 Monte Carlo methods in finance

Monte Carlo methods are used in 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 their average value over the range of resultant outcomes.^{[1]}^{[2]} The advantage of Monte Carlo methods over other techniques increases as the dimensions (sources of uncertainty) of the problem increase.
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]}
This article discusses typical financial problems in which Monte Carlo methods are used. It also touches on the use of socalled "quasirandom" methods such as the use of Sobol sequences.
Contents
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.
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]} In terms of financial theory, this, essentially, is an application of risk neutral valuation;^{[6]} see also risk neutrality.
Some examples:
 In Corporate Finance,^{[7]}^{[8]}^{[9]} project finance ^{[7]} and real options analysis,^{[1]} Monte Carlo Methods are used by financial analysts who wish to construct "stochastic" or probabilistic financial models as opposed to the traditional static and deterministic models. Here, in order to analyze the characteristics of a project’s net present value (NPV), the cash flow components that are (heavily ^{[9]}) impacted by uncertainty are modeled, incorporating any correlation, mathematically reflecting their "random characteristics". Then, these results are combined in a histogram of NPV (i.e. the project’s probability distribution), and the average NPV of the potential investment  as well as its volatility and other sensitivities  is observed. This distribution allows, for example, for an estimate of the probability that the project has a net present value greater than zero (or any other value).^{[10]} See further under Corporate finance.
 In valuing an option on equity, the simulation generates several thousand possible (but random) price paths for the underlying share, with the associated exercise value (i.e. "payoff") of the option for each path. These payoffs are then averaged and discounted to today, and this result is the value of the option today;^{[11]} see Monte Carlo methods for option pricing for discussion as to further  and more complex  option modelling.
 To value bonds, and bond options,^{[12]} the underlying source of uncertainty which is simulated is the short rate  the annualized interest rate at which an entity can borrow money for a given period of time; see Shortrate model. For each possible evolution of interest rates we observe a different yield curve and a different resultant bond price. To determine the bond value, these bond prices are then averaged; to value the bond option, as for equity options, the corresponding exercise values are averaged and present valued. A similar approach is used in valuing swaps and swaptions.^{[13]}
 Monte Carlo Methods are used for portfolio evaluation.^{[14]} Here, for each sample, the correlated behaviour of the factors impacting the component instruments is simulated over time, the resultant value of each instrument is calculated, and the portfolio value is then observed. As for corporate finance, above, the various portfolio values are then combined in a histogram, and the statistical characteristics of the portfolio are observed, and the portfolio assessed as required. A similar approach is used in calculating value at risk.^{[15]}^{[16]}
 Monte Carlo Methods are used for personal financial planning.^{[17]}^{[18]} For instance, by simulating the overall market, the chances of a 401(k) allowing for retirement on a target income can be calculated. As appropriate, the worker in question can then take greater risks with the retirement portfolio or start saving more money.
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 arbitragefree 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 timeconsuming and computationally intensive.
Monte Carlo methods can deal with derivatives which have path dependent payoffs in a fairly straight forward manner. On the other hand Finite Difference (PDE) solvers struggle with path dependence.
American options
MonteCarlo 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 BlackScholes PDE approach these prices are easily obtained, because the simulation runs backwards from the expiry date. In MonteCarlo 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).
Stochastic Grid Method (SGM) proposed by Jain and Oosterlee provides a general framework for pricing high dimensional American options (see link to original paper).
Monte Carlo methods
Mathematically
The fundamental theorem of arbitragefree 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 riskneutral 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 riskneutral probability space is and that we have a derivative H that depends on a set of underlying instruments S_{1},...,S_{n}. Then given a sample ω 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 riskfree rate. I.e. the derivative has value:
where DF_{T} is the discount factor corresponding to the riskfree 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 BlackScholes 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 δt, and approximate the Brownian motion over the interval dt by a single normal variable of mean 0 and variance δt. This leads to a sample path of
for each k between 1 and M. Here each ε_{i} 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 ω corresponds to a set and
We obtain the MonteCarlo 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 Europeanstyle 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 timeconsuming 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 riskmanagement activity, uncertainty on the price of a portfolio of derivatives and/or on its risks can lead to suboptimal riskmanagement 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 . Not only does this reduce the number of normal samples to be taken to generate N paths, but also, under same conditions, 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 atthemoney call and an atthemoney 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 blackbox 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 weightchanging technique described above. This results in a risk that will be much more stable than the one obtained through the blackbox approach.
Quasirandom (lowdiscrepancy) methods
Main article: QuasiMonte Carlo methods in financeInstead of generating sample paths randomly, it is possible to systematically (and in fact completely deterministically, despite the "quasirandom" in the name) select points in a probability spaces so as to optimally "fill up" the space. The selection of points is a lowdiscrepancy sequence such as a Sobol sequence. Taking averages of derivative payoffs at points in a lowdiscrepancy sequence is often more efficient than taking averages of payoffs at random points.
Notes
 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.
 More general processes, such as Lévy processes, are also sometimes used. These may also be simulated.
See also
References
Notes
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} "Real Options with Monte Carlo Simulation". http://www.pucrio.br/marco.ind/montecarlo.html. Retrieved 20100924.
 ^ "Monte Carlo Simulation". Palisade Corporation. 2010. http://www.palisade.com/risk/monte_carlo_simulation.asp. Retrieved 20100924.
 ^ "Risk Analysis in Capital Investment". Harvard Business Review. date: Sep 01, 1979. pp. 12. http://hbr.org/product/riskanalysisincapitalinvestmentharvardbusine/an/79504PDFENG. Retrieved 20100924.
 ^ Boyle, Phelim P.. "Options: A Monte Carlo approach". Journal of Financial Economics, Volume (Year): 4 (1977), Issue (Month): 3 (May). pp. 323–338. http://ideas.repec.org/a/eee/jfinec/v4y1977i3p323338.html. Retrieved 20100924.
 ^ "Monte Carlo Simulation : Financial Mathematics Glossary KO". Global Derivatives. 2009. http://www.globalderivatives.com/index.php?option=com_content&task=view&id=21. Retrieved 20100924.
 ^ "FAQ Number 4 : Does RiskNeutral Valuation Mean that Investors Are RiskNeutral? What Is the Difference Between Real Simulation and RiskNeutral Simulation?". http://www.pucrio.br/marco.ind/faq4.html. Retrieved 20100924.
 ^ ^{a} ^{b} Savvakis C. Savvides, Cyprus Development Bank  Project Financing Division. "Risk Analysis in Investment Appraisal". Project Appraisal Journal, Vol. 9, No. 1, March 1994. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=265905. Retrieved 20100924.
 ^ David Shimko, President, Asset Deployment, USA. "Quantifying Corporate Financial Risk". qfinance.com. http://www.qfinance.com/financialriskmanagementbestpractice/quantifyingcorporatefinancialrisk?full. Retrieved 20110114.
 ^ ^{a} ^{b} Marius Holtan, Onward Inc. (20020531). "Using simulation to calculate the NPV of a project". http://www.investmentscience.com/Content/howtoArticles/simulation.pdf. Retrieved 20100924.
 ^ [1]
 ^ [2]^{[dead link]}
 ^ Peter Carr, Guang Yang (February 26, 1998). "Simulating American Bond Options in an HJM Framework". http://www.math.nyu.edu/research/carrp/papers/pdf/hjm.pdf. Retrieved 20100924.
 ^ Carlos Blanco, Josh Gray and Marc Hazzard. "Alternative Valuation Methods for Swaptions: The Devil is in the Details". http://www.fea.com/resources/pdf/swaptions.pdf. Retrieved 20100924.
 ^ Martin Haugh (Fall 2004). "The Monte Carlo Framework, Examples from Finance and Generating Correlated Random Variables". http://www.columbia.edu/~mh2078/MCS04/MCS_framework_FEegs.pdf. Retrieved 20100924.
 ^ "Monte Carlo ValueatRisk". Contingency Analysis. 2004. http://www.riskglossary.com/link/monte_carlo_transformation.htm. Retrieved 20100924.
 ^ David Harper,CFA, FRM. "An Introduction To Value at Risk (VAR)". Investopedia. http://www.investopedia.com/articles/04/092904.asp. Retrieved 20100924.
 ^ Christopher Farrell (January 22, 2001). "A Better Way to Size Up Your Nest Egg : Monte Carlo models simulate all kinds of scenarios". Bloomberg Businessweek. http://www.businessweek.com/2001/01_04/b3716156.htm. Retrieved 20100924.
 ^ John Norstad (February 2, 2005). "Financial Planning Using Random Walks". http://homepage.mac.com/j.norstad/finance/finplan.pdf. Retrieved 20100924.
Articles
 Boyle, P., Broadie, M. and Glasserman, P. Monte Carlo Methods for Security Pricing. Journal of Economic Dynamics and Control, Volume 21, Issues 89, Pages 12671321
 Rubinstein, Samorodnitsky, Shaked. Antithetic Variates, Multivariate Dependence and Simulation of Stochastic Systems. Managemen Science, Vol. 31, No. 1, Jan 1985, pages 66–67
Books
 Peter E. Kloeden and Eckhard Platen (1992). Numerical Solution of Stochastic Differential Equations. Springer  Verlag.
 Bruno Dupire (1998). Monte Carlo:methodologies and applications for pricing and risk management. Risk.
 John C. Hull (2000). Options, futures and other derivatives (4th ed.). Prentice Hall. ISBN 0130158224.
 Damiano Brigo, Fabio Mercurio (2001). Interest Rate Models  Theory and Practice with Smile, Inflation and Credit (2nd ed. 2006 ed.). Springer Verlag. ISBN 9783540221494.
 Peter Jäckel (2002). Monte Carlo methods in finance. John Wiley and Sons. ISBN 047149741X.
 Paul Glasserman (2003). Monte Carlo methods in financial engineering. SpringerVerlag. ISBN 0387004513.
 Daniel J. Duffy and Joerg Kienitz (2009). Monte Carlo Frameworks: Building Customisable Highperformance C++ Applications. Wiley. ISBN 0470060698.
Software
 Fairmat (freeware) modeling and pricing complex options
 MG Soft, (freeware) valuation and Greeks of vanilla and exotic options
 SimulAr Free Monte Carlo simulation Excel Addin
External links
General
 MonteCarlo Simulation in Finance, globalderivatives.com
 Monte Carlo Method, riskglossary.com
 The Monte Carlo Framework, Examples from Finance, Martin Haugh, Columbia University
 Monte Carlo techniques applied to finance, Simon Leger
Derivative valuation
 Monte Carlo Simulation, Prof. Don M. Chance, Louisiana State University
 Option pricing by simulation, Bernt Arne Ødegaard, Norwegian School of Management
 Applications of Monte Carlo Methods in Finance: Option Pricing, Y. Lai and J. Spanier, Claremont Graduate University
 Monte Carlo Derivative valuation, contd., Timothy L. Krehbiel, Oklahoma State University–Stillwater
 Pricing complex options using a simple Monte Carlo Simulation, Peter Fink  reprint at quantnotes.com
 LeastSquares MonteCarlo for American options by Carriere, 1996, ideas.repec.org
 LeastSquares MonteCarlo for American options by Longstaff and Schwartz, 2001, repositories.cdlib.org
 Using simulation for option pricing, John Charnes
 Stochastic Grid Method for Pricing HighDimensional American Options, S. Jain and C.W. Oosterlee
Corporate Finance
 Real Options with Monte Carlo Simulation, Marco Dias, Pontifícia Universidade Católica do Rio de Janeiro
 Using simulation to calculate the NPV of a project, investmentscience.com
 Simulations, Decision Trees and Scenario Analysis in Valuation Prof. Aswath Damodaran, Stern School of Business
 The Monte Carlo method in Excel Prof. André Farber Solvay Business School
 Sales Forecasting, vertex42.com
 Pricing using Monte Carlo simulation, a practical example, Prof. Giancarlo Vercellino
Value at Risk and portfolio analysis
 Monte Carlo ValueatRisk, riskglossary.com
Personal finance
 A Better Way to Size Up Your Nest Egg, Businessweek Online: January 22, 2001
 Online Monte Carlo retirement planner with source code, Jim Richmond, 2006
 Free spreadsheetbased retirement calculator and Monte Carlo simulator, by Eric C., 2008
 Financial Planning Using Random Walks, John Norstad, 2005
 Retirement Calculator by VestingPoint.com
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