Black–Scholes model

Introducing auxiliary variables allows for the formula to be simplified and reformulated in a form that can be more convenient (this is a special case of the Black '76 formula):

${\displaystyle {\begin{aligned}C(F,\tau )&=D\left[N(d_{+})F-N(d_{-})K\right]\\d_{+}&={\frac {1}{\sigma {\sqrt {\tau }}}}\left[\ln \left({\frac {F}{K}}\right)+{\frac {1}{2}}\sigma ^{2}\tau \right]\\d_{-}&=d_{+}-\sigma {\sqrt {\tau }}\end{aligned}}}$

where:

${\displaystyle D=e^{-r\tau }}$ is the discount factor

${\displaystyle F=e^{r\tau }S={\frac {S}{D}}}$ is the forward price of the underlying asset, and ${\displaystyle S=DF}$

Given put–call parity, which is expressed in these terms as:

${\displaystyle C-P=D(F-K)=S-DK}$

the price of a put option is:

${\displaystyle P(F,\tau )=D\left[N(-d_{-})K-N(-d_{+})F\right]}$

It is possible to have intuitive interpretations of the Black–Scholes formula, with the main subtlety being the interpretation of the ${\displaystyle N(d_{\pm })}$ (and a fortiori ${\displaystyle d_{\pm }}$) terms, particularly ${\displaystyle d_{+}}$ and why there are two different terms.[14]

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

Thus the formula:

${\displaystyle C=D\left[N(d_{+})F-N(d_{-})K\right]}$

breaks up as:

${\displaystyle C=DN(d_{+})F-DN(d_{-})K,}$

where ${\displaystyle DN(d_{+})F}$ is the present value of an asset-or-nothing call and ${\displaystyle DN(d_{-})K}$ 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 ${\displaystyle N(d_{+})~F}$ is the future value of an asset-or-nothing call and ${\displaystyle N(d_{-})~K}$ is the future value of a cash-or-nothing call. In risk-neutral terms, these are the expected value of the asset and the expected value of the cash in the risk-neutral measure.

A naive, and slightly incorrect, interpretation of these terms is that ${\displaystyle N(d_{+})F}$ is the probability of the option expiring in the money ${\displaystyle N(d_{+})}$, multiplied by the value of the underlying at expiry F, while ${\displaystyle N(d_{-})K}$ is the probability of the option expiring in the money ${\displaystyle N(d_{-}),}$ multiplied by the value of the cash at expiry K. This interpretation is incorrect because either both binaries expire in the money or both expire out of the money (either cash is exchanged for the asset or it is not), but the probabilities ${\displaystyle N(d_{+})}$ and ${\displaystyle N(d_{-})}$ are not equal. In fact, ${\displaystyle d_{\pm }}$ can be interpreted as measures of moneyness (in standard deviations) and ${\displaystyle N(d_{\pm })}$ as probabilities of expiring ITM (percent moneyness), in the respective numéraire, as discussed below. Simply put, the interpretation of the cash option, ${\displaystyle N(d_{-})K}$, is correct, as the value of the cash is independent of movements of the underlying asset, and thus can be interpreted as a simple product of "probability times value", while the ${\displaystyle N(d_{+})F}$ is more complicated, as the probability of expiring in the money and the value of the asset at expiry are not independent.[14] 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 ${\displaystyle d_{\pm }}$ instead of the ${\textstyle {\frac {1}{2}}\sigma ^{2}}$ term there is ${\textstyle \left(r\pm {\frac {1}{2}}\sigma ^{2}\right)\tau ,}$ which can be interpreted as a drift factor (in the risk-neutral measure for appropriate numéraire). The use of d₋ for moneyness rather than the standardized moneyness ${\textstyle m={\frac {1}{\sigma {\sqrt {\tau }}}}\ln \left({\frac {F}{K}}\right)}$ – in other words, the reason for the ${\textstyle {\frac {1}{2}}\sigma ^{2}}$ 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 applied to geometric Brownian motion. In addition, another way to see that the naive interpretation is incorrect is that replacing ${\displaystyle N(d_{+})}$ by ${\displaystyle N(d_{-})}$ in the formula yields a negative value for out-of-the-money call options.[14]^: 6

In detail, the terms ${\displaystyle N(d_{1}),N(d_{2})}$ are the probabilities of the option expiring in-the-money under the equivalent exponential martingale probability measure (numéraire=stock) and the equivalent martingale probability measure (numéraire=risk free asset), respectively.[14] The risk neutral probability density for the stock price ${\displaystyle S_{T}\in (0,\infty )}$ is

${\displaystyle p(S,T)={\frac {N^{\prime }[d_{2}(S_{T})]}{S_{T}\sigma {\sqrt {T}}}}}$

where ${\displaystyle d_{2}=d_{2}(K)}$ is defined as above.

Specifically, ${\displaystyle N(d_{2})}$ is the probability that the call will be exercised provided one assumes that the asset drift is the risk-free rate. ${\displaystyle N(d_{1})}$, however, does not lend itself to a simple probability interpretation. ${\displaystyle SN(d_{1})}$ is correctly interpreted as the present value, using the risk-free interest rate, of the expected asset price at expiration, given that the asset price at expiration is above the exercise price.[15] For related discussion – and graphical representation – see Datar–Mathews method for real option valuation.

The equivalent martingale probability measure is also called the risk-neutral probability measure. Note that both of these are probabilities in a measure theoretic sense, and neither of these is the true probability of expiring in-the-money 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.

A standard derivation for solving the Black–Scholes PDE is given in the article Black–Scholes equation.

The Feynman–Kac formula says that the solution to this type of PDE, when discounted appropriately, is actually a martingale. 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.[14] Note the 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 "risk neutral valuation" under Rational pricing as well as section "Derivatives pricing: the Q world" under Mathematical finance; for details, once again, see Hull.[16]^{: 307–309}

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 ${\displaystyle [t,t+dt]}$ is then modelled as:

${\displaystyle qS_{t}\,dt}$

for some constant ${\displaystyle q}$ (the dividend yield).

Under this formulation the arbitrage-free price implied by the Black–Scholes model can be shown to be:

${\displaystyle C(S_{t},t)=e^{-r(T-t)}[FN(d_{1})-KN(d_{2})]\,}$

and

${\displaystyle P(S_{t},t)=e^{-r(T-t)}[KN(-d_{2})-FN(-d_{1})]\,}$

where now

${\displaystyle F=S_{t}e^{(r-q)(T-t)}\,}$

is the modified forward price that occurs in the terms ${\displaystyle d_{1},d_{2}}$:

${\displaystyle d_{1}={\frac {1}{\sigma {\sqrt {T-t}}}}\left[\ln \left({\frac {S_{t}}{K}}\right)+\left(r-q+{\frac {1}{2}}\sigma ^{2}\right)(T-t)\right]}$

and

${\displaystyle d_{2}=d_{1}-\sigma {\sqrt {T-t}}={\frac {1}{\sigma {\sqrt {T-t}}}}\left[\ln \left({\frac {S_{t}}{K}}\right)+\left(r-q-{\frac {1}{2}}\sigma ^{2}\right)(T-t)\right]}$.[18]

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 ${\displaystyle \delta }$ of the stock price is paid out at pre-determined times ${\displaystyle t_{1},t_{2},\ldots ,t_{n}}$. The price of the stock is then modelled as:

${\displaystyle S_{t}=S_{0}(1-\delta )^{n(t)}e^{ut+\sigma W_{t}}}$

where ${\displaystyle n(t)}$ is the number of dividends that have been paid by time ${\displaystyle t}$.

The price of a call option on such a stock is again:

${\displaystyle C(S_{0},T)=e^{-rT}[FN(d_{1})-KN(d_{2})]\,}$

where now

${\displaystyle F=S_{0}(1-\delta )^{n(T)}e^{rT}\,}$

is the forward price for the dividend paying stock.

The problem of finding the price of an American option is related to the optimal stopping problem of finding the time to execute the option. Since the American option can be exercised at any time before the expiration date, the Black–Scholes equation becomes a variational inequality of the form:

${\displaystyle {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV\leq 0}$[19]

together with ${\displaystyle V(S,t)\geq H(S)}$ where ${\displaystyle H(S)}$ denotes the payoff at stock price ${\displaystyle S}$ and the terminal condition: ${\displaystyle V(S,T)=H(S)}$.

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;[20][21] see also Black's approximation.

Barone-Adesi and Whaley[22] 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, ${\displaystyle s*}$, such that one is indifferent between early exercise and holding to maturity.[23][24]

Bjerksund and Stensland[25] 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 ${\displaystyle S-X}$, otherwise the option "boils down to: (i) a European 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.[26]

Despite the lack of a general analytical solution for American put options, it is possible to derive such a formula for the case of a perpetual option - meaning that the option never expires (i.e., ${\displaystyle T\rightarrow \infty }$).[27] In this case, the time decay of the option is equal to zero, which leads to the Black–Scholes PDE becoming an ODE:

$${\displaystyle {1 \over {2}}\sigma ^{2}S^{2}{d^{2}V \over {dS^{2}}}+(r-q)S{dV \over {dS}}-rV=0}$$

Let ${\displaystyle S_{-}}$ denote the lower exercise boundary, below which is optimal for exercising the option. The boundary conditions are:

$${\displaystyle V(S_{-})=K-S_{-},\quad V_{S}(S_{-})=-1,\quad V(S)\leq K}$$

The solutions to the ODE are a linear combination of any two linearly independent solutions:

$${\displaystyle V(S)=A_{1}S^{\lambda _{1}}+A_{2}S^{\lambda _{2}}}$$

For ${\displaystyle S_{-}\leq S}$, substitution of this solution into the ODE for ${\displaystyle i={1,2}}$ yields:

$${\displaystyle \left[{1 \over {2}}\sigma ^{2}\lambda _{i}(\lambda _{i}-1)+(r-q)\lambda _{i}-r\right]S^{\lambda _{i}}=0}$$

Rearranging the terms gives:

$${\displaystyle {1 \over {2}}\sigma ^{2}\lambda _{i}^{2}+\left(r-q-{1 \over {2}}\sigma ^{2}\right)\lambda _{i}-r=0}$$

Using the quadratic formula, the solutions for ${\displaystyle \lambda _{i}}$ are:

$${\displaystyle {\begin{aligned}\lambda _{1}&={-\left(r-q-{1 \over {2}}\sigma ^{2}\right)+{\sqrt {\left(r-q-{1 \over {2}}\sigma ^{2}\right)^{2}+2\sigma ^{2}r}} \over {\sigma ^{2}}}\\\lambda _{2}&={-\left(r-q-{1 \over {2}}\sigma ^{2}\right)-{\sqrt {\left(r-q-{1 \over {2}}\sigma ^{2}\right)^{2}+2\sigma ^{2}r}} \over {\sigma ^{2}}}\end{aligned}}}$$

In order to have a finite solution for the perpetual put, since the boundary conditions imply upper and lower finite bounds on the value of the put, it is necessary to set ${\displaystyle A_{1}=0}$, leading to the solution ${\displaystyle V(S)=A_{2}S^{\lambda _{2}}}$. From the first boundary condition, it is known that:

$${\displaystyle V(S_{-})=A_{2}(S_{-})^{\lambda _{2}}=K-S_{-}\implies A_{2}={K-S_{-} \over {(S_{-})^{\lambda _{2}}}}}$$

Therefore, the value of the perpetual put becomes:

$${\displaystyle V(S)=(K-S_{-})\left({S \over {S_{-}}}\right)^{\lambda _{2}}}$$

The second boundary condition yields the location of the lower exercise boundary:

$${\displaystyle V_{S}(S_{-})=\lambda _{2}{K-S_{-} \over {S_{-}}}=-1\implies S_{-}={\lambda _{2}K \over {\lambda _{2}-1}}}$$

To conclude, for ${\textstyle S\geq S_{-}={\lambda _{2}K \over {\lambda _{2}-1}}}$, the perpetual American put option is worth:

$${\displaystyle V(S)={K \over {1-\lambda _{2}}}\left({\lambda _{2}-1 \over {\lambda _{2}}}\right)^{\lambda _{2}}\left({S \over {K}}\right)^{\lambda _{2}}}$$

By solving the Black–Scholes differential equation with the Heaviside function as a boundary condition, one ends up with the pricing of options that pay one unit above some predefined strike price and nothing below.[28]

In fact, the Black–Scholes formula for the price of a vanilla call option (or put option) can be interpreted by decomposing a call option into an asset-or-nothing call option minus a cash-or-nothing call option, and similarly for a put—the binary options are easier to analyze, and correspond to the two terms in the Black–Scholes formula.

This pays out one unit of cash if the spot is above the strike at maturity. Its value is given by:

${\displaystyle C=e^{-r(T-t)}N(d_{2}).\,}$

This pays out one unit of cash if the spot is below the strike at maturity. Its value is given by:

${\displaystyle P=e^{-r(T-t)}N(-d_{2}).\,}$

This pays out one unit of asset if the spot is above the strike at maturity. Its value is given by:

${\displaystyle C=Se^{-q(T-t)}N(d_{1}).\,}$

This pays out one unit of asset if the spot is below the strike at maturity. Its value is given by:

${\displaystyle P=Se^{-q(T-t)}N(-d_{1}),}$

Denoting by S the FOR/DOM exchange rate (i.e., 1 unit of foreign currency is worth S units of domestic currency) one can observe that paying out 1 unit of the domestic currency if the spot at maturity is above or below the strike is exactly like a cash-or nothing call and put respectively. Similarly, paying out 1 unit of the foreign currency if the spot at maturity is above or below the strike is exactly like an asset-or nothing call and put respectively. Hence by taking ${\displaystyle r_{FOR}}$, the foreign interest rate, ${\displaystyle r_{DOM}}$, the domestic interest rate, and the rest as above, the following results can be obtained:

In the case of a digital call (this is a call FOR/put DOM) paying out one unit of the domestic currency gotten as present value:

${\displaystyle C=e^{-r_{DOM}T}N(d_{2})\,}$

In the case of a digital put (this is a put FOR/call DOM) paying out one unit of the domestic currency gotten as present value:

${\displaystyle P=e^{-r_{DOM}T}N(-d_{2})\,}$

In the case of a digital call (this is a call FOR/put DOM) paying out one unit of the foreign currency gotten as present value:

${\displaystyle C=Se^{-r_{FOR}T}N(d_{1})\,}$

In the case of a digital put (this is a put FOR/call DOM) paying out one unit of the foreign currency gotten as present value:

${\displaystyle P=Se^{-r_{FOR}T}N(-d_{1})\,}$

In the standard Black–Scholes model, one can interpret the premium of the binary option in the risk-neutral world as the expected value = probability of being in-the-money * unit, discounted to the present value. The Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset. Market makers adjust for such skewness by, instead of using a single standard deviation for the underlying asset ${\displaystyle \sigma }$ across all strikes, incorporating a variable one ${\displaystyle \sigma (K)}$ where volatility depends on strike price, thus incorporating the volatility skew into account. The skew matters because it affects the binary considerably more than the regular options.

A binary call option is, at long expirations, similar to a tight call spread using two vanilla options. One can model the value of a binary cash-or-nothing option, C, at strike K, as an infinitesimally tight spread, where ${\displaystyle C_{v}}$ is a vanilla European call:[29][30]

${\displaystyle C=\lim _{\epsilon \to 0}{\frac {C_{v}(K-\epsilon )-C_{v}(K)}{\epsilon }}}$

Thus, the value of a binary call is the negative of the derivative of the price of a vanilla call with respect to strike price:

${\displaystyle C=-{\frac {dC_{v}}{dK}}}$

When one takes volatility skew into account, ${\displaystyle \sigma }$ is a function of ${\displaystyle K}$:

${\displaystyle C=-{\frac {dC_{v}(K,\sigma (K))}{dK}}=-{\frac {\partial C_{v}}{\partial K}}-{\frac {\partial C_{v}}{\partial \sigma }}{\frac {\partial \sigma }{\partial K}}}$

The first term is equal to the premium of the binary option ignoring skew:

${\displaystyle -{\frac {\partial C_{v}}{\partial K}}=-{\frac {\partial (SN(d_{1})-Ke^{-r(T-t)}N(d_{2}))}{\partial K}}=e^{-r(T-t)}N(d_{2})=C_{\text{no skew}}}$

${\displaystyle {\frac {\partial C_{v}}{\partial \sigma }}}$ is the Vega of the vanilla call; ${\displaystyle {\frac {\partial \sigma }{\partial K}}}$ is sometimes called the "skew slope" or just "skew". If the skew is typically negative, the value of a binary call will be higher when taking skew into account.

${\displaystyle C=C_{\text{no skew}}-{\text{Vega}}_{v}\cdot {\text{Skew}}}$

Since a binary call is a mathematical derivative of a vanilla call with respect to strike, the price of a binary call has the same shape as the delta of a vanilla call, and the delta of a binary call has the same shape as the gamma of a vanilla call.

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 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 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.

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".[34] 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 Black–Scholes 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 alternative approaches developed here, see Financial economics § Challenges and criticism.

Black–Scholes cannot be applied directly to bond securities because of 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.[35] See Bond option § Valuation.

In practice, interest rates are not constant—they vary by tenor (coupon frequency), 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 long-dated options. This is simply like the interest rate and bond price relationship which is inversely related.

Taking a short stock position, as inherent in the derivation, is not typically free of cost; equivalently, it is 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.

• Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy. 81 (3): 637–654. doi:10.1086/260062. S2CID 154552078. (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. doi:10.2307/3003143. hdl:10338.dmlcz/135817. JSTOR 3003143.
• Hull, John C. (1997). Options, Futures, and Other Derivatives. Prentice Hall. ISBN 0-13-601589-1.

• Bernstein, Peter (1992). Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press. ISBN 0-02-903012-9.
• Derman, Emanuel. "My Life as a Quant" John Wiley & Sons, Inc. 2004. ISBN 0471394203
• MacKenzie, Donald (2003). "An Equation and its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics" (PDF). Social Studies of Science. 33 (6): 831–868. doi:10.1177/0306312703336002. hdl:20.500.11820/835ab5da-2504-4152-ae5b-139da39595b8. S2CID 15524084.
• 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. CiteSeerX 10.1.1.461.4099. doi:10.1086/374404. S2CID 145805302.
• MacKenzie, Donald (2006). An Engine, not a Camera: How Financial Models Shape Markets. MIT Press. ISBN 0-262-13460-8.
• Mandelbrot & Hudson, "The (Mis)Behavior of Markets" Basic Books, 2006. ISBN 9780465043552
• 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.
• Taleb, Nassim. "Dynamic Hedging" John Wiley & Sons, Inc. 1997. ISBN 0471152803
• Thorp, Ed. "A Man for all Markets" Random House, 2017. ISBN 9781400067961

• Haug, E. G (2007). "Option Pricing and Hedging from Theory to Practice". Derivatives: Models on Models. Wiley. 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.
• Triana, Pablo (2009). Lecturing Birds on Flying: Can Mathematical Theories Destroy the Financial Markets?. Wiley. ISBN 978-0-470-40675-5. The book takes a critical look at the Black, Scholes and Merton 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

• Solution of the Black–Scholes Equation Using the Green's Function, Prof. Dennis Silverman
• The Black–Scholes Equation Expository article by mathematician Terence Tao.

• Black–Scholes in Multiple Languages
• Black–Scholes in Java -moving to link below-
• Black–Scholes in Java
• Chicago Option Pricing Model (Graphing Version)
• Black–Scholes–Merton Implied Volatility Surface Model (Java)
• Online Black–Scholes Calculator

• 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 TV-programme on the so-called Midas formula and the bankruptcy of Long-Term Capital Management (LTCM)
• BBC News Magazine Black–Scholes: The maths formula linked to the financial crash (April 27, 2012 article)