Computational Finance

Computational finance

Computational finance Computational finance or financial engineering is a cross-disciplinary field which relies on mathematical finance, numerical methods, computational intelligence and computer simulations to make trading, hedging and investment decisions, as well as facilitating the risk management of those decisions. Utilizing various methods, practitioners of computational finance aim to precisely determine the financial risk that certain financial instruments create. 1) 2) 3) 4) 5)

Mathematical finance Numerical methods Computer simulations Computational intelligence Financial risk

History Generally, individuals who fill positions in computational finance are known as “quants”, referring to the quantitative skills necessary to perform the job. Specifically, knowledge of the C++ programming language, as well as of the mathematical subfields of: stochastic calculus, multivariate calculus, linear algebra, differential equations, probability theory and statistical inference are often entry level requisites for such a position. C++ has become the dominant language for two main reasons: the computationally intensive nature of many algorithms, and the focus on libraries rather than applications. Computational finance was traditionally populated by Ph.Ds in finance, physics and mathematics who moved into the field from more pure, academic backgrounds (either directly from graduate school, or after teaching or research). However, as the actual use of computers has become essential to rapidly carrying out computational finance decisions, a background in computer programming has become useful, and hence many computer programmers enter the field either from Ph.D. programs or from other fields of software engineering. In recent years, advanced computational methods, such as neural network and evolutionary computation have opened new doors in computational finance. Practitioners of computational finance have come from the fields of signal processing and computational fluid dynamics and artificial intelligence. Today, all full service institutional finance firms employ computational finance professionals in their banking and finance operations (as opposed to being ancillary information technology specialists), while there are many other boutique firms ranging from 20 or fewer employees to several thousand that specialize in quantitative trading alone. JPMorgan Chase & Co. was one of the first firms to create a large derivatives business and employ computational finance (including through the formation of RiskMetrics), while D. E. Shaw & Co. is probably the oldest and largest quant fund (Citadel Investment Group is a major rival).

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Introduction One of the most common applications of computational finance is within the arena of investment banking. Because of the sheer amount of funds involved in this type of situation, computational finance comes to the fore as one of the tools used to evaluate every potential investment, whether it be something as simple as a new start-up company or a well established fund. Computational finance can help prevent the investment of large amounts of funding in something that simply does not appear to have much of a future. Another area where computational finance comes into play is the world of financial risk management. Stockbrokers, stockholders, and anyone who chooses to invest in any type of investment can benefit from using the basic principles of computational finance as a way of managing an individual portfolio. Running the numbers for individual investors, just alike for larger concerns, can often make it clear what risks are associated with any given investment opportunity. The result can often be an individual who is able to sidestep a bad opportunity, and live to invest another day in something that will be worthwhile in the long run. In the business world, the use of computational finance can often come into play when the time to engage in some form of corporate strategic planning arrives. For instance, reorganizing the operating structure of a company in order to maximize profits may look very good at first glance, but running the data through a process of computational finance may in fact uncover some drawbacks to the current plan that were not readily visible before. Being aware of the complete and true expenses associated with the restructure may prove to be more costly than anticipated, and in the long run not as productive as was originally hoped. Computational finance can help get past the hype and provide some realistic views of what could happen, before any corporate strategy is implemented.

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Computational finance

Quantitative analysis A quantitative analysis is working in finance using numerical or quantitative techniques. Similar work is done in most other modern industries, the work is called quantitative analysis. In the investment industry, people who perform quantitative analysis are frequently called quants. Although the original quants were concerned with risk management and derivatives pricing, the meaning of the term has expanded over time to include those individuals involved in almost any application of mathematics in finance. An example is statistical arbitrage. History Quantitative finance started in the U.S. in the 1930s as some astute investors began using mathematical formulae to price stocks and bonds. Robert C. Merton, a pioneer of quantitative analysis, introduced stochastic calculus into the study of finance. Harry Markowitz's 1952 Ph.D thesis "Portfolio Selection" was one of the first papers to formally adapt mathematical concepts to finance. Markowitz formalized a notion of mean return and covariances for common stocks which allowed him to quantify the concept of "diversification" in a market. He showed how to compute the mean return and variance for a given portfolio and argued that investors should hold only those portfolios whose variance is minimal among all portfolios with a given mean return. Although the language of finance now involves Itō calculus, minimization of risk in a quantifiable manner underlies much of the modern theory. In 1969 Robert Merton introduced stochastic calculus into the study of finance. Merton was motivated by the desire to understand how prices are set in financial markets, which is the classical economics question of "equilibrium," and in later papers he used the machinery of stochastic calculus to begin investigation of this issue. At the same time as Merton's work and with Merton's assistance, Fischer Black and Myron Scholes were developing their option pricing formula, which led to winning the 1997 Nobel Prize in Economics. It provided a solution for a practical problem, that of finding a fair price for a European call option, i.e., the right to buy one share of a given stock at a specified price and time. Such options are frequently purchased by investors as a risk-hedging device. In 1981, Harrison and Pliska used the general theory of continuous-time stochastic processes to put the Black-Scholes option pricing formula on a solid theoretical basis, and as a result, showed how to price numerous other "derivative" securities.

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Quantitative and Computational Finance ‘Quantitative Finance’ as a branch of modern finance is one of the fastest growing areas within the corporate world. Together with the sophistication and complexity of modern financial products, this exciting discipline continues to act as the motivating factor for new mathematical models and the subsequent development of associated computational schemes. Alternative names for this subject area are Mathematical Finance, Financial Mathematics or Financial Engineering. This is a course in the applied aspects of mathematical finance, in particular derivative pricing. The necessary understanding of products and markets required will be covered during the course. The overall theme of the course is to develop the Partial Differential Equation (PDE) approach to the pricing of options. As well as a two hour examination during the summer term, students will undertake a short computing project where they will use numerical and computational techniques to perform derivative pricing.

Simulation Methods in Finance Financial Products and Markets

Brief introduction to Stochastic Differential Equations (SDEs) – drift, diffusion, Itô’s Lemma. The statistics of random number generation in Excel. Simulating asset price SDEs in Excel. Introduction to the financial markets and the products which are traded in them: Equities, indices, foreign exchange, fixed income world and commodities. Options contracts and strategies for speculation and hedging.

Black-Scholes framework

Similarity reduction and fundamental solution for the heat equation. Black-Scholes PDE: simple European calls and puts; put-call parity. The PDE for pricing commodity and currency options. Discontinuous payoffs – Binary and Digital options. The greeks: theta, delta, gamma, vega & rho and their role in hedging.

Computational Finance

Solving the pricing PDEs numerically using Explicit, Implicit and Crank-Nicholson Finite Difference Schemes. Stability criteria. Monte Carlo Technique for derivative pricing.

Fixed-Income Products

Introduction to the properties and features of fixed income products; yield, duration & convexity. Stochastic interest rate models: stochastic differential equation for the spot interest rate; bond pricing PDE; popular models for the spot rate (Vasicek, CIR and Hull & White); solutions of the bond pricing equation;

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Fixed Income The fixed-income market demands a vast selection of investment options with a variety of credit quality, maturities, and yields to meet investors' objectives. To accomplish this, fixed income groups frequently create and modify mathematical models to calculate bond pricing, perform yield analysis, calculate cash flows, and develop hedging strategies. Fixed-income research groups use the thousands of prewritten math and graphics functions in MathWorks products to access bond data, perform statistical analysis, calculate spreads, determine bond and derivative pricing, perform sensitivity analyses, and run Monte Carlo simulations. Advanced graphics and rendering capabilities in MATLAB make reviewing cash flows, visualizing decision trees, plotting spot and forward curves, and creating deployable interactive 2- and 3-D models easy.

Equity Smart security investing requires in-depth research and analysis. Measuring all the influencing factors is an essential part of risk management. As a result, research groups continually create and modify mathematical models to calculate stock value, review forecasts, and develop innovative risk strategies. Equity research groups use the thousands of math and graphics functions in MathWorks products to access stock data, perform statistical analysis, determine derivatives pricing, perform sensitivity analyses, and run Monte Carlo simulations. The graphics capabilities in MATLAB offer a variety of ways to review time series data, visualize portfolio risks and returns, and create forecasting graphs.

Investment Management and Trading To meet the investment needs of individuals, institutions, and governments, investment firms need to deliver a wide range of investment opportunities with risk-adjusted performance and consistent returns over time. To accomplish this, financial professionals need to develop and use mathematical models to optimize portfolios and develop trading strategies and systems that can respond to market conditions. Investment management and trading research groups use the thousands of math and graphics functions in MathWorks products to easily access securities data, perform statistical analysis, determine pricing, conduct sensitivity and principal component analyses, and implement buy and sell criteria. The graphics capabilities in MATLAB offer a variety of ways to easily review time series data, visualize portfolio risks and returns, and create forecasting graphs. With

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Computational finance MathWorks deployment tools, you can easily compile and integrate your MATLAB algorithms into your system.

Mathematical and statistical approaches According to Fund of Funds analyst Fred Gehm, "There are two types of quantitative analysis and, therefore, two types of quants. One type works primarily with mathematical models and the other primarily with statistical models. While there is no logical reason why one person can't do both kinds of work, this doesn’t seem to happen, perhaps because these types demand different skill sets and, much more important, different psychologies." A typical problem for a numerically oriented quantitative analyst would be to develop a model for pricing and managing a complex derivative product. A typical problem for statistically oriented quantitative analyst would be to develop a model for deciding which stocks are relatively expensive and which stocks are relatively cheap. The model might include a company's book value to price ratio, its trailing earnings to price ratio and other accounting factors. An investment manager might implement this analysis by buying the underpriced stocks, selling the overpriced stocks or both. One of the principal mathematical tools of quantitative finance is stochastic calculus. According to a July 2008 Aite Group report, today quants often use alpha generation platforms to help them develop financial models. These software solutions enable quants to centralize and streamline the alpha generation process.

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Computational finance

Areas of application Areas where computational finance techniques are employed include: • • • • • • • • • • • • • • • • • •

Investment banking Forecasting Risk Management software Corporate strategic planning Securities trading and financial risk management Derivatives trading and risk management Investment management Pension scheme Insurance policy Mortgage agreement Lottery design Islamic banking Currency peg Gold and commodity valuation Collateralised debt obligation Credit default swap Bargaining Market mechanism design

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Computational finance

Classification of method Computational finance or financial engineering is a cross-disciplinary field which relies on mathematical finance, numerical methods, computational intelligence and computer simulations to make trading, hedging and investment decisions, as well as facilitating the risk management of those decisions. Utilizing various methods, practitioners of computational finance aim to precisely determine the financial risk that certain financial instruments create. 1) 2) 3) 4) 5)

Mathematical finance Numerical methods Computer simulations Computational intelligence Financial risk

Mathematical finance Mathematical finance comprises the branches of applied mathematics concerned with the financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive, and extend, the mathematical or numerical models suggested by financial economics. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the fair value of derivatives of the stock (see: Valuation of options). In terms of practice, mathematical finance also overlaps heavily with the field of computational finance (also known as financial engineering). Arguably, these are largely synonymous, although the latter focuses on application, while the former focuses on modeling and derivation (see: Quantitative analyst). The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance.

History The history of mathematical finance starts with the theory of portfolio optimization by Harold Markowitz on using mean-variance estimates of portfolios to judge investment strategies, causing a shift away from the concept of trying to identify the best individual stock for investment. Using a linear regression strategy to understand and quantify the risk (i.e. variance) and return (i.e. mean) of an entire portfolio of stocks and bonds, an optimization strategy was used to

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Computational finance choose a portfolio with largest mean return subject to acceptable levels of variance in the return. Simultaneously, William Sharpe developed the mathematics of determining the correlation between each stock and the market. For their pioneering work, Markowitz and Sharpe, along with Merton Miller, shared the 1990 Nobel Prize in economics, for the first time ever awarded for a work in finance. The portfolio-selection work of Markowitz and Sharpe introduced mathematics to the “black art” of investment management. With time, the mathematics has become more sophisticated. Thanks to Robert Merton and Paul Samuelson, oneperiod 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.

INTRODUCTION: The mathematical formulation of problems arising in science, engineering, economics and finance involving rate of change w.r.t. one independent variable is governed by ordinary differential equations. Solutions of “real life” problems often require developing and applying numerical/computational techniques to model complex physical situations, which otherwise are not possible to solve by analytical means. The choice of such a technique depends on how accurate a solution is required, posing a multitude of several other factors including computing time constraint and stability of a method. Once a suitable numerical technique has been applied and the problem is transformed into an algorithmic form, one can use the powerful computational facilities available.

GOALS: •

Develop / derive techniques to solve an initial value problem ( IVP) or boundary value problem (BVP ) and calculate the associated local/global truncation errors.

•

Identify and apply an economical and efficient method to get a numerical solution of an ODE or a system of ODEs.

•

Examine the stability of various numerical schemes.

•

Compare numerical solutions of some easy differential equations with their analytical counterparts.

•

Gain experience of ODE solvers through MATLAB.

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Applied mathematics Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.

Divisions of applied mathematics There is no consensus of what the various branches of applied mathematics are. Such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees. Historically, applied mathematics consisted principally of applied analysis, most notably differential equations, approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis), and applied probability. These areas of mathematics were intimately tied to the development of Newtonian Physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a legacy as well; until the early 20th century subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments. Today, the term applied mathematics is used in a broader sense. It includes the classical areas above, as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptology), though they are not generally considered to be part of the field of applied mathematics per se. Sometimes the term applicable mathematics is used to distinguish between the traditional field of applied mathematics and the many more areas of mathematics that are applicable to real-world problems. Mathematicians distinguish between applied mathematics, which is concerned with mathematical methods, and the applications of mathematics within science and engineering. A biologist using a population model and applying known mathematics would not be doing applied mathematics, but rather using it. However, nonmathematicians do not usually draw this distinction. The use of mathematics to solve industrial problems is called industrial mathematics. Industrial mathematics is sometimes split in two branches: techno-mathematics (covering problems coming from technology) and econo-mathematics (for problems in economy and finance). The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics, computational science, and

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Computational finance computational engineering, which use high performance computing for the simulation of phenomena and solution of problems in the sciences and engineering. These are often considered interdisciplinary programs.

Utility of applied mathematics Historically, mathematics was most important in the natural sciences and engineering. However, after World War II, fields outside of the physical sciences have spawned the creation of new areas of mathematics, such as game theory, which grew out of economic considerations, or neural networks, which arose out of the study of the brain in neuroscience, or bioinformatics, from the importance of analyzing large data sets in biology. The advent of the computer has created new applications, both in studying and using the new computer technology itself (computer science, which uses combinatorics, formal logic, and lattice theory), as well as using computers to study problems arising in other areas of science (computational science), and of course studying the mathematics of computation (numerical analysis). Statistics is probably the most widespread application of mathematics in the social sciences, but other areas of mathematics are proving increasingly useful in these disciplines, especially in economics and management science.

Other mathematical sciences (associated with applied mathematics) Applied mathematics is closely related to other mathematical sciences. Scientific computing Scientific computing includes applied mathematics (especially numerical analysis), computing science (especially high-performance computing, and mathematical modelling in a scientific discipline. Computer Science Computer science relies on logic and combinatorics. Operations research and management science Operations research and management science are often taught in faculties of engineering, business, public policy. Statistics Applied mathematics has substantial overlap with the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions.

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Computational finance Statistical theory relies on probability and decision theory, and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorics. Applied mathematicians and statisticians often work in a department of mathematical sciences (particularly at colleges and small universities). Statisticians have long complained that many mathematics departments have assigned mathematicians (without statistical competence) to teach statistics courses, effectively giving "double blind" courses. Examing data from 2000, Schaeffer and Stasny reported By far the majority of instructors within statistics departments have at least a master’s degree in statistics or biostatistics (about 89% for doctoral departments and about 79% for master’s departments). In doctoral mathematics departments, however, only about 58% of statistics course instructors had at least a master’s degree in statistics or biostatistics as their highest degree earned. In master’s-level mathematics departments, the corresponding percentage was near 44%, and in bachelor’s-level departments only 19% of statistics course instructors had at least a master’s degree in statistics or biostatistics as their highest degree earned. As we expected, a large majority of instructors in statistics departments (83% for doctoral departments and 62% for master’s departments) held doctoral degrees in either statistics or biostatistics. The comparable percentages for instructors of statistics in mathematics departments were about 52% and 38%. This unprofessional conduct violates the "Statement on Professional Ethics" of the American Association of University Professors (which has been affirmed by many colleges and universities in the USA) and the ethical codes of the International Statistical Institute and the American Statistical Association. The principle that statistics-instructors should have statistical competence has been affirmed by the guidelines of the Mathematical Association of America, which has been endorsed by the American Statistical Association. Actuarial science Actuarial science uses probability, statistics, and economic theory. Other disciplines The line between applied mathematics and specific areas of application is often blurred. Many universities teach mathematical and statistical courses outside of the respective departments, in departments and areas including business and economics, engineering, physics, psychology, biology, computer science, and mathematical physics.

Mathematical tools

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Computational finance • • • • • • • • • •

• • • •

• • • • • • • • • • •

• • •

Asymptotic analysis Calculus Copulas Differential equation Ergodic theory Gaussian copulas Numerical analysis Real analysis Probability Probability distribution o Binomial distribution o Log-normal distribution Expected value Value at risk Risk-neutral measure Stochastic calculus o Brownian motion o Lévy process Itô's lemma Fourier transform Girsanov's theorem Radon-Nikodym derivative Monte Carlo method Quantile function Partial differential equations o Heat equation Martingale representation theorem Feynman Kac Formula Stochastic differential equations Volatility o ARCH model o GARCH model Stochastic volatility Mathematical model Numerical method o Numerical partial differential equations  Crank-Nicolson method  Finite difference method

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Computational finance

Derivatives pricing • •

• •

•

The Brownian Motion Model of Financial Markets Rational pricing assumptions o Risk neutral valuation o Arbitrage-free pricing Futures o Futures contract pricing Options o Put–call parity (Arbitrage relationships for options) o Intrinsic value, Time value o Moneyness o Pricing models  Black–Scholes model  Black model  Binomial options model  Monte Carlo option model  Implied volatility, Volatility smile  SABR Volatility Model  Markov Switching Multifractal  The Greeks o Optimal stopping (Pricing of American options) Interest rate derivatives o Short rate model  Hull-White model  Cox-Ingersoll-Ross model  Chen model o LIBOR Market Model o Heath-Jarrow-Morton framework

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Areas of application • • • • • • • • • • •

Computational finance Quantitative Behavioral Finance Derivative (finance), list of derivatives topics Modeling and analysis of financial markets International Swaps and Derivatives Association Fundamental financial concepts - topics Model (economics) List of finance topics List of economics topics, List of economists List of accounting topics Statistical Finance

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Computational finance

Numerical analysis Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in carpentry and construction. In a rectangular wall section that is 2.40 meter by 3.75 meter, a diagonal beam has to be 4.45 meters long. Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation to , modern numerical analysis does not seek exact answers, because exact answers are impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in the movement of heavenly bodies (planets, stars and galaxies); optimization occurs in portfolio management; numerical linear algebra is essential to quantitative psychology; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology. Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. The interpolation algorithms nevertheless may be used as part of the software for solving differential equations.

General introduction The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following. • • •

•

Advanced numerical methods are essential in making numerical weather prediction feasible. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically. Hedge funds (private investment funds) use tools from all fields of numerical analysis to calculate the value of stocks and derivatives more precisely than other market participants.

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Computational finance •

•

Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. This field is also called operations research. Insurance companies use numerical programs for actuarial analysis.

The rest of this section outlines several important themes of numerical analysis.

History The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method. To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy. The mechanical calculator was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done.

Direct and iterative methods Direct vs. iterative methods Consider the problem of solving 3x3+4=28 for the unknown quantity x. Direct Method 3x3 + 4 = 28. Subtract 4 3x3 = 24. Divide by 3 x3 = 8. Take cube roots x = 2.

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Computational finance SHOULD BE NOTED THAT THE BISECTION METHOD BELOW ISN'T EXACTLY WHAT IS DESCRIBED ON THE BISECTION PAGE For the iterative method, apply the bisection method to f(x) = 3x3 - 24. The initial values are a = 0, b = 3, f(a) = -24, f(b) = 57.

a 0 1.5 1.5 1.875

Iterative Method b mid 3 1.5 3 2.25 2.25 1.875 2.25 2.0625

f(mid) -13.875 10.17... -4.22... 2.32...

We conclude from this table that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2. Discretization and numerical integration In a two hour race, we have measured the speed of the car at three instants and recorded them in the following table. Time 0:20 1:00 1:40 km/h 140 150 180 A discretization would be to say that the speed of the car was constant from 0:00 to 0:40, then from 0:40 to 1:20 and finally from 1:20 to 2:00. For instance, the total distance traveled in the first 40 minutes is approximately (2/3h x 140 km/h)=93.3 km. This would allow us to estimate the total distance traveled as 93.3 km + 100 km + 120 km = 313.3 km, which is an example of numerical integration (see below) using a Riemann sum, because displacement is the integral of velocity. Ill posed problem: Take the function f(x) = 1/(x − 1). Note that f(1.1) = 10 and f(1.001) = 1000: a change in x of less than 0.1 turns into a change in f(x) of nearly 1000. Evaluating f(x) near x = 1 is an ill-conditioned problem. Well-posed problem: By contrast, the function is continuous and so evaluating it is well-posed, at least for x being not close to zero.

Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in infinite precision arithmetic. Examples include Gaussian elimination, the QR factorization method for solving systems of linear equations, and the simplex

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Computational finance method of linear programming. In practice, finite precision is used and the result is an approximation of the true solution (assuming stability). In contrast to direct methods, iterative methods are not expected to terminate in a number of steps. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence criterion is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the bisection method, and Jacobi iteration. In computational matrix algebra, iterative methods are generally needed for large problems. Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. GMRES and the conjugate gradient method. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method.

Discretization Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called discretization. For example, the solution of a differential equation is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum.

The generation and propagation of errors The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem. Round-off Round-off errors arise because it is impossible to represent all real numbers exactly on a finite-state machine (which is what all practical digital computers are). Truncation and discretization error Truncation errors are committed when an iterative method is terminated and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. For instance, in the iteration in the sidebar to compute the solution of 3x3 + 4 = 28, after 10 or so iterations, we conclude that the root is roughly 1.99 (for example). We therefore have a truncation error of 0.01.

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Computational finance Once an error is generated, it will generally propagate through the calculation. For instance, we have already noted that the operation + on a calculator (or a computer) is inexact. It follows that a calculation of the type a+b+c+d+e is even more inexact. Numerical stability and well-posed problems Numerical stability is an important notion in numerical analysis. An algorithm is called numerically stable if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if the problem is well-conditioned, meaning that the solution changes by only a small amount if the problem data are changed by a small amount. To the contrary, if a problem is ill-conditioned, then any small error in the data will grow to be a large error. Both the original problem and the algorithm used to solve that problem can be well-conditioned and/or ill-conditioned, and any combination is possible. So an algorithm that solves a well-conditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem. For instance, computing the square root of 2 (which is roughly 1.41421) is a well-posed problem. Many algorithms solve this problem by starting with an initial approximation x1 to , for instance x1=1.4, and then computing improved guesses x2, x3, etc... One such method is the famous Babylonian method, which is given by xk+1 = xk/2 + 1/xk. Another iteration, which we will call Method X, is given by xk + 1 = (xk2−2)2 + xk. We have calculated a few iterations of each scheme in table form below, with initial guesses x1 = 1.4 and x1 = 1.42. Babylonian

Babylonian

Method X

Method X

x1 = 1.4

x1 = 1.42

x1 = 1.4

x1 = 1.42

x2 = 1.41...

x2 = 1.41...

x2 = 1.40

x2 = 1.42

x3 = 1.41…

x3 = 1.41...

x3 = 1.40...

x3 = 1.42...

...

...

X1000000 = 1.41421...

x28 = 7280.2284...

Observe that the Babylonian method converges fast regardless of the initial guess, whereas Method X converges extremely slowly with initial guess 1.4 and diverges for initial guess 1.42. Hence, the Babylonian method is numerically stable, while Method X is numerically unstable.

Areas of study

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Computational finance The field of numerical analysis is divided into different disciplines according to the problem that is to be solved.

Computing values of functions Interpolation: We have observed the temperature to vary from 20 degrees Celsius at 1:00 to 14 degrees at 3:00. A linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at 1:30pm. Extrapolation: If the gross domestic product of a country has been growing an average of 5% per year and was 100 billion dollars last year, we might extrapolate that it will be 105 billion dollars this year. Regression: In linear regression, given n points, we compute a line that passes as close as possible to those n points. Optimization: Say you sell lemonade at a lemonade stand, and notice that at $1, you can sell 197 glasses of lemonade per day, and that for each increase of $0.01, you will sell one less lemonade per day. If you could charge $1.485, you would maximize your profit, but due to the constraint of having to charge a whole cent amount, charging $1.49 per glass will yield the maximum income of $220.52 per day. Differential equation: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the Euler method for solving an ordinary differential equation. One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control round-off errors arising from the use of floating point arithmetic.

Interpolation, extrapolation, and regression Interpolation solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points? A very simple method is to use linear interpolation, which assumes that the unknown function is linear between every pair of successive points. This can be generalized to polynomial interpolation,

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Computational finance which is sometimes more accurate but suffers from Runge's phenomenon. Other interpolation methods use localized functions like splines or wavelets. Extrapolation is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points. Regression is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function. The least squares-method is one popular way to achieve this.

Areas of application • • • • •

Scientific computing List of numerical analysis topics Gram-Schmidt process Numerical differentiation Symbolic-numeric computation

General • • • • • • • •

numerical-methods.com numericalmathematics.com Numerical Recipes "Alternatives to Numerical Recipes" Scientific computing FAQ Numerical analysis DMOZ category Numerical Computing Resources on the Internet - maintained by Indiana University Stat/Math Center Numerical Methods Resources

Software Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free alternative is the GNU Scientific Library. There are several popular numerical computing applications such as MATLAB, S-PLUS, LabVIEW, and IDL and Scilab as well as free and open source alternatives such as FreeMat, GNU Octave (similar to Matlab), IT++ (a C++ library), R (similar to S-PLUS) and certain variants of Python. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude.

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Computational finance Many computer algebra systems such as Mathematica also benefit from the availability of arbitrary precision arithmetic which can provide more accurate results. Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis.

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Computational intelligence Computational intelligence (CI) is an offshoot of artificial intelligence. As an alternative to GOFAI it rather relies on heuristic algorithms such as in fuzzy systems, neural networks and evolutionary computation. In addition, computational intelligence also embraces techniques that use Swarm intelligence, Fractals and Chaos Theory, Artificial immune systems, Wavelets, etc. Computational intelligence combines elements of learning, adaptation, evolution and Fuzzy logic (rough sets) to create programs that are, in some sense, intelligent. Computational intelligence research does not reject statistical methods, but often gives a complementary view (as is the case with fuzzy systems). Artificial neural networks is a branch of computational intelligence that is closely related to machine learning. Computational intelligence is further closely associated with soft computing, connectionist systems and cybernetics.

Artificial intelligence Artificial Intelligence (AI) is the intelligence of machines and the branch of computer science which aims to create it. Major AI textbooks define the field as "the study and design of intelligent agents," where an intelligent agent is a system that perceives its environment and takes actions which maximize its chances of success. John McCarthy, who coined the term in 1956, defines it as "the science and engineering of making intelligent machines." The field was founded on the claim that a central property of human beings, intelligence—the sapience of Homo sapiens—can be so precisely described that it can be simulated by a machine. This raises philosophical issues about the nature of the mind and limits of scientific hubris, issues which have been addressed by myth, fiction and philosophy since antiquity. Artificial intelligence has been the subject of breathtaking optimism, has suffered stunning setbacks and, today, has become an essential part of the technology industry, providing the heavy lifting for many of the most difficult problems in computer science. AI research is highly technical and specialized, so much so that some critics decry the "fragmentation" of the field. Subfields of AI are organized around particular problems, the application of particular tools and around longstanding theoretical differences of opinion. The central problems of AI include such traits as reasoning, knowledge, planning, learning, communication, perception and the ability to move and manipulate objects. General intelligence (or "strong AI") is still a long-term goal of (some) research.

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Perspectives on CI CI in myth, fiction and speculation Thinking machines and artificial beings appear in Greek myths, such as Talos of Crete, the golden robots of Hephaestus and Pygmalion's Galatea. Human likenesses believed to have intelligence were built in many ancient societies; some of the earliest being the sacred statues worshipped in Egypt and Greece, and including the machines of Yan Shi, Hero of Alexandria, Al-Jazari or Wolfgang von Kempelen. It was widely believed that artificial beings had been created by Geber, Judah Loew and Paracelsus. Stories of these creatures and their fates discuss many of the same hopes, fears and ethical concerns that are presented by artificial intelligence. Mary Shelley's Frankenstein, considers a key issue in the ethics of artificial intelligence: if a machine can be created that has intelligence, could it also feel? If it can feel, does it have the same rights as a human being? The idea also appears in modern science fiction: the film Artificial Intelligence: A.I. considers a machine in the form of a small boy which has been given the ability to feel human emotions, including, tragically, the capacity to suffer. This issue, now known as "robot rights", is currently being considered by, for example, California's Institute for the Future, although many critics believe that the discussion is premature. Another issue explored by both science fiction writers and futurists is the impact of artificial intelligence on society. In fiction, AI has appeared as a servant (R2D2 in Star Wars), a law enforcer (K.I.T.T. "Knight Rider"), a comrade (Lt. Commander Data in Star Trek), a conqueror (The Matrix), a dictator (With Folded Hands), an exterminator (Terminator, Battlestar Galactica), an extension to human abilities (Ghost in the Shell) and the saviour of the human race (R. Daneel Olivaw in the Foundation Series). Academic sources have considered such consequences as: a decreased demand for human labor, the enhancement of human ability or experience, and a need for redefinition of human identity and basic values. Several futurists argue that artificial intelligence will transcend the limits of progress and fundamentally transform humanity. Ray Kurzweil has used Moore's law (which describes the relentless exponential improvement in digital technology with uncanny accuracy) to calculate that desktop computers will have the same processing power as human brains by the year 2029, and that by 2045 artificial intelligence will reach a point where it is able to improve itself at a rate that far exceeds anything conceivable in the past, a scenario that science fiction writer Vernor Vinge named the "technological singularity". Edward Fredkin argues that "artificial intelligence is the next stage in evolution," an idea first proposed by Samuel Butler's "Darwin among the Machines" (1863), and expanded upon by George Dyson in his book of the same name in 1998. Several futurists and science fiction writers have predicted that human beings and machines will merge in the future into cyborgs that are more capable and powerful than either. This idea, called transhumanism, which has roots in Aldous Huxley and Robert Ettinger, is now associated with robot designer Hans Moravec, cyberneticist Kevin Warwick

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Computational finance and inventor Ray Kurzweil. Transhumanism has been illustrated in fiction as well, for example in the manga Ghost in the Shell and the science fiction series Dune. Pamela McCorduck writes that these scenarios are expressions of an ancient human desire to, as she calls it, "forge the gods." History of CI research In the middle of the 20th century, a handful of scientists began a new approach to building intelligent machines, based on recent discoveries in neurology, a new mathematical theory of information, an understanding of control and stability called cybernetics, and above all, by the invention of the digital computer, a machine based on the abstract essence of mathematical reasoning. The field of modern AI research was founded at a conference on the campus of Dartmouth College in the summer of 1956. Those who attended would become the leaders of AI research for many decades, especially John McCarthy, Marvin Minsky, Allen Newell and Herbert Simon, who founded AI laboratories at MIT, CMU and Stanford. They and their students wrote programs that were, to most people, simply astonishing: computers were solving word problems in algebra, proving logical theorems and speaking English. By the middle 60s their research was heavily funded by the U.S. Department of Defense, and they were optimistic about the future of the new field: • •

1965, H. A. Simon: "[M]achines will be capable, within twenty years, of doing any work a man can do" 1967, Marvin Minsky: "Within a generation ... the problem of creating 'artificial intelligence' will substantially be solved."

These predictions, and many like them, would not come true. They had failed to recognize the difficulty of some of the problems they faced. In 1974, in response to the criticism of England's Sir James Lighthill and ongoing pressure from Congress to fund more productive projects, the U.S. and British governments cut off all undirected, exploratory research in AI. This was the first AI winter. In the early 80s, AI research was revived by the commercial success of expert systems, a form of AI program that simulated the knowledge and analytical skills of one or more human experts. By 1985 the market for AI had reached more than a billion dollars, and governments around the world poured money back into the field. However, just a few years later, beginning with the collapse of the Lisp Machine market in 1987, AI once again fell into disrepute, and a second, longer lasting AI winter began. In the 90s and early 21st century, AI achieved its greatest successes, albeit somewhat behind the scenes. Artificial intelligence is used for logistics, data mining, medical diagnosis and many other areas throughout the technology industry. The success was due to several factors: the incredible power of computers today (see Moore's law), a greater emphasis on solving specific subproblems, the creation of new ties between AI and other fields working on

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Computational finance similar problems, and above all a new commitment by researchers to solid mathematical methods and rigorous scientific standards. Philosophy of AI Artificial intelligence, by claiming to be able to recreate the capabilities of the human mind, is both a challenge and an inspiration for philosophy. Are there limits to how intelligent machines can be? Is there an essential difference between human intelligence and artificial intelligence? Can a machine have a mind and consciousness? A few of the most influential answers to these questions are given below. Turing's "polite convention" If a machine acts as intelligently as a human being, then it is as intelligent as a human being. Alan Turing theorized that, ultimately, we can only judge the intelligence of a machine based on its behavior. This theory forms the basis of the Turing test. The Dartmouth proposal "Every aspect of learning or any other feature of intelligence can be so precisely described that a machine can be made to simulate it." This assertion was printed in the proposal for the Dartmouth Conference of 1956, and represents the position of most working AI researchers. Newell and Simon's physical symbol system hypothesis "A physical symbol system has the necessary and sufficient means of general intelligent action." This statement claims that the essence of intelligence is symbol manipulation. Hubert Dreyfus argued that, on the contrary, human expertise depends on unconscious instinct rather than conscious symbol manipulation and on having a "feel" for the situation rather than explicit symbolic knowledge. Gödel's incompleteness theorem A formal system (such as a computer program) can not prove all true statements. Roger Penrose is among those who claim that Gödel's theorem limits what machines can do. Searle's strong AI hypothesis "The appropriately programmed computer with the right inputs and outputs would thereby have a mind in exactly the same sense human beings have minds." Searle counters this assertion with his Chinese room argument, which asks us to look inside the computer and try to find where the "mind" might be.

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Computational finance The artificial brain argument The brain can be simulated. Hans Moravec, Ray Kurzweil and others have argued that it is technologically feasible to copy the brain directly into hardware and software, and that such a simulation will be essentially identical to the original.

CI research In the 21st century, AI research has become highly specialized and technical. It is deeply divided into subfields that often fail to communicate with each other. Subfields have grown up around particular institutions, the work of particular researchers, particular problems (listed below), long standing differences of opinion about how AI should be done (listed as "approaches" below) and the application of widely differing tools (see tools of AI, below). Problems of AI The problem of simulating (or creating) intelligence has been broken down into a number of specific sub-problems. These consist of particular traits or capabilities that researchers would like an intelligent system to display. The traits described below have received the most attention. Deduction, reasoning, problem solving Early AI researchers developed algorithms that imitated the step-by-step reasoning that human beings use when they solve puzzles, play board games or make logical deductions. By the late 80s and 90s, AI research had also developed highly successful methods for dealing with uncertain or incomplete information, employing concepts from probability and economics. For difficult problems, most of these algorithms can require enormous computational resources — most experience a "combinatorial explosion": the amount of memory or computer time required becomes astronomical when the problem goes beyond a certain size. The search for more efficient problem solving algorithms is a high priority for AI research. Human beings solve most of their problems using fast, intuitive judgments rather than the conscious, step-by-step deduction that early AI research was able to model. AI has made some progress at imitating this kind of "sub-symbolic" problem solving: embodied approaches emphasize the importance of sensorimotor skills to higher reasoning; neural net research attempts to simulate the structures inside human and animal brains that gives rise to this skill. Knowledge representation Knowledge representation and knowledge engineering are central to AI research. Many of the problems machines are expected to solve will require extensive

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Computational finance knowledge about the world. Among the things that AI needs to represent are: objects, properties, categories and relations between objects; situations, events, states and time; causes and effects; knowledge about knowledge (what we know about what other people know); and many other, less well researched domains. A complete representation of "what exists" is an ontology (borrowing a word from traditional philosophy), of which the most general are called upper ontologies. Among the most difficult problems in knowledge representation are: Default reasoning and the qualification problem Many of the things people know take the form of "working assumptions." For example, if a bird comes up in conversation, people typically picture an animal that is fist sized, sings, and flies. None of these things are true about all birds. John McCarthy identified this problem in 1969 as the qualification problem: for any commonsense rule that AI researchers care to represent, there tend to be a huge number of exceptions. Almost nothing is simply true or false in the way that abstract logic requires. AI research has explored a number of solutions to this problem. The breadth of commonsense knowledge The number of atomic facts that the average person knows is astronomical. Research projects that attempt to build a complete knowledge base of commonsense knowledge (e.g., Cyc) require enormous amounts of laborious ontological engineering — they must be built, by hand, one complicated concept at a time. The subsymbolic form of some commonsense knowledge Much of what people know isn't represented as "facts" or "statements" that they could actually say out loud. For example, a chess master will avoid a particular chess position because it "feels too exposed" or an art critic can take one look at a statue and instantly realize that it is a fake. These are intuitions or tendencies that are represented in the brain non-consciously and sub-symbolically. Knowledge like this informs, supports and provides a context for symbolic, conscious knowledge. As with the related problem of sub-symbolic reasoning, it is hoped that situated AI or computational intelligence will provide ways to represent this kind of knowledge.

Planning Intelligent agents must be able to set goals and achieve them. They need a way to visualize the future (they must have a representation of the state of the world and

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Computational finance be able to make predictions about how their actions will change it) and be able to make choices that maximize the utility (or "value") of the available choices. In some planning problems, the agent can assume that it is the only thing acting on the world and it can be certain what the consequences of its actions may be. However, if this is not true, it must periodically check if the world matches its predictions and it must change its plan as this becomes necessary, requiring the agent to reason under uncertainty. Multi-agent planning uses the cooperation and competition of many agents to achieve a given goal. Emergent behavior such as this is used by evolutionary algorithms and swarm intelligence. Learning Machine learning has been central to AI research from the beginning. Unsupervised learning is the ability to find patterns in a stream of input. Supervised learning includes both classification (be able to determine what category something belongs in, after seeing a number of examples of things from several categories) and regression (given a set of numerical input/output examples, discover a continuous function that would generate the outputs from the inputs). In reinforcement learning the agent is rewarded for good responses and punished for bad ones. These can be analyzed in terms of decision theory, using concepts like utility. The mathematical analysis of machine learning algorithms and their performance is a branch of theoretical computer science known as computational learning theory. Natural language processing Natural language processing gives machines the ability to read and understand the languages that the human beings speak. Many researchers hope that a sufficiently powerful natural language processing system would be able to acquire knowledge on its own, by reading the existing text available over the internet. Some straightforward applications of natural language processing include information retrieval (or text mining) and machine translation. Motion and manipulation The field of robotics is closely related to AI. Intelligence is required for robots to be able to handle such tasks as object manipulation and navigation, with subproblems of localization (knowing where you are), mapping (learning what is around you) and motion planning (figuring out how to get there). Perception Machine perception is the ability to use input from sensors (such as cameras, microphones, sonar and others more exotic) to deduce aspects of the world.

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Computational finance Computer vision is the ability to analyze visual input. A few selected subproblems are speech recognition, facial recognition and object recognition. Social intelligence Emotion and social skills play two roles for an intelligent agent: •

•

It must be able to predict the actions of others, by understanding their motives and emotional states. (This involves elements of game theory, decision theory, as well as the ability to model human emotions and the perceptual skills to detect emotions.) For good human-computer interaction, an intelligent machine also needs to display emotions — at the very least it must appear polite and sensitive to the humans it interacts with. At best, it should have normal emotions itself.

Creativity A sub-field of AI addresses creativity both theoretically (from a philosophical and psychological perspective) and practically (via specific implementations of systems that generate outputs that can be considered creative). General intelligence Most researchers hope that their work will eventually be incorporated into a machine with general intelligence (known as strong AI), combining all the skills above and exceeding human abilities at most or all of them. A few believe that anthropomorphic features like artificial consciousness or an artificial brain may be required for such a project. Many of the problems above are considered AI-complete: to solve one problem, you must solve them all. For example, even a straightforward, specific task like machine translation requires that the machine follow the author's argument (reason), know what it's talking about (knowledge), and faithfully reproduce the author's intention (social intelligence). Machine translation, therefore, is believed to be AI-complete: it may require strong AI to be done as well as humans can do it.

Approaches to CI There is no established unifying theory or paradigm that guides AI research. Researchers disagree about many issues. A few of the most long standing questions that have remained unanswered are these: Can intelligence be reproduced using high-level symbols, similar to words and ideas? Or does it require "sub-symbolic" processing? Should artificial intelligence simulate natural intelligence, by studying human psychology or animal neurobiology? Or is human biology as irrelevant to AI research as bird biology is to aeronautical engineering? Can intelligent behavior be described using simple, elegant principles (such as

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Computational finance logic or optimization)? Or does artificial intelligence necessarily require solving many unrelated problems Cybernetics and brain simulation The human brain provides inspiration for artificial intelligence researchers, however there is no consensus on how closely it should be simulated. In the 1940s and 1950s, a number of researchers explored the connection between neurology, information theory, and cybernetics. Some of them built machines that used electronic networks to exhibit rudimentary intelligence, such as W. Grey Walter's turtles and the Johns Hopkins Beast. Many of these researchers gathered for meetings of the Teleological Society at Princeton University and the Ratio Club in England. By 1960, this approach was largely abandoned, although elements of it would be revived in the 1980s. Traditional symbolic AI When access to digital computers became possible in the middle 1950s, AI research began to explore the possibility that human intelligence could be reduced to symbol manipulation. The research was centered in three institutions: CMU, Stanford and MIT, and each one developed its own style of research. John Haugeland named these approaches to AI "good old fashioned AI" or "GOFAI". Cognitive simulation Economist Herbert Simon and Alan Newell studied human problem solving skills and attempted to formalize them, and their work laid the foundations of the field of artificial intelligence, as well as cognitive science, operations research and management science. Their research team performed psychological experiments to demonstrate the similarities between human problem solving and the programs (such as their "General Problem Solver") they were developing. This tradition, centered at Carnegie Mellon University would eventually culminate in the development of the Soar architecture in the middle 80s. Logical AI Unlike Newell and Simon, John McCarthy felt that machines did not need to simulate human thought, but should instead try to find the essence of abstract reasoning and problem solving, regardless of whether people used the same algorithms. His laboratory at Stanford (SAIL) focused on using formal logic to solve a wide variety of problems, including knowledge representation, planning and learning. Logic was also focus of the work at the University of Edinburgh and elsewhere in Europe which led to the development of the programming language Prolog and the science of logic programming. "Scruffy" symbolic AI

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Computational finance Researchers at MIT (such as Marvin Minsky and Seymour Papert) found that solving difficult problems in vision and natural language processing required ad-hoc solutions – they argued that there was no simple and general principle (like logic) that would capture all the aspects of intelligent behavior. Roger Schank described their "anti-logic" approaches as "scruffy" (as opposed to the "neat" paradigms at CMU and Stanford). Commonsense knowledge bases (such as Doug Lenat's Cyc) are an example of "scruffy" AI, since they must be built by hand, one complicated concept at a time. Knowledge based AI When computers with large memories became available around 1970, researchers from all three traditions began to build knowledge into AI applications. This "knowledge revolution" led to the development and deployment of expert systems (introduced by Edward Feigenbaum), the first truly successful form of AI software. The knowledge revolution was also driven by the realization that enormous amounts of knowledge would be required by many simple AI applications. Sub-symbolic AI During the 1960s, symbolic approaches had achieved great success at simulating high-level thinking in small demonstration programs. Approaches based on cybernetics or neural networks were abandoned or pushed into the background. By the 1980s, however, progress in symbolic AI seemed to stall and many believed that symbolic systems would never be able to imitate all the processes of human cognition, especially perception, robotics, learning and pattern recognition. A number of researchers began to look into "sub-symbolic" approaches to specific AI problems. Bottom-up, embodied, situated, behavior-based or nouvelle AI Researchers from the related field of robotics, such as Rodney Brooks, rejected symbolic AI and focused on the basic engineering problems that would allow robots to move and survive. Their work revived the nonsymbolic viewpoint of the early cybernetics researchers of the 50s and reintroduced the use of control theory in AI. These approaches are also conceptually related to the embodied mind thesis.

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Computational finance Computational Intelligence Interest in neural networks and "connectionism" was revived by David Rumelhart and others in the middle 1980s. These and other sub-symbolic approaches, such as fuzzy systems and evolutionary computation, are now studied collectively by the emerging discipline of computational intelligence. Statistical AI In the 1990s, AI researchers developed sophisticated mathematical tools to solve specific subproblems. These tools are truly scientific, in the sense that their results are both measurable and verifiable, and they have been responsible for many of AI's recent successes. The shared mathematical language has also permitted a high level of collaboration with more established fields (like mathematics, economics or operations research). Russell & Norvig (2003) describe this movement as nothing less than a "revolution" and "the victory of the neats." Integrating the approaches Intelligent agent paradigm An intelligent agent is a system that perceives its environment and takes actions which maximizes its chances of success. The simplest intelligent agents are programs that solve specific problems. The most complicated intelligent agents are rational, thinking human beings. The paradigm gives researchers license to study isolated problems and find solutions that are both verifiable and useful, without agreeing on one single approach. An agent that solves a specific problem can use any approach that works — some agents are symbolic and logical, some are sub-symbolic neural networks and others may use new approaches. The paradigm also gives researchers a common language to communicate with other fields—such as decision theory and economics— that also use concepts of abstract agents. The intelligent agent paradigm became widely accepted during the 1990s. An agent architecture or cognitive architecture Researchers have designed systems to build intelligent systems out of interacting intelligent agents in a multi-agent system. A system with both symbolic and sub-symbolic components is a hybrid intelligent system, and the study of such systems is artificial intelligence systems integration. A hierarchical control system provides a bridge between sub-symbolic AI at its lowest, reactive levels and traditional symbolic AI at its highest levels, where relaxed time constraints permit planning and world modelling. Rodney Brooks' subsumption architecture was an early proposal for such a hierarchical system.

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Tools of CI research In the course of 50 years of research, AI has developed a large number of tools to solve the most difficult problems in computer science. A few of the most general of these methods are discussed below. Search and optimization Many problems in AI can be solved in theory by intelligently searching through many possible solutions: Reasoning can be reduced to performing a search. For example, logical proof can be viewed as searching for a path that leads from premises to conclusions, where each step is the application of an inference rule. Planning algorithms search through trees of goals and subgoals, attempting to find a path to a target goal, a process called means-ends analysis. Robotics algorithms for moving limbs and grasping objects use local searches in configuration space. Many learning algorithms use search algorithms based on optimization. Simple exhaustive searches are rarely sufficient for most real world problems: the search space (the number of places to search) quickly grows to astronomical numbers. The result is a search that is too slow or never completes. The solution, for many problems, is to use "heuristics" or "rules of thumb" that eliminate choices that are unlikely to lead to the goal (called "pruning the search tree"). Heuristics supply the program with a "best guess" for what path the solution lies on. A very different kind of search came to prominence in the 1990s, based on the mathematical theory of optimization. For many problems, it is possible to begin the search with some form of a guess and then refine the guess incrementally until no more refinements can be made. These algorithms can be visualized as blind hill climbing: we begin the search at a random point on the landscape, and then, by jumps or steps, we keep moving our guess uphill, until we reach the top. Other optimization algorithms are simulated annealing, beam search and random optimization. Evolutionary computation uses a form of optimization search. For example, they may begin with a population of organisms (the guesses) and then allow them to mutate and recombine, selecting only the fittest to survive each generation (refining the guesses). Forms of evolutionary computation include swarm intelligence algorithms (such as ant colony or particle swarm optimization) and evolutionary algorithms (such as genetic algorithms and genetic programming). Logic Logic was introduced into AI research by John McCarthy in his 1958 Advice Taker proposal. In 1963, J. Alan Robinson discovered a simple, complete and entirely algorithmic method for logical deduction which can easily be performed by digital computers. However, a naive implementation of the algorithm quickly leads to a combinatorial explosion or an infinite loop. In 1974, Robert Kowalski

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Computational finance suggested representing logical expressions as Horn clauses (statements in the form of rules: "if p then q"), which reduced logical deduction to backward chaining or forward chaining. This greatly alleviated (but did not eliminate) the problem. Logic is used for knowledge representation and problem solving, but it can be applied to other problems as well. For example, the satplan algorithm uses logic for planning, and inductive logic programming is a method for learning. There are several different forms of logic used in AI research. • •

•

•

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Propositional or sentential logic is the logic of statements which can be true or false. First-order logic also allows the use of quantifiers and predicates, and can express facts about objects, their properties, and their relations with each other. Fuzzy logic, a version of first-order logic which allows the truth of a statement to be represented as a value between 0 and 1, rather than simply True (1) or False (0). Fuzzy systems can be used for uncertain reasoning and have been widely used in modern industrial and consumer product control systems. Default logics, non-monotonic logics and circumscription are forms of logic designed to help with default reasoning and the qualification problem. Several extensions of logic have been designed to handle specific domains of knowledge, such as: description logics; situation calculus, event calculus and fluent calculus (for representing events and time); causal calculus; belief calculus; and modal logics.

Probabilistic methods for uncertain reasoning Many problems in AI (in reasoning, planning, learning, perception and robotics) require the agent to operate with incomplete or uncertain information. Starting in the late 80s and early 90s, Judea Pearl and others championed the use of methods drawn from probability theory and economics to devise a number of powerful tools to solve these problems. Bayesian networks are a very general tool that can be used for a large number of problems: reasoning (using the Bayesian inference algorithm), learning (using the expectation-maximization algorithm), planning (using decision networks) and perception (using dynamic Bayesian networks). Probabilistic algorithms can also be used for filtering, prediction, smoothing and finding explanations for streams of data, helping perception systems to analyze processes that occur over time (e.g., hidden Markov models or Kalman filters). A key concept from the science of economics is "utility": a measure of how valuable something is to an intelligent agent. Precise mathematical tools have been developed that analyze how an agent can make choices and plan, using

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Computational finance decision theory, decision analysis, information value theory. These tools include models such as Markov decision processes, dynamic decision networks, game theory and mechanism design. Classifiers and statistical learning methods The simplest AI applications can be divided into two types: classifiers ("if shiny then diamond") and controllers ("if shiny then pick up"). Controllers do however also classify conditions before inferring actions, and therefore classification forms a central part of many AI systems. Classifiers are functions that use pattern matching to determine a closest match. They can be tuned according to examples, making them very attractive for use in AI. These examples are known as observations or patterns. In supervised learning, each pattern belongs to a certain predefined class. A class can be seen as a decision that has to be made. All the observations combined with their class labels are known as a data set. When a new observation is received, that observation is classified based on previous experience. A classifier can be trained in various ways; there are many statistical and machine learning approaches. A wide range of classifiers are available, each with its strengths and weaknesses. Classifier performance depends greatly on the characteristics of the data to be classified. There is no single classifier that works best on all given problems; this is also referred to as the "no free lunch" theorem. Various empirical tests have been performed to compare classifier performance and to find the characteristics of data that determine classifier performance. Determining a suitable classifier for a given problem is however still more an art than science. The most widely used classifiers are the neural network, kernel methods such as the support vector machine, k-nearest neighbor algorithm, Gaussian mixture model, naive Bayes classifier, and decision tree. The performance of these classifiers have been compared over a wide range of classification tasks in order to find data characteristics that determine classifier performance. Neural networks A neural network is an interconnected group of nodes, akin to the vast network of neurons in the human brain. The study of artificial neural networks began in the decade before the field AI research was founded. In the 1960s Frank Rosenblatt developed an important early version, the perceptron. Paul Werbos developed the backpropagation algorithm for multilayer perceptrons in 1974, which led to a renaissance in neural network research and connectionism in general in the middle 1980s. The Hopfield net, a form of attractor network, was first described by John Hopfield in 1982.

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Computational finance Common network architectures which have been developed include the feedforward neural network, the radial basis network, the Kohonen selforganizing map and various recurrent neural networks. Neural networks are applied to the problem of learning, using such techniques as Hebbian learning, competitive learning and the relatively new architectures of Hierarchical Temporal Memory and Deep Belief Networks. Control theory Control theory, the grandchild of cybernetics, has many important applications, especially in robotics. Specialized languages AI researchers have developed several specialized languages for AI research: •

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•

•

IPL was the first language developed for artificial intelligence. It includes features intended to support programs that could perform general problem solving, including lists, associations, schemas (frames), dynamic memory allocation, data types, recursion, associative retrieval, functions as arguments, generators (streams), and cooperative multitasking. Lisp is a practical mathematical notation for computer programs based on lambda calculus. Linked lists are one of Lisp languages' major data structures, and Lisp source code is itself made up of lists. As a result, Lisp programs can manipulate source code as a data structure, giving rise to the macro systems that allow programmers to create new syntax or even new domain-specific programming languages embedded in Lisp. There are many dialects of Lisp in use today. Prolog is a declarative language where programs are expressed in terms of relations, and execution occurs by running queries over these relations. Prolog is particularly useful for symbolic reasoning, database and language parsing applications. Prolog is widely used in AI today. STRIPS is a language for expressing automated planning problem instances. It expresses an initial state, the goal states, and a set of actions. For each action preconditions (what must be established before the action is performed) and postconditions (what is established after the action is performed) are specified. Planner is a hybrid between procedural and logical languages. It gives a procedural interpretation to logical sentences where implications are interpreted with pattern-directed inference.

AI applications are also often written in standard languages like C++ and languages designed for mathematics, such as MATLAB and Lush.

Evaluating artificial intelligence How can one determine if an agent is intelligent? In 1950, Alan Turing proposed a general procedure to test the intelligence of an agent now known as the Turing

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Computational finance test. This procedure allows almost all the major problems of artificial intelligence to be tested. However, it is a very difficult challenge and at present all agents fail. Artificial intelligence can also be evaluated on specific problems such as small problems in chemistry, hand-writing recognition and game-playing. Such tests have been termed subject matter expert Turing tests. Smaller problems provide more achievable goals and there are an ever-increasing number of positive results. The broad classes of outcome for an AI test are: • • • •

optimal: it is not possible to perform better strong super-human: performs better than all humans super-human: performs better than most humans sub-human: performs worse than most humans

For example, performance at checkers (draughts) is optimal, performance at chess is super-human and nearing strong super-human, and performance at many everyday tasks performed by humans is sub-human.

Applications of artificial intelligence Artificial intelligence has successfully been used in a wide range of fields including medical diagnosis, stock trading, robot control, law, scientific discovery, video games and toys. Frequently, when a technique reaches mainstream use it is no longer considered artificial intelligence, sometimes described as the AI effect. It may also become integrated into artificial life.

Areas of application • • • • • • • • • • • • • • • • • •

Simulated annealing Machine learning Artificial immune systems Expert systems Hybrid intelligent systems Hybrid logic Simulated reality Soft computing Bayesian networks Chaos theory Ant colony optimization Particle swarm optimisation Cognitive robotics Developmental robotics Evolutionary robotics Intelligent agents Knowledge-Based Engineering Type-2 fuzzy sets and systems

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Remotely related topics: • • • • • •

Bioinformatics and Bioengineering Computational finance and Computational economics Intelligent systems Emergence Data mining Concept mining

Software • •

Computational Intelligence Library (CILib) OAT (Optimization Algorithm Toolkit): A set of standard computational intelligence optimization algorithms and problems in Java.

Organizations • •

IEEE Computational Intelligence Society The Computational Intelligence and Machine Learning Virtual Community

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Computer simulation A computer simulation, a computer model or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system. Computer simulations have become a useful part of mathematical modeling of many natural systems in physics (computational physics), chemistry and biology, human systems in economics, psychology, and social science and in the process of engineering new technology, to gain insight into the operation of those systems, or to observe their behavior. Computer simulations vary from computer programs that run a few minutes, to network-based groups of computers running for hours, to ongoing simulations that run for days. The scale of events being simulated by computer simulations has far exceeded anything possible (or perhaps even imaginable) using the traditional paper-and-pencil mathematical modeling: over 10 years ago, a desertbattle simulation, of one force invading another, involved the modeling of 66,239 tanks, trucks and other vehicles on simulated terrain around Kuwait, using multiple supercomputers in the DoD High Performance Computer Modernization Program; a 1-billion-atom model of material deformation (2002); a 2.64-millionatom model of the complex maker of protein in all organisms, a ribosome, in 2005; and the Blue Brain project at EPFL (Switzerland), began in May 2005, to create the first computer simulation of the entire human brain, right down to the molecular level.

History Computer simulation was developed hand-in-hand with the rapid growth of the computer, following its first large-scale deployment during the Manhattan Project in World War II to model the process of nuclear detonation. It was a simulation of 12 hard spheres using a Monte Carlo algorithm. Computer simulation is often used as an adjunct to, or substitution for, modeling systems for which simple closed form analytic solutions are not possible. There are many different types of computer simulation; the common feature they all share is the attempt to generate a sample of representative scenarios for a model in which a complete enumeration of all possible states of the model would be prohibitive or impossible. Computer models were initially used as a supplement for other arguments, but their use later became rather widespread.

Data preparation The data input/output for the simulation can be either through formatted textfiles or a pre- and postprocessor.

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Types Computer models can be classified according to several independent pairs of attributes, including: • • • •

Stochastic or deterministic (and as a special case of deterministic, chaotic) for examples of stochastic vs. deterministic simulations Steady-state or dynamic Continuous or discrete (and as an important special case of discrete, discrete event or DE models) Local or distributed.

These attributes may be arbitrarily combined to form terminology that describes simulation types, such as "continuous dynamic simulations" or "discrete dynamic simulations." For example: •

• • •

•

Steady-state models use equations defining the relationships between elements of the modeled system and attempt to find a state in which the system is in equilibrium. Such models are often used in simulating physical systems, as a simpler modeling case before dynamic simulation is attempted. Dynamic simulations model changes in a system in response to (usually changing) input signals. Stochastic models use random number generators to model chance or random events; A discrete event simulation (DES) manages events in time. Most computer, logic-test and fault-tree simulations are of this type. In this type of simulation, the simulator maintains a queue of events sorted by the simulated time they should occur. The simulator reads the queue and triggers new events as each event is processed. It is not important to execute the simulation in real time. It's often more important to be able to access the data produced by the simulation, to discover logic defects in the design, or the sequence of events. A continuous dynamic simulation performs numerical solution of differential-algebraic equations or differential equations (either partial or ordinary). Periodically, the simulation program solves all the equations, and uses the numbers to change the state and output of the simulation. Applications include flight simulators, construction and management simulation games, chemical process modeling, and simulations of electrical circuits. Originally, these kinds of simulations were actually implemented on analog computers, where the differential equations could be represented directly by various electrical components such as op-amps. By the late 1980s, however, most "analog" simulations were run on conventional digital computers that emulate the behavior of an analog computer.

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•

A special type of discrete simulation which does not rely on a model with an underlying equation, but can nonetheless be represented formally, is agent-based simulation. In agent-based simulation, the individual entities (such as molecules, cells, trees or consumers) in the model are represented directly (rather than by their density or concentration) and possess an internal state and set of behaviors or rules which determine how the agent's state is updated from one time-step to the next. Distributed models run on a network of interconnected computers, possibly through the Internet. Simulations dispersed across multiple host computers like this are often referred to as "distributed simulations". There are several standards for distributed simulation, including Aggregate Level Simulation Protocol (ALSP), Distributed Interactive Simulation (DIS), the High Level Architecture (simulation) (HLA) and the Test and Training Enabling Architecture (TENA).

CGI computer simulation Formerly, the output data from a computer simulation was sometimes presented in a table, or a matrix, showing how data was affected by numerous changes in the simulation parameters. The use of the matrix format was related to traditional use of the matrix concept in mathematical models; however, psychologists and others noted that humans could quickly perceive trends by looking at graphs or even moving-images or motion-pictures generated from the data, as displayed by computer-generated-imagery (CGI) animation. Although observers couldn't necessarily read out numbers, or spout math formulas, from observing a moving weather chart, they might be able to predict events (and "see that rain was headed their way"), much faster than scanning tables of rain-cloud coordinates. Such intense graphical displays, which transcended the world of numbers and formulae, sometimes also led to output that lacked a coordinate grid or omitted timestamps, as if straying too far from numeric data displays. Today, weather forecasting models tend to balance the view of moving rain/snow clouds against a map that uses numeric coordinates and numeric timestamps of events. Similarly, CGI computer simulations of CAT scans can simulate how a tumor might shrink or change, during an extended period of medical treatment, presenting the passage of time as a spinning view of the visible human head, as the tumor changes. Other applications of CGI computer simulations are being developed to graphically display large amounts of data, in motion, as changes occur during a simulation run.

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Computer simulation in science Generic examples of types of computer simulations in science, which are derived from an underlying mathematical description: •

•

a numerical simulation of differential equations which cannot be solved analytically, theories which involve continuous systems such as phenomena in physical cosmology, fluid dynamics (e.g. climate models, roadway noise models, roadway air dispersion models), continuum mechanics and chemical kinetics fall into this category. a stochastic simulation, typically used for discrete systems where events occur probabilistically, and which cannot be described directly with differential equations (this is a discrete simulation in the above sense). Phenomena in this category include genetic drift, biochemical or gene regulatory networks with small numbers of molecules. (see also: Monte Carlo method).

Specific examples of computer simulations follow: •

•

•

• • •

•

statistical simulations based upon an agglomeration of a large number of input profiles, such as the forecasting of equilibrium temperature of receiving waters, allowing the gamut of meteorological data to be input for a specific locale. This technique was developed for thermal pollution forecasting . agent based simulation has been used effectively in ecology, where it is often called individual based modeling and has been used in situations for which individual variability in the agents cannot be neglected, such as population dynamics of salmon and trout (most purely mathematical models assume all trout behave identically). time stepped dynamic model. In hydrology there are several such hydrology transport models such as the SWMM and DSSAM Models developed by the U.S. Environmental Protection Agency for river water quality forecasting. computer simulations have also been used to formally model theories of human cognition and performance, e.g. ACT-R computer simulation using molecular modeling for drug discovery Computational fluid dynamics simulations are used to simulate the behaviour of flowing air, water and other fluids. There are one-, two- and three- dimensional models used. A one dimensional model might simulate the effects of water hammer in a pipe. A two-dimensional model might be used to simulate the drag forces on the cross-section of an aeroplane wing. A three-dimensional simulation might estimate the heating and cooling requirements of a large building. An understanding of statistical thermodynamic molecular theory is fundamental to the appreciation of molecular solutions. Development of the Potential Distribution Theorem (PDT) allows one to simplify this complex subject to down-to-earth presentations of molecular theory.

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Simulation versus modeling Traditionally, forming large models (spelt 'modeling' in American English) of systems has been via a mathematical model, which attempts to find analytical solutions to problems and thereby enable the prediction of the behavior of the system from a set of parameters and initial conditions. While computer simulations might use some algorithms from purely mathematical models, computers can combine simulations with reality or actual events, such as generating input responses, to simulate test subjects who are no longer present. Whereas the missing test subjects are being modeled/simulated, the system they use could be the actual equipment, revealing performance limits or defects in long-term use by the simulated users. Note that the term computer simulation is broader than computer modeling, which implies that all aspects are being modeled in the computer representation. However, computer simulation also includes generating inputs from simulated users to run actual computer software or equipment, with only part of the system being modeled: an example would be flight simulators which can run machines as well as actual flight software. Computer simulations are used in many fields, including science, technology, entertainment, and business planning and scheduling.

Simulation environments for physics and engineering Graphical environments to design simulations have been developed. Special care was taken to handle events (situations in which the simulation equations are not valid and have to be changed). The open project Open Source Physics was started to develop reusable libraries for simulations in Java, together with Easy Java Simulations, a complete graphical environment that generates code based on these libraries.

Computer simulation in practical contexts Computer simulations are used in a wide variety of practical contexts, such as: • • • • • • • • •

analysis of air pollutant dispersion using atmospheric dispersion modeling design of complex systems such as aircraft and also logistics systems. design of Noise barriers to effect roadway noise mitigation flight simulators to train pilots weather forecasting forecasting of prices on financial markets (for example Adaptive Modeler) behavior of structures (such as buildings and industrial parts) under stress and other conditions design of industrial processes, such as chemical processing plants Strategic Management and Organizational Studies

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•

Reservoir simulation for the petroleum engineering to model the subsurface reservoir Process Engineering Simulation tools. Robot simulators for the design of robots and robot control algorithms Traffic engineering to plan or redesign parts of the street network from single junctions over cities to a national highway network, see for example VISSIM. modeling car crashes to test safety mechanisms in new vehicle models

The reliability and the trust people put in computer simulations depends on the validity of the simulation model, therefore verification and validation are of crucial importance in the development of computer simulations. Another important aspect of computer simulations is that of reproducibility of the results, meaning that a simulation model should not provide a different answer for each execution. Although this might seem obvious, this is a special point of attention in stochastic simulations, where random numbers should actually be semi-random numbers. An exception to reproducibility are human in the loop simulations such as flight simulations and computer games. Here a human is part of the simulation and thus influences the outcome in a way that is hard if not impossible to reproduce exactly. Vehicle manufacturers make use of computer simulation to test safety features in new designs. By building a copy of the car in a physics simulation environment, they can save the hundreds of thousands of dollars that would otherwise be required to build a unique prototype and test it. Engineers can step through the simulation milliseconds at a time to determine the exact stresses being put upon each section of the prototype. Computer graphics can be used to display the results of a computer simulation. Animations can be used to experience a simulation in real-time e.g. in training simulations. In some cases animations may also be useful in faster than real-time or even slower than real-time modes. For example, faster than real-time animations can be useful in visualizing the buildup of queues in the simulation of humans evacuating a building. Furthermore, simulation results are often aggregated into static images using various ways of scientific visualization. In debugging, simulating a program execution under test (rather than executing natively) can detect far more errors than the hardware itself can detect and, at the same time, log useful debugging information such as instruction trace, memory alterations and instruction counts. This technique can also detect buffer overflow and similar "hard to detect" errors as well as produce performance information and tuning data.

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Pitfalls Although sometimes ignored in computer simulations, it is very important to perform sensitivity analysis to ensure that the accuracy of the results are properly understood. For example, the probabilistic risk analysis of factors determining the success of an oilfield exploration program involves combining samples from a variety of statistical distributions using the Monte Carlo method. If, for instance, one of the key parameters (i.e. the net ratio of oil-bearing strata) is known to only one significant figure, then the result of the simulation might not be more precise than one significant figure, although it might (misleadingly) be presented as having four significant figures.

Areas of application • • • • • • • • • • • • • • • • • • • • • •

ACT-R Articulatory synthesis Artificial life CAVE Computer-aided design Computer simulation and organizational studies Dry Lab Earth Simulator Emulator Experiment in silico Global climate model Ice sheet model List of computer simulation software Mathematical model MapleSim Molecular dynamics SimApp Simcyp Simulator Simulated reality Social simulation Solver (computer science) Web based simulation

Organizations • • • • • •

EUROSIM - Federation of European Simulation Societies Institute for Simulation and Training, University of Central Florida Simulation Interoperability Standards Organization The Society for Modeling and Simulation International (Formerly the Society of Computer Simulation) United States Defense Modeling and Simulation Office The System Dynamics Society

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The Computational Modelling Group at Cambridge University's Department of Chemical Engineering Liophant Simulation United Simulation Team - Genoa University High Performance Systems Group at the University of Warwick, UK

Education • • • • •

Simulation-An Enabling Technology in Software Engineering Sabanci University School of Languages Podcasts: Computer Simulation by Prof. David M. Goldsman IMTEK Mathematica Supplement (IMS) (some Mathematica-specific tutorials here) The Creative Learning Exchange McLeod Institute of Simulation Science

Examples • • • • • • • • • • • • •

WARPP Distributed/Parallel System Simulation Toolkit written by the High Performance Systems Group at the University of Warwick A portfolio of free public simulations from the University of Florida Integrated Land Use, Transportation, Environment, (ILUTE) Modeling System Nanorobotics Simulation - Computational Nanomechatronics Lab. at Center for Automation in Nanobiotech (CAN) Online traffic simulation Adaptive Modeler - simulation models for price forecasting of financial markets Shakemovie Caltech's Online Seismic Event Simulation DIG - Demographics, Investment and Company Growth Simulation Global Politics Simulation Industrial & Educational Examples of Modelling & Simulation Generalized online simulation utility Catchment Modelling Toolkit Cellular Automata for Simulation in Games

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Financial Risk Risk Management is the identification, assessment, and prioritization of risks followed by coordinated and economical application of resources to minimize, monitor, and control the probability and/or impact of unfortunate events.. Risks can come from uncertainty in financial markets, project failures, legal liabilities, credit risk, accidents, natural causes and disasters as well as deliberate attacks from an adversary. Several risk management standards have been developed including the Project Management Institute, the National Institute of Science and Technology, actuarial societies, and ISO standards. 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. For the most part, these methodologies consist of the following elements, performed, more or less, in the following order. 1. identify, characterize, and assess threats 2. assess the vulnerability of critical assets to specific threats 3. determine the risk (i.e. the expected consequences of specific types of attacks on specific assets) 4. identify ways to reduce those risks 5. prioritize risk reduction measures based on a strategy The strategies to manage risk include transferring the risk to another party, avoiding the risk, reducing the negative effect of the risk, and accepting some or all of the consequences of a particular risk.

Introduction This section provides an introduction to the principles of risk management. The vocabulary of risk management is defined in ISO Guide 73, "Risk management. Vocabulary". In ideal risk management, a prioritization process is followed whereby the risks with the greatest loss and the greatest probability of occurring are handled first, and risks with lower probability of occurrence and lower loss are handled in descending order. In practice the process can be very difficult, and balancing between risks with a high probability of occurrence but lower loss versus a risk with high loss but lower probability of occurrence can often be mishandled. Intangible risk management identifies a new type of a risk that has a 100% probability of occurring but is ignored by the organization due to a lack of identification ability. For example, when deficient knowledge is applied to a situation, a knowledge risk materialises. Relationship risk appears when ineffective collaboration occurs. Process-engagement risk may be an issue when ineffective operational procedures are applied. These risks directly reduce the

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Computational finance productivity of knowledge workers, decrease cost effectiveness, profitability, service, quality, reputation, brand value, and earnings quality. Intangible risk management allows risk management to create immediate value from the identification and reduction of risks that reduce productivity. Risk management also faces difficulties allocating resources. This is the idea of opportunity cost. Resources spent on risk management could have been spent on more profitable activities. Again, ideal risk management minimizes spending while maximizing the reduction of the negative effects of risks.

Principles of risk management The International Organization for Standardization identifies the following principles of risk management: • • • • • • • • • • •

Risk management should create value. Risk management should be an integral part of organizational processes. Risk management should be part of decision making. Risk management should explicitly address uncertainty. Risk management should be systematic and structured. Risk management should be based on the best available information. Risk management should be tailored. Risk management should take into account human factors. Risk management should be transparent and inclusive. Risk management should be dynamic, iterative and responsive to change. Risk management should be capable of continual improvement and enhancement.

Process According to the standard ISO/DIS 31000 "Risk management -- Principles and guidelines on implementation", the process of risk management consists of several steps as follows:

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Establishing the context Establishing the context involves 1. Identification of risk in a selected domain of interest 2. Planning the remainder of the process. 3. Mapping out the following: o the social scope of risk management o the identity and objectives of stakeholders o the basis upon which risks will be evaluated, constraints. 4. Defining a framework for the activity and an agenda for identification. 5. Developing an analysis of risks involved in the process. 6. Mitigation of risks using available technological, human and organizational resources.

Identification After establishing 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. Hence, risk identification can start with the source of problems, or with the problem itself. •

Source analysis Risk sources may be internal or external to the system that is the target of risk management.

Examples of risk sources are: stakeholders of a project, employees of a company or the weather over an airport. •

Problem analysis Risks are related to identified threats. For example: the threat of losing money, the threat of abuse of privacy information or the threat of accidents and casualties. The threats may exist with various entities, most important with shareholders, customers and legislative bodies such as the government.

When either source or problem 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; privacy information may be stolen by employees even within a closed network; lightning striking a Boeing 747 during takeoff may make all people onboard immediate casualties. The chosen method of identifying risks may depend on culture, industry practice and compliance. The identification methods are formed by templates or the development of templates for identifying source, problem or event. Common risk identification methods are:

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•

•

•

•

Objectives-based risk identification Organizations and project teams have objectives. Any event that may endanger achieving an objective partly or completely is identified as risk. Scenario-based risk identification In scenario analysis different scenarios are created. The scenarios may be the alternative ways to achieve an objective, or an analysis of the interaction of forces in, for example, a market or battle. Any event that triggers an undesired scenario alternative is identified as risk - see Futures Studies for methodology used by Futurists. Taxonomy-based risk identification The taxonomy in taxonomy-based risk identification is a breakdown of possible risk sources. Based on the taxonomy and knowledge of best practices, a questionnaire is compiled. The answers to the questions reveal risks. Taxonomy-based risk identification in software industry can be found in CMU/SEI-93-TR-6. Common-risk checking In several industries lists with known risks are available. Each risk in the list can be checked for application to a particular situation. An example of known risks in the software industry is the Common Vulnerability and Exposures list found at http://cve.mitre.org. Risk charting (risk mapping) This method combines the above approaches by listing Resources at risk, Threats to those resources Modifying Factors which may increase or decrease the risk and Consequences it is wished to avoid. Creating a matrix under these headings enables a variety of approaches. One can begin with resources and consider the threats they are exposed to and the consequences of each. Alternatively one can start with the threats and examine which resources they would affect, or one can begin with the consequences and determine which combination of threats and resources would be involved to bring them about.

Assessment Once risks have been identified, they must then be assessed as to their potential severity of 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 the case of the probability of an unlikely event occurring. Therefore, in the assessment process it is critical to make the best educated guesses possible in order to properly prioritize the implementation of the risk management plan. The fundamental difficulty in risk assessment is determining the rate of occurrence since statistical information is not available on all kinds of past incidents. Furthermore, evaluating the severity of the consequences (impact) is often quite difficult for immaterial assets. Asset valuation is another question that needs to be addressed. Thus, best educated opinions and available statistics are the primary sources of information. Nevertheless, risk assessment should produce such information for the management of the organization that the primary risks are easy to understand and that the risk management decisions may be prioritized.

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Computational finance 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 of occurrence multiplied by the impact of the event equals risk 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 the findings of risk assessments in financial 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 (annualised loss expectancy) and compares the expected loss value to the security control implementation costs (cost-benefit analysis).

Potential risk treatments Once risks have been identified and assessed, all techniques to manage the risk fall into one or more of these four major categories: • • • •

Avoidance (eliminate) Reduction (mitigate) Transfer (outsource or insure) Retention (accept and budget)

Ideal use of these strategies may not be possible. Some of them may involve trade-offs that are not acceptable to the organization or person making the risk management decisions. Another source, from the US Department of Defense, 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. Risk avoidance Includes not performing an activity that could carry risk. An example would be not buying a property or business in order to not take on the liability that comes with it. Another would be not flying in order to not take the risk that the airplane were to be hijacked. Avoidance may seem 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.

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Computational finance Risk reduction Involves methods that reduce 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. Risk management may also take the form of a set policy, such as only allow the use of secured IM platforms (like Brosix) and not allowing personal IM platforms (like AIM) to be used in order to reduce the risk of data leaks. Modern software development methodologies reduce risk by developing and delivering software incrementally. Early methodologies suffered from 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 reduction if the outsourcer can demonstrate higher capability at managing or reducing risks. In this case companies outsource only some of their departmental needs. For example, 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 call center. Risk retention Involves accepting the loss when it occurs. True self insurance falls in this category. Risk retention 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 they either 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 the loss attributed by 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 it would hinder the goals of the organization too much. Risk transfer In the terminology of practitioners and scholars alike, the purchase of an insurance contract is often described as 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

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Computational finance 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 policy holder namely the person who has been in the accident. The insurance policy simply provides that if an accident (the event) occurs involving the policy holder then some compensation may be payable to the policy holder that is commensurate to the suffering/damage. Some ways of managing risk fall into multiple categories. Risk retention pools are technically retaining the risk for the group, but spreading it over 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 up front, but instead losses are assessed to all members of the group.

Create a risk-management plan Select appropriate controls or countermeasures to measure each risk. Risk mitigation needs to be approved by the appropriate level of management. For example, 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 a schedule for control implementation and responsible persons for those actions. 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 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 Follow all of the planned methods for mitigating the effect of the risks. Purchase insurance policies for the risks that have been decided to be transferred to an insurer, avoid all risks that can be avoided without sacrificing the entity's goals, reduce others, and retain the rest.

Review and evaluation of the plan 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 the risks being faced.

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Computational finance Risk analysis results and management plans should be updated periodically. There are two primary reasons for this: 1. to evaluate whether the previously selected security controls are still applicable and effective, and 2. to evaluate the possible risk level changes in the business environment. For example, information risks are a good example of rapidly changing business environment.

Limitations If risks are improperly assessed and prioritized, time can be wasted in dealing with risk of losses that are not likely to occur. Spending too much time assessing and managing unlikely risks can divert resources that could be used more profitably. Unlikely events do occur but if the risk is unlikely enough to occur it may be better to simply retain the risk and deal with the result if the loss does in fact occur. Qualitative risk assessment is subjective and lack consistency. The primary justification for a formal risk assessment process is legal and bureaucratic. Prioritizing too highly the risk management processes could keep an organization from ever completing a project or even getting started. This is especially true if other work is suspended until the risk management process is considered complete. It is also important to keep in mind the distinction between risk and uncertainty. Risk can be measured by impacts x probability.

Areas of risk management As applied to corporate finance, risk management is the technique for measuring, monitoring and controlling the financial or operational risk on a firm's balance sheet. See value at risk. The Basel II framework breaks risks into market risk (price risk), credit risk and operational risk and also specifies methods for calculating capital requirements for each of these components.

Enterprise risk management In enterprise risk management, a risk is defined as a possible event or circumstance that can have negative influences on the enterprise in question. Its impact can be on the very existence, the resources (human and capital), the products and services, or the customers of the enterprise, as well as external impacts on society, markets, or the environment. In a financial institution, enterprise risk management is normally thought of as the combination of credit

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Computational finance risk, interest rate risk or asset liability management, market risk, and operational risk. In the more general case, every probable risk can have a pre-formulated plan to deal with its possible consequences (to ensure contingency if the risk becomes a liability). From the information above and the average cost per employee over time, or cost accrual ratio, a project manager can estimate: •

•

•

the cost associated with the risk if it arises, estimated by multiplying employee costs per unit time by the estimated time lost (cost impact, C where C = cost accrual ratio * S). the probable increase in time associated with a risk (schedule variance due to risk, Rs where Rs = P * S): o Sorting on this value puts the highest risks to the schedule first. This is intended to cause the greatest risks to the project to be attempted first so that risk is minimized as quickly as possible. o This is slightly misleading as schedule variances with a large P and small S and vice versa are not equivalent. (The risk of the RMS Titanic sinking vs. the passengers' meals being served at slightly the wrong time). the probable increase in cost associated with a risk (cost variance due to risk, Rc where Rc = P*C = P*CAR*S = P*S*CAR) o sorting on this value puts the highest risks to the budget first. o see concerns about schedule variance as this is a function of it, as illustrated in the equation above.

Risk in a project or process can be due either to Special Cause Variation or Common Cause Variation and requires appropriate treatment. That is to re-iterate the concern about extremal cases not being equivalent in the list immediately above.

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Risk-management activities as applied to project management • •

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Planning how risk will be managed in the particular project. Plan should include risk management tasks, responsibilities, activities and budget. Assigning a risk officer - a team member other than a project manager who is responsible for foreseeing potential project problems. Typical characteristic of risk officer is a healthy skepticism. Maintaining live project risk database. Each risk should have the following attributes: opening date, title, short description, probability and importance. Optionally a risk may have an assigned person responsible for its resolution and a date by which the risk must be resolved. Creating anonymous risk reporting channel. Each team member should have possibility to report risk that he foresees in the project. Preparing mitigation plans for risks that are chosen to be mitigated. The purpose of the mitigation plan is to describe how this particular risk will be handled – what, when, by who and how will it be done to avoid it or minimize consequences if it becomes a liability. Summarizing planned and faced risks, effectiveness of mitigation activities, and effort spent for the risk management.

Risk management and business continuity Risk management is simply a practice of systematically selecting cost effective approaches for minimising the effect of threat realization to the organization. All risks can never be fully avoided or mitigated simply because of financial and practical limitations. Therefore all organizations have to accept some level of residual risks. Whereas risk management tends to be preemptive, business continuity planning (BCP) was invented to deal with the consequences of realised residual risks. The necessity to have BCP in place arises because even very unlikely events will occur if given enough time. Risk management and BCP are often mistakenly seen as rivals or overlapping practices. In fact these processes are so tightly tied together that such separation seems artificial. For example, the risk management process creates important inputs for the BCP (assets, impact assessments, cost estimates etc). Risk management also proposes applicable controls for the observed risks. Therefore, risk management covers several areas that are vital for the BCP process. However, the BCP process goes beyond risk management's preemptive approach and moves on from the assumption that the disaster will realize at some point.

Risk Communication Risk communication refers to the idea that people are uncomfortable talking about risk. People tend to put off admitting that risk is involved, as well as

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Benefits and Barriers of Risk Communication "Some of the Benefits of risk communication include, improved collective and individual decision making. Both the purpose of the exchange, and the nature of the information have an impact on the benefits. Depending on the situation, personal and community anxieties about environmental health risks can be reduced or increased. For example, a goal might be raising concern about radon and prompting action."

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Areas of application • • • • • • • • • • •

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Active Agenda Benefit risk Business continuity planning Chief Risk Officer Corporate governance Cost overrun Cost risk Critical chain Earned value management Enterprise Risk Management Environmental Risk Management Authority Event chain methodology Financial risk management Futures Studies Hazard prevention

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Hazop • • • • •

Insurance International Risk Governance Council ISDA Legal Risk List of finance topics List of project management topics Managerial risk accounting Megaprojects Megaprojects and risk Occupational Health and Safety Operational risk management Optimism bias Outsourcing Precautionary principle Process Safety Management Project management Public Entity Risk Institute Reference class forecasting Risk

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Risk homeostasis Risk Management Agency Risk Management Authority Risk Management Information Systems Risk Management Research Programme Risk register Roy's safety-first criterion Society for Risk Analysis Timeboxing Social Risk Management Substantial equivalence Uncertainty Value at risk Viable System Model Vulnerability assessment

Risk analysis (engineering)

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Conclusion This topic lays the mathematical foundations for careers in a number of areas in the financial world. In particular, this suitable for novice quantitative analysts and developers who are working in quantitative finance. This topic is also suitable for IT personnel who wish to develop their mathematical skills. Computational finance or financial engineering is a cross-disciplinary field which relies on mathematical finance, numerical methods, computational intelligence and computer simulations to make trading, hedging and investment decisions, as well as facilitating the risk management of those decisions. Utilizing various methods, practitioners of computational finance aim to precisely determine the financial risk that certain financial instruments create. Mathematics

In this part of the topic we discuss a number of concepts and methods that are concerned with variables, functions and transformations defined on finite or infinite discrete sets. In particular, linear algebra will be important because of its role in numerical analysis in general and quantitative finance in particular. We also introduce probability theory and statistics as well as a number of sections on numerical analysis. The latter group is of particular importance when we approximate differential equations using the finite difference method. Numerical Methods

The goal of this part of the topic is to develop robust, efficient and accurate numerical schemes that allow us to produce algorithms in applications. These methods lie at the heart of computational finance and a good understanding of how to use them is vital if you wish to create applications. In general, the methods approximate equations and models defined in a continuous, infinite-dimensional space by models that are defined on a finite-dimensional space.

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