parameters from market data]]
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| As above, the '''Black–Scholes equation''' is a [[partial differential equation]], which describes the price of the option over time. The equation is:
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| :<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 = 0</math>
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| The key financial insight behind the equation is that one can perfectly [[hedge (finance)|hedge]] the option by buying and selling the [[underlying]] asset in just the right way and consequently "eliminate risk".{{citation needed|date=November 2013}} This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula (see the [[Black scholes#Black–Scholes formula|next section]]).
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| ==Black–Scholes formula==
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| [[File:European Call Surface.png|thumbnail|A European call valued using the Black-Scholes pricing equation for varying asset price S and time-to-expiry T. In this particular example, the strike price is set to unity.]]
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| 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 [[Black–Scholes#The Black–Scholes equation and its derivation|as above]]; this follows since the formula can be obtained [[Equation solving#Differential equations|by solving]] the equation for the corresponding terminal and boundary conditions.
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| The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:
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| :<math>\begin{align}
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| C(S, t) &= N(d_1)S - N(d_2) Ke^{-r(T - t)} \\
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| d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\
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| d_2 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S}{K}\right) + \left(r - \frac{\sigma^2}{2}\right)(T - t)\right] \\
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| &= d_1 - \sigma\sqrt{T - t}
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| \end{align}</math>
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| The price of a corresponding [[put option]] based on [[put–call parity]] is:
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| :<math>\begin{align}
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| P(S, t) &= Ke^{-r(T - t)} - S + C(S, t) \\
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| &= N(-d_2) Ke^{-r(T - t)} - N(-d_1) S
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| \end{align}\,</math>
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| For both, as [[Black–Scholes#Notation|above]]:
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| * <math>N(\cdot)</math> is the [[cumulative distribution function]] of the [[standard normal distribution]]
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| * <math>T - t</math> is the time to maturity
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| * <math>S</math> is the [[spot price]] of the underlying asset
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| * <math>K</math> is the strike price
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| * <math>r</math> is the [[risk free rate]] (annual rate, expressed in terms of [[continuous compounding]])
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| * <math>\sigma</math> is the [[volatility (finance)|volatility]] of returns of the underlying asset
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| ===Alternative formulation===
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| Introducing some auxiliary variables allows the formula to be simplified and reformulated in a form that is often more convenient (this is a special case of the [[Black model|Black '76 formula]]):
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| :<math>\begin{align}
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| C(F, \tau) &= D \left( N(d_+) F - N(d_-) K \right) \\
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| d_\pm &=
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| \frac{1}{\sigma\sqrt{\tau}}\left[\ln\left(\frac{F}{K}\right) \pm \frac{1}{2}\sigma^2\tau\right] \\
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| d_\pm &= d_\mp \pm \sigma\sqrt{\tau}
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| \end{align}</math>
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| The auxiliary variables are:
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| * <math>\tau = T - t</math> is the time to expiry (remaining time, backwards time)
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| * <math>D = e^{-r\tau}</math> is the [[discount factor]]
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| * <math>F = e^{r\tau} S = \frac{S}{D}</math> is the [[forward price]] of the underlying asset, and <math>S = DF</math>
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| with ''d''<sub>+</sub> = ''d''<sub>1</sub> and ''d''<sub>−</sub> = ''d''<sub>2</sub> to clarify notation.
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| Given put-call parity, which is expressed in these terms as:
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| :<math>C - P = D(F - K) = S - D K</math>
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| the price of a put option is:
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| :<math>P(F, \tau) = D \left[ N(-d_-) K - N(-d_+) F \right]</math>
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| ===Interpretation===
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| The Black–Scholes formula can be interpreted fairly easily, with the main subtlety the interpretation of the <math>N(d_\pm)</math> (and a fortiori <math>d_\pm</math>) terms, particularly <math>d_+</math> and why there are two different terms.<ref name="Nielsen"/>
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| 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 equal the value of the binary call options. These binary options are much less frequently traded than vanilla call options, but are easier to analyze.
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| Thus the formula:
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| :<math>C = D \left[ N(d_+) F - N(d_-) K \right]</math>
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| breaks up as:
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| :<math>C = D N(d_+) F - D N(d_-) K</math>
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| 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.
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| The naive, and not quite correct, 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>, times 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> times the value of the cash at expiry ''K.'' This is obviously incorrect, as either both binaries expire in the money or both expire out of the money (either cash is exchanged for 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, 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.
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| If one uses spot ''S'' instead of forward ''F,'' in <math>d_\pm</math> instead of the <math>\frac{1}{2}\sigma^2</math> term there is <math>\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>m = \frac{1}{\sigma\sqrt{\tau}}\ln\left(\frac{F}{K}\right) </math> – in other words, the reason for the <math>\frac{1}{2}\sigma^2</math> factor – 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 ''N''(''d''<sub>+</sub>) by ''N''(''d''<sub>−</sub>) in the formula yields a negative value for out-of-the-money call options.<ref name="Nielsen"/>{{rp|6}}
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| In detail, the terms <math>N(d_1), N(d_2)</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
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| :<math>p(S, T) = \frac{N^\prime [d_2(S_T)]}{S_T \sigma\sqrt{T}}</math>
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| where <math>d_2 = d_2(K)</math> is defined as above.
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| Specifically, <math>N(d_2)</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_1)</math>, however, does not lend itself to a simple probability interpretation. <math>SN(d_1)</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 web|author=Don Chance|date=June 3, 2011|url=http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN99-02.pdf|format=PDF|title=Derivation and Interpretation of the Black–Scholes Model|accessdate=March 27, 2012}}</ref> For related discussion – and graphical representation – see section [[Datar–Mathews method for real option valuation#Interpretation|"Interpretation"]] under [[Datar–Mathews method for real option valuation]].
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| 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]].
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| ====Derivations====
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| {{See also|Martingale pricing}}
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| A standard derivation for solving the Black–Scholes PDE is given in the article [[Black-Scholes equation]].
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| 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 journal |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 | journal = Revue Finance (Journal of the French Finance Association) | volume = 14 | number = [http://www.affi.asso.fr/TPL_CODE/TPL_REVUE/PAR_TPL_IDENTIFIANT/53/193-publications.htm 1] | pages = 95–106 | accessdate = Dec 8, 2012 | postscript =, earlier circulated as [[INSEAD]] Working Paper [http://librarycatalogue.insead.edu/bib/972 92/71/FIN] (1992); [http://www.ltnielsen.com/papers/understanding-nd1-and-nd2-risk-adjusted-probabilities-in-the-black-scholes-model abstract] and link to article, [http://www.affi.asso.fr/TPL_CODE/TPL_REVUEARTICLEDOWNLOAD/PAR_TPL_IDENTIFIANT/187/193-publications.htm published article].}}</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 detail, once again, see Hull.<ref name="Hull">{{Cite book|last=Hull |first=John C. |year=2008| edition=7 |title=Options, Futures and Other Derivatives |publisher=[[Prentice Hall]] |isbn=0-13-505283-1}}</ref>{{rp|307–309}}
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| ==The Greeks==
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| "[[Greeks (finance)|The Greeks]]" measure the sensitivity of the value of a derivative or a portfolio to changes in parameter value(s) while holding the other parameters fixed. They are [[partial derivatives]] of the price with respect to the parameter values. One Greek, "gamma" (as well as others not listed here) is a partial derivative of another Greek, "delta" in this case.
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| The Greeks are important not only in the mathematical theory of finance, but also for those actively trading. Financial institutions will typically set (risk) limit values for each of the Greeks that their traders must not exceed. Delta is the most important Greek since this usually confers the largest risk. Many traders will zero their delta at the end of the day if they are not speculating and following a delta-neutral hedging approach as defined by Black–Scholes.
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| The Greeks for Black–Scholes are given in [[Closed-form expression|closed form]] below. They can be obtained by [[Differentiation (mathematics)|differentiation]] of the Black–Scholes formula.<ref>Although with significant algebra; see, for example, Hong-Yi Chen, Cheng-Few Lee and Weikang Shih (2010).
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| [https://docs.google.com/viewer?a=v&q=cache:ai5xEtbLxCIJ:centerforpbbefr.rutgers.edu/TaipeiPBFR%26D/01-16-09%2520papers/5-4%2520Greek%2520letters.doc+Derivations+and+Applications+of+Greek+Letters+%E2%80%93+Review+and+Integration&hl=en&pid=bl&srcid=ADGEEShU4q28apOYjO-BmqXOJTOHj2BG0BgnxtLn-ccCfh27FYlCDla0nspYCidFFFWiPfYjM2PTT0_109Lth79rFwKsenMFpawjU9BtpBSQO81hUj0OjG3owSKTyv6-VTziJ6tq5CNb&sig=AHIEtbREe6Jg8SlzylhuYC9xEoG0eG3dGg Derivations and Applications of Greek Letters: Review and Integration], ''Handbook of Quantitative Finance and Risk Management'', III:491–503.</ref>
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| {| class="wikitable" border="1"
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| |-
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| ! colspan=2 | !! Calls !! Puts
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| |- style="text-align:center"
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| ! Delta || <math>\frac{\partial C}{\partial S}</math>
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| | <math>N(d_1)\,</math> || <math>-N(-d_1) = N(d_1) - 1\,</math>
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| |- style="text-align:center"
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| ! Gamma || <math>\frac{\partial^{2} C}{\partial S^{2}}</math>
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| | colspan="2" | <math>\frac{N'(d_1)}{S\sigma\sqrt{T - t}}\,</math>
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| |- style="text-align:center"
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| ! Vega || <math>\frac{\partial C}{\partial \sigma}</math>
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| | colspan="2" | <math>S N'(d_1) \sqrt{T-t}\,</math>
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| |- style="text-align:center"
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| ! Theta || <math>\frac{\partial C}{\partial t}</math>
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| | <math>-\frac{S N'(d_1) \sigma}{2 \sqrt{T - t}} - rKe^{-r(T - t)}N(d_2)\,</math>
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| | <math>-\frac{S N'(d_1) \sigma}{2 \sqrt{T - t}} + rKe^{-r(T - t)}N(-d_2)\,</math>
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| |- style="text-align:center"
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| ! Rho || <math>\frac{\partial C}{\partial r}</math>
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| | <math> K(T - t)e^{-r(T - t)}N( d_2)\,</math>
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| | <math>-K(T - t)e^{-r(T - t)}N(-d_2)\,</math>
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| |}
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| Note that from the formulas, it is clear that the gamma is the same value for calls and puts and so too is the vega the same value for calls and put options. This can be seen directly from [[put–call parity]], since the difference of a put and a call is a forward, which is linear in ''S'' and independent of ''σ'' (so a forward has zero gamma and zero vega).
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| In practice, some sensitivities are usually quoted in scaled-down terms, to match the scale of likely changes in the parameters. For example, rho is often reported divided by 10,000 (1 basis point rate change), vega by 100 (1 vol point change), and theta by 365 or 252 (1 day decay based on either calendar days or trading days per year).
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| (Vega is of course not a letter in the Greek alphabet; the name arises from reading the Greek letter ν (nu) as a V.)
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| ==Extensions of the model==
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| 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]]).
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| ===Instruments paying continuous yield dividends===
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| 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.
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| The dividend payment paid over the time period <math>[t, t + dt)</math> is then modelled as
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| :<math>qS_t\,dt</math>
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| for some constant <math>q</math> (the [[dividend yield]]).
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| Under this formulation the arbitrage-free price implied by the Black–Scholes model can be shown to be
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| :<math>C(S_0, t) = e^{-r(T - t)}[FN(d_1) - KN(d_2)]\,</math>
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| and
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| :<math>P(S_0, t) = e^{-r(T - t)}[KN(-d_2) - FN(-d_1)]\,</math>
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| where now
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| :<math>F = S_0 e^{(r - q)(T - t)}\,</math>
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| is the modified forward price that occurs in the terms <math>d_1, d_2</math>:
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| :<math>d_1 = \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{F}{K}\right) + \frac{1}{2}\sigma^2(T - t)\right]</math>
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| and
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| :<math>d_2 = d_1 - \sigma\sqrt{T - t}</math>
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| <ref>http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node9.html</ref>
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| Extending the Black Scholes formula Adjusting for payouts of the underlying.
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| ===Instruments paying discrete proportional dividends===
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| 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.
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| 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</math>. The price of the stock is then modelled as
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| :<math>S_t = S_0(1 - \delta)^{n(t)}e^{ut + \sigma W_t}</math>
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| where <math>n(t)</math> is the number of dividends that have been paid by time <math>t</math>.
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| The price of a call option on such a stock is again
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| :<math>C(S_0, T) = e^{-rT}[FN(d_1) - KN(d_2)]\,</math>
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| where now
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| :<math>F = S_{0}(1 - \delta)^{n(T)}e^{rT}\,</math>
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| is the forward price for the dividend paying stock.
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| === American options ===
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| 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 an inequality of the form
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| :<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|accessdate=May 5, 2012}}</ref>
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| With the terminal and (free) boundary conditions: <math>V(S, T) = H(S)</math> and <math>V(S, t) \geq H(S)</math> where <math>H(S)</math> denotes the payoff at stock price <math>S</math>
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| 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|accessdate=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|accessdate=May 16, 2012}}</ref>
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| Barone-Adesi and Whaley<ref>{{cite journal|authors= Giovanni Barone-Adesi and Robert E Whaley|title=Efficient analytic approximation of American option values|journal=Journal of Finance|volume=42 (2)|date=June 1987|pages=301–20|url=http://ideas.repec.org/a/bla/jfinan/v42y1987i2p301-20.html}}</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 algorithm|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|accessdate=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|accessdate=June 25, 2012}}</ref>
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| 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>
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| ==Black–Scholes in practice==
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| [[File:Crowd outside nyse.jpg|thumb|The normality assumption of the Black–Scholes model does not capture extreme movements such as [[stock market crash]]es.]]
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| The Black–Scholes model disagrees with reality in a number of ways, some significant. It is widely employed as a useful approximation, but proper application requires understanding its limitations – blindly following the model exposes the user to unexpected risk. <ref>Yalincak, Hakan, "Criticism of the Black-Scholes Model: But Why Is It Still Used? (The Answer is Simpler than the Formula)" <<http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2115141>></ref>
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| Among the most significant limitations are:
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| * the underestimation of extreme moves, yielding [[tail risk]], which can be hedged with [[out-of-the-money]] options;
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| * the assumption of instant, cost-less trading, yielding [[liquidity risk]], which is difficult to hedge;
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| * the assumption of a stationary process, yielding [[volatility risk]], which can be hedged with volatility hedging;
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| * the assumption of continuous time and continuous trading, yielding gap risk, which can be hedged with Gamma hedging.
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| In short, while in the Black–Scholes model one can perfectly hedge options by simply Delta hedging, in practice there are many other sources of risk.
| |
| | |
| Results using the Black–Scholes model differ from real world prices because of simplifying assumptions of the model. One significant limitation is that in reality security prices do not follow a strict stationary [[Log-normal distribution|log-normal]] process, nor is the risk-free interest actually known (and is not constant over time). The variance has been observed to be non-constant leading to models such as [[GARCH]] to model volatility changes. Pricing discrepancies between empirical and the Black–Scholes model have long been observed in options that are far [[out-of-the-money]], corresponding to extreme price changes; such events would be very rare if returns were lognormally distributed, but are observed much more often in practice.
| |
| | |
| Nevertheless, Black–Scholes pricing is widely used in practice,<ref name="bodie-kane-marcus"/>{{rp|751}}<ref name = "Wilmott Defence"/> because it is:
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| * easy to calculate
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| * a useful approximation, particularly when analyzing the direction in which prices move when crossing critical points
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| * a robust basis for more refined models
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| * reversible, as the model's original output, price, can be used as an input and one of the other variables solved for; the implied volatility calculated in this way is often used to quote option prices (that is, as a ''quoting convention'')
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| The first point is self-evidently useful. The others can be further discussed:
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| Useful approximation: although volatility is not constant, results from the model are often helpful in setting up hedges in the correct proportions to minimize risk. Even when the results are not completely accurate, they serve as a first approximation to which adjustments can be made.
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| Basis for more refined models: The Black–Scholes model is ''robust'' in that it can be adjusted to deal with some of its failures. Rather than considering some parameters (such as volatility or interest rates) as ''constant,'' one considers them as ''variables,'' and thus added sources of risk. This is reflected in the [[Greeks (finance)|Greeks]] (the change in option value for a change in these parameters, or equivalently the partial derivatives with respect to these variables), and hedging these Greeks mitigates the risk caused by the non-constant nature of these parameters. Other defects cannot be mitigated by modifying the model, however, notably tail risk and liquidity risk, and these are instead managed outside the model, chiefly by minimizing these risks and by [[stress testing]].
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| Explicit modeling: this feature mean that, rather than ''assuming'' a volatility ''a priori'' and computing prices from it, one can use the model to solve for volatility, which gives the [[implied volatility]] of an option at given prices, durations and exercise prices. Solving for volatility over a given set of durations and strike prices one can construct an [[volatility surface|implied volatility surface]]. In this application of the Black–Scholes model, a [[coordinate transformation]] from the ''price domain'' to the ''volatility domain'' is obtained. Rather than quoting option prices in terms of dollars per unit (which are hard to compare across strikes and [[Tenor (finance)|tenors]]), option prices can thus be quoted in terms of implied volatility, which leads to trading of volatility in option markets.
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| | |
| ===The volatility smile===
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| {{Main|Volatility smile}}
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| One of the attractive features of the Black–Scholes model is that the parameters in the model (other than the volatility) — the time to maturity, the strike, the risk-free interest rate, and the current underlying price – are unequivocally observable. All other things being equal, an option's theoretical value is a [[Monotonic function|monotonic increasing function]] of implied volatility.
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| | |
| By computing the implied volatility for traded options with different strikes and maturities, the Black–Scholes model can be tested. If the Black–Scholes model held, then the implied volatility for a particular stock would be the same for all strikes and maturities. In practice, the [[volatility smile|volatility surface]] (the 3D graph of implied volatility against strike and maturity) is not flat.
| |
| | |
| The typical shape of the implied volatility curve for a given maturity depends on the underlying instrument. Equities tend to have skewed curves: compared to [[at-the-money]], implied volatility is substantially higher for low strikes, and slightly lower for high strikes. Currencies tend to have more symmetrical curves, with implied volatility lowest [[at-the-money]], and higher volatilities in both wings. Commodities often have the reverse behavior to equities, with higher implied volatility for higher strikes.
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| | |
| Despite the existence of the volatility smile (and the violation of all the other assumptions of the Black–Scholes model), the Black–Scholes PDE and Black–Scholes formula are still used extensively in practice. A typical approach is to regard the volatility surface as a fact about the market, and use an implied volatility from it in a Black–Scholes valuation model. This has been described as using "the wrong number in the wrong formula to get the right price."<ref>{{cite book|author=Riccardo Rebonato|authorlink=Riccardo Rebonato|year=1999|title=Volatility and correlation in the pricing of equity, FX and interest-rate options|publisher=Wiley|isbn=0-471-89998-4}}</ref> This approach also gives usable values for the hedge ratios (the Greeks).
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| Even when more advanced models are used, traders prefer to think in terms of volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on.
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| | |
| ===Valuing bond options===
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| Black–Scholes cannot be applied directly to [[bond (finance)|bond securities]] because of [[pull to par|pull-to-par]]. As the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its volatility, and the simple Black–Scholes model does not reflect this process. A large number of extensions to Black–Scholes, beginning with the [[Black model]], have been used to deal with this phenomenon.<ref>{{cite journal|first=Andrew|last=Kalotay|authorlink=Andrew Kalotay|url=http://kalotay.com/sites/default/files/private/BlackScholes.pdf|format=PDF|title=The Problem with Black, Scholes et al.|journal=Derivatives Strategy|date=November 1995}}</ref> See [[Bond option#Valuation|Bond option: Valuation]].
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| | |
| ===Interest-rate curve===
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| In practice, interest rates are not constant – they vary by [[Tenor (finance)|tenor]], giving an [[yield curve|interest rate curve]] which may be interpolated to pick an appropriate rate to use in the Black–Scholes formula. Another consideration is that interest rates vary over time. This volatility may make a significant contribution to the price, especially of long-dated options.This is simply like the interest rate and bond price relationship which is inversely related.
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| ===Short stock rate===
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| It is not free to take a [[short (finance)|short stock]] position. Similarly, it may be possible to lend out a long stock position for a small fee. In either case, this can be treated as a continuous dividend for the purposes of a Black–Scholes valuation, provided that there is no glaring asymmetry between the short stock borrowing cost and the long stock lending income.{{Citation needed|date=April 2012}}
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| ==Criticism==
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| [[Espen Gaarder Haug]] and [[Nassim Nicholas Taleb]] argue that the Black–Scholes model merely recast existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk", to make them more compatible with mainstream [[neoclassical economics|neoclassical economic]] theory.<ref>[[Espen Gaarder Haug]] and [[Nassim Nicholas Taleb]] (2011). [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1012075 Option Traders Use (very) Sophisticated Heuristics, Never the Black–Scholes–Merton Formula]. ''Journal of Economic Behavior and Organization'', Vol. 77, No. 2, 2011</ref> They also assert that Boness in 1964 had already published a formula that is "actually identical" to the Black–Scholes call option pricing equation.<ref>Boness, A James, 1964, Elements of a theory of stock-option value, Journal of Political Economy, 72,
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| 163-175.</ref> [[Edward O. Thorp|Edward Thorp]] also claims to have guessed the Black–Scholes formula in 1967 but kept it to himself to make money for his investors.<ref name="thorpe">[http://edwardothorp.com/sitebuildercontent/sitebuilderfiles/thorpwilmottqfinrev2003.pdf A Perspective on Quantitative Finance: Models for Beating the Market], Quantitative Finance Review, 2003. Also see [http://edwardothorp.com/sitebuildercontent/sitebuilderfiles/optiontheory.doc Option Theory Part 1] by Edward Thorpe</ref> [[Emanuel Derman]] and Nassim Taleb have also criticized dynamic hedging and state that a number of researchers had put forth similar models prior to Black and Scholes.<ref>[[Emanuel Derman]] and [[Nassim Taleb]] (2005). [http://www.ederman.com/new/docs/qf-Illusions-dynamic.pdf The illusions of dynamic replication], ''Quantitative Finance'', Vol. 5, No. 4, August 2005, 323–326</ref> In response, [[Paul Wilmott]] has defended the model.<ref name = "Wilmott Defence">[[Paul Wilmott]] (2008): [http://www.wilmott.com/blogs/paul/index.cfm/2008/4/29/Science-in-Finance-IX-In-defence-of-Black-Scholes-and-Merton In defence of Black Scholes and Merton], [http://www.wilmott.com/blogs/paul/index.cfm/2008/7/23/Science-in-Finance-X-Dynamic-hedging-and-further-defence-of-BlackScholes Dynamic hedging and further defence of Black-Scholes]</ref><ref>See also: Doriana Ruffinno and Jonathan Treussard (2006). [http://wayback.archive.org/web/*/http://www.bu.edu/econ/workingpapers/papers/RuffinoTreussardDT.pdf ''Derman and Taleb’s The Illusions of Dynamic Replication: A Comment''], WP2006-019, [[Boston University]] - Department of Economics.</ref>
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| | |
| British mathematician [[Ian Stewart (mathematician)|Ian Stewart]] published a criticism in which he suggested that "the equation itself wasn't the real problem" and he stated a possible role as "one ingredient in a rich stew of financial irresponsibility, political ineptitude, perverse incentives and lax regulation" due to its abuse in the financial industry.<ref name="IanStewart">[[Ian Stewart (mathematician)|Ian Stewart]] (2012) [http://www.guardian.co.uk/science/2012/feb/12/black-scholes-equation-credit-crunch The mathematical equation that caused the banks to crash], [[The Observer]], February 12.</ref>
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| ==See also==
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| *[[Binomial options model]], which is a discrete [[numerical method]] for calculating option prices.
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| *[[Black model]], a variant of the Black–Scholes option pricing model.
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| *[[Black Shoals]], a financial art piece.
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| *[[Brownian model of financial markets]]
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| *[[Financial mathematics]], which contains a list of related articles.
| |
| *[[Heat equation]], to which the Black–Scholes PDE can be transformed.
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| *[[Jump diffusion]]
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| *[[Monte Carlo option model]], using [[simulation]] in the valuation of options with complicated features.
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| *[[Real options analysis]]
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| *[[Stochastic volatility]]
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| <!-- *[[Fuzzy pay-off method for real option valuation]]-->
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| | |
| ==Notes==
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| {{Reflist|group="Notes"}}
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| ==References==
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| {{Reflist|30em}}
| |
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| ===Primary references===
| |
| *{{cite journal|title=The Pricing of Options and Corporate Liabilities|last=Black|first=Fischer|coauthors=Myron Scholes|journal=Journal of Political Economy|year=1973|volume=81|issue=3|pages=637–654|doi=10.1086/260062}} [http://links.jstor.org/sici?sici=0022-3808%28197305%2F06%2981%3A3%3C637%3ATPOOAC%3E2.0.CO%3B2-P] (Black and Scholes' original paper.)
| |
| *{{cite journal|title=Theory of Rational Option Pricing|last=Merton|first=Robert C.|journal=Bell Journal of Economics and Management Science|year=1973|volume=4|issue=1|pages=141–183|doi=10.2307/3003143|publisher=The RAND Corporation|jstor=3003143}} [http://links.jstor.org/sici?sici=0005-8556%28197321%294%3A1%3C141%3ATOROP%3E2.0.CO%3B2-0&origin=repec]
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| * {{cite book|title=Options, Futures, and Other Derivatives|last=Hull|first=John C.|authorlink=John C. Hull|year=1997|isbn=0-13-601589-1|publisher=Prentice Hall}}
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| | |
| ===Historical and sociological aspects===
| |
| * {{cite book|title=Capital Ideas: The Improbable Origins of Modern Wall Street|last=Bernstein|first=Peter|authorlink=Peter L. Bernstein|year=1992|isbn=0-02-903012-9|publisher=The Free Press}}
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| *{{cite journal|title=An Equation and its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics|last=MacKenzie|first=Donald|journal=Social Studies of Science|year=2003|volume=33|issue=6|pages=831–868|doi=10.1177/0306312703336002}} [http://sss.sagepub.com/cgi/content/abstract/33/6/831]
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| *{{cite journal|title=Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange|last=MacKenzie|first=Donald|coauthors=Yuval Millo|journal=American Journal of Sociology|year=2003|volume=109|issue=1|pages=107–145|doi=10.1086/374404}} [http://www.journals.uchicago.edu/AJS/journal/issues/v109n1/060259/brief/060259.abstract.html]
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| * {{cite book|title=An Engine, not a Camera: How Financial Models Shape Markets|last=MacKenzie|first=Donald|
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| isbn=0-262-13460-8|publisher=MIT Press|year=2006}}
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| * Szpiro, George G. ''Pricing the Future: Finance, Physics, and the 300-Year Journey to the Black-Scholes Equation; A Story of Genius and Discovery'' (New York: Basic, 2011) 298 pp.
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| | |
| ===Further reading===
| |
| *{{cite book|last=Haug, E. G|title=Derivatives: Models on Models|publisher=Wiley|year=2007|chapter=Option Pricing and Hedging from Theory to Practice|isbn=978-0-470-01322-9}} The book gives a series of historical references supporting the theory that option traders use much more robust hedging and pricing principles than the Black, Scholes and Merton model.
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| *{{cite book|last=Triana|first=Pablo|title=Lecturing Birds on Flying: Can Mathematical Theories Destroy the Financial Markets?|publisher=Wiley|year=2009|isbn=978-0-470-40675-5}} The book takes a critical look at the Black, Scholes and Merton model.
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| ==External links==
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| | |
| ===Discussion of the model===
| |
| *[http://www.mayin.org/ajayshah/PDFDOCS/Shah1997_bms.pdf Ajay Shah. Black, Merton and Scholes: Their work and its consequences. Economic and Political Weekly, XXXII(52):3337–3342, December 1997]
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| *[http://www.guardian.co.uk/science/2012/feb/12/black-scholes-equation-credit-crunch The mathematical equation that caused the banks to crash] by [[Ian Stewart (mathematician)|Ian Stewart]] in [[The Observer]], February 12, 2012
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| *[http://www.ederman.com/new/docs/risk-non_continuous_hedge.pdf When You Cannot Hedge Continuously: The Corrections to Black–Scholes], [[Emanuel Derman]]
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| *[https://www.tastytrade.com/tt/shows/soom The Skinny On Options] TastyTrade Show (archives)
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| | |
| ===Derivation and solution===
| |
| *[http://www.sjsu.edu/faculty/watkins/blacksch.htm Derivation of the Black–Scholes Equation for Option Value], Prof. Thayer Watkins
| |
| *[http://www.physics.uci.edu/%7Esilverma/bseqn/bs/bs.html Solution of the Black–Scholes Equation Using the Green's Function], Prof. Dennis Silverman
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| *[http://homepages.nyu.edu/~sl1544/KnownClosedForms.pdf Solution via risk neutral pricing or via the PDE approach using Fourier transforms] (includes discussion of other option types), Simon Leger
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| *[http://www.black-scholes.co.uk/assumptions-for-black-scholes-model.html Assumptions for Black Scholes Model], black-scholes.co.uk
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| *[http://planetmath.org/encyclopedia/AnalyticSolutionOfBlackScholesPDE.html Step-by-step solution of the Black–Scholes PDE], planetmath.org.
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| *[http://terrytao.wordpress.com/2008/07/01/the-black-scholes-equation/ The Black–Scholes Equation] Expository article by mathematician [[Terence Tao]]. <!--this article is also in Tao's book "Poincaré's Legacies" ISBN 978-0-8218-4885-2.-->
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| ===Computer implementations===
| |
| * [http://www.quantcalc.net/BSV.html Calculator for vanilla call and put based on Black-Sholes model]
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| *[http://www.espenhaug.com/black_scholes.html Black–Scholes in Multiple Languages]
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| *[http://sourceforge.net/projects/chipricingmodel/ Chicago Option Pricing Model (Graphing Version)]
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| *[https://github.com/OpenGamma/OG-Platform/blob/master/projects/OG-Analytics/src/main/java/com/opengamma/analytics/financial/model/volatility/surface/BlackScholesMertonImpliedVolatilitySurfaceModel.java Black-Scholes-Merton Implied Volatility Surface Model (Java)]
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| * [http://black-scholes.com/ Black-Scholes Calculator]
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| * [https://leventozturk.com/engineering/Black_Scholes/ Online Black-Scholes Calculator]
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| ===Historical===
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| *[http://www.pbs.org/wgbh/nova/stockmarket/ Trillion Dollar Bet]—Companion Web site to a Nova episode originally broadcast on February 8, 2000. ''"The film tells the fascinating story of the invention of the Black–Scholes Formula, a mathematical Holy Grail that forever altered the world of finance and earned its creators the 1997 Nobel Prize in Economics."''
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| *[http://www.bbc.co.uk/science/horizon/1999/midas.shtml BBC Horizon] A TV-programme on the so-called [[Midas formula]] and the bankruptcy of [[Long-Term Capital Management]] ([[LTCM]])
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| *[http://www.bbc.co.uk/news/magazine-17866646 BBC News Magazine] Black–Scholes: The maths formula linked to the financial crash (April 27, 2012 article)
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| {{Derivatives market}}
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| {{DEFAULTSORT:Black-Scholes Model}}
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