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Black–Scholes model
The Black–Scholes /ˌblæk ˈʃoʊlz/^{[1]} or Black–Scholes–Merton model is a mathematical model of a financial market containing certain derivative investment instruments. From the model, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of Europeanstyle 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.^{[2]} lt is widely used, although often with adjustments and corrections, by options market participants.^{[3]}^{:751} Many empirical tests have shown that the Black–Scholes price is "fairly close" to the observed prices, although there are wellknown discrepancies such as the "option smile".^{[3]}^{:770–771}
The Black–Scholes model 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 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 the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk. This type of hedging is called delta hedging and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds.
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 for their work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.^{[4]}
The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management. It is the insights of the model, as exemplified in the BlackScholes formula, that are frequently used by market participants, as distinguished from the actual prices. These insights include noarbitrage bounds and riskneutral pricing. The BlackScholes 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.
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 OTC derivatives.
Contents
The BlackScholes world
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.
Now we make assumptions on the assets (which explain their names):
 (riskless rate) The rate of return on the riskless asset is constant and thus called the riskfree interest rate.
 (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 timevarying, we can deduce a suitably modified Black–Scholes formula quite simply, as long as the volatility is not random).
 The stock does not pay a dividend.^{[Notes 1]}
Assumptions on the market:
 There is no arbitrage opportunity (i.e., there is no way to make a riskless profit).
 It is possible to borrow and lend any amount, even fractional, of cash at the riskless rate.
 It is possible to buy and sell any amount, even fractional, of the stock (this includes short selling).
 The above transactions do not incur any fees or costs (i.e., frictionless market).
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 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".^{[5]} 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.
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]}, transaction costs and taxes (Ingersoll, 1976)^{[citation needed]}, and dividend payout.^{[6]}
Notation
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).
 <math>V(S, t)</math>, the price of a derivative as a function of time and stock price.
 <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.
 <math>K</math>, the strike price of the option.
 <math>r</math>, the annualized riskfree interest rate, continuously compounded (the force of interest).
 <math>\mu</math>, the drift rate of <math>S</math>, annualized.
 <math>\sigma</math>, the standard deviation of the stock's returns; this is the square root of the quadratic variation of the stock's log price process.
 <math>t</math>, a time in years; we generally use: now=0, expiry=T.
 <math>\Pi</math>, the value of a portfolio.
Finally we will use <math>N(x)</math> to denote the standard normal cumulative distribution function,
 <math>N(x) = \frac{1}{\sqrt{2\pi}}\int_{\infty}^{x} e^{\frac{z^2}{2}}\, dz</math>.
<math>N'(x)</math> will denote the standard normal probability density function,
 <math>N'(x) = \frac{1}{\sqrt{2\pi}} e^{\frac{x^2}{2}} </math>
The Black–Scholes equation
As above, the Black–Scholes equation is a partial differential equation, which describes the price of the option over time. The equation is:
 <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>
The key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk".^{[citation needed]} This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula (see the next section).
BlackScholes formula
The Black–Scholes formula calculates the price of European put and call options. This price is consistent with the Black–Scholes equation as above; this follows since the formula can be obtained by solving the equation for the corresponding terminal and boundary conditions.
The value of a call option for a nondividendpaying underlying stock in terms of the Black–Scholes parameters is:
 <math>\begin{align}
C(S, t) &= N(d_1)S  N(d_2) Ke^{r(T  t)} \\ 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] \\ 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] \\ &= d_1  \sigma\sqrt{T  t}
\end{align}</math>
The price of a corresponding put option based on put–call parity is:
 <math>\begin{align}
P(S, t) &= Ke^{r(T  t)}  S + C(S, t) \\ &= N(d_2) Ke^{r(T  t)}  N(d_1) S
\end{align}\,</math>
For both, as above:
 <math>N(\cdot)</math> is the cumulative distribution function of the standard normal distribution
 <math>T  t</math> is the time to maturity
 <math>S</math> is the spot price of the underlying asset
 <math>K</math> is the strike price
 <math>r</math> is the risk free rate (annual rate, expressed in terms of continuous compounding)
 <math>\sigma</math> is the volatility of returns of the underlying asset
Alternative formulation
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 '76 formula):
 <math>\begin{align}
C(F, \tau) &= D \left( N(d_+) F  N(d_) K \right) \\ d_\pm &= \frac{1}{\sigma\sqrt{\tau}}\left[\ln\left(\frac{F}{K}\right) \pm \frac{1}{2}\sigma^2\tau\right] \\ d_\pm &= d_\mp \pm \sigma\sqrt{\tau}
\end{align}</math>
The auxiliary variables are:
 <math>\tau = T  t</math> is the time to expiry (remaining time, backwards time)
 <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>
with d_{+} = d_{1} and d_{−} = d_{2} to clarify notation.
Given putcall 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
The Black–Scholes formula can be interpreted fairly handily, 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.^{[7]}
The formula can be interpreted by first decomposing a call option into the difference of two binary options: an assetornothing call minus a cashornothing call (long an assetornothing call, short a cashornothing call). A call option exchanges cash for an asset at expiry, while an assetornothing call just yields the asset (with no cash in exchange) and a cashornothing 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.
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 assetornothing call and <math>D N(d_) K</math> is the present value of a cashornothing 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 assetornothing call and <math>N(d_) ~ K</math> is the future value of a cashornothing call. In riskneutral terms, these are the expected value of the asset and the expected value of the cash in the riskneutral measure.
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.^{[7]} 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>\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 riskneutral measure for appropriate numéraire). The use of d_{−} 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 lognormal distribution; it is the same factor as in 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_{+}) by N(d_{−}) in the formula yields a negative value for outofthemoney call options.^{[7]}^{:6}
In detail, the terms <math>N(d_1), N(d_2)</math> are the probabilities of the option expiring inthemoney under the equivalent exponential martingale probability measure (numéraire=stock) and the equivalent martingale probability measure (numéraire=risk free asset), respectively.^{[7]} 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_2(S_T)]}{S_T \sigma\sqrt{T}}</math>
where <math>d_2 = d_2(K)</math> is defined as above.
Specifically, <math>N(d_2)</math> is the probability that the call will be exercised provided one assumes that the asset drift is the riskfree 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 riskfree interest rate, of the expected asset price at expiration, given that the asset price at expiration is above the exercise price.^{[8]} For related discussion – and graphical representation – see section "Interpretation" under Datar–Mathews method for real option valuation.
The equivalent martingale probability measure is also called the riskneutral probability measure. Note that both of these are probabilities in a measure theoretic sense, and neither of these is the true probability of expiring inthemoney under the 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
A standard derivation for solving the Black–Scholes PDE is given in the article BlackScholes equation.
The FeynmanKac formula says that the solution to this type of PDE, when discounted appropriately, is actually a 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.^{[7]} Note the expectation of the option payoff is not done under the real world probability measure, but an artificial riskneutral measure, which differs from the real world measure. For the underlying logic see section "risk neutral valuation" under Rational pricing as well as section "Derivatives pricing: the Q world" under Mathematical finance; for detail, once again, see Hull.^{[9]}^{:307–309}
The Greeks
"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.
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 deltaneutral hedging approach as defined by Black–Scholes.
The Greeks for Black–Scholes are given in closed form below. They can be obtained by differentiation of the Black–Scholes formula.^{[10]}
Calls  Puts  

Delta  <math>\frac{\partial C}{\partial S}</math>  <math>N(d_1)\,</math>  <math>N(d_1) = N(d_1)  1\,</math> 
Gamma  <math>\frac{\partial^{2} C}{\partial S^{2}}</math>  <math>\frac{N'(d_1)}{S\sigma\sqrt{T  t}}\,</math>  
Vega  <math>\frac{\partial C}{\partial \sigma}</math>  <math>S N'(d_1) \sqrt{Tt}\,</math>  
Theta  <math>\frac{\partial C}{\partial t}</math>  <math>\frac{S N'(d_1) \sigma}{2 \sqrt{T  t}}  rKe^{r(T  t)}N(d_2)\,</math>  <math>\frac{S N'(d_1) \sigma}{2 \sqrt{T  t}} + rKe^{r(T  t)}N(d_2)\,</math> 
Rho  <math>\frac{\partial C}{\partial r}</math>  <math> K(T  t)e^{r(T  t)}N( d_2)\,</math>  <math>K(T  t)e^{r(T  t)}N(d_2)\,</math> 
Note that from the formulae, 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).
In practice, some sensitivities are usually quoted in scaleddown 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).
(Vega is not a letter in the Greek alphabet; the name arises from reading the Greek letter ν (nu) as a V.)
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, closedform solutions are available if the dividend is a known proportion of the stock price. 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 lattices and 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 arbitragefree price implied by the Black–Scholes model can be shown to be
 <math>C(S_0, t) = e^{r(T  t)}[FN(d_1)  KN(d_2)]\,</math>
and
 <math>P(S_0, t) = e^{r(T  t)}[KN(d_2)  FN(d_1)]\,</math>
where now
 <math>F = S_0 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{F}{K}\right) +(r  q + \frac{1}{2}\sigma^2)(T  t)\right]</math>
and
 <math>d_2 = d_1  \sigma\sqrt{T  t}</math>
^{[11]} Extending the Black Scholes formula Adjusting for payouts of the underlying.
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 predetermined times <math>t_1, t_2, \ldots</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 an 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>^{[12]}
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>.
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 RollGeskeWhaley method provides a solution for an American call with one dividend.^{[13]}^{[14]}
BaroneAdesi and Whaley^{[15]} 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 finding the critical value, <math>s*</math>, such that one is indifferent between early exercise and holding to maturity.^{[16]}^{[17]}
Bjerksund and Stensland^{[18]} 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 upandout call option… and (ii) a rebate that is received at the knockout 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 BaroneAdesi and Whaley.^{[19]}
Black–Scholes in practice
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.^{[20]} Among the most significant limitations are:
 the underestimation of extreme moves, yielding tail risk, which can be hedged with outofthemoney options;
 the assumption of instant, costless trading, yielding liquidity risk, which is difficult to hedge;
 the assumption of a stationary process, yielding volatility risk, which can be hedged with volatility hedging;
 the assumption of continuous time and continuous trading, yielding gap risk, which can be hedged with Gamma hedging.
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 lognormal process, nor is the riskfree interest actually known (and is not constant over time). The variance has been observed to be nonconstant 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 outofthemoney, 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,^{[3]}^{:751}^{[21]} because it is:
 easy to calculate
 a useful approximation, particularly when analyzing the direction in which prices move when crossing critical points
 a robust basis for more refined models
 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)
The first point is selfevidently useful. The others can be further discussed:
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.
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 (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 nonconstant 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.
Explicit modeling: this feature means 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 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 tenors), option prices can thus be quoted in terms of implied volatility, which leads to trading of volatility in option markets.
The volatility smile
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 riskfree interest rate, and the current underlying price) are unequivocally observable. All other things being equal, an option's theoretical value is a monotonic increasing function of implied volatility.
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 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 atthemoney, 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 atthemoney, and higher volatilities in both wings. Commodities often have the reverse behavior to equities, with higher implied volatility for higher strikes.
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."^{[22]} This approach also gives usable values for the hedge ratios (the Greeks). Even when more advanced models are used, traders prefer to think in terms of BlackScholes implied volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on. For a discussion as to the various alternate approaches developed here, see Financial economics #Challenges and criticism.
Valuing bond options
Black–Scholes cannot be applied directly to bond securities because of pulltopar. 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.^{[23]} See Bond option: Valuation.
Interestrate curve
In practice, interest rates are not constant – they vary by tenor, giving an 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 longdated options.This is simply like the interest rate and bond price relationship which is inversely related.
Short stock rate
It is not free to take a 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]}
Criticism
Espen Gaarder Haug and Nassim Nicholas Taleb argue that the Black–Scholes model merely recasts existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk", to make them more compatible with mainstream neoclassical economic theory.^{[24]} They also assert that Boness in 1964 had already published a formula that is "actually identical" to the Black–Scholes call option pricing equation.^{[25]} Edward Thorp also claims to have guessed the Black–Scholes formula in 1967 but kept it to himself to make money for his investors.^{[26]} 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.^{[27]} In response, Paul Wilmott has defended the model.^{[21]}^{[28]}
British 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.^{[29]}
See also
 Binomial options model, which is a discrete numerical method for calculating option prices.
 Black model, a variant of the Black–Scholes option pricing model.
 Black Shoals, a financial art piece.
 Brownian model of financial markets
 Financial mathematics, which contains a list of related articles.
 Heat equation, to which the Black–Scholes PDE can be transformed.
 Jump diffusion
 Monte Carlo option model, using simulation in the valuation of options with complicated features.
 Real options analysis
 Stochastic volatility
Notes
 ↑ Although the original model assumed no dividends, trivial extensions to the model can accommodate a continuous dividend yield factor.
References
 ↑ "Scholes". Retrieved March 26, 2012.
 ↑ MacKenzie, Donald (2006). An Engine, Not a Camera: How Financial Models Shape Markets. Cambridge, MA: MIT Press. ISBN 0262134608.
 ↑ ^{3.0} ^{3.1} ^{3.2} Bodie, Zvi; Alex Kane; Alan J. Marcus (2008). Investments (7th ed.). New York: McGrawHill/Irwin. ISBN 9780073269672.
 ↑ "Nobel Prize Foundation, 1997 Press release". October 14, 1997. Retrieved March 26, 2012.
 ↑ Black, Fischer; Scholes, Myron. "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): 637–654. doi:10.1086/260062.
 ↑ Merton, Robert. "Theory of Rational Option Pricing". Bell Journal of Economics and Management Science 4 (1): 141–183. doi:10.2307/3003143.
 ↑ ^{7.0} ^{7.1} ^{7.2} ^{7.3} ^{7.4} Nielsen, Lars Tyge (1993). "Understanding N(d_{1}) and N(d_{2}): RiskAdjusted Probabilities in the BlackScholes Model" (PDF). Revue Finance (Journal of the French Finance Association) 14 (1): 95–106. Retrieved Dec 8, 2012, earlier circulated as INSEAD Working Paper 92/71/FIN (1992); abstract and link to article, published article.
 ↑ Don Chance (June 3, 2011). "Derivation and Interpretation of the Black–Scholes Model" (PDF). Retrieved March 27, 2012.
 ↑ Hull, John C. (2008). Options, Futures and Other Derivatives (7 ed.). Prentice Hall. ISBN 0135052831.
 ↑ Although with significant algebra; see, for example, HongYi Chen, ChengFew Lee and Weikang Shih (2010). Derivations and Applications of Greek Letters: Review and Integration, Handbook of Quantitative Finance and Risk Management, III:491–503.
 ↑ http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node9.html
 ↑ André Jaun. "The BlackScholes equation for American options". Retrieved May 5, 2012.
 ↑ Bernt Ødegaard (2003). "Extending the Black Scholes formula". Retrieved May 5, 2012.
 ↑ Don Chance (2008). "ClosedForm American Call Option Pricing: RollGeskeWhaley" (PDF). Retrieved May 16, 2012.
 ↑ Giovanni BaroneAdesi and Robert E Whaley (June 1987). "Efficient analytic approximation of American option values". Journal of Finance 42 (2): 301–20. doi:10.2307/2328254.
 ↑ Bernt Ødegaard (2003). "A quadratic approximation to American prices due to BaroneAdesi and Whaley". Retrieved June 25, 2012.
 ↑ Don Chance (2008). "Approximation Of American Option Values: BaroneAdesiWhaley" (PDF). Retrieved June 25, 2012.
 ↑ Petter Bjerksund and Gunnar Stensland, 2002. Closed Form Valuation of American Options
 ↑ American options
 ↑ Yalincak, Hakan, "Criticism of the BlackScholes Model: But Why Is It Still Used? (The Answer is Simpler than the Formula)" <<http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2115141>>
 ↑ ^{21.0} ^{21.1} Paul Wilmott (2008): In defence of Black Scholes and Merton, Dynamic hedging and further defence of BlackScholes
 ↑ Riccardo Rebonato (1999). Volatility and correlation in the pricing of equity, FX and interestrate options. Wiley. ISBN 0471899984.
 ↑ Kalotay, Andrew (November 1995). "The Problem with Black, Scholes et al." (PDF). Derivatives Strategy.
 ↑ Espen Gaarder Haug and Nassim Nicholas Taleb (2011). Option Traders Use (very) Sophisticated Heuristics, Never the Black–Scholes–Merton Formula. Journal of Economic Behavior and Organization, Vol. 77, No. 2, 2011
 ↑ Boness, A James, 1964, Elements of a theory of stockoption value, Journal of Political Economy, 72, 163175.
 ↑ A Perspective on Quantitative Finance: Models for Beating the Market, Quantitative Finance Review, 2003. Also see Option Theory Part 1 by Edward Thorpe
 ↑ Emanuel Derman and Nassim Taleb (2005). The illusions of dynamic replication, Quantitative Finance, Vol. 5, No. 4, August 2005, 323–326
 ↑ See also: Doriana Ruffinno and Jonathan Treussard (2006). Derman and Taleb’s The Illusions of Dynamic Replication: A Comment, WP2006019, Boston University  Department of Economics.
 ↑ Ian Stewart (2012) The mathematical equation that caused the banks to crash, The Observer, February 12.
Primary references
 Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): 637–654. doi:10.1086/260062. [1] (Black and Scholes' original paper.)
 Merton, Robert C. (1973). "Theory of Rational Option Pricing". Bell Journal of Economics and Management Science (The RAND Corporation) 4 (1): 141–183. JSTOR 3003143. doi:10.2307/3003143. [2]
 Hull, John C. (1997). Options, Futures, and Other Derivatives. Prentice Hall. ISBN 0136015891.
Historical and sociological aspects
 Bernstein, Peter (1992). Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press. ISBN 0029030129.
 MacKenzie, Donald (2003). "An Equation and its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics". Social Studies of Science 33 (6): 831–868. doi:10.1177/0306312703336002. [3]
 MacKenzie, Donald; Yuval Millo (2003). "Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange". American Journal of Sociology 109 (1): 107–145. doi:10.1086/374404. [4]
 MacKenzie, Donald (2006). An Engine, not a Camera: How Financial Models Shape Markets. MIT Press. ISBN 0262134608.
 Szpiro, George G. Pricing the Future: Finance, Physics, and the 300Year Journey to the BlackScholes Equation; A Story of Genius and Discovery (New York: Basic, 2011) 298 pp.
Further reading
 Haug, E. G (2007). "Option Pricing and Hedging from Theory to Practice". Derivatives: Models on Models. Wiley. ISBN 9780470013229. 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.
 Triana, Pablo (2009). Lecturing Birds on Flying: Can Mathematical Theories Destroy the Financial Markets?. Wiley. ISBN 9780470406755. The book takes a critical look at the Black, Scholes and Merton model.
External links
Discussion of the model
 Ajay Shah. Black, Merton and Scholes: Their work and its consequences. Economic and Political Weekly, XXXII(52):3337–3342, December 1997
 The mathematical equation that caused the banks to crash by Ian Stewart in The Observer, February 12, 2012
 When You Cannot Hedge Continuously: The Corrections to Black–Scholes, Emanuel Derman
 The Skinny On Options TastyTrade Show (archives)
Derivation and solution
 Derivation of the Black–Scholes Equation for Option Value, Prof. Thayer Watkins
 Solution of the Black–Scholes Equation Using the Green's Function, Prof. Dennis Silverman
 Solution via risk neutral pricing or via the PDE approach using Fourier transforms (includes discussion of other option types), Simon Leger
 Assumptions for Black Scholes Model, blackscholes.co.uk
 Stepbystep solution of the Black–Scholes PDE, planetmath.org.
 The Black–Scholes Equation Expository article by mathematician Terence Tao.
Computer implementations
 Calculator for vanilla call and put based on BlackSholes model
 Black–Scholes in Multiple Languages
 BlackScholes in Java moving to link below
 BlackScholes in Java
 Chicago Option Pricing Model (Graphing Version)
 BlackScholesMerton Implied Volatility Surface Model (Java)
 Online BlackScholes Calculator
 Online financial calculator with BlackScholes
Historical
 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."
 BBC Horizon A TVprogramme on the socalled Midas formula and the bankruptcy of LongTerm Capital Management (LTCM)
 BBC News Magazine Black–Scholes: The maths formula linked to the financial crash (April 27, 2012 article)
