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Black–Scholes model

The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The equation and model are named after economists Fischer Black and Myron Scholes. Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited.

The main principle behind the model is to hedge the option by buying and selling the underlying asset in a specific way to eliminate risk. This type of hedging is called "continuously revised delta hedging" and is the basis of more complicated hedging strategies such as those used by investment banks and hedge funds.

The model is widely used, although often with some adjustments, by options market participants. 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. The insights of the model, as exemplified by the Black–Scholes formula, are frequently used by market participants, as distinguished from the actual prices. These insights include no-arbitrage bounds and risk-neutral pricing (thanks to continuous revision). Further, the Black–Scholes equation, a partial differential equation that governs the price of the option, enables pricing using numerical methods when an explicit formula is not possible.

The Black–Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, though it can be found from the price of other options. Since the option value (whether put or call) 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.

History
Economists Fischer Black and Myron Scholes demonstrated in 1968 that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument. They based their thinking on work previously done by market researchers and practitioners including Louis Bachelier, Sheen Kassouf and Edward O. Thorp. Black and Scholes then attempted to apply the formula to the markets, but incurred financial losses, due to a lack of risk management in their trades. In 1970, they decided to return to the academic environment. After three years of efforts, the formula—named in honor of them for making it public—was finally published in 1973 in an article titled "The Pricing of Options and Corporate Liabilities", in the Journal of Political Economy. 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".

The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world.

Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for their work, the committee citing their discovery of the risk neutral dynamic revision as a breakthrough that separates the option from the risk of the underlying security. Although ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.

Fundamental hypotheses
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.

The following assumptions are made about the assets (which relate to the names of the assets):
 * Riskless rate: The rate of return on the riskless asset is constant and thus called the risk-free interest rate.
 * Random walk: The instantaneous log return of the stock price is an infinitesimal random walk with drift; more precisely, the stock price follows a geometric Brownian motion, and it is assumed that the drift and volatility of the motion are constant. If drift and volatility are time-varying, a suitably modified Black–Scholes formula can be deduced, as long as the volatility is not random.
 * The stock does not pay a dividend.

The assumptions about the market are:
 * No arbitrage opportunity (i.e., there is no way to make a riskless profit).
 * Ability to borrow and lend any amount, even fractional, of cash at the riskless rate.
 * Ability 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, suppose there is a derivative security also trading in this market. It is specified that this security will have a certain payoff at a specified date in the future, depending on the values taken by the stock up to that date. Even though the path the stock price will take in the future is unknown, the derivative's price can be determined at the current time. 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". Their dynamic hedging strategy led to a partial differential equation which governs 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), transaction costs and taxes (Ingersoll, 1976), and dividend payout.

Notation
The notation used in the analysis of the Black-Scholes model is defined as follows (definitions grouped by subject):

General and market related:
 * $$t$$ is a time in years; with $$ t = 0 $$ generally representing the present year.
 * $$r$$ is the annualized risk-free interest rate, continuously compounded (also known as the force of interest).

Asset related:
 * $$S(t)$$ is the price of the underlying asset at time t, also denoted as $$S_t$$.
 * $$\mu$$ is the drift rate of $$S$$, annualized.
 * $$\sigma$$ is the standard deviation of the stock's returns. This is the square root of the quadratic variation of the stock's log price process, a measure of its volatility.

Option related:
 * $$V(S, t)$$ is the price of the option as a function of the underlying asset S at time t, in particular:
 * $$C(S, t)$$ is the price of a European call option and
 * $$P(S, t)$$ is the price of a European put option.
 * $$T$$ is the time of option expiration.
 * $$\tau$$ is the time until maturity: $$\tau = T - t$$.
 * $$K$$ is the strike price of the option, also known as the exercise price.

$$N(x)$$ denotes the standard normal cumulative distribution function:
 * $$N(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-z^2/2}\, dz.$$

$$N'(x)$$ denotes the standard normal probability density function:
 * $$N'(x) = \frac{dN(x)}{dx} = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}. $$

Black–Scholes equation
The Black–Scholes equation is a parabolic partial differential equation that describes the price $$ V(S, t) $$ of the option, where $$S$$ is the price of the underlying and $$t$$ is time:

 * $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$

A key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset and the bank account asset (cash) in such a way as to "eliminate risk". This implies that there is a unique price for the option given by the Black–Scholes formula (see the next section).

Black–Scholes formula

The Black–Scholes formula calculates the price of European put and call options. This price is consistent with the Black–Scholes equation. This follows since the formula can be obtained by solving the equation for the corresponding terminal and boundary conditions:

 * $$\begin{align}

& C(0, t) = 0\text{ for all }t \\ & C(S, t) \rightarrow S - K \text{ as }S \rightarrow \infty \\ & C(S, T) = \max\{S - K, 0\} \end{align}$$

The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:
 * $$\begin{align}

C(S_t, t) &= N(d_+)S_t - N(d_-)Ke^{-r(T - t)} \\ d_+ &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\ d_- &= d_+ - \sigma\sqrt{T - t} \\ \end{align}$$

The price of a corresponding put option based on put–call parity with discount factor $$e^{-r(T-t)}$$ is:
 * $$\begin{align}

P(S_t, t) &= Ke^{-r(T - t)} - S_t + C(S_t, t) \\ &= N(-d_-) Ke^{-r(T - t)} - N(-d_+) S_t \end{align}\,$$

Alternative formulation
Introducing auxiliary variables allows for the formula to be simplified and reformulated in a form that can be more convenient (this is a special case of the Black '76 formula):
 * $$\begin{align}

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

where:

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

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

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

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

Interpretation
It is possible to have intuitive interpretations of the Black–Scholes formula, with the main subtlety being the interpretation of $$d_\pm$$ and why there are two different terms.

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:
 * $$C = D \left[ N(d_+) F - N(d_-) K \right]$$

breaks up as:
 * $$C = D N(d_+) F - D N(d_-) K,$$

where $$D N(d_+) F$$ is the present value of an asset-or-nothing call and $$D N(d_-) K$$ 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 $$N(d_+) ~ F$$ is the future value of an asset-or-nothing call and $$N(d_-) ~ K$$ 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 $$N(d_+) F$$ is the probability of the option expiring in the money $$N(d_+)$$, multiplied by the value of the underlying at expiry F, while $$N(d_-) K$$ is the probability of the option expiring in the money $$N(d_-),$$ 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 $$N(d_+)$$ and $$N(d_-)$$ are not equal. In fact, $$d_\pm$$ can be interpreted as measures of moneyness (in standard deviations) and $$N(d_\pm)$$ as probabilities of expiring ITM (percent moneyness), in the respective numéraire, as discussed below. Simply put, the interpretation of the cash option, $$N(d_-) K$$, 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 $$N(d_+) F$$ is more complicated, as the probability of expiring in the money and the value of the asset at expiry are not independent. 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 $$d_\pm$$ instead of the $\frac{1}{2}\sigma^2$ term there is $\left(r \pm \frac{1}{2}\sigma^2\right)\tau,$  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 $m = \frac{1}{\sigma\sqrt{\tau}}\ln\left(\frac{F}{K}\right)$ –  in other words, the reason for the $\frac{1}{2}\sigma^2$  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 $$N(d_+)$$ by $$N(d_-)$$ in the formula yields a negative value for out-of-the-money call options.

In detail, the terms $$N(d_+), N(d_-)$$ 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. The risk neutral probability density for the stock price $$S_T \in (0, \infty)$$ is
 * $$p(S, T) = \frac{N^\prime [d_-(S_T)]}{S_T \sigma\sqrt{T}}$$

where $$d_- = d_-(K)$$ is defined as above.

Specifically, $$N(d_-)$$ is the probability that the call will be exercised provided one assumes that the asset drift is the risk-free rate. $$N(d_+)$$, however, does not lend itself to a simple probability interpretation. $$SN(d_+)$$ 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. 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.

Derivations
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. 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.

The Options Greeks
"The Greeks" measure the sensitivity of the value of a derivative product or a financial portfolio to changes in parameter values 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 on the direction of the market and following a delta-neutral hedging approach as defined by Black–Scholes. When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio's gamma, as this will ensure that the hedge will be effective over a wider range of underlying price movements.

The Greeks for Black–Scholes are given in closed form below. They can be obtained by differentiation of the Black–Scholes formula.

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 puts 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). N' is the standard normal probability density function.

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).

Note that "Vega" is not a letter in the Greek alphabet; the name arises from misreading the Greek letter nu (variously rendered as $$\nu$$, $ν$, and ν) 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, closed-form 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 $$[t, t + dt]$$ is then modelled as:
 * $$qS_t\,dt$$

for some constant $$q$$ (the dividend yield).

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

and
 * $$P(S_t, t) = e^{-r(T - t)}[KN(-d_2) - FN(-d_1)]\,$$

where now
 * $$F = S_t e^{(r - q)(T - t)}\,$$

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

and
 * $$d_2 = d_1 - \sigma\sqrt{T - t} = \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r - q - \frac{1}{2}\sigma^2\right)(T - t)\right]$$.

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 $$\delta$$ of the stock price is paid out at pre-determined times $$t_1, t_2, \ldots, t_n $$. The price of the stock is then modelled as:
 * $$S_t = S_0(1 - \delta)^{n(t)}e^{ut + \sigma W_t}$$

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

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

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

is the forward price for the dividend paying stock.

American options
The problem of finding the price of an American option is related to the optimal stopping problem of finding the time to execute the option. Since the American option can be exercised at any time before the expiration date, the Black–Scholes equation becomes a variational inequality of the form:
 * $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV \leq 0$$

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

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; see also Black's approximation.

Barone-Adesi and Whaley 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, $$s*$$, such that one is indifferent between early exercise and holding to maturity.

Bjerksund and Stensland 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 $$S - X$$, 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.

Perpetual put
Despite the lack of a general analytical solution for American put options, it is possible to derive such a formula for the case of a perpetual option – meaning that the option never expires (i.e., $$T\rightarrow \infty$$). In this case, the time decay of the option is equal to zero, which leads to the Black–Scholes PDE becoming an ODE:$${1\over{2}}\sigma^{2}S^{2}{d^{2}V\over{dS^{2}}} + (r-q)S{dV\over{dS}} - rV = 0$$Let $$S_{-}$$ denote the lower exercise boundary, below which is optimal for exercising the option. The boundary conditions are:$$V(S_{-}) = K-S_{-}, \quad V_{S}(S_{-}) = -1, \quad V(S) \leq K$$The solutions to the ODE are a linear combination of any two linearly independent solutions:$$V(S) = A_{1}S^{\lambda_{1}} + A_{2}S^{\lambda_{2}}$$For $$S_{-} \leq S$$, substitution of this solution into the ODE for $$i = {1,2}$$ yields:$$\left[ {1\over{2}}\sigma^{2}\lambda_{i}(\lambda_{i}-1) + (r-q)\lambda_{i} - r \right]S^{\lambda_{i}} = 0$$Rearranging the terms gives:$${1\over{2}}\sigma^{2}\lambda_{i}^{2} + \left(r-q - {1\over{2}} \sigma^{2}\right)\lambda_{i} - r = 0$$Using the quadratic formula, the solutions for $$\lambda_{i}$$ are:$$\begin{aligned} \lambda_{1} &= {-\left(r-q-{1\over{2}}\sigma^{2} \right ) + \sqrt{\left(r-q-{1\over{2}}\sigma^{2} \right )^{2} + 2\sigma^{2}r}\over{\sigma^{2}}} \\ \lambda_{2} &= {-\left(r-q-{1\over{2}}\sigma^{2} \right ) - \sqrt{\left(r-q-{1\over{2}}\sigma^{2} \right )^{2} + 2\sigma^{2}r}\over{\sigma^{2}}} \end{aligned}$$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 $$A_{1} = 0$$, leading to the solution $$V(S) = A_{2}S^{\lambda_{2}}$$. From the first boundary condition, it is known that:$$V(S_{-}) = A_{2}(S_{-})^{\lambda_{2}} = K-S_{-} \implies A_{2} = {K-S_{-}\over{(S_{-})^{\lambda_{2}}}}$$Therefore, the value of the perpetual put becomes:$$V(S) = (K-S_{-})\left( {S\over{S_{-}}} \right)^{\lambda_{2}}$$The second boundary condition yields the location of the lower exercise boundary:$$V_{S}(S_{-}) = \lambda_{2}{K-S_{-}\over{S_{-}}} = -1 \implies S_{-} = {\lambda_{2}K\over{\lambda_{2}-1}}$$To conclude, for $S \geq S_{-} = {\lambda_{2}K\over{\lambda_{2}-1}}$, the perpetual American put option is worth:$$V(S) = {K\over{1-\lambda_{2}}} \left( {\lambda_{2}-1\over{\lambda_{2}}}\right)^{\lambda_{2}} \left( {S\over{K}} \right)^{\lambda_{2}}$$

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

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

Cash-or-nothing call
This pays out one unit of cash if the spot is above the strike at maturity. Its value is given by:
 * $$ C =e^{-r (T-t)}N(d_2). \,$$

Cash-or-nothing put
This pays out one unit of cash if the spot is below the strike at maturity. Its value is given by:
 * $$ P = e^{-r (T-t)}N(-d_2). \,$$

Asset-or-nothing call
This pays out one unit of asset if the spot is above the strike at maturity. Its value is given by:
 * $$ C = Se^{-q (T-t)}N(d_1). \,$$

Asset-or-nothing put
This pays out one unit of asset if the spot is below the strike at maturity. Its value is given by:
 * $$ P = Se^{-q (T-t)}N(-d_1),$$

Foreign Exchange (FX)
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 $$r_{f}$$, the foreign interest rate, $$r_{d}$$, 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:
 * $$ C = e^{-r_{d} T}N(d_2) \,$$

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:
 * $$ P = e^{-r_{d}T}N(-d_2) \,$$

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:
 * $$ C = Se^{-r_{f} T}N(d_1) \,$$

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:
 * $$ P = Se^{-r_{f}T}N(-d_1) \,$$

Skew
In the standard Black–Scholes model, one can interpret the premium of the binary option in the risk-neutral world as the expected value = probability of being in-the-money * unit, discounted to the present value. The Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset. Market makers adjust for such skewness by, instead of using a single standard deviation for the underlying asset $$\sigma$$ across all strikes, incorporating a variable one $$\sigma(K)$$ 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 $$C_v$$ is a vanilla European call:
 * $$ C = \lim_{\epsilon \to 0} \frac{C_v(K-\epsilon) - C_v(K)}{\epsilon} $$

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:
 * $$ C = -\frac{dC_v}{dK} $$

When one takes volatility skew into account, $$\sigma$$ is a function of $$K$$:
 * $$ C = -\frac{dC_v(K,\sigma(K))}{dK} = -\frac{\partial C_v}{\partial K} - \frac{\partial C_v}{\partial \sigma} \frac{\partial \sigma}{\partial K}$$

The first term is equal to the premium of the binary option ignoring skew:
 * $$ -\frac{\partial C_v}{\partial K} = -\frac{\partial (S N(d_1) - Ke^{-r(T-t)} N(d_2))}{\partial K} = e^{-r (T-t)} N(d_2) = C_\text{no skew}$$

$$\frac{\partial C_v}{\partial \sigma}$$ is the Vega of the vanilla call; $$\frac{\partial \sigma}{\partial K}$$ 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.
 * $$ C = C_\text{no skew} - \text{Vega}_v \cdot \text{Skew}$$

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

Black–Scholes in practice
The assumptions of the Black–Scholes model are not all empirically valid. The model is widely employed as a useful approximation to reality, but proper application requires understanding its limitations – blindly following the model exposes the user to unexpected risk. Among the most significant limitations are:

 * the underestimation of extreme moves, yielding tail risk, which can be hedged with out-of-the-money options;
 * the assumption of instant, cost-less 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;
 * the model tends to underprice deep out-of-the-money options and overprice deep in-the-money options.

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 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, 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 self-evidently 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 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.

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, durations and coupon frequencies), 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 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". 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.

Valuing bond options
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. See.

Interest rate curve
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.

Short stock rate
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.

Criticism and comments
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. They also assert that Boness in 1964 had already published a formula that is "actually identical" to the Black–Scholes call option pricing equation. Edward Thorp also claims to have guessed the Black–Scholes formula in 1967 but kept it to himself to make money for his investors. Emanuel Derman and Taleb have also criticized dynamic hedging and state that a number of researchers had put forth similar models prior to Black and Scholes. In response, Paul Wilmott has defended the model.

In his 2008 letter to the shareholders of Berkshire Hathaway, Warren Buffett wrote: "I believe the Black–Scholes formula, even though it is the standard for establishing the dollar liability for options, produces strange results when the long-term variety are being valued... The Black–Scholes formula has approached the status of holy writ in finance ... If the formula is applied to extended time periods, however, it can produce absurd results. In fairness, Black and Scholes almost certainly understood this point well. But their devoted followers may be ignoring whatever caveats the two men attached when they first unveiled the formula."

British mathematician Ian Stewart, author of the 2012 book entitled In Pursuit of the Unknown: 17 Equations That Changed the World, said that Black–Scholes had "underpinned massive economic growth" and the "international financial system was trading derivatives valued at one quadrillion dollars per year" by 2007. He said that the Black–Scholes equation was the "mathematical justification for the trading"—and therefore—"one ingredient in a rich stew of financial irresponsibility, political ineptitude, perverse incentives and lax regulation" that contributed to the financial crisis of 2007–08. He clarified that "the equation itself wasn't the real problem", but its abuse in the financial industry.

The Black–Scholes model assumes positive underlying prices; if the underlying has a negative price, the model does not work directly. When dealing with options whose underlying can go negative, practitioners may use a different model such as the Bachelier model or simply add a constant offset to the prices.

Primary references

 * (Black and Scholes' original paper.)
 *

Historical and sociological aspects

 * Derman, Emanuel. "My Life as a Quant" John Wiley & Sons, Inc. 2004. ISBN 0-471-39420-3
 *
 *
 * Mandelbrot & Hudson, "The (Mis)Behavior of Markets" Basic Books, 2006. ISBN 978-0-465-04355-2
 * 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 0-471-15280-3
 * Thorp, Ed. "A Man for all Markets" Random House, 2017. ISBN 978-1-4000-6796-1
 * Taleb, Nassim. "Dynamic Hedging" John Wiley & Sons, Inc. 1997. ISBN 0-471-15280-3
 * Thorp, Ed. "A Man for all Markets" Random House, 2017. ISBN 978-1-4000-6796-1

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

Derivation and solution

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

Computer implementations

 * 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

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 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)

title: Black–Scholes model


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Fischer Black

Fischer Sheffey Black (January 11, 1938 – August 30, 1995) was an American economist, best known as one of the authors of the Black–Scholes equation.

Background
Fischer Sheffey Black was born on January 11, 1938. He graduated from Harvard College with a major in physics in 1959 and received a PhD in applied mathematics from Harvard University in 1964. He was initially expelled from the PhD program due to his inability to settle on a thesis topic, having switched from physics to mathematics, then to computers and artificial intelligence. Black joined the consultancy Bolt, Beranek and Newman, working on a system for artificial intelligence. He spent a summer developing his ideas at the RAND corporation. He became a student of MIT professor Marvin Minsky, and was later able to submit his research for completion of the Harvard PhD.

Black joined Arthur D. Little, where he was first exposed to economic and financial consulting and where he met his future collaborator Jack Treynor. In 1971, he began to work at the University of Chicago. He later left the University of Chicago in 1975 to work at the MIT Sloan School of Management. In 1984, he joined Goldman Sachs where he worked until death.

Economic career
Black began thinking seriously about monetary policy around 1970 and found, at this time, that the big debate in this field was between Keynesians and monetarists. The Keynesians (under the leadership of Franco Modigliani) believe there is a natural tendency of the credit markets toward instability, toward boom and bust, and they assign to both monetary and fiscal policy roles in damping down this cycle, working toward the goal of smooth sustainable growth. In the Keynesian view, central bankers have to have discretionary powers to fulfill their role properly. Monetarists, under the leadership of Milton Friedman, believe that discretionary central banking is the problem, not the solution. Friedman believed that the growth of the money supply could and should be set at a constant rate, say 3% a year, to accommodate predictable growth in real GDP.

On the basis of the capital asset pricing model, Black concluded that discretionary monetary policy could not do the good that Keynesians wanted it to do. He concluded that monetary policy should be passive within an economy. But he also concluded that it could not do the harm monetarists feared it would do. Black said in a letter to Friedman, in January 1972:

"In the U.S. economy, much of the public debt is in the form of Treasury bills. Each week, some of these bills mature, and new bills are sold. If the Federal Reserve System tries to inject money into the private sector, the private sector will simply turn around and exchange its money for Treasury bills at the next auction. If the Federal Reserve withdraws money, the private sector will allow some of its Treasury bills to mature without replacing them."

In 1973, Black, along with Myron Scholes, published the paper 'The Pricing of Options and Corporate Liabilities' in 'The Journal of Political Economy'. This was his most famous work and included the Black–Scholes equation.

In March 1976, Black proposed that human capital and business have "ups and downs that are largely unpredictable [...] because of basic uncertainty about what people will want in the future and about what the economy will be able to produce in the future. If future tastes and technology were known, profits and wages would grow smoothly and surely over time." A boom is a period when technology matches well with demand. A bust is a period of mismatch. This view made Black an early contributor to real business cycle theory.

Economist Tyler Cowen has argued that Black's work on monetary economics and business cycles can be used to explain the Great Recession.

Black's works on monetary theory, business cycles and options are parts of his vision of a unified framework. He once stated: "I like the beauty and symmetry in Mr. Treynor's equilibrium models so much that I started designing them myself. I worked on models in several areas:""Monetary theory, Business cycles, Options and warrants""For 20 years, I have been struggling to show people the beauty in these models to pass on knowledge I received from Mr. Treynor.""In monetary theory --- the theory of how money is related to economic activity --- I am still struggling. In business cycle theory --- the theory of fluctuation in the economy --- I am still struggling. In options and warrants, though, people see the beauty."It can be shown that the mathematical techniques developed in the option theory can be extended to provide a mathematical analysis of monetary theory and business cycles as well.

Business Cycles and Equilibrium (1987)
Fischer Black has published many academic articles, including his most known book, Business Cycles and Equilibrium. In this book, Black proposes at the beginning of the book to imagine a world where money does not exist. With its theory that economic and financial markets are in a continual equilibrium-is one of his books that still rings true today, given the current economic crisis. Building upon these statements, Black creates models as well as challenges monetary theorists, especially those who subscribe to the ideas of the quantity theory of money and liquidity of money. Banks are the main institutions of monetary transactions in Black's book, to which he also states that money is an endogenous resource (contrary to monetarists who believe money to be an exogenous resource), provided by banks due to profit maximization. Controversial statements such as "Monetary and exchange rate policies accomplish almost nothing, and fiscal policies are unimportant in causing or changing business cycles" have made Black enemies with Keynesians and Monetarists alike.

Illness and death
In early 1994, Black was diagnosed with throat cancer. Surgery at first appeared successful, and Black was well enough to attend the annual meeting of the International Association of Financial Engineers that October, where he received their award as Financial Engineer of the Year. However, the cancer returned, and Black died in August 1995.

Posthumous recognition
The Nobel Prize is not given posthumously, so it was not awarded to Black in 1997 when his co-author Myron Scholes received the honor for their landmark work on option pricing along with Robert C. Merton, another pioneer in the development of valuation of stock options. However, when announcing the award that year, the Nobel committee did prominently mention Black's key role.

Black has also received recognition as the co-author of the Black–Derman–Toy interest rate derivatives model, which was developed for in-house use by Goldman Sachs in the 1980s but eventually published. He also co-authored the Black–Litterman model on global asset allocation while at Goldman Sachs.

The advisory board of The Journal of Performance Measurement inducted Black into the Performance & Risk Measurement Hall of Fame in 2017. The announcement appears in the Winter 2016/2017 issue of the journal. The Hall of Fame recognizes individuals who have made significant contributions to investment performance and risk measurement.

Fischer Black Prize
In 2002, the American Finance Association established the biennially awarded Fischer Black Prize in memory of Fischer Black. The award is given to a young researcher whose body of work "best exemplifies the Fischer Black hallmark of developing original research that is relevant to finance practice".

Selected bibliography

 * F. Black, Myron Scholes, & Michael Jensen, "The Capital-Asset Pricing Model: Some empirical tests", in Jensen, editor, Studies in the Theory of Capital Markets (1972).
 * F. Black, "Active and Passive Monetary Policy in a Neoclassical Model", The Journal of Finance, Vol. 27, No. 4 (Sep., 1972), pp. 801–814.
 * Fischer Black & Myron Scholes, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy (1973).
 * F. Black & M. Scholes, "The Effects of Dividend Yield and Dividend Policy on Common Stock Prices and Returns", Journal of Financial Economics (1974).
 * F. Black, "Fact and Fantasy in the Use of Options", Financial Analysts Journal 31, pp36–41, 61–72 (July/August 1975).
 * F. Black, "The Pricing of Commodity Contracts", 1976, Journal of Financial Economics.
 * F. Black, "Noise", Journal of Finance, vol. 41, pp. 529–543 (1986).
 * Fischer Black, Business Cycles and Equilibrium, Basil Blackwell, 1987. ISBN 0470499176
 * F. Black, E. Derman, & W. Toy, "A One-Factor Model of Interest Rates and its Application to Treasury Bond Options", Financial Analyst Journal (1990).
 * F. Black & R. Litterman, "Global Portfolio Optimization", Financial Analysts Journal vol. 48, no. 5, pp. 28–43 (1992).
 * F. Black, "Beta and Return", Journal of Portfolio Management, vol. 20 (1), pp. 8–18 (1993).
 * F. Black, "Interest Rates as Options", Journal of Finance, vol. 50, pp. 1371–1376 (1995).
 * Fischer Black, Exploring General Equilibrium, MIT Press, 1995. ISBN 0262514095

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Fischer Black


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User talk:Eric Hanson

Welcome to Wikipedia! By the way, when signing your contribs in talk pages, you can type three tildes, which are automatically replaced by a link to your user page. (Four tildes, and it includes the time and date.) You can customize how the name appears in your preferences. --Brion 06:25 Oct 20, 2002 (UTC)

title: User talk:Eric Hanson


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Myron Scholes

Myron Samuel Scholes (born July 1, 1941) is a Canadian–American financial economist. Scholes is the Frank E. Buck Professor of Finance, Emeritus, at the Stanford Graduate School of Business, Nobel Laureate in Economic Sciences, and co-originator of the Black–Scholes options pricing model. Scholes is currently the chairman of the Board of Economic Advisers of Stamos Capital Partners. Previously he served as the chairman of Platinum Grove Asset Management and on the Dimensional Fund Advisors board of directors, American Century Mutual Fund board of directors and the Cutwater Advisory Board. He was a principal and limited partner at Long-Term Capital Management (LTCM), a highly leveraged hedge fund that collapsed in 1998, and a managing director at Salomon Brothers. Other positions Scholes held include the Edward Eagle Brown Professor of Finance at the University of Chicago, senior research fellow at the Hoover Institution, director of the Center for Research in Security Prices, and professor of finance at MIT's Sloan School of Management. Scholes earned his PhD at the University of Chicago.

In 1997, Scholes – together with Robert C. Merton – was awarded the Nobel Memorial Prize in Economic Sciences for a method to determine the value of derivatives. The model provides a conceptual framework for valuing options, such as calls or puts, and is referred to as the Black–Scholes model.

Early life and education
Scholes was born to a Jewish family on July 1, 1941, in Timmins, Ontario, where his family had moved during the Great Depression. In 1951 the family moved to Hamilton, Ontario. Scholes was a good student although fighting with his impaired vision starting with his teens until finally getting an operation when he was twenty-six. Through his family, he became interested in economics early, as he helped with his uncles' businesses and his parents helped him open an account for investing in the stock market while he was in high school.

After his mother died from cancer, Scholes remained in Hamilton for undergraduate studies and earned a Bachelor's degree in economics from McMaster University in 1962. One of his professors at McMaster introduced him to the works of George Stigler and Milton Friedman, two University of Chicago economists who would later both win Nobel prizes in economics. After receiving his B.A. he decided to enroll in graduate studies in economics at the University of Chicago. Here, Scholes was a colleague with Michael Jensen and Richard Roll, and he had the opportunity to study with Eugene Fama and Merton Miller, researchers who were developing the relatively new field of financial economics. He earned his MBA at the Booth School of Business in 1964 and his Ph.D. in 1969 with a dissertation written under the supervision of Eugene Fama and Merton Miller.

Academic career
In 1968, after finishing his dissertation, Scholes took an academic position at the MIT Sloan School of Management. Here he met Fischer Black, who was a consultant for Arthur D. Little at the time, and Robert C. Merton, who joined MIT in 1970. For the following years Scholes, Black and Merton undertook groundbreaking research in asset pricing, including the work on their famous option pricing model. At the same time, Scholes continued collaborating with Merton Miller and Michael Jensen. In 1973 he decided to move to the University of Chicago Booth School of Business, looking forward to work closely with Eugene Fama, Merton Miller and Fischer Black, who had taken his first academic position at Chicago in 1972 (although he moved two years later to MIT). While at Chicago, Scholes also started working closely with the Center for Research in Security Prices, helping to develop and analyze its famous database of high frequency stock market data.

In 1981 he moved to Stanford University, where he remained until he retired from teaching in 1996. Since then he holds the position of Frank E. Buck Professor of Finance Emeritus at Stanford. While at Stanford his research interest concentrated on the economics of investment banking and tax planning in corporate finance.

In 1997 he shared the Nobel Memorial Prize in Economics with Robert C. Merton "for a new method to determine the value of derivatives". Fischer Black, who co-authored with them the work that was awarded, had died in 1995 and thus was not eligible for the prize.

In 2012, he authored an article entitled 'Not All Growth Is Good' in The 4% Solution: Unleashing the Economic Growth America Needs, published by the George W. Bush Presidential Center.

Investment activity
In 1990 Scholes became more involved directly with the financial markets. He went to Salomon Brothers as a special consultant, then becoming a managing director and co-head of its fixed-income-derivative group. In 1994 Scholes joined several colleagues, including John Meriwether, the former vice-chairman and head of bond trading at Salomon Brothers, and his future Nobel Memorial Prize co-winner Robert C. Merton, and co-founded a hedge fund called Long-Term Capital Management (LTCM). The fund, which started operations with $1 billion of investor capital, performed extremely well in the first years, realizing annualized returns of over 40%. However, following the 1997 Asian financial crisis and the 1998 Russian financial crisis the highly leveraged fund in 1998 lost $4.6 billion in less than four months and collapsed abruptly, becoming one of the most prominent examples of risk potential in the investment industry.

LTCM brought legal problems for Scholes in 2005 in the case of Long-Term Capital Holdings v. United States. The firm's corporate structure and accounting had established an offshore tax shelter to avoid taxes on investment profits. Courts disallowed the firm's claim of $40 million in tax savings, finding it based on formal accounting losses of $106 million that represented no economic substance.

Subsequent to LTCM, in 1999 Scholes joined Oak Hill Capital, the private equity firm led by Robert Bass. There, Scholes and his LTCM colleague, Chi-Fu Huang, launched a new hedge fund, Oak Hill Platinum Partners.

Other interests and roles
Scholes is also chief investment strategist at Janus Henderson, a role he held at legacy firm Janus Capital Group since 2014. Janus Capital merged with Henderson Group in 2017 to form Janus Henderson. In this role, he leads the firm's evolving asset allocation product development efforts and partners with the investment team contributing macro insights and quantitative analysis specific to hedging, risk management and disciplined portfolio construction. He also serves on the boards of the Chicago Mercantile Exchange and Dimensional Fund Advisors. Scholes is currently the chairman of the Board of Economic Advisers of Stamos Capital Partners.

title: Myron Scholes


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Short (finance)

In finance, being short in an asset means investing in such a way that the investor will profit if the market value of the asset falls. This is the opposite of the more common long position, where the investor will profit if the market value of the asset rises. An investor that sells an asset short is, as to that asset, a short seller.

There are a number of ways of achieving a short position. The most basic is physical selling short or short-selling, by which the short seller borrows an asset (often a security such as a share of stock or a bond) and quickly selling it. The short seller must later buy the same amount of the asset to return it to the lender. If the market value of the asset has fallen in the meantime, the short seller will have made a profit equal to the difference. Conversely, if the price has risen then the short seller will bear a loss. The short seller usually must pay handling fee to borrow the asset (charged at a particular rate over time, similar to an interest payment) and reimburse the lender for any cash return (such as a dividend) that was paid on the asset while borrowed.

A short position can also be created through a futures contract, forward contract, or option contract, by which the short seller assumes an obligation or right to sell an asset at a future date at a price stated in the contract. If the price of the asset falls below the contract price, the short seller can buy it at the lower market value and immediately sell it at the higher price specified in the contract. A short position can also be achieved through certain types of swap, such as a contract for difference. This is an agreements between two parties to pay each other the difference if the price of an asset rises or falls, under which the party that will benefit if the price falls will have a short position.

Because a short seller can incur a liability to the lender if the price rises, and because a short sale is normally done through a stockbroker, a short seller is typically required to post margin to its broker as collateral to ensure that any such liabilities can be met, and to post additional margin if losses begin to accrue. For analogous reasons, short positions in derivatives also usually involve the posting of margin with the counterparty. Any failure to post margin promptly would prompt the broker or counterparty to close the position.

Short selling is a common practice in public securities, futures, and currency markets that are fungible and reasonably liquid.

A short sale may have a variety of objectives. Speculators may sell short hoping to realize a profit on an instrument that appears overvalued, just as long investors or speculators hope to profit from a rise in the price of an instrument that appears undervalued. Alternatively, traders or fund managers may use offsetting short positions to hedge certain risks that exist in a long position or a portfolio.

Research indicates that banning short selling is ineffective and has negative effects on markets. Nevertheless, short selling is subject to criticism and periodically faces hostility from society and policymakers.

Physical shorting with borrowed securities
To profit from a decrease in the price of a security, a short seller can borrow the security and sell it, expecting that it will be cheaper to repurchase in the future. When the seller decides that the time is right (or when the lender recalls the securities), the seller buys the same number of equivalent securities and returns them to the lender. The act of buying back the securities that were sold short is called covering the short, covering the position or simply covering. A short position can be covered at any time before the securities are due to be returned. Once the position is covered, the short seller is not affected by subsequent rises or falls in the price of the securities, for it already holds the securities that it will return to the lender.

The process relies on the fact that the securities (or the other assets being sold short) are fungible. An investor therefore "borrows" securities in the same sense as one borrows a $10 bill, where the legal ownership of the money is transferred to the borrower and it can be freely disposed of, and different bank notes or coins can be returned to the lender. This can be contrasted with the sense in which one borrows a bicycle, where the ownership of the bicycle does not change and the same bicycle must be returned, not merely one that is the same model.

Because the price of a share is theoretically unlimited, the potential losses of a short-seller are also theoretically unlimited.

Worked example of a profitable short sale
Shares in ACME Inc. currently trade at $10 per share.

 * 1) A short seller borrows from a lender 100 shares of ACME Inc., and immediately sells them for a total of $1,000.
 * 2) Subsequently, the price of the shares falls to $8 per share.
 * 3) Short seller now buys 100 shares of ACME Inc. for $800.
 * 4) Short seller returns the shares to the lender, who must accept the return of the same number of shares as was lent despite the fact that the market value of the shares has decreased.
 * 5) Short seller keeps as its profit the $200 difference between the price at which the short seller sold the borrowed shares and the lower price at which the short seller purchased the equivalent shares (minus borrowing fees paid to the lender).

Worked example of a loss-making short sale
Shares in ACME Inc. currently trade at $10 per share.

 * 1) A short seller borrows 100 shares of ACME Inc., and sells them for a total of $1,000.
 * 2) Subsequently, the price of the shares rises to $25 per share.
 * 3) Short seller is required to return the shares, and is compelled to buy 100 shares of ACME Inc. for $2,500.
 * 4) Short seller returns the shares to the lender, who accepts the return of the same number of shares as was lent.
 * 5) Short seller incurs as a loss the $1,500 difference between the price at which he sold the borrowed shares and the higher price at which the short seller had to purchase the equivalent shares (plus any borrowing fees).

Synthetic shorting with derivatives
"Shorting" or "going short" (and sometimes also "short selling") also refer more broadly to any transaction used by an investor to profit from the decline in price of a borrowed asset or financial instrument. Derivatives contracts that can be used in this way include futures, options, and swaps. These contracts are typically cash-settled, meaning that no buying or selling of the asset in question is actually involved in the contract, although typically one side of the contract will be a broker that will effect a back-to-back sale of the asset in question in order to hedge their position.

History
The practice of short selling was likely invented in 1609 by Dutch businessman Isaac Le Maire, a sizeable shareholder of the Dutch East India Company (Vereenigde Oostindische Compagnie or VOC in Dutch). Short selling can exert downward pressure on the underlying stock, driving down the price of shares of that security. This, combined with the seemingly complex and hard-to-follow tactics of the practice, has made short selling a historical target for criticism. At various times in history, governments have restricted or banned short selling.

The London banking house of Neal, James, Fordyce and Down collapsed in June 1772, precipitating a major crisis that included the collapse of almost every private bank in Scotland, and a liquidity crisis in the two major banking centres of the world, London and Amsterdam. The bank had been speculating by shorting East India Company stock on a massive scale, and apparently using customer deposits to cover losses. It was perceived as having a magnifying effect in the violent downturn in the Dutch tulip market in the eighteenth century. In another well-referenced example, George Soros became notorious for "breaking the Bank of England" on Black Wednesday of 1992, when he sold short more than $10 billion worth of pounds sterling.

The term short was in use from at least the mid-nineteenth century. It is commonly understood that the word "short" (i.e. 'lacking') is used because the short seller is in a deficit position with his brokerage house. Jacob Little, known as The Great Bear of Wall Street, began shorting stocks in the United States in 1822.

Short sellers were blamed for the Wall Street Crash of 1929. Regulations governing short selling were implemented in the United States in 1929 and in 1940. Political fallout from the 1929 crash led Congress to enact a law banning short sellers from selling shares during a downtick; this was known as the uptick rule and was in effect until 3 July 2007, when it was removed by the Securities and Exchange Commission (SEC Release No. 34-55970). President Herbert Hoover condemned short sellers and even J. Edgar Hoover said he would investigate short sellers for their role in prolonging the Depression. A few years later, in 1949, Alfred Winslow Jones founded a fund (that was unregulated) that bought stocks while selling other stocks short, hence hedging some of the market risk, and the hedge fund was born.

Negative news, such as litigation against a company, may also entice professional traders to sell the stock short in hope of the stock price going down.

During the dot-com bubble, shorting a start-up company could backfire since it could be taken over at a price higher than the price at which speculators shorted; short-sellers were forced to cover their positions at acquisition prices, while in many cases the firm often overpaid for the start-up.

Naked short selling restrictions
During the 2008 financial crisis, critics argued that investors taking large short positions in struggling financial firms like Lehman Brothers, HBOS and Morgan Stanley created instability in the stock market and placed additional downward pressure on prices. In response, a number of countries introduced restrictive regulations on short-selling in 2008 and 2009. Naked short selling is the practice of short-selling a tradable asset without first borrowing the security or ensuring that the security can be borrowed – it was this practice that was commonly restricted. Investors argued that it was the weakness of financial institutions, not short-selling, that drove stocks to fall. In September 2008, the Securities Exchange Commission in the United States abruptly banned short sales, primarily in financial stocks, to protect companies under siege in the stock market. That ban expired several weeks later as regulators determined the ban was not stabilizing the price of stocks.

Temporary short-selling bans were also introduced in the United Kingdom, Germany, France, Italy and other European countries in 2008 to minimal effect. Australia moved to ban naked short selling entirely in September 2008. Germany placed a ban on naked short selling of certain euro zone securities in 2010. Spain, Portugal and Italy introduced short selling bans in 2011 and again in 2012.

During the COVID-19 pandemic, shorting was severely restricted or temporarily banned, with European market watchdogs tightening the rules on short selling "in an effort to stem the historic losses arising from the coronavirus pandemic".

Worldwide, economic regulators seem inclined to restrict short selling to decrease potential downward price cascades. Investors continue to argue this only contributes to market inefficiency.

Mechanism
A short seller typically borrows through a broker, who is usually holding the securities for another investor who owns the securities; the broker himself seldom purchases the securities to lend to the short seller. The lender does not lose the right to sell the securities while they have been lent, as the broker usually holds a large pool of such securities for a number of investors which, as such securities are fungible, can instead be transferred to any buyer. In most market conditions there is a ready supply of securities to be borrowed, held by pension funds, mutual funds and other investors.

Shorting stock in the U.S.
To sell stocks short in the U.S., the seller must arrange for a broker-dealer to confirm that it can deliver the shorted securities. This is referred to as a locate. Brokers have a variety of means to borrow stocks to facilitate locates and make good on delivery of the shorted security.

The vast majority of stocks borrowed by U.S. brokers come from loans made by the leading custody banks and fund management companies (see list below). Institutions often lend out their shares to earn extra money on their investments. These institutional loans are usually arranged by the custodian who holds the securities for the institution. In an institutional stock loan, the borrower puts up cash collateral, typically 102% of the value of the stock. The cash collateral is then invested by the lender, who often rebates part of the interest to the borrower. The interest that is kept by the lender is the compensation to the lender for the stock loan.

Brokerage firms can also borrow stocks from the accounts of their own customers. Typical margin account agreements give brokerage firms the right to borrow customer shares without notifying the customer. In general, brokerage accounts are only allowed to lend shares from accounts for which customers have debit balances, meaning they have borrowed from the account. SEC Rule 15c3-3 imposes such severe restrictions on the lending of shares from cash accounts or excess margin (fully paid for) shares from margin accounts that most brokerage firms do not bother except in rare circumstances. (These restrictions include that the broker must have the express permission of the customer and provide collateral or a letter of credit.)

Most brokers allow retail customers to borrow shares to short a stock only if one of their own customers has purchased the stock on margin. Brokers go through the "locate" process outside their own firm to obtain borrowed shares from other brokers only for their large institutional customers.

Stock exchanges such as the NYSE or the NASDAQ typically report the "short interest" of a stock, which gives the number of shares that have been legally sold short as a percent of the total float. Alternatively, these can also be expressed as the short interest ratio, which is the number of shares legally sold short as a multiple of the average daily volume. These can be useful tools to spot trends in stock price movements but for them to be reliable, investors must also ascertain the number of shares brought into existence by naked shorters. Speculators are cautioned to remember that for every share that has been shorted (owned by a new owner), a 'shadow owner' exists (i.e., the original owner) who also is part of the universe of owners of that stock, i.e., despite having no voting rights, he has not relinquished his interest and some rights in that stock.

Securities lending
When a security is sold, the seller is contractually obliged to deliver it to the buyer. If a seller sells a security short without owning it first, the seller must borrow the security from a third party to fulfill its obligation. Otherwise, the seller fails to deliver, the transaction does not settle, and the seller may be subject to a claim from its counterparty. Certain large holders of securities, such as a custodian or investment management firm, often lend out these securities to gain extra income, a process known as securities lending. The lender receives a fee for this service. Similarly, retail investors can sometimes make an extra fee when their broker wants to borrow their securities. This is only possible when the investor has full title of the security, so it cannot be used as collateral for margin buying.

Sources of short interest data
Time delayed short interest data (for legally shorted shares) is available in a number of countries, including the US, the UK, Hong Kong, and Spain. The number of stocks being shorted on a global basis has increased in recent years for various structural reasons (e.g., the growth of 130/30 type strategies, short or bear ETFs). The data is typically delayed; for example, the NASDAQ requires its broker-dealer member firms to report data on the 15th of each month, and then publishes a compilation eight days later.

Some market data providers (like Data Explorers and SunGard Financial Systems ) believe that stock lending data provides a good proxy for short interest levels (excluding any naked short interest). SunGard provides daily data on short interest by tracking the proxy variables based on borrowing and lending data it collects.

Short selling terms
Days to Cover (DTC) is the relationship between the number of shares in a given equity that has been legally short-sold and the number of days of typical trading that it would require to 'cover' all legal short positions outstanding. For example, if there are ten million shares of XYZ Inc. that are currently legally short-sold and the average daily volume of XYZ shares traded each day is one million, it would require ten days of average trading for all legal short positions to be covered (10 million / 1 million).

Short Interest relates the number of shares in a given equity that have been legally shorted divided by the total shares outstanding for the company, usually expressed as a percent. For example, if there are ten million shares of XYZ Inc. that are currently legally short-sold, and the total number of shares issued by the company is one hundred million, the Short Interest is 10% (10 million / 100 million). If, however, shares are being created through naked short selling, "fails" data must be accessed to assess accurately the true level of short interest.

Borrow cost is the fee paid to a securities lender for borrowing the stock or other security. The cost of borrowing the stock is usually negligible compared to fees paid and interest accrued on the margin account – in 2002, 91% of stocks could be shorted for less than a 1% fee per annum, generally lower than interest rates earned on the margin account. However, certain stocks become "hard to borrow" as stockholders willing to lend their stock become more difficult to locate. The cost of borrowing these stocks can become significant – in February 2001, the cost to borrow (short) Krispy Kreme stock reached an annualized 55%, indicating that a short seller would need to pay the lender more than half the price of the stock over the course of the year, essentially as interest for borrowing a stock in limited supply. This has important implications for derivatives pricing and strategy, for the borrow cost itself can become a significant convenience yield for holding the stock (similar to additional dividend) – for instance, put–call parity relationships are broken and the early exercise feature of American call options on non-dividend paying stocks can become rational to exercise early, which otherwise would not be economical.

Major lenders

 * State Street Corporation (Boston, United States)
 * Merrill Lynch (New Jersey, United States)
 * JP Morgan Chase (New York, United States)
 * Northern Trust (Chicago, United States)
 * Fortis (Amsterdam, Netherlands, now defunct)
 * ABN AMRO (Amsterdam, Netherlands, formerly Fortis)
 * Citibank (New York, United States)
 * Bank of New York Mellon Corporation (New York, United States)
 * UBS AG (Zurich, Switzerland)
 * Barclays (London, United Kingdom)

Naked short selling
A naked short sale occurs when a security is sold short without borrowing the security within a set time (for example, three days in the US.) This means that the buyer of such a short is buying the short-seller's promise to deliver a share, rather than buying the share itself. The short-seller's promise is known as a hypothecated share.

When the holder of the underlying stock receives a dividend, the holder of the hypothecated share would receive an equal dividend from the short seller.

Naked shorting has been made illegal except where allowed under limited circumstances by market makers. It is detected by the Depository Trust & Clearing Corporation (in the US) as a "failure to deliver" or simply "fail." While many fails are settled in a short time, some have been allowed to linger in the system.

In the US, arranging to borrow a security before a short sale is called a locate. In 2005, to prevent widespread failure to deliver securities, the U.S. Securities and Exchange Commission (SEC) put in place Regulation SHO, intended to prevent speculators from selling some stocks short before doing a locate. More stringent rules were put in place in September 2008, ostensibly to prevent the practice from exacerbating market declines. These rules were made permanent in 2009.

Fees
When a broker facilitates the delivery of a client's short sale, the client is charged a fee for this service, usually a standard commission similar to that of purchasing a similar security.

If the short position begins to move against the holder of the short position (i.e., the price of the security begins to rise), money is removed from the holder's cash balance and moved to their margin balance. If short shares continue to rise in price, and the holder does not have sufficient funds in the cash account to cover the position, the holder begins to borrow on margin for this purpose, thereby accruing margin interest charges. These are computed and charged just as for any other margin debit. Therefore, only margin accounts can be used to open a short position.

When a security's ex-dividend date passes, the dividend is deducted from the shortholder's account and paid to the person from whom the stock is borrowed.

For some brokers, the short seller may not earn interest on the proceeds of the short sale or use it to reduce outstanding margin debt. These brokers may not pass this benefit on to the retail client unless the client is very large. The interest is often split with the lender of the security.

Dividends and voting rights
Where shares have been shorted and the company that issues the shares distributes a dividend, the question arises as to who receives the dividend. The new buyer of the shares, who is the holder of record and holds the shares outright, receives the dividend from the company. However, the lender, who may hold its shares in a margin account with a prime broker and is unlikely to be aware that these particular shares are being lent out for shorting, also expects to receive a dividend. The short seller therefore pays the lender an amount equal to the dividend to compensate—though technically, as this payment does not come from the company, it is not a dividend. The short seller is therefore said to be short the dividend.

A similar issue comes up with the voting rights attached to the shorted shares. Unlike a dividend, voting rights cannot legally be synthesized and so the buyer of the shorted share, as the holder of record, controls the voting rights. The owner of a margin account from which the shares were lent agreed in advance to relinquish voting rights to shares during the period of any short sale.

As noted earlier, victims of naked shorting sometimes report that the number of votes cast is greater than the number of shares issued by the company.

Markets
Transactions in financial derivatives such as options and futures have the same name but have different overlaps, one notable overlap is having an equal "negative" amount in the position. However, the practice of a short position in derivatives is completely different. Derivatives are contracts between two parties, a buyer and seller. Each trade results in a "long" (buyer's position) and a "short" (seller's position).

Futures and options contracts
When trading futures contracts, being 'short' means having the legal obligation to deliver something at the expiration of the contract, although the holder of the short position may alternately buy back the contract prior to expiration instead of making delivery. Short futures transactions are often used by producers of a commodity to fix the future price of goods they have not yet produced. Shorting a futures contract is sometimes also used by those holding the underlying asset (i.e. those with a long position) as a temporary hedge against price declines. Shorting futures may also be used for speculative trades, in which case the investor is looking to profit from any decline in the price of the futures contract prior to expiration.

An investor can also purchase a put option, giving that investor the right (but not the obligation) to sell the underlying asset (such as shares of stock) at a fixed price. In the event of a market decline, the option holder may exercise these put options, obliging the counterparty to buy the underlying asset at the agreed upon (or "strike") price, which would then be higher than the current quoted spot price of the asset.

Currency
Selling short on the currency markets is different from selling short on the stock markets. Currencies are traded in pairs, each currency being priced in terms of another. In this way, selling short on the currency markets is identical to going long on stocks.

Novice traders or stock traders can be confused by the failure to recognize and understand this point: a contract is always long in terms of one medium and short another.

When the exchange rate has changed, the trader buys the first currency again; this time he gets more of it, and pays back the loan. Since he got more money than he had borrowed initially, he makes money. The reverse can also occur.

An example of this is as follows: Let us say a trader wants to trade with the US dollar and the Indian rupee currencies. Assume that the current market rate is US$1 to Rs. 50 and the trader borrows Rs. 100. With this, he buys US$2. If the next day, the conversion rate becomes US$1 to Rs. 51, then the trader sells his US$2 and gets Rs. 102. He returns Rs. 100 and keeps the Rs. 2 profit (minus fees).

One may also take a short position in a currency using futures or options; the preceding method is used to bet on the spot price, which is more directly analogous to selling a stock short.

Risks
Note: this section does not apply to currency markets.

Short selling is sometimes referred to as a "negative income investment strategy" because there is no potential for dividend income or interest income. Stock is held only long enough to be sold pursuant to the contract, and one's return is therefore limited to short term capital gains, which are taxed as ordinary income. For this reason, buying shares (called "going long") has a very different risk profile from selling short. Furthermore, a "long's" losses are limited because the price can only go down to zero, but gains are not, as there is no limit, in theory, on how high the price can go. On the other hand, the short seller's possible gains are limited to the original price of the stock, which can only go down to zero, whereas the loss potential, again in theory, has no limit. For this reason, short selling probably is most often used as a hedge strategy to manage the risks of long investments.

Many short sellers place a stop order with their stockbroker after selling a stock short—an order to the brokerage to cover the position if the price of the stock should rise to a certain level. This is to limit the loss and avoid the problem of unlimited liability described above. In some cases, if the stock's price skyrockets, the stockbroker may decide to cover the short seller's position immediately and without his consent to guarantee that the short seller can make good on his debt of shares.

Short sellers must be aware of the potential for a short squeeze. When the price of a stock rises significantly, some people who are shorting the stock cover their positions to limit their losses (this may occur in an automated way if the short sellers had stop-loss orders in place with their brokers); others may be forced to close their position to meet a margin call; others may be forced to cover, subject to the terms under which they borrowed the stock, if the person who lent the stock wishes to sell and take a profit. Since covering their positions involves buying shares, the short squeeze causes an ever further rise in the stock's price, which in turn may trigger additional covering. Because of this, most short sellers restrict their activities to heavily traded stocks, and they keep an eye on the "short interest" levels of their short investments. Short interest is defined as the total number of shares that have been legally sold short, but not covered. A short squeeze can be deliberately induced. This can happen when large investors (such as companies or wealthy individuals) notice significant short positions, and buy many shares, with the intent of selling the position at a profit to the short sellers, who may be panicked by the initial uptick or who are forced to cover their short positions to avoid margin calls.

Another risk is that a given stock may become "hard to borrow". As defined by the SEC and based on lack of availability, a broker may charge a hard to borrow fee daily, without notice, for any day that the SEC declares a share is hard to borrow. Additionally, a broker may be required to cover a short seller's position at any time ("buy in"). The short seller receives a warning from the broker that he is "failing to deliver" stock, which leads to the buy-in.

Because short sellers must eventually deliver the shorted securities to their broker, and need money to buy them, there is a credit risk for the broker. The penalties for failure to deliver on a short selling contract inspired financier Daniel Drew to warn: "He who sells what isn't his'n, Must buy it back or go to pris'n." To manage its own risk, the broker requires the short seller to keep a margin account, and charges interest of between 2% and 8% depending on the amounts involved.

In 2011, the eruption of the massive China stock frauds on North American equity markets brought a related risk to light for the short seller. The efforts of research-oriented short sellers to expose these frauds eventually prompted NASDAQ, NYSE and other exchanges to impose sudden, lengthy trading halts that froze the values of shorted stocks at artificially high values. Reportedly in some instances, brokers charged short sellers excessively large amounts of interest based on these high values as the shorts were forced to continue their borrowings at least until the halts were lifted.

Short sellers tend to temper overvaluation by selling into exuberance. Likewise, short sellers are said to provide price support by buying when negative sentiment is exacerbated after a significant price decline. Short selling can have negative implications if it causes a premature or unjustified share price collapse when the fear of cancellation due to bankruptcy becomes contagious.

Hedging
Hedging often represents a means of minimizing the risk from a more complex set of transactions. Examples of this are:
 * A farmer who has just planted their wheat wants to lock in the price at which they can sell after the harvest. The farmer would take a short position in wheat futures.
 * A market maker in corporate bonds is constantly trading bonds when clients want to buy or sell. This can create substantial bond positions.  The largest risk is that interest rates overall move.  The trader can hedge this risk by selling government bonds short against his long positions in corporate bonds.  In this way, the risk that remains is credit risk of the corporate bonds.
 * An options trader may short shares to remain delta neutral so that they are not exposed to risk from price movements in the stocks that underlie their options.

Arbitrage
A short seller may be trying to benefit from market inefficiencies arising from the mispricing of certain products. Examples of this are
 * An arbitrageur who buys long futures contracts on a US Treasury security, and sells short the underlying US Treasury security.

Against the box
One variant of selling short involves a long position. "Selling short against the box" consists of holding a long position on which the shares have already risen, whereupon one then enters a short sell order for an equal number of shares. The term box alludes to the days when a safe deposit box was used to store (long) shares. The purpose of this technique is to lock in paper profits on the long position without having to sell that position (and possibly incur taxes if said position has appreciated). Once the short position has been entered, it serves to balance the long position taken earlier. Thus, from that point in time, the profit is locked in (less brokerage fees and short financing costs), regardless of further fluctuations in the underlying share price. For example, one can ensure a profit in this way, while delaying sale until the subsequent tax year.

U.S. investors considering entering into a "short against the box" transaction should be aware of the tax consequences of this transaction. Unless certain conditions are met, the IRS deems a "short against the box" position to be a "constructive sale" of the long position, which is a taxable event. These conditions include a requirement that the short position be closed out within 30 days of the end of the year and that the investor must hold their long position, without entering into any hedging strategies, for a minimum of 60 days after the short position has been closed.

United States
The Securities Exchange Act of 1934 gave the Securities and Exchange Commission the power to regulate short sales. The first official restriction on short selling came in 1938, when the SEC adopted a rule (known as the uptick rule) that a short sale could only be made when the price of a particular stock was higher than the previous trade price. The uptick rule aimed to prevent short sales from causing or exacerbating market price declines. In January 2005, The Securities and Exchange Commission enacted Regulation SHO to target abusive naked short selling. Regulation SHO was the SEC's first update to short selling restrictions since the uptick rule in 1938.

The regulation contains two key components: the "locate" and the "close-out". The locate component attempts to reduce failure to deliver securities by requiring a broker possess or have arranged to possess borrowed shares. The close out component requires that a broker be able to deliver the shares that are to be shorted. In the US, initial public offers (IPOs) cannot be sold short for a month after they start trading. This mechanism is in place to ensure a degree of price stability during a company's initial trading period. However, some brokerage firms that specialize in penny stocks (referred to colloquially as bucket shops) have used the lack of short selling during this month to pump and dump thinly traded IPOs. Canada and other countries do allow selling IPOs (including U.S. IPOs) short.

The Securities and Exchange Commission initiated a temporary ban on short selling of 799 financial stocks from 19 September 2008 until 2 October 2008. Greater penalties for naked shorting, by mandating delivery of stocks at clearing time, were also introduced. Some state governors have been urging state pension bodies to refrain from lending stock for shorting purposes. An assessment of the effect of the temporary ban on short-selling in the United States and other countries in the wake of the financial crisis showed that it had only "little impact" on the movements of stocks, with stock prices moving in the same way as they would have moved anyhow, but the ban reduced volume and liquidity.

Europe, Australia and China
In the UK, the Financial Services Authority had a moratorium on short selling of 29 leading financial stocks, effective from 2300 GMT on 19 September 2008 until 16 January 2009. After the ban was lifted, John McFall, chairman of the Treasury Select Committee, House of Commons, made clear in public statements and a letter to the FSA that he believed it ought to be extended. Between 19 and 21 September 2008, Australia temporarily banned short selling, and later placed an indefinite ban on naked short selling. Australia's ban on short selling was further extended for another 28 days on 21 October 2008. Also during September 2008, Germany, Ireland, Switzerland and Canada banned short selling of leading financial stocks, and France, the Netherlands and Belgium banned naked short selling of leading financial stocks. By contrast with the approach taken by other countries, Chinese regulators responded by allowing short selling, along with a package of other market reforms. Short selling was completely allowed on 31 March 2010, limited to " for large blue chip stocks with good earnings performance and little price volatility." However, in 2015, short selling was effectively banned due to legislative restrictions on borrowing stocks following the stock market crash the same year.

Views of short selling
Advocates of short selling argue that the practice is an essential part of the price discovery mechanism. Financial researchers at Duke University said in a study that short interest is an indicator of poor future stock performance (the self-fulfilling aspect) and that short sellers exploit market mistakes about firms' fundamentals.

Such noted investors as Seth Klarman and Warren Buffett have said that short sellers help the market. Klarman argued that short sellers are a useful counterweight to the widespread bullishness on Wall Street, while Buffett believes that short sellers are useful in uncovering fraudulent accounting and other problems at companies.

Shortseller James Chanos received widespread publicity when he was an early critic of the accounting practices of Enron. Chanos responds to critics of short-selling by pointing to the critical role they played in identifying problems at Enron, Boston Market and other "financial disasters" over the years. In 2011, research oriented short sellers were widely acknowledged for exposing the China stock frauds.

Commentator Jim Cramer has expressed concern about short selling and started a petition calling for the reintroduction of the uptick rule. Books like Don't Blame the Shorts by Robert Sloan and Fubarnomics by Robert E. Wright suggest Cramer exaggerated the costs of short selling and underestimated the benefits, which may include the ex ante identification of asset bubbles.

Individual short sellers have been subject to criticism and even litigation. Manuel P. Asensio, for example, engaged in a lengthy legal battle with the pharmaceutical manufacturer Hemispherx Biopharma.

Several studies of the effectiveness of short selling bans indicate that short selling bans do not contribute to more moderate market dynamics.

General and cited references

 * Fleckner, Andreas M. "Regulating Trading Practices" in The Oxford Handbook of Financial Regulation (Oxford: Oxford University Press, 2015). ISBN 978-0-19-968720-6.
 * Sloan, Robert. Don't Blame the Shorts: Why Short Sellers Are Always Blamed for Market Crashes and Why History Is Repeating Itself (New York: McGraw-Hill Professional, 2009). ISBN 978-0-07-163686-5.
 * Wright, Robert E. Fubarnomics: A Lighthearted, Serious Look at America's Economic Ills (Buffalo, N.Y.: Prometheus, 2010). ISBN 978-1-61614-191-2.

title: Short (finance)