Mathematical finance , also known as quantitative finance and financial mathematics , is a field of applied mathematics , concerned with mathematical modeling in the financial field.
111-407: In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. Mathematical finance overlaps heavily with the fields of computational finance and financial engineering . The latter focuses on applications and modeling, often with the help of stochastic asset models , while
222-706: A professional role , a risk manager will "oversee the organization's comprehensive insurance and risk management program, assessing and identifying risks that could impede the reputation, safety, security, or financial success of the organization", and then develop plans to minimize and / or mitigate any negative (financial) outcomes. Risk Analysts support the technical side of the organization's risk management approach: once risk data has been compiled and evaluated, analysts share their findings with their managers, who use those insights to decide among possible solutions. See also Chief Risk Officer , internal audit , and Financial risk management § Corporate finance . Risk
333-590: A property or business to avoid legal liability is one such example. Avoiding airplane flights for fear of hijacking . Avoidance may seem like the answer to all risks, but avoiding risks also means losing out on the potential gain that accepting (retaining) the risk may have allowed. Not entering a business to avoid the risk of loss also avoids the possibility of earning profits. Increasing risk regulation in hospitals has led to avoidance of treating higher risk conditions, in favor of patients presenting with lower risk. Risk reduction or "optimization" involves reducing
444-420: A random walk in which the short-term changes had a finite variance . This causes longer-term changes to follow a Gaussian distribution . The theory remained dormant until Fischer Black and Myron Scholes , along with fundamental contributions by Robert C. Merton , applied the second most influential process, the geometric Brownian motion , to option pricing . For this M. Scholes and R. Merton were awarded
555-413: A "transfer of risk." However, technically speaking, the buyer of the contract generally retains legal responsibility for the losses "transferred", meaning that insurance may be described more accurately as a post-event compensatory mechanism. For example, a personal injuries insurance policy does not transfer the risk of a car accident to the insurance company. The risk still lies with the policyholder namely
666-416: A balance between negative risk and the benefit of the operation or activity; and between risk reduction and effort applied. By effectively applying Health, Safety and Environment (HSE) management standards, organizations can achieve tolerable levels of residual risk . Modern software development methodologies reduce risk by developing and delivering software incrementally. Early methodologies suffered from
777-473: A basic tool for measurement and computation in many areas of science and engineering; in these contexts log x still often means the base ten logarithm. In mathematics log x usually means to the natural logarithm (base e ). In computer science and information theory, log often refers to binary logarithms (base 2). The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by
888-517: A company may outsource only its software development, the manufacturing of hard goods, or customer support needs to another company, while handling the business management itself. This way, the company can concentrate more on business development without having to worry as much about the manufacturing process, managing the development team, or finding a physical location for a center. Also, implanting controls can also be an option in reducing risk. Controls that either detect causes of unwanted events prior to
999-444: A discipline in the 1970s, following the work of Fischer Black , Myron Scholes and Robert Merton on option pricing theory. Mathematical investing originated from the research of mathematician Edward Thorp who used statistical methods to first invent card counting in blackjack and then applied its principles to modern systematic investing. The subject has a close relationship with the discipline of financial economics , which
1110-447: A great aid to calculations before the invention of computers. Given a positive real number b such that b ≠ 1 , the logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to yield x . In other words, the logarithm of x to base b is the unique real number y such that b y = x {\displaystyle b^{y}=x} . The logarithm
1221-484: A higher probability but lower loss, versus a risk with higher loss but lower probability. Opportunity cost represents a unique challenge for risk managers. It can be difficult to determine when to put resources toward risk management and when to use those resources elsewhere. Again, ideal risk management optimises resource usage (spending, manpower etc), and also minimizes the negative effects of risks. Opportunities first appear in academic research or management books in
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#17327757804591332-541: A portfolio. Increasingly, elements of this process are automated; see Outline of finance § Quantitative investing for a listing of relevant articles. For their pioneering work, Markowitz and Sharpe , along with Merton Miller , shared the 1990 Nobel Memorial Prize in Economic Sciences , for the first time ever awarded for a work in finance. The portfolio-selection work of Markowitz and Sharpe introduced mathematics to investment management . With time,
1443-446: A product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p -th power of a number is p times the logarithm of the number itself; the logarithm of a p -th root is the logarithm of the number divided by p . The following table lists these identities with examples. Each of the identities can be derived after substitution of
1554-614: A profit in the short-run, this type of modeling is often in conflict with a central tenet of modern macroeconomics, the Lucas critique - or rational expectations - which states that observed relationships may not be structural in nature and thus may not be possible to exploit for public policy or for profit unless we have identified relationships using causal analysis and econometrics . Mathematical finance models do not, therefore, incorporate complex elements of human psychology that are critical to modeling modern macroeconomic movements such as
1665-417: A schedule for control implementation and responsible persons for those actions. There are four basic steps of risk management plan, which are threat assessment, vulnerability assessment, impact assessment and risk mitigation strategy development. According to ISO/IEC 27001 , the stage immediately after completion of the risk assessment phase consists of preparing a Risk Treatment Plan, which should document
1776-457: A similar relationship is used to define the price of new derivatives. The main quantitative tools necessary to handle continuous-time Q-processes are Itô's stochastic calculus , simulation and partial differential equations (PDEs). Risk and portfolio management aims to model the statistically derived probability distribution of the market prices of all the securities at a given future investment horizon. This "real" probability distribution of
1887-477: Is ISO Guide 31073:2022 , "Risk management — Vocabulary". Ideally in risk management, a prioritization process is followed. Whereby the risks with the greatest loss (or impact) and the greatest probability of occurring are handled first. Risks with lower probability of occurrence and lower loss are handled in descending order. In practice the process of assessing overall risk can be tricky, and organisation has to balance resources used to mitigate between risks with
1998-533: Is log b y . Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of f ( x ) evaluates to ln( b ) b by the properties of the exponential function , the chain rule implies that the derivative of log b x is given by d d x log b x = 1 x ln b . {\displaystyle {\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}.} That is,
2109-604: Is a positive real number . (If b is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.) One of the main historical motivations of introducing logarithms is the formula log b ( x y ) = log b x + log b y , {\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,} by which tables of logarithms allow multiplication and division to be reduced to addition and subtraction,
2220-427: Is a viable strategy for small risks where the cost of insuring against the risk would be greater over time than the total losses sustained. All risks that are not avoided or transferred are retained by default. This includes risks that are so large or catastrophic that either they cannot be insured against or the premiums would be infeasible. War is an example since most property and risks are not insured against war, so
2331-445: Is called a " martingale ". A martingale does not reward risk. Thus the probability of the normalized security price process is called "risk-neutral" and is typically denoted by the blackboard font letter " Q {\displaystyle \mathbb {Q} } ". The relationship ( 1 ) must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time. The quants who operate in
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#17327757804592442-426: Is called the base- b logarithm function or logarithmic function (or just logarithm ). The function log b x can also be essentially characterized by the product formula log b ( x y ) = log b x + log b y . {\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y.} More precisely,
2553-525: Is concerned with much of the underlying theory that is involved in financial mathematics. While trained economists use complex economic models that are built on observed empirical relationships, in contrast, mathematical finance analysis will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. See: Valuation of options ; Financial modeling ; Asset pricing . The fundamental theorem of arbitrage-free pricing
2664-511: Is defined as the possibility that an event will occur that adversely affects the achievement of an objective. Uncertainty, therefore, is a key aspect of risk. Risk management appears in scientific and management literature since the 1920s. It became a formal science in the 1950s, when articles and books with "risk management" in the title also appear in library searches. Most of research was initially related to finance and insurance. One popular standard clarifying vocabulary used in risk management
2775-539: Is denoted " log b x " (pronounced as "the logarithm of x to base b ", "the base- b logarithm of x ", or most commonly "the log, base b , of x "). An equivalent and more succinct definition is that the function log b is the inverse function to the function x ↦ b x {\displaystyle x\mapsto b^{x}} . Several important formulas, sometimes called logarithmic identities or logarithmic laws , relate logarithms to one another. The logarithm of
2886-451: Is determining the rate of occurrence since statistical information is not available on all kinds of past incidents and is particularly scanty in the case of catastrophic events, simply because of their infrequency. Furthermore, evaluating the severity of the consequences (impact) is often quite difficult for intangible assets. Asset valuation is another question that needs to be addressed. Thus, best educated opinions and available statistics are
2997-491: Is exactly one real number x such that b x = y {\displaystyle b^{x}=y} . We let log b : R > 0 → R {\displaystyle \log _{b}\colon \mathbb {R} _{>0}\to \mathbb {R} } denote the inverse of f . That is, log b y is the unique real number x such that b x = y {\displaystyle b^{x}=y} . This function
3108-405: Is frequently used in computer science . Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators , scientists, engineers, surveyors , and others to perform high-accuracy computations more easily. Using logarithm tables , tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This
3219-432: Is greater than one. In that case, log b ( x ) is an increasing function . For b < 1 , log b ( x ) tends to minus infinity instead. When x approaches zero, log b x goes to minus infinity for b > 1 (plus infinity for b < 1 , respectively). Analytic properties of functions pass to their inverses. Thus, as f ( x ) = b is a continuous and differentiable function , so
3330-531: Is known, the events that a source may trigger or the events that can lead to a problem can be investigated. For example: stakeholders withdrawing during a project may endanger funding of the project; confidential information may be stolen by employees even within a closed network; lightning striking an aircraft during takeoff may make all people on board immediate casualties. The chosen method of identifying risks may depend on culture, industry practice and compliance. The identification methods are formed by templates or
3441-414: Is often used in place of risk-sharing in the mistaken belief that you can transfer a risk to a third party through insurance or outsourcing. In practice, if the insurance company or contractor go bankrupt or end up in court, the original risk is likely to still revert to the first party. As such, in the terminology of practitioners and scholars alike, the purchase of an insurance contract is often described as
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3552-494: Is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results. Today many universities offer degree and research programs in mathematical finance. There are two separate branches of finance that require advanced quantitative techniques: derivatives pricing, and risk and portfolio management. One of the main differences is that they use different probabilities such as
3663-575: Is possible because the logarithm of a product is the sum of the logarithms of the factors: log b ( x y ) = log b x + log b y , {\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,} provided that b , x and y are all positive and b ≠ 1 . The slide rule , also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler , who connected them to
3774-403: Is related to the number of decimal digits of a positive integer x : The number of digits is the smallest integer strictly bigger than log 10 ( x ) . For example, log 10 (5986) is approximately 3.78 . The next integer above it is 4, which is the number of digits of 5986. Both the natural logarithm and the binary logarithm are used in information theory , corresponding to
3885-485: Is the inverse function of exponentiation with base b . That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x . For example, since 1000 = 10 , the logarithm base 10 {\displaystyle 10} of 1000 is 3 , or log 10 (1000) = 3 . The logarithm of x to base b is denoted as log b ( x ) , or without parentheses, log b x . When
3996-483: Is therefore difficult or impossible to predict. A common error in risk assessment and management is to underestimate the wildness of risk, assuming risk to be mild when in fact it is wild, which must be avoided if risk assessment and management are to be valid and reliable, according to Mandelbrot. According to the standard ISO 31000 , "Risk management – Guidelines", the process of risk management consists of several steps as follows: This involves: After establishing
4107-411: Is written as f ( x ) = b . When b is positive and unequal to 1, we show below that f is invertible when considered as a function from the reals to the positive reals. Let b be a positive real number not equal to 1 and let f ( x ) = b . It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from
4218-651: The International Organization for Standardization . The history of logarithms in seventeenth-century Europe saw the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio ( Description of the Wonderful Canon of Logarithms ). Prior to Napier's invention, there had been other techniques of similar scopes, such as
4329-695: The Project Management Institute , the National Institute of Standards and Technology , actuarial societies, and International Organization for Standardization . Methods, definitions and goals vary widely according to whether the risk management method is in the context of project management , security , engineering , industrial processes , financial portfolios , actuarial assessments , or public health and safety . Certain risk management standards have been criticized for having no measurable improvement on risk, whereas
4440-439: The acidity of an aqueous solution . Logarithms are commonplace in scientific formulae , and in measurements of the complexity of algorithms and of geometric objects called fractals . They help to describe frequency ratios of musical intervals , appear in formulas counting prime numbers or approximating factorials , inform some models in psychophysics , and can aid in forensic accounting . The concept of logarithm as
4551-431: The decimal number system: log 10 ( 10 x ) = log 10 10 + log 10 x = 1 + log 10 x . {\displaystyle \log _{10}\,(\,10\,x\,)\ =\;\log _{10}10\ +\;\log _{10}x\ =\ 1\,+\,\log _{10}x\,.} Thus, log 10 ( x )
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4662-413: The exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms. Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms , mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for
4773-510: The function now known as the natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by Grégoire de Saint-Vincent , a Belgian Jesuit residing in Prague. Archimedes had written The Quadrature of the Parabola in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that
4884-579: The intermediate value theorem . Now, f is strictly increasing (for b > 1 ), or strictly decreasing (for 0 < b < 1 ), is continuous, has domain R {\displaystyle \mathbb {R} } , and has range R > 0 {\displaystyle \mathbb {R} _{>0}} . Therefore, f is a bijection from R {\displaystyle \mathbb {R} } to R > 0 {\displaystyle \mathbb {R} _{>0}} . In other words, for each positive real number y , there
4995-520: The prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600. Napier coined the term for logarithm in Middle Latin, logarithmus , literally meaning ' ratio-number ' , derived from the Greek logos ' proportion, ratio, word ' + arithmos ' number ' . The common logarithm of a number is the index of that power of ten which equals
5106-932: The self-fulfilling panic that motivates bank runs . Risk management Risk management is the identification, evaluation, and prioritization of risks , followed by the minimization, monitoring, and control of the impact or probability of those risks occurring. Risks can come from various sources (i.e, threats ) including uncertainty in international markets , political instability , dangers of project failures (at any phase in design, development, production, or sustaining of life-cycles), legal liabilities , credit risk , accidents , natural causes and disasters , deliberate attack from an adversary, or events of uncertain or unpredictable root-cause . There are two types of events wiz. Risks and Opportunities. Negative events can be classified as risks while positive events are classified as opportunities. Risk management standards have been developed by various institutions, including
5217-407: The slope of the tangent touching the graph of the base- b logarithm at the point ( x , log b ( x )) equals 1/( x ln( b )) . The derivative of ln( x ) is 1/ x ; this implies that ln( x ) is the unique antiderivative of 1/ x that has the value 0 for x = 1 . It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of
5328-400: The x - and the y -coordinates (or upon reflection at the diagonal line x = y ), as shown at the right: a point ( t , u = b ) on the graph of f yields a point ( u , t = log b u ) on the graph of the logarithm and vice versa. As a consequence, log b ( x ) diverges to infinity (gets bigger than any given number) if x grows to infinity, provided that b
5439-407: The 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables. A deeper study of logarithms requires the concept of a function . A function is a rule that, given one number, produces another number. An example is the function producing the x -th power of b from any real number x , where the base b is a fixed number. This function
5550-1048: The 1990s. The first PMBoK Project Management Body of Knowledge draft of 1987 doesn't mention opportunities at all. Modern project management school recognize the importance of opportunities. Opportunities have been included in project management literature since the 1990s, e.g. in PMBoK, and became a significant part of project risk management in the years 2000s, when articles titled "opportunity management" also begin to appear in library searches. Opportunity management thus became an important part of risk management. Modern risk management theory deals with any type of external events, positive and negative. Positive risks are called opportunities . Similarly to risks, opportunities have specific mitigation strategies: exploit, share, enhance, ignore. In practice, risks are considered "usually negative". Risk-related research and practice focus significantly more on threats than on opportunities. This can lead to negative phenomena such as target fixation . For
5661-487: The 1997 Nobel Memorial Prize in Economic Sciences . Black was ineligible for the prize because he died in 1995. The next important step was the fundamental theorem of asset pricing by Harrison and Pliska (1981), according to which the suitably normalized current price P 0 of security is arbitrage-free, and thus truly fair only if there exists a stochastic process P t with constant expected value which describes its future evolution: A process satisfying ( 1 )
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#17327757804595772-403: The Q world of derivatives pricing are specialists with deep knowledge of the specific products they model. Securities are priced individually, and thus the problems in the Q world are low-dimensional in nature. Calibration is one of the main challenges of the Q world: once a continuous-time parametric process has been calibrated to a set of traded securities through a relationship such as ( 1 ),
5883-483: The acceptance technique, the business intentionally assumes risks without financial protections in the hopes that possible gains will exceed prospective losses. The transfer approach shields the business from losses by shifting risks to a third party, frequently in exchange for a fee, while the third-party benefits from the project. By choosing not to participate in high-risk ventures, the avoidance strategy avoids losses but also loses out on possibilities. Last but not least,
5994-407: The advance of science, especially astronomy . They were critical to advances in surveying , celestial navigation , and other domains. Pierre-Simon Laplace called logarithms As the function f ( x ) = b is the inverse function of log b x , it has been called an antilogarithm . Nowadays, this function is more commonly called an exponential function . A key tool that enabled
6105-502: The appropriate level of management. For instance, a risk concerning the image of the organization should have top management decision behind it whereas IT management would have the authority to decide on computer virus risks. The risk management plan should propose applicable and effective security controls for managing the risks. For example, an observed high risk of computer viruses could be mitigated by acquiring and implementing antivirus software. A good risk management plan should contain
6216-474: The areas surrounding the improved traffic capacity. Over time, traffic thereby increases to fill available capacity. Turnpikes thereby need to be expanded in a seemingly endless cycles. There are many other engineering examples where expanded capacity (to do any function) is soon filled by increased demand. Since expansion comes at a cost, the resulting growth could become unsustainable without forecasting and management. The fundamental difficulty in risk assessment
6327-425: The base is clear from the context or is irrelevant it is sometimes written log x . The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics because of its very simple derivative . The binary logarithm uses base 2 and
6438-451: The base is given by: b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},} which can be seen from taking the defining equation x = b log b x = b y {\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y . {\displaystyle {\tfrac {1}{y}}.} Among all choices for
6549-405: The base, three are particularly common. These are b = 10 , b = e (the irrational mathematical constant e ≈ 2.71828183 ), and b = 2 (the binary logarithm ). In mathematical analysis , the logarithm base e is widespread because of analytical properties explained below. On the other hand, base 10 logarithms (the common logarithm ) are easy to use for manual calculations in
6660-457: The case of an unlikely event, the probability of occurrence of which is unknown. Therefore, in the assessment process it is critical to make the best educated decisions in order to properly prioritize the implementation of the risk management plan . Even a short-term positive improvement can have long-term negative impacts. Take the "turnpike" example. A highway is widened to allow more traffic. More traffic capacity leads to greater development in
6771-453: The common logarithms of trigonometric functions . Another critical application was the slide rule , a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule , was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to
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#17327757804596882-504: The confidence in estimates and decisions seems to increase. Strategies to manage threats (uncertainties with negative consequences) typically include avoiding the threat, reducing the negative effect or probability of the threat, transferring all or part of the threat to another party, and even retaining some or all of the potential or actual consequences of a particular threat. The opposite of these strategies can be used to respond to opportunities (uncertain future states with benefits). As
6993-429: The consequences occurring during use of the product, or detection of the root causes of unwanted failures that the team can then avoid. Controls may focus on management or decision-making processes. All these may help to make better decisions concerning risk. Briefly defined as "sharing with another party the burden of loss or the benefit of gain, from a risk, and the measures to reduce a risk." The term 'risk transfer'
7104-451: The context, the next step in the process of managing risk is to identify potential risks. Risks are about events that, when triggered, cause problems or benefits. Hence, risk identification can start with the source of problems and those of competitors (benefit), or with the problem's consequences. Some examples of risk sources are: stakeholders of a project, employees of a company or the weather over an airport. When either source or problem
7215-460: The customers of the enterprise, as well as external impacts on society, markets, or the environment. There are various defined frameworks here, where every probable risk can have a pre-formulated plan to deal with its possible consequences (to ensure contingency if the risk becomes a liability ). Managers thus analyze and monitor both the internal and external environment facing the enterprise, addressing business risk generally, and any impact on
7326-435: The decisions about how each of the identified risks should be handled. Mitigation of risks often means selection of security controls , which should be documented in a Statement of Applicability, which identifies which particular control objectives and controls from the standard have been selected, and why. Implementation follows all of the planned methods for mitigating the effect of the risks. Purchase insurance policies for
7437-434: The development of templates for identifying source, problem or event. Common risk identification methods are: Once risks have been identified, they must then be assessed as to their potential severity of impact (generally a negative impact, such as damage or loss) and to the probability of occurrence. These quantities can be either simple to measure, in the case of the value of a lost building, or impossible to know for sure in
7548-400: The differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here: For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until
7659-426: The enterprise achieving its strategic goals . ERM thus overlaps various other disciplines - operational risk management , financial risk management etc. - but is differentiated by its strategic and long-term focus. ERM systems usually focus on safeguarding reputation, acknowledging its significant role in comprehensive risk management strategies. Logarithm In mathematics , the logarithm to base b
7770-433: The fact that they only delivered software in the final phase of development; any problems encountered in earlier phases meant costly rework and often jeopardized the whole project. By developing in iterations, software projects can limit effort wasted to a single iteration. Outsourcing could be an example of risk sharing strategy if the outsourcer can demonstrate higher capability at managing or reducing risks. For example,
7881-545: The field notably by Paul Wilmott , and by Nassim Nicholas Taleb , in his book The Black Swan . Taleb claims that the prices of financial assets cannot be characterized by the simple models currently in use, rendering much of current practice at best irrelevant, and, at worst, dangerously misleading. Wilmott and Emanuel Derman published the Financial Modelers' Manifesto in January 2009 which addresses some of
7992-565: The findings of risk assessments in financial, market, or schedule terms. Robert Courtney Jr. (IBM, 1970) proposed a formula for presenting risks in financial terms. The Courtney formula was accepted as the official risk analysis method for the US governmental agencies. The formula proposes calculation of ALE (annualized loss expectancy) and compares the expected loss value to the security control implementation costs ( cost–benefit analysis ). Planning for risk management uses four essential techniques. Under
8103-433: The following formula: log b x = log k x log k b . {\displaystyle \log _{b}x={\frac {\log _{k}x}{\log _{k}b}}.} Typical scientific calculators calculate the logarithms to bases 10 and e . Logarithms with respect to any base b can be determined using either of these two logarithms by
8214-475: The former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing , which relies on statistical and numerical models (and lately machine learning ) as opposed to traditional fundamental analysis when managing portfolios . French mathematician Louis Bachelier 's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical finance. But mathematical finance emerged as
8325-416: The impact of the event equals risk magnitude." Risk mitigation measures are usually formulated according to one or more of the following major risk options, which are: Later research has shown that the financial benefits of risk management are less dependent on the formula used but are more dependent on the frequency and how risk assessment is performed. In business it is imperative to be able to present
8436-482: The inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses in public-key cryptography . Addition , multiplication , and exponentiation are three of
8547-408: The length of the time interval to a power a bit more than 1/2. Large changes up or down are more likely than what one would calculate using a Gaussian distribution with an estimated standard deviation . But the problem is that it does not solve the problem as it makes parametrization much harder and risk control less reliable. Perhaps more fundamental: though mathematical finance models may generate
8658-462: The log base 2 ; and in photography rescaled base 2 logarithms are used to measure exposure values , light levels , exposure times , lens apertures , and film speeds in "stops". The abbreviation log x is often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial. Common logarithms (base 10), historically used in logarithm tables and slide rules, are
8769-433: The logarithm definitions x = b log b x {\displaystyle x=b^{\,\log _{b}x}} or y = b log b y {\displaystyle y=b^{\,\log _{b}y}} in the left hand sides. The logarithm log b x can be computed from the logarithms of x and b with respect to an arbitrary base k using
8880-434: The logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in prosthaphaeresis , leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by Christiaan Huygens , and James Gregory . The notation Log y
8991-528: The logarithm to any base b > 1 is the only increasing function f from the positive reals to the reals satisfying f ( b ) = 1 and f ( x y ) = f ( x ) + f ( y ) . {\displaystyle f(xy)=f(x)+f(y).} As discussed above, the function log b is the inverse to the exponential function x ↦ b x {\displaystyle x\mapsto b^{x}} . Therefore, their graphs correspond to each other upon exchanging
9102-926: The lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis , which relies on trigonometric identities . Calculations of powers and roots are reduced to multiplications or divisions and lookups by c d = ( 10 log 10 c ) d = 10 d log 10 c {\displaystyle c^{d}=\left(10^{\,\log _{10}c}\right)^{d}=10^{\,d\log _{10}c}} and c d = c 1 d = 10 1 d log 10 c . {\displaystyle {\sqrt[{d}]{c}}=c^{\frac {1}{d}}=10^{{\frac {1}{d}}\log _{10}c}.} Trigonometric calculations were facilitated by tables that contained
9213-450: The loss attributed to war is retained by the insured. Also any amounts of potential loss (risk) over the amount insured is retained risk. This may also be acceptable if the chance of a very large loss is small or if the cost to insure for greater coverage amounts is so great that it would hinder the goals of the organization too much. Select appropriate controls or countermeasures to mitigate each risk. Risk mitigation needs to be approved by
9324-1221: The mantissa, as the characteristic can be easily determined by counting digits from the decimal point. The characteristic of 10 · x is one plus the characteristic of x , and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by log 10 3542 = log 10 ( 1000 ⋅ 3.542 ) = 3 + log 10 3.542 ≈ 3 + log 10 3.54 {\displaystyle {\begin{aligned}\log _{10}3542&=\log _{10}(1000\cdot 3.542)\\&=3+\log _{10}3.542\\&\approx 3+\log _{10}3.54\end{aligned}}} Greater accuracy can be obtained by interpolation : log 10 3542 ≈ 3 + log 10 3.54 + 0.2 ( log 10 3.55 − log 10 3.54 ) {\displaystyle \log _{10}3542\approx {}3+\log _{10}3.54+0.2(\log _{10}3.55-\log _{10}3.54)} The value of 10 can be determined by reverse look up in
9435-457: The market parameters. See Financial risk management § Investment management . Much effort has gone into the study of financial markets and how prices vary with time. Charles Dow , one of the founders of Dow Jones & Company and The Wall Street Journal , enunciated a set of ideas on the subject which are now called Dow Theory . This is the basis of the so-called technical analysis method of attempting to predict future changes. One of
9546-462: The market prices is typically denoted by the blackboard font letter " P {\displaystyle \mathbb {P} } ", as opposed to the "risk-neutral" probability " Q {\displaystyle \mathbb {Q} } " used in derivatives pricing. Based on the P distribution, the buy-side community takes decisions on which securities to purchase in order to improve the prospective profit-and-loss profile of their positions considered as
9657-509: The mathematics has become more sophisticated. Thanks to Robert Merton and Paul Samuelson, one-period models were replaced by continuous time, Brownian-motion models , and the quadratic utility function implicit in mean–variance optimization was replaced by more general increasing, concave utility functions. Furthermore, in recent years the focus shifted toward estimation risk, i.e., the dangers of incorrectly assuming that advanced time series analysis alone can provide completely accurate estimates of
9768-447: The most fundamental arithmetic operations. The inverse of addition is subtraction , and the inverse of multiplication is division . Similarly, a logarithm is the inverse operation of exponentiation . Exponentiation is when a number b , the base , is raised to a certain power y , the exponent , to give a value x ; this is denoted b y = x . {\displaystyle b^{y}=x.} For example, raising 2 to
9879-597: The most part, these methods consist of the following elements, performed, more or less, in the following order: The Risk management knowledge area, as defined by the Project Management Body of Knowledge PMBoK, consists of the following processes: The International Organization for Standardization (ISO) identifies the following principles for risk management: Benoit Mandelbrot distinguished between "mild" and "wild" risk and argued that risk assessment and management must be fundamentally different for
9990-545: The most serious concerns. Bodies such as the Institute for New Economic Thinking are now attempting to develop new theories and methods. In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate. In the 1960s it was discovered by Benoit Mandelbrot that changes in prices do not follow a Gaussian distribution , but are rather modeled better by Lévy alpha- stable distributions . The scale of change, or volatility, depends on
10101-425: The number. Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the "order of a number". The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation. Some of these methods used tables derived from trigonometric identities. Such methods are called prosthaphaeresis . Invention of
10212-806: The organization or person making the risk management decisions. Another source, from the US Department of Defense (see link), Defense Acquisition University , calls these categories ACAT, for Avoid, Control, Accept, or Transfer. This use of the ACAT acronym is reminiscent of another ACAT (for Acquisition Category) used in US Defense industry procurements, in which Risk Management figures prominently in decision making and planning. Similarly to risks, opportunities have specific mitigation strategies: exploit, share, enhance, ignore. This includes not performing an activity that could present risk. Refusing to purchase
10323-400: The person who has been in the accident. The insurance policy simply provides that if an accident (the event) occurs involving the policyholder then some compensation may be payable to the policyholder that is commensurate with the suffering/damage. Methods of managing risk fall into multiple categories. Risk-retention pools are technically retaining the risk for the group, but spreading it over
10434-414: The power of 3 gives 8 : 2 3 = 8. {\displaystyle 2^{3}=8.} The logarithm of base b is the inverse operation, that provides the output y from the input x . That is, y = log b x {\displaystyle y=\log _{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b
10545-424: The practical use of logarithms was the table of logarithms . The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained the common logarithms of all integers in the range from 1 to 1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed
10656-475: The previous formula: log b x = log 10 x log 10 b = log e x log e b . {\displaystyle \log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log _{e}b}}.} Given a number x and its logarithm y = log b x to an unknown base b ,
10767-508: The primary sources of information. Nevertheless, risk assessment should produce such information for senior executives of the organization that the primary risks are easy to understand and that the risk management decisions may be prioritized within overall company goals. Thus, there have been several theories and attempts to quantify risks. Numerous different risk formulae exist, but perhaps the most widely accepted formula for risk quantification is: "Rate (or probability) of occurrence multiplied by
10878-458: The reduction approach lowers risks by implementing strategies like insurance, which provides protection for a variety of asset classes and guarantees reimbursement in the event of losses. Once risks have been identified and assessed, all techniques to manage the risk fall into one or more of these four major categories: Ideal use of these risk control strategies may not be possible. Some of them may involve trade-offs that are not acceptable to
10989-407: The risk-neutral probability (or arbitrage-pricing probability), denoted by "Q", and the actual (or actuarial) probability, denoted by "P". The goal of derivatives pricing is to determine the fair price of a given security in terms of more liquid securities whose price is determined by the law of supply and demand . The meaning of "fair" depends, of course, on whether one considers buying or selling
11100-413: The risks being faced. Risk analysis results and management plans should be updated periodically. There are two primary reasons for this: Enterprise risk management (ERM) defines risk as those possible events or circumstances that can have negative influences on the enterprise in question, where the impact can be on the very existence, the resources (human and capital), the products and services, or
11211-406: The risks that it has been decided to transferred to an insurer, avoid all risks that can be avoided without sacrificing the entity's goals, reduce others, and retain the rest. Initial risk management plans will never be perfect. Practice, experience, and actual loss results will necessitate changes in the plan and contribute information to allow possible different decisions to be made in dealing with
11322-1046: The same table, since the logarithm is a monotonic function . The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms. The product cd or quotient c / d came from looking up the antilogarithm of the sum or difference, via the same table: c d = 10 log 10 c 10 log 10 d = 10 log 10 c + log 10 d {\displaystyle cd=10^{\,\log _{10}c}\,10^{\,\log _{10}d}=10^{\,\log _{10}c\,+\,\log _{10}d}} and c d = c d − 1 = 10 log 10 c − log 10 d . {\displaystyle {\frac {c}{d}}=cd^{-1}=10^{\,\log _{10}c\,-\,\log _{10}d}.} For manual calculations that demand any appreciable precision, performing
11433-414: The security. Examples of securities being priced are plain vanilla and exotic options , convertible bonds , etc. Once a fair price has been determined, the sell-side trader can make a market on the security. Therefore, derivatives pricing is a complex "extrapolation" exercise to define the current market value of a security, which is then used by the sell-side community. Quantitative derivatives pricing
11544-447: The severity of the loss or the likelihood of the loss from occurring. For example, sprinklers are designed to put out a fire to reduce the risk of loss by fire. This method may cause a greater loss by water damage and therefore may not be suitable. Halon fire suppression systems may mitigate that risk, but the cost may be prohibitive as a strategy . Acknowledging that risks can be positive or negative, optimizing risks means finding
11655-481: The tenets of "technical analysis" is that market trends give an indication of the future, at least in the short term. The claims of the technical analysts are disputed by many academics. Over the years, increasingly sophisticated mathematical models and derivative pricing strategies have been developed, but their credibility was damaged by the financial crisis of 2007–2010 . Contemporary practice of mathematical finance has been subjected to criticism from figures within
11766-426: The two types of risk. Mild risk follows normal or near-normal probability distributions , is subject to regression to the mean and the law of large numbers , and is therefore relatively predictable. Wild risk follows fat-tailed distributions , e.g., Pareto or power-law distributions , is subject to regression to the tail (infinite mean or variance, rendering the law of large numbers invalid or ineffective), and
11877-488: The use of nats or bits as the fundamental units of information, respectively. Binary logarithms are also used in computer science , where the binary system is ubiquitous; in music theory , where a pitch ratio of two (the octave ) is ubiquitous and the number of cents between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 cents per semitone in conventional equal temperament ), or equivalently
11988-455: The values of log 10 x for any number x in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of x can be separated into an integer part and a fractional part , known as the characteristic and mantissa . Tables of logarithms need only include
12099-416: The whole group involves transfer among individual members of the group. This is different from traditional insurance, in that no premium is exchanged between members of the group upfront, but instead, losses are assessed to all members of the group. Risk retention involves accepting the loss, or benefit of gain, from a risk when the incident occurs. True self-insurance falls in this category. Risk retention
12210-673: Was adopted by Leibniz in 1675, and the next year he connected it to the integral ∫ d y y . {\textstyle \int {\frac {dy}{y}}.} Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in 1714 that log ( cos θ + i sin θ ) = i θ . {\displaystyle \log(\cos \theta +i\sin \theta )=i\theta .} By simplifying difficult calculations before calculators and computers became available, logarithms contributed to
12321-541: Was initiated by Louis Bachelier in The Theory of Speculation ("Théorie de la spéculation", published 1900), with the introduction of the most basic and most influential of processes, Brownian motion , and its applications to the pricing of options. Brownian motion is derived using the Langevin equation and the discrete random walk . Bachelier modeled the time series of changes in the logarithm of stock prices as
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