Monte Carlo methods in finance

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Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes. [1] [2] This is usually done by help of stochastic asset models. The advantage of Monte Carlo methods over other techniques increases as the dimensions (sources of uncertainty) of the problem increase.

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.

Corporate finance area of finance dealing with the sources of funding and the capital structure of corporations

Corporate finance is an area of finance that deals with sources of funding, the capital structure of corporations, the actions that managers take to increase the value of the firm to the shareholders, and the tools and analysis used to allocate financial resources. The primary goal of corporate finance is to maximize or increase shareholder value. Although it is in principle different from managerial finance which studies the financial management of all firms, rather than corporations alone, the main concepts in the study of corporate finance are applicable to the financial problems of all kinds of firms.

Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. 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 corresponding value of derivatives of the stock. The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results.

Contents

Monte Carlo methods were first introduced to finance in 1964 by David B. Hertz through his Harvard Business Review article, [3] discussing their application in Corporate Finance. In 1977, Phelim Boyle pioneered the use of simulation in derivative valuation in his seminal Journal of Financial Economics paper. [4]

David Bendel Hertz was an operations research practitioner and academic, known for various contributions to the discipline, and specifically, and more widely, for pioneering the use of Monte Carlo methods in finance. He developed innovative modeling approaches for the solution of complex management issues. His earliest publications added insights to the industrial process of research and development.

<i>Harvard Business Review</i> journal

Harvard Business Review (HBR) is a general management magazine published by Harvard Business Publishing, a wholly owned subsidiary of Harvard University. HBR is published six times a year and is headquartered in Brighton, Massachusetts.

Phelim P. Boyle, is an Irish economist and distinguished professor and actuary, and a pioneer of quantitative finance. He is best known for initiating the use of Monte Carlo methods in option pricing.

This article discusses typical financial problems in which Monte Carlo methods are used. It also touches on the use of so-called "quasi-random" methods such as the use of Sobol sequences.

Sobol sequence

Sobol sequences are an example of quasi-random low-discrepancy sequences. They were first introduced by the Russian mathematician Ilya M. Sobol in 1967.

Overview

The Monte Carlo method encompasses any technique of statistical sampling employed to approximate solutions to quantitative problems. [5] Essentially, the Monte Carlo method solves a problem by directly simulating the underlying (physical) process and then calculating the (average) result of the process. [1] This very general approach is valid in areas such as physics, chemistry, computer science etc.

Physics Study of the fundamental properties of matter and energy

Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Chemistry is the scientific discipline involved with elements and compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they undergo during a reaction with other substances.

Computer science Study of the theoretical foundations of information and computation

Computer science is the study of processes that interact with data and that can be represented as data in the form of programs. It enables the use of algorithms to manipulate, store, and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems.

Computer science is no more about computers than astronomy is about telescopes.

In finance, the Monte Carlo method is used to simulate the various sources of uncertainty that affect the value of the instrument, portfolio or investment in question, and to then calculate a representative value given these possible values of the underlying inputs. [1] ("Covering all conceivable real world contingencies in proportion to their likelihood." [6] ) In terms of financial theory, this, essentially, is an application of risk neutral valuation; [7] see also risk neutrality.

Finance Academic discipline studying businesses and investments

Finance is a field that is concerned with the allocation (investment) of assets and liabilities over space and time, often under conditions of risk or uncertainty. Finance can also be defined as the art of money management. Participants in the market aim to price assets based on their risk level, fundamental value, and their expected rate of return. Finance can be split into three sub-categories: public finance, corporate finance and personal finance.

Financial instrument monetary contract between parties

Financial instruments are monetary contracts between parties. They can be created, traded, modified and settled. They can be cash (currency), evidence of an ownership interest in an entity (share), or a contractual right to receive or deliver cash (bond).

In finance, a portfolio is a collection of investments held by an investment company, hedge fund, financial institution or individual.

Some examples:

Project finance is the long-term financing of infrastructure and industrial projects based upon the projected cash flows of the project rather than the balance sheets of its sponsors. Usually, a project financing structure involves a number of equity investors, known as 'sponsors', a 'syndicate' of banks or other lending institutions that provide loans to the operation. They are most commonly non-recourse loans, which are secured by the project assets and paid entirely from project cash flow, rather than from the general assets or creditworthiness of the project sponsors, a decision in part supported by financial modeling. The financing is typically secured by all of the project assets, including the revenue-producing contracts. Project lenders are given a lien on all of these assets and are able to assume control of a project if the project company has difficulties complying with the loan terms.

A financial analyst, securities analyst, research analyst, equity analyst, investment analyst, or rating analyst is a person who performs financial analysis for external or internal financial clients as a core part of the job.

Stochastic refers to a randomly determined process. The word first appeared in English to describe a mathematical object called a stochastic process, but now in mathematics the terms stochastic process and random process are considered interchangeable. The word, with its current definition meaning random, came from German, but it originally came from Greek στόχος (stókhos), meaning 'aim, guess'.

Although Monte Carlo methods provide flexibility, and can handle multiple sources of uncertainty, the use of these techniques is nevertheless not always appropriate. In general, simulation methods are preferred to other valuation techniques only when there are several state variables (i.e. several sources of uncertainty). [1] These techniques are also of limited use in valuing American style derivatives. See below.

Applicability

Level of complexity

Many problems in mathematical finance entail the computation of a particular integral (for instance the problem of finding the arbitrage-free value of a particular derivative). In many cases these integrals can be valued analytically, and in still more cases they can be valued using numerical integration, or computed using a partial differential equation (PDE). However, when the number of dimensions (or degrees of freedom) in the problem is large, PDEs and numerical integrals become intractable, and in these cases Monte Carlo methods often give better results.

For more than three or four state variables, formulae such as Black–Scholes (i.e. analytic solutions) do not exist, while other numerical methods such as the Binomial options pricing model and finite difference methods face several difficulties and are not practical. In these cases, Monte Carlo methods converge to the solution more quickly than numerical methods, require less memory and are easier to program. For simpler situations, however, simulation is not the better solution because it is very time-consuming and computationally intensive.

Monte Carlo methods can deal with derivatives which have path dependent payoffs in a fairly straightforward manner. On the other hand, Finite Difference (PDE) solvers struggle with path dependence.

American options

Monte-Carlo methods are harder to use with American options. This is because, in contrast to a partial differential equation, the Monte Carlo method really only estimates the option value assuming a given starting point and time.

However, for early exercise, we would also need to know the option value at the intermediate times between the simulation start time and the option expiry time. In the Black–Scholes PDE approach these prices are easily obtained, because the simulation runs backwards from the expiry date. In Monte-Carlo this information is harder to obtain, but it can be done for example using the least squares algorithm of Carriere (see link to original paper) which was made popular a few years later by Longstaff and Schwartz (see link to original paper).

Monte Carlo methods

Mathematically

The fundamental theorem of arbitrage-free pricing states that the value of a derivative is equal to the discounted expected value of the derivative payoff where the expectation is taken under the risk-neutral measure [1]. An expectation is, in the language of pure mathematics, simply an integral with respect to the measure. Monte Carlo methods are ideally suited to evaluating difficult integrals (see also Monte Carlo method).

Thus if we suppose that our risk-neutral probability space is and that we have a derivative H that depends on a set of underlying instruments . Then given a sample from the probability space the value of the derivative is . Today's value of the derivative is found by taking the expectation over all possible samples and discounting at the risk-free rate. I.e. the derivative has value:

where is the discount factor corresponding to the risk-free rate to the final maturity date T years into the future.

Now suppose the integral is hard to compute. We can approximate the integral by generating sample paths and then taking an average. Suppose we generate N samples then

which is much easier to compute.

Sample paths for standard models

In finance, underlying random variables (such as an underlying stock price) are usually assumed to follow a path that is a function of a Brownian motion 2. For example, in the standard Black–Scholes model, the stock price evolves as

To sample a path following this distribution from time 0 to T, we chop the time interval into M units of length , and approximate the Brownian motion over the interval by a single normal variable of mean 0 and variance . This leads to a sample path of

for each k between 1 and M. Here each is a draw from a standard normal distribution.

Let us suppose that a derivative H pays the average value of S between 0 and T then a sample path corresponds to a set and

We obtain the Monte-Carlo value of this derivative by generating N lots of M normal variables, creating N sample paths and so N values of H, and then taking the average. Commonly the derivative will depend on two or more (possibly correlated) underlyings. The method here can be extended to generate sample paths of several variables, where the normal variables building up the sample paths are appropriately correlated.

It follows from the central limit theorem that quadrupling the number of sample paths approximately halves the error in the simulated price (i.e. the error has order convergence in the sense of standard deviation of the solution).

In practice Monte Carlo methods are used for European-style derivatives involving at least three variables (more direct methods involving numerical integration can usually be used for those problems with only one or two underlyings. See Monte Carlo option model.

Greeks

Estimates for the "Greeks" of an option i.e. the (mathematical) derivatives of option value with respect to input parameters, can be obtained by numerical differentiation. This can be a time-consuming process (an entire Monte Carlo run must be performed for each "bump" or small change in input parameters). Further, taking numerical derivatives tends to emphasize the error (or noise) in the Monte Carlo value - making it necessary to simulate with a large number of sample paths. Practitioners regard these points as a key problem with using Monte Carlo methods.

Variance reduction

Square root convergence is slow, and so using the naive approach described above requires using a very large number of sample paths (1 million, say, for a typical problem) in order to obtain an accurate result. Remember that an estimator for the price of a derivative is a random variable, and in the framework of a risk-management activity, uncertainty on the price of a portfolio of derivatives and/or on its risks can lead to suboptimal risk-management decisions.

This state of affairs can be mitigated by variance reduction techniques.

Antithetic paths

A simple technique is, for every sample path obtained, to take its antithetic path that is given a path to also take . Since the variables and form an antithetic pair, a large value of one is accompanied by a small value of the other. This suggests that an unusually large or small output computed from the first path may be balanced by the value computed from the antithetic path, resulting in a reduction in variance. [23] Not only does this reduce the number of normal samples to be taken to generate N paths, but also, under same conditions, such as negative correlation between two estimates, reduces the variance of the sample paths, improving the accuracy.

Control variate method

It is also natural to use a control variate. Let us suppose that we wish to obtain the Monte Carlo value of a derivative H, but know the value analytically of a similar derivative I. Then H* = (Value of H according to Monte Carlo) + B*[(Value of I analytically) (Value of I according to same Monte Carlo paths)] is a better estimate, where B is covar(H,I)/var(H).

The intuition behind that technique, when applied to derivatives, is the following: note that the source of the variance of a derivative will be directly dependent on the risks (e.g. delta, vega) of this derivative. This is because any error on, say, the estimator for the forward value of an underlier, will generate a corresponding error depending on the delta of the derivative with respect to this forward value. The simplest example to demonstrate this consists in comparing the error when pricing an at-the-money call and an at-the-money straddle (i.e. call+put), which has a much lower delta.

Therefore, a standard way of choosing the derivative I consists in choosing a replicating portfolios of options for H. In practice, one will price H without variance reduction, calculate deltas and vegas, and then use a combination of calls and puts that have the same deltas and vegas as control variate.

Importance sampling

Importance sampling consists of simulating the Monte Carlo paths using a different probability distribution (also known as a change of measure) that will give more likelihood for the simulated underlier to be located in the area where the derivative's payoff has the most convexity (for example, close to the strike in the case of a simple option). The simulated payoffs are then not simply averaged as in the case of a simple Monte Carlo, but are first multiplied by the likelihood ratio between the modified probability distribution and the original one (which is obtained by analytical formulas specific for the probability distribution). This will ensure that paths whose probability have been arbitrarily enhanced by the change of probability distribution are weighted with a low weight (this is how the variance gets reduced).

This technique can be particularly useful when calculating risks on a derivative. When calculating the delta using a Monte Carlo method, the most straightforward way is the black-box technique consisting in doing a Monte Carlo on the original market data and another one on the changed market data, and calculate the risk by doing the difference. Instead, the importance sampling method consists in doing a Monte Carlo in an arbitrary reference market data (ideally one in which the variance is as low as possible), and calculate the prices using the weight-changing technique described above. This results in a risk that will be much more stable than the one obtained through the black-box approach.

Quasi-random (low-discrepancy) methods

Instead of generating sample paths randomly, it is possible to systematically (and in fact completely deterministically, despite the "quasi-random" in the name) select points in a probability spaces so as to optimally "fill up" the space. The selection of points is a low-discrepancy sequence such as a Sobol sequence. Taking averages of derivative payoffs at points in a low-discrepancy sequence is often more efficient than taking averages of payoffs at random points.

Notes

  1. Frequently it is more practical to take expectations under different measures, however these are still fundamentally integrals, and so the same approach can be applied.
  2. More general processes, such as Lévy processes, are also sometimes used. These may also be simulated.

See also

Related Research Articles

The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return. The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is widely used, although often with some adjustments, by options market participants.

In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting.

In mathematical finance, a risk-neutral measure is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free.

Rational pricing is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features. The first application to option pricing was by Phelim Boyle in 1977. In 1996, M. Broadie and P. Glasserman showed how to price Asian options by Monte Carlo. In 2001 F. A. Longstaff and E. S. Schwartz developed a practical Monte Carlo method for pricing American-style options.

In finance, delta neutral describes a portfolio of related financial securities, in which the portfolio value remains unchanged when small changes occur in the value of the underlying security. Such a portfolio typically contains options and their corresponding underlying securities such that positive and negative delta components offset, resulting in the portfolio's value being relatively insensitive to changes in the value of the underlying security.

Generalized least squares

In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordinary least squares and weighted least squares can be statistically inefficient, or even give misleading inferences. GLS was first described by Alexander Aitken in 1934.

Martingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. options, futures, interest rate derivatives, credit derivatives, etc.

Uncertainty quantification (UQ) is the science of quantitative characterization and reduction of uncertainties in both computational and real world applications. It tries to determine how likely certain outcomes are if some aspects of the system are not exactly known. An example would be to predict the acceleration of a human body in a head-on crash with another car: even if we exactly knew the speed, small differences in the manufacturing of individual cars, how tightly every bolt has been tightened, etc., will lead to different results that can only be predicted in a statistical sense.

The following outline is provided as an overview of and topical guide to finance:

In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especially in the interest rate derivative markets. It was developed by Patrick S. Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward.

Option (finance) Right to buy or sell a certain thing at a later date at an agreed price

In finance, an option is a contract which gives the buyer the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price prior to or on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed.

The topological derivative is, conceptually, a derivative of a shape functional with respect to infinitesimal changes in its topology, such as adding an infinitesimal hole or crack. When used in higher dimensions than one, the term topological gradient is also used to name the first-order term of the topological asymptotic expansion, dealing only with infinitesimal singular domain perturbations. It has applications in shape optimization, topology optimization, image processing and mechanical modeling.

High-dimensional integrals in hundreds or thousands of variables occur commonly in finance. These integrals have to be computed numerically to within a threshold . If the integral is of dimension then in the worst case, where one has a guarantee of error at most , the computational complexity is typically of order . That is, the problem suffers the curse of dimensionality. In 1977 P. Boyle, University of Waterloo, proposed using Monte Carlo (MC) to evaluate options. Starting in early 1992, J. F. Traub, Columbia University, and a graduate student at the time, S. Paskov, used quasi-Monte Carlo (QMC) to price a Collateralized mortgage obligation with parameters specified by Goldman Sachs. Even though it was believed by the world's leading experts that QMC should not be used for high-dimensional integration, Paskov and Traub found that QMC beat MC by one to three orders of magnitude and also enjoyed other desirable attributes. Their results were first published in 1995. Today QMC is widely used in the financial sector to value financial derivatives; see list of books below.

In statistics, the antithetic variates method is a variance reduction technique used in Monte Carlo methods. Considering that the error reduction in the simulated signal has a square root convergence, a very large number of sample paths is required to obtain an accurate result. The antithetic variates method reduces the variance of the simulation results.

Credit valuation adjustment (CVA) is the difference between the risk-free portfolio value and the true portfolio value that takes into account the possibility of a counterparty’s default. In other words, CVA is the market value of counterparty credit risk. This price depends on counterparty credit spreads as well as on the market risk factors that drive derivatives’ values and, therefore, exposure. CVA is one of a family of related valuation adjustments, collectively xVA; for further context here see Financial economics #Derivative pricing.

Biology Monte Carlo methods (BioMOCA) have been developed at the University of Illinois at Urbana-Champaign to simulate ion transport in an electrolyte environment through ion channels or nano-pores embedded in membranes. It is a 3-D particle-based Monte Carlo simulator for analyzing and studying the ion transport problem in ion channel systems or similar nanopores in wet/biological environments. The system simulated consists of a protein forming an ion channel (or an artificial nanopores like a Carbon Nano Tube, CNT), with a membrane (i.e. lipid bilayer) that separates two ion baths on either side. BioMOCA is based on two methodologies, namely the Boltzmann transport Monte Carlo (BTMC) and particle-particle-particle-mesh (P3M). The first one uses Monte Carlo method to solve the Boltzmann equation, while the later splits the electrostatic forces into short-range and long-range components.

The Datar–Mathews method is a method for real options valuation. The method provides an easy way to determine the real option value of a project simply by using the average of positive outcomes for the project. The method can be understood as an extension of the net present value (NPV) multi-scenario Monte Carlo model with an adjustment for risk aversion and economic decision-making. The method uses information that arises naturally in a standard discounted cash flow (DCF), or NPV, project financial valuation. It was created in 2000 by Professor Vinay Datar, Seattle University, and Scott H. Mathews, Technical Fellow, The Boeing Company.

In mathematics, the walk-on-spheres method (WoS) is a numerical probabilistic algorithm, or Monte-Carlo method, used mainly in order to approximate the solutions of some specific boundary value problem for partial differential equations (PDEs). The WoS method was first introduced by Mervin E. Muller in 1956 to solve Laplace's equation, and was since then generalized to other problems.

References

Notes

  1. 1 2 3 4 5 "Real Options with Monte Carlo Simulation" . Retrieved 2010-09-24.
  2. "Monte Carlo Simulation". Palisade Corporation. 2010. Retrieved 2010-09-24.
  3. "Risk Analysis in Capital Investment". Harvard Business Review. Sep 1, 1979. p. 12. Retrieved 2010-09-24.
  4. Boyle, Phelim P. (1977). "Options: A Monte Carlo approach". Journal of Financial Economics. Journal of Financial Economics, Volume (Year): 4 (1977), Issue (Month): 3 (May). 4 (3): 323–338. doi:10.1016/0304-405X(77)90005-8 . Retrieved 2010-09-24.
  5. "Monte Carlo Simulation : Financial Mathematics Glossary K-O". Global Derivatives. 2009. Retrieved 2010-09-24.
  6. The Flaw of Averages Archived 2011-12-07 at the Wayback Machine , Prof. Sam Savage, Stanford University.
  7. "FAQ Number 4 : Does Risk-Neutral Valuation Mean that Investors Are Risk-Neutral? What Is the Difference Between Real Simulation and Risk-Neutral Simulation?". Archived from the original on 2010-07-16. Retrieved 2010-09-24.
  8. 1 2 Savvakis C. Savvides, Cyprus Development Bank - Project Financing Division (1994). "Risk Analysis in Investment Appraisal". Project Appraisal Journal, Vol. 9, No. 1, March 1994. SSRN   265905 .
  9. David Shimko, President, Asset Deployment, USA. "Quantifying Corporate Financial Risk". qfinance.com. Retrieved 2011-01-14.CS1 maint: Multiple names: authors list (link)
  10. 1 2 Marius Holtan; Onward Inc. (2002-05-31). "Using simulation to calculate the NPV of a project" (PDF). Retrieved 2010-09-24.
  11. "Introduction".
  12. TEACHING NOTE 96-03: MONTE CARLO SIMULATION
  13. Peter Carr; Guang Yang (February 26, 1998). "Simulating American Bond Options in an HJM Framework" (PDF). Retrieved 2010-09-24.
  14. Carlos Blanco, Josh Gray & Marc Hazzard. "Alternative Valuation Methods for Swaptions: The Devil is in the Details" (PDF). Retrieved 2010-09-24.
  15. Ammann, Manuel; Kind, Axel; Wilde, Christian (2007). "Simulation-Based Pricing of Convertible Bonds" (PDF). Journal of Empirical Finance. doi:10.2139/ssrn.762804.
  16. Frank J. Fabozzi: Valuation of fixed income securities and derivatives, pg. 138
  17. Donald R. van Deventer (Kamakura Corporation): Pitfalls in Asset and Liability Management: One Factor Term Structure Models
  18. Martin Haugh (Fall 2004). "The Monte Carlo Framework, Examples from Finance and Generating Correlated Random Variables" (PDF). Archived from the original (PDF) on 2012-01-05. Retrieved 2010-09-24.
  19. "Monte Carlo Value-at-Risk". Contingency Analysis. 2004. Retrieved 2010-09-24.
  20. David Harper,CFA, FRM. "An Introduction To Value at Risk (VAR)". Investopedia. Retrieved 2010-09-24.CS1 maint: Multiple names: authors list (link)
  21. Christopher Farrell (January 22, 2001). "A Better Way to Size Up Your Nest Egg : Monte Carlo models simulate all kinds of scenarios". Bloomberg Businessweek. Retrieved 2010-09-24.
  22. John Norstad (February 2, 2005). "Financial Planning Using Random Walks" (PDF). Retrieved 2010-09-24.
  23. Glasserman, P. (2004). Monte Carlo methods in financial engineering. New York: Springer. p. 205.

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