Editing Monte Carlo methods in finance (section)

==Overview==
The [[Monte Carlo method]] encompasses any technique of statistical sampling employed to approximate solutions to quantitative problems.<ref>{{cite web | title = Monte Carlo Simulation : Financial Mathematics Glossary K-O | url = http://www.global-derivatives.com/index.php?option=com_content&task=view&id=21 | publisher = Global Derivatives | year = 2009 | access-date = 2010-09-24}}</ref>  Essentially, the Monte Carlo method solves a problem by directly [[Simulating#Computer simulation|simulating]] the underlying (physical) process and then calculating the (average) result of the process.<ref name="puc"/> 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 [[financial instrument|instrument]], [[portfolio (finance)|portfolio]] or [[investment]] in question, and to then calculate a representative value given these possible values of the underlying inputs.<ref name="puc"/> ("Covering all conceivable real world contingencies in proportion to their likelihood."<ref name="savage">[http://www.analycorp.com/uncertainty/flawarticle.htm The Flaw of Averages] {{webarchive|url=https://web.archive.org/web/20111207025740/http://www.analycorp.com/uncertainty/flawarticle.htm |date=2011-12-07 }}, Prof. Sam Savage, [[Stanford University]].</ref>) In terms of [[financial economics|financial theory]], this, essentially, is an application of  [[Rational pricing#Risk neutral valuation|risk neutral valuation]];<ref name="puc2">{{cite web | title = FAQ Number 4 : Does Risk-Neutral Valuation Mean that Investors Are Risk-Neutral? What Is the Difference Between Real Simulation and Risk-Neutral Simulation? | url = http://www.puc-rio.br/marco.ind/faq4.html | access-date = 2010-09-24 | url-status = dead | archive-url = https://web.archive.org/web/20100716195259/http://www.puc-rio.br/marco.ind/faq4.html | archive-date = 2010-07-16 }}</ref> see also [[risk neutrality]].

Applications:

* In [[Corporate Finance]],<ref name="Savvides">{{cite journal | title = Risk Analysis in Investment Appraisal | author = Savvakis C. Savvides, Cyprus Development Bank - Project Financing Division | ssrn =  265905 | publisher = Project Appraisal Journal, Vol. 9, No. 1, March 1994 | year = 1994 }}</ref><ref name=" Shimko ">{{cite web | title = Quantifying Corporate Financial Risk | author = David Shimko, President, Asset Deployment, USA | url = http://www.qfinance.com/financial-risk-management-best-practice/quantifying-corporate-financial-risk?full | publisher = qfinance.com | access-date = 2011-01-14 | archive-url = https://web.archive.org/web/20100717072252/http://www.qfinance.com/financial-risk-management-best-practice/quantifying-corporate-financial-risk?full | archive-date = 2010-07-17 | url-status = dead }}</ref><ref name="Holtan">{{cite web | title = Using simulation to calculate the NPV of a project  |author1=Marius Holtan |author2=Onward Inc. | url =  http://www.investmentscience.com/Content/howtoArticles/simulation.pdf | date = 2002-05-31 | access-date = 2010-09-24 }}</ref> [[project finance]]<ref name="Savvides"/> and [[real options analysis]],<ref name="puc"/> Monte Carlo Methods are used by [[financial analyst]]s who wish to construct "[[stochastic]]" or [[probabilistic]] financial models as opposed to the traditional static and [[Deterministic system (mathematics)|deterministic]] models. Here, in order to analyze the characteristics of a project’s [[net present value]] (NPV), the cash flow components that are (heavily<ref name="Holtan"/>) impacted by [[uncertainty]] are modeled, incorporating any [[correlation]] between these, 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 (finance)|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).<ref>{{Cite web | url=http://www.simularsoft.com.ar/SimulAr1e.htm | title=Introduction}}</ref> See [[Corporate finance#Quantifying uncertainty|further]] under Corporate finance.

* In valuing an [[option (finance)|option on equity]], the simulation generates several thousand possible (but random) price paths for the underlying share, with the associated [[Exercise (options)|exercise]] [[Option time value#Intrinsic value|value]] (i.e. "payoff") of the option for each path. These payoffs are then averaged and [[present value|discounted]] to today, and this result is the value of the option today.<ref>[http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-03.pdf TEACHING NOTE 96-03: MONTE CARLO SIMULATION] [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-03.pdf]</ref> Note that whereas equity options are more commonly valued using other [[Valuation_of_options#Pricing_models|pricing models]] such as [[Lattice model (finance)|lattice based models]], for [[Path dependence|path dependent]] [[exotic derivatives]] – such as [[Asian options]] – simulation is the valuation method most commonly employed; see [[Monte Carlo methods for option pricing]] for discussion as to further – and more [[exotic option|complex]] – option modelling.

* To value [[fixed income|fixed income instruments]] and [[interest rate derivatives]] the underlying source of uncertainty which is simulated is the [[Short-rate model#The short rate|short rate]] – the annualized [[interest rate]] at which an entity can borrow money for a given period of time;  see [[Short-rate model#Particular short-rate models|Short-rate model]]. For example, for [[bond (finance)|bonds]], and [[bond option]]s,<ref name="hjm">{{cite web | title = Simulating American Bond Options in an HJM Framework |author1=Peter Carr |author2=Guang Yang | url = http://www.math.nyu.edu/research/carrp/papers/pdf/hjm.pdf | date = February 26, 1998 | access-date = 2010-09-24 }}</ref> under each possible evolution of [[interest rate]]s we observe a different [[yield curve]] and a different [[Bond valuation#Arbitrage-free pricing approach|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 (options)|exercise value]]s are averaged and present valued. A similar approach is used in valuing [[swap (finance)|swaps]], [[swaption]]s,<ref name="Swaptions">{{cite web | title = Alternative Valuation Methods for Swaptions: The Devil is in the Details | author1 = Carlos Blanco, Josh Gray | author2 = Marc Hazzard | name-list-style = amp | url = http://www.fea.com/resources/pdf/swaptions.pdf | access-date = 2010-09-24 | archive-url = https://web.archive.org/web/20071202214825/http://www.fea.com/resources/pdf/swaptions.pdf | archive-date = 2007-12-02 | url-status = dead }}</ref> and [[convertible bond]]s.<ref>{{cite journal |last1=Ammann |first1=Manuel |last2=Kind |first2=Axel |last3=Wilde |first3=Christian |title=Simulation-Based Pricing of Convertible Bonds |journal=Journal of Empirical Finance |date=2007 |doi=10.2139/ssrn.762804 |s2cid=18764314 |url=https://www.alexandria.unisg.ch/38377/1/PubsAmmann2007_Convertibles_JEF.pdf }}</ref>  As for equity, for path dependent [[interest rate derivative]]s – such as [[Collateralized mortgage obligation|CMOs]] – simulation is the ''primary'' technique employed;<ref>[[Frank J. Fabozzi]]: [https://books.google.com/books?id=wF8yVzLI6EYC&dq=cmo+valuation+fabozzi+simulation&pg=PA138 ''Valuation of fixed income securities and derivatives'', pg. 138]</ref> (Note that "to create realistic interest rate simulations" [[Short-rate model#Multi-factor short-rate models|Multi-factor short-rate models]] are sometimes employed.<ref>Donald R. van Deventer (Kamakura Corporation): [http://www.kamakuraco.com/Blog/tabid/231/EntryId/347/Pitfalls-in-Asset-and-Liability-Management-One-Factor-Term-Structure-Models.aspx Pitfalls in Asset and Liability Management: One Factor Term Structure Models] {{Webarchive|url=https://web.archive.org/web/20120403233145/http://www.kamakuraco.com/Blog/tabid/231/EntryId/347/Pitfalls-in-Asset-and-Liability-Management-One-Factor-Term-Structure-Models.aspx |date=2012-04-03 }}</ref>)

* Monte Carlo Methods are used for [[Portfolio (finance)|portfolio]] evaluation.<ref name="MCS">{{cite web | title = The Monte Carlo Framework, Examples from Finance and Generating Correlated Random Variables | author = Martin Haugh | url = http://www.columbia.edu/~mh2078/MCS04/MCS_framework_FEegs.pdf | date = Fall 2004 | access-date = 2010-09-24 | url-status = dead | archive-url = https://web.archive.org/web/20120105084424/http://www.columbia.edu/~mh2078/MCS04/MCS_framework_FEegs.pdf | archive-date = 2012-01-05 }}</ref> Here, for each sample, the [[correlation|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 [[Descriptive statistics|statistical characteristics]] of the portfolio are observed, and the portfolio assessed as required. Here analysts may apply [[Principal component analysis]], where through [[Principal component analysis#Dimensionality reduction|dimensionality reduction]], a limited set of factors may be simulated instead of each of the individual sources of uncertainty.  

*A similar approach is used in calculating [[Value at risk#Common VaR calculation models|value at risk]],<ref name="VAR">{{cite web | title = Monte Carlo Value-at-Risk | url = http://www.riskglossary.com/link/monte_carlo_transformation.htm | publisher = Contingency Analysis | year = 2004 | access-date = 2010-09-24 }}</ref><ref name="Investopedia">{{cite web | title = An Introduction To Value at Risk (VAR) | author = David Harper,CFA, FRM | url = http://www.investopedia.com/articles/04/092904.asp | publisher = Investopedia | access-date = 2010-09-24 }}</ref>,or "VaR", an estimate of how much a position, [[Trading_room#Organization|"desk"]], or other area might lose with a given probability (or [[confidence level]]) and in a set time period.  A typical application of VaR is in [[investment banking]], where the bank holds [[economic capital|economic “risk capital”]] corresponding to the estimated number; see {{slink|Financial risk management#Banking}}. VaR is also used in [[Financial_risk_management#Investment_management|portfolio risk management]], where, as above, simulation allows the fund manager to estimate losses at a given horizon and confidence level,<ref>[https://financetrain.com/value-at-risk-of-a-portfolio Value at Risk of a Portfolio], financetrain.com</ref> and to then hedge as / if appropriate.

*[[Valuation_of_options#Post_crisis|Post crisis]], banks will make various “valuation adjustments” - collectively [[XVA]] - when assessing the value of derivative contracts that they have entered into. The purpose of these is twofold: primarily to hedge for possible losses due to [[counterparty credit risk|the other parties' failures to pay]] amounts due on the derivative contracts ([[credit valuation adjustment]]); but also to determine ([[XVA#Accounting_impact|and hedge]]) the amount of capital required under [[Basel III|the bank capital adequacy rules]]. These are calculated under a simulation framework as the [[risk-neutral|risk-neutral expectation]] value of the possible loss or other impact. <ref name="Hull_White_2">[[John C. Hull (economist)|John C. Hull]] and Alan White (2014). [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2212953 Collateral and Credit Issues in Derivatives Pricing]. Rotman School of Management Working Paper No. 2212953</ref> See {{slink|Credit valuation adjustment#Calculation}}.

*[[Structurer]]s use simulation to [[Structured product #Product design and manufacture|estimate the likely payout]] - and possibility of losses - of their bespoke [[structured note]] or other [[structured product]], typically comprising several component securities. <ref>{{cite web | title = Risk analysis of structured products | url = https://www.math.kth.se/matstat/seminarier/reports/M-exjobb09/090615a.pdf | publisher = [[KTH Royal Institute of Technology]] | year = 2009 | access-date = 2021-11-23|author = Jonas Larsson}}</ref>  

* Monte Carlo Methods are used for [[personal financial planning]].<ref name="businessweek">{{cite magazine | title = A Better Way to Size Up Your Nest Egg : ''Monte Carlo'' models simulate all kinds of scenarios | author = Christopher Farrell | url = http://www.businessweek.com/2001/01_04/b3716156.htm | archive-url = https://web.archive.org/web/20010123221600/http://www.businessweek.com/2001/01_04/b3716156.htm | url-status = dead | archive-date = January 23, 2001 | magazine = Bloomberg Businessweek | date = January 22, 2001 | access-date = 2010-09-24 }}</ref><ref name="finplan">{{cite web | title = Financial Planning Using Random Walks | author = John Norstad | url = http://homepage.mac.com/j.norstad/finance/finplan.pdf | date = February 2, 2005 | access-date = 2010-09-24 }}</ref> 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.

* [[Discrete event simulation]] can be used in evaluating a proposed [[capital investment]]'s impact on existing operations. Here, a "current state" model is constructed.  Once operating correctly, having been [[model validation|tested and validated]] against historical data, the simulation is altered to reflect the proposed capital investment. This "future state" model is then used to assess the investment, by evaluating the improvement in performance (i.e. return) relative to the cost (via histogram as above); it may also be used in [[stress testing]] the design. See {{slink|Discrete event simulation#Evaluating capital investment decisions}}.

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).<ref name="puc"/> These techniques are also of limited use in valuing American style derivatives. See below.