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for OTC derivatives. Economists Fischer Black and Myron Scholes demonstrated in that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument. 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 , they decided to return to the academic environment. 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 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. 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.

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". 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, , [ citation needed ] transaction costs and taxes Ingersoll, , [ citation needed ] and dividend payout.

The notation used in the analysis of the Black-Scholes model is defined as follows definitions grouped by subject :. The Black—Scholes equation is a parabolic partial differential equation , which describes the price of the option over time.

The equation is:. 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". 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 :. The value of a call option for a non-dividend-paying underlying stock in terms of the Black—Scholes parameters is:.

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 :.

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. 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. In risk-neutral terms, these are the expected value of the asset and the expected value of the cash in the risk-neutral measure.

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

To calculate the probability under the real "physical" probability measure, additional information is required—the drift term in the physical measure, or equivalently, the market price of risk. A standard derivation for solving the Black—Scholes PDE is given in the article Black—Scholes equation.

The Feynman—Kac formula says that the solution to this type of PDE, when discounted appropriately, is actually a martingale. Thus the option price is the expected value of the discounted payoff of the option.

Computing the option price via this expectation is the risk neutrality approach and can be done without knowledge of PDEs. 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.

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, 1 basis point rate change , vega by 1 vol point change , and theta by or 1 day decay based on either calendar days or trading days per year.

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.

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. Under this formulation the arbitrage-free price implied by the Black—Scholes model can be shown to be:. 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. The price of the stock is then modelled as:. 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:. In general this inequality does not have a closed form solution, though an American call with no dividends is equal to a European call and the Roll—Geske—Whaley method provides a solution for an American call with one dividend; [20] [21] see also Black's approximation.

Barone-Adesi and Whaley [22] is a further approximation formula. Here, the stochastic differential equation which is valid for the value of any derivative is split into two components: the European option value and the early exercise premium. With some assumptions, a quadratic equation that approximates the solution for the latter is then obtained. Bjerksund and Stensland [25] provide an approximation based on an exercise strategy corresponding to a trigger price.

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.

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. 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.

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

This pays out one unit of asset if the spot is above the strike at maturity. This pays out one unit of asset if the spot is below the strike at maturity. 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. The Black—Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset.

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. 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:. If the skew is typically negative, the value of a binary call will be higher when taking skew into account. 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.

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. 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. 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.

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". Even when more advanced models are used, traders prefer to think in terms of Black—Scholes implied volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on. For a discussion as to the various alternative approaches developed here, see Financial economics § Challenges and criticism.

Black—Scholes cannot be applied directly to bond securities because of pull-to-par. As the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its volatility, and the simple Black—Scholes model does not reflect this process. A large number of extensions to Black—Scholes, beginning with the Black model , have been used to deal with this phenomenon. 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.

Below is list of command-line options recognized by the ImageMagick command-line tools. If you want a description of a particular option, click on the option name in the navigation bar above and you will go right to it. Unless otherwise noted, each option is recognized by the commands: convert and mogrify. A Gaussian operator of the given radius and standard deviation sigma is used. If sigma is not given it defaults to 1.

The sigma value is the important argument, and determines the actual amount of blurring that will take place. The radius is only used to determine the size of the array which holds the calculated Gaussian distribution.

It should be an integer. If not given, or set to zero, IM will calculate the largest possible radius that will provide meaningful results for the Gaussian distribution. See Image Geometry for complete details about the geometry argument.

The -adaptive-resize option defaults to data-dependent triangulation. Use the -filter to choose a different resampling algorithm.

Offsets, if present in the geometry string, are ignored, and the -gravity option has no effect. This option is enabled by default. An attempt is made to save all images of an image sequence into the given output file.

However, some formats, such as JPEG and PNG, do not support more than one image per file, and in that case ImageMagick is forced to write each image as a separate file. As such, if more than one image needs to be written, the filename given is modified by adding a -scene number before the suffix, in order to make distinct names for each image. As an example, the command. will create a sequence of 17 images the two given plus 15 more created by -morph , named: my00morph.

jpg, my01morph. jpg, my02morph. In summary, ImageMagick tries to write all images to one file, but will save to multiple files, if any of the following conditions exist Set the drawing transformation matrix for combined rotating and scaling.

This option sets a transformation matrix, for use by subsequent -draw or -transform options. The matrix entries are entered as comma-separated numeric values either in quotes or without spaces. Internally, the transformation matrix has 3x3 elements, but three of them are omitted from the input because they are constant. The new transformed coordinates x' , y' of a pixel at position x , y in the original image are calculated using the following matrix equation. The size of the resulting image is that of the smallest rectangle that contains the transformed source image.

The parameters t x and t y subsequently shift the image pixels so that those that are moved out of the image area are cut off. The transformation matrix complies with the left-handed pixel coordinate system: positive x and y directions are rightward and downward, resp.

If the translation coefficients t x and t y are omitted they default to 0,0. Therefore, four parameters suffice for rotation and scaling without translation. Scaling by the factors s x and s y in the x and y directions, respectively, is accomplished with the following. See -transform , and the -distort method ' Affineprojection for more information. Translation by a displacement t x , t y is accomplished like so:.

The cumulative effect of a sequence of -affine transformations can be accomplished by instead by a single -affine operation using the matrix equal to the product of the matrices of the individual transformations. An attempt is made to detect near-singular transformation matrices.

If the matrix determinant has a sufficiently small absolute value it is rejected. Used to set a flag on an image indicating whether or not to use existing alpha channel data, to create an alpha channel, or to perform other operations on the alpha channel. Choose the argument type from the list below. This is a convenience for annotating an image with text. For more precise control over text annotations, use -draw.

The values Xdegrees and Ydegrees control the shears applied to the text, while t x and t y are offsets that give the location of the text relative any -gravity setting and defaults to the upper left corner of the image. Using -annotate degrees or -annotate degrees x degrees produces an unsheared rotation of the text.

The direction of the rotation is positive, which means a clockwise rotation if degrees is positive. This conforms to the usual mathematical convention once it is realized that the positive y —direction is conventionally considered to be downward for images.

The new transformed coordinates x' , y' of a pixel at position x , y in the image are calculated using the following matrix equation. If t x and t y are omitted, they default to 0.

This makes the bottom-left of the text becomes the upper-left corner of the image, which is probably undesirable. Adding a -gravity option in this case leads to nice results.

Text is any UTF-8 encoded character sequence. If text is of the form ' mytext. txt', the text is read from the file mytext. Text in a file is taken literally; no embedded formatting characters are recognized. By default, objects e. text, lines, polygons, etc. are antialiased when drawn. This will then reduce the number of colors added to an image to just the colors being directly drawn.

That is, no mixed colors are added when drawing such objects. This option creates a single longer image, by joining all the current images in sequence top-to-bottom. If they are not of the same width, narrower images are padded with the current -background color setting, and their position relative to each other can be controlled by the current -gravity setting. For more flexible options, including the ability to add space between images, use -smush.

Use this option to supply a password for decrypting a PDF that has been encrypted using Microsoft Crypto API MSC API. The encrypting using the MSC API is not supported.

For a different encryption method, see -encipher and -decipher. This works well for real-life images with little or no extreme dark and light areas, but tend to fail for images with large amounts of bright sky or dark shadows. It also does not work well for diagrams or cartoon like images. It uses the -channel setting, including the ' sync ' flag for channel synchronization , to determine which color values is used and modified.

As the default -channel setting is ' RGB,sync ', channels are modified together by the same gamma value, preserving colors. This is a 'perfect' image normalization operator.

It finds the exact minimum and maximum color values in the image and then applies a -level operator to stretch the values to the full range of values. On the other hand it is the right operator to use for color stretching gradient images being used to generate Color lookup tables, distortion maps, or other 'mathematically' defined images.

The operator is very similar to the -normalize , -contrast-stretch , and -linear-stretch operators, but without 'histogram binning' or 'clipping' problems that these operators may have. That is -auto-level is the perfect or ideal version these operators. It uses the -channel setting, including the special ' sync ' flag for channel synchronization , to determine which color values are used and modified. Adjusts an image so that its orientation is suitable for viewing i.

top-left orientation. This operator reads and resets the EXIF image profile setting 'Orientation' and then performs the appropriate 90 degree rotation on the image to orient the image, for correct viewing.

This EXIF profile setting is usually set using a gravity sensor in digital camera, however photos taken directly downward or upward may not have an appropriate value. Also images that have been orientation 'corrected' without reseting this setting, may be 'corrected' again resulting in a incorrect result.

If the EXIF profile was previously stripped, the -auto-orient operator will do nothing. The computed threshold is returned as the auto-threshold:verbose image property.

This backdrop covers the entire workstation screen and is useful for hiding other X window activity while viewing the image. The color of the backdrop is specified as the background color. The color is specified using the format described under the -fill option. The default background color if none is specified or found in the image is white.

Repeat the entire command for the given number of iterations and report the user-time and elapsed time. For instance, consider the following command and its output.

Modify the benchmark with the -duration to run the benchmark for a fixed number of seconds and -concurrent to run the benchmark in parallel requires the OpenMP feature. In this example, 5 iterations were completed at 2. This option shifts the output of -convolve so that positive and negative results are relative to the specified bias value. This is important for non-HDRI compilations of ImageMagick when dealing with convolutions that contain negative as well as positive values.

This is especially the case with convolutions involving high pass filters or edge detection. Without an output bias, the negative values are clipped at zero. See the discussion on HDRI implementations of ImageMagick on the page High Dynamic-Range Images. For more about HDRI go the ImageMagick Usage pages or this Wikipedia entry. A non-linear, edge-preserving, and noise-reducing smoothing filter for images.

It replaces the intensity of each pixel with a weighted average of intensity values from nearby pixels. This weight is based on a Gaussian distribution.

ISO except that if the current locale is UTF-8 then the package code is translated to UTF-8 for syntax checking, so it is strongly recommended to check in a UTF-8 locale. The format and filename are platform-specific; for example, a binary package for Windows is usually supplied as a. The angle given is the angle toward which the image is blurred. For example knitr version 1. Authority control : National libraries Germany Israel United States Latvia Czech Republic. The Duality Principle , or De Morgan's laws , can be understood as asserting that complementing all three ports of an AND gate converts it to an OR gate and vice versa, as shown in Figure 4 below.

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