View source for Monte Carlo methods for option pricing

{{Short description|Model in mathematical finance}}
In [[mathematical finance]], a '''Monte Carlo option model''' uses [[Monte Carlo method]]s<ref group="Notes">Although the term 'Monte Carlo method' was coined by [[Stanislaw Ulam]] in the 1940s, some trace such methods to the 18th century French naturalist [[Georges-Louis Leclerc, Comte de Buffon|Buffon]], and a question he asked about the results of dropping a needle randomly on a striped floor or table. See [[Buffon's needle]].</ref> to calculate the value of an [[Option (finance)|option]] with multiple sources of uncertainty or with complicated features.<ref name="Marco Dias"/> The first application to option pricing was by [[Phelim Boyle]] in 1977 (for [[European option]]s).  In 1996, M. Broadie and P. Glasserman showed how to price [[Asian option]]s by Monte Carlo.  An important development was the introduction in 1996 by Carriere of Monte Carlo methods for options with [[Option_style#American_and_European_options|early exercise features]].

==Methodology==
As [[Valuation_of_options|is standard]], Monte Carlo valuation relies on [[Rational_pricing#Risk_neutral_valuation|risk neutral valuation]].<ref name="Marco Dias">Marco Dias: [http://marcoagd.usuarios.rdc.puc-rio.br/faq4.html Real Options with Monte Carlo Simulation]</ref> Here the price of the option is its [[present value|discounted]] [[expected value]]; see [[risk neutrality]] and [[Rational pricing#Risk neutral valuation|rational pricing]]. The technique applied then, is (1) to generate a large number of possible, but [[random]], price paths for the [[underlying]] (or underlyings) via [[simulation]], and (2) to then calculate the associated [[Exercise (options)|exercise]] [[Option time value#Intrinsic value|value]] (i.e. "payoff") of the option for each path. (3) These payoffs are then averaged and (4) discounted to today. This result is the value of the option.<ref name="Don Chance">Don Chance: [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-03.pdf Teaching Note 96-03: Monte Carlo Simulation]</ref>

This approach, although relatively straightforward, allows for increasing complexity:

* An [[option (finance)|option on equity]] may be modelled with one source of uncertainty: the price of the underlying [[stock]] in question.<ref name="Don Chance"/> Here the price of the [[underlying instrument]] <math> \ S_t \,</math> is usually modelled such that it follows a [[geometric Brownian motion]] with constant drift <math> \mu \,</math> and [[Volatility (finance)|volatility]] <math> \sigma \,</math>. So: <math> dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \, </math>, where <math> dW_t \,</math> is found via a [[random sampling]] from a [[normal distribution]]; see [[Black–Scholes#The model|further]] under [[Black–Scholes]]. Since the underlying random process is the same, for enough price paths, the value of a [[european option]] here should be [[Convergence (mathematics)|the same as under Black–Scholes]]. More generally though, simulation is employed for [[Path dependence|path dependent]] [[exotic derivatives]], such as [[Asian options]].

* In other cases, the source of uncertainty may be at a remove. For example, for [[bond option]]s<ref>Peter Carr and Guang Yang: [http://www.math.nyu.edu/research/carrp/papers/pdf/hjm.pdf Simulating American Bond Options in an HJM Framework]</ref> the underlying is a [[Bond (finance)|bond]], but the source of uncertainty is the annualized [[interest rate]] (i.e. the [[Short-rate model#The short rate|short rate]]). Here, for each randomly generated [[yield curve]] we observe a different [[Bond valuation#Arbitrage-free pricing approach|resultant bond price]] on the option's exercise date; this bond price is then the input for the determination of the option's payoff. The same approach is used in valuing [[swaption]]s,<ref>Carlos Blanco, Josh Gray and Marc Hazzard: [http://www.fea.com/resources/pdf/swaptions.pdf Alternative Valuation Methods for Swaptions: The Devil is in the Details] {{Webarchive|url=https://web.archive.org/web/20071202214825/http://www.fea.com/resources/pdf/swaptions.pdf |date=2007-12-02 }}</ref> where the value of the underlying [[swap (finance)|swap]] is also a function of the evolving interest rate. (Whereas these options are more commonly valued using [[Lattice model (finance)|lattice based models]], as above, 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>) For the models used to simulate the interest-rate see [[Short-rate model#Particular short-rate models|further]] under [[Short-rate model]]; "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>  To apply simulation here, the analyst must first "calibrate" the model parameters, such that bond prices produced by the model [[curve fitting|best fit]] observed market prices.

* Monte Carlo Methods allow for a [[Joint probability|compounding in the uncertainty]].<ref name="Cortazar et al">Gonzalo Cortazar, Miguel Gravet and Jorge Urzua: [http://www.realoptions.org/papers2005/Cortazar_GU052RealOptionsParis.pdf The valuation of multidimensional American real options using the LSM simulation method]</ref> For example, where the underlying is denominated in a foreign currency, an additional source of uncertainty will be the [[exchange rate]]: the underlying price and the exchange rate must be separately simulated and then combined to determine the value of the underlying in the local currency. In all such models, [[correlation]] between the underlying sources of risk is also incorporated; see {{slink|Cholesky decomposition#Monte Carlo simulation}}. Further complications, such as the impact of [[commodity markets|commodity prices]] or [[inflation]] on the underlying, can also be introduced. Since simulation can accommodate complex problems of this sort, it is often used in analysing [[real options]]<ref name="Marco Dias"/> where management's decision at any point is a function of multiple underlying variables.

* Simulation can similarly be used to value options where the payoff depends on the value of multiple underlying assets<ref>global-derivatives.com: [http://www.global-derivatives.com/index.php?option=com_content&task=view&id=26#MCS Basket Options – Simulation]</ref> such as a [[Basket option]] or [[Rainbow option]]. Here, correlation between asset returns is likewise incorporated.<ref name="Rubinstein1991">Rubinstein, Mark. "Somewhere over the rainbow." Risk 4.11 (1991): 61-63.</ref>

* As required, Monte Carlo simulation can be used with any type of [[probability distribution]], including changing distributions: the modeller is not limited to [[normal distribution|normal]] or [[log-normal distribution|log-normal]] returns;<ref name="Tanenbaum"/> see for example [[Datar–Mathews method for real option valuation]]. Additionally, the [[stochastic process]] of the underlying(s) may be specified so as to exhibit [[jump process|jumps]] or [[mean reverting process|mean reversion]] or both; this feature makes simulation the primary valuation method applicable to [[energy derivative]]s.<ref>Les Clewlow, Chris Strickland and Vince Kaminski: [http://www.erasmusenergy.com/downloadattachment.php?aId=4b0d2207d4169ee155591c70efa19c63&articleId=139 Extending mean-reversion jump diffusion]</ref> Further, some models even allow for (randomly) varying [[Statistical parameter|statistical]] (and other) [[parameter]]s of the sources of uncertainty. For example, in models incorporating [[stochastic volatility]], the [[Volatility (finance)|volatility]] of the underlying changes with time; see [[Heston model]].{{Citation needed|date=September 2019}}

==Least Square Monte Carlo==
Least Square Monte Carlo is a technique for valuing early-exercise options (i.e. [[Bermudan option|Bermudan]] or [[American option]]s). It was first introduced by Jacques Carriere in 1996.<ref name="carriere">{{Cite journal|title=Valuation of the early-exercise price for options using simulations and nonparametric regression|last=Carriere|first=Jacques|date=1996|journal=Insurance: Mathematics and Economics |doi=10.1016/S0167-6687(96)00004-2|volume=19|pages=19–30}}</ref>

It is based on the iteration of a two step procedure:

* First, a [[backward induction]] process is performed in which a value is recursively assigned to every state at every timestep. The value is defined as the [[least squares regression]] against market price of the option value at that [[State prices|state]] and time (-step). Option value for this regression is defined as the value of exercise possibilities (dependent on market price) plus the value of the timestep value which that exercise would result in (defined in the previous step of the process).<ref name=":0">{{Cite web|url=https://people.math.ethz.ch/~hjfurrer/teaching/LongstaffSchwartzAmericanOptionsLeastSquareMonteCarlo.pdf|title=Valuing American Options by Simulation: A Simple Least-Squares Approach|last=Longstaff|first=Francis|access-date=18 December 2019}}</ref>
* Secondly, when all states are valued for every timestep, the value of the option is calculated by moving through the timesteps and states by making an optimal decision on option exercise at every step on the hand of a price path and the value of the state that would result in. This second step can be done with multiple price paths to add a stochastic effect to the procedure.<ref name="carriere" />

==Application==
As can be seen, Monte Carlo Methods are particularly useful in the valuation of options with multiple sources of uncertainty or with complicated features, which would make them difficult to value through a straightforward [[Black–Scholes]]-style or [[BOPM|lattice based]] computation. The technique is thus widely used in valuing path dependent structures like [[Lookback option|lookback-]] and [[Asian option]]s<ref name="Tanenbaum">Rich Tanenbaum: [http://www.savvysoft.com/treevsmontecarlo.htm Battle of the Pricing Models: Trees vs Monte Carlo]</ref> and in [[real options analysis]].<ref name="Marco Dias"/><ref name="Cortazar et al"/> Additionally, as above, the modeller is not limited as to the probability distribution assumed.<ref name="Tanenbaum"/>

Conversely, however, if an [[Closed-form expression|analytical technique]] for valuing the option exists—or even a [[Numerical methods|numeric technique]], such as a (modified) [[binomial options pricing model|pricing tree]]<ref name="Tanenbaum"/>—Monte Carlo methods will usually be too slow to be competitive. They are, in a sense, a method of last resort;<ref name="Tanenbaum"/> see [[Monte Carlo methods in finance#Level of complexity|further]] under [[Monte Carlo methods in finance]]. With faster computing capability this computational constraint is less of a concern.{{According to whom|date=September 2019}}

==See also==
*[[Monte Carlo methods in finance]]
*[[Quasi-Monte Carlo methods in finance]] 
*[[Stochastic modelling (insurance)]] 
*[[Stochastic asset model]]

==References==
'''Notes'''
{{Reflist|group="Notes"}}
'''Sources'''
{{Reflist}}
'''Primary references'''
*{{cite journal|last1=Boyle |first1=Phelim P. |url=http://ideas.repec.org/a/eee/jfinec/v4y1977i3p323-338.html |access-date=June 28, 2012 |title=Options: A Monte Carlo Approach |journal=Journal of Financial Economics |volume=4 |number=3 |year=1977 |pages=323–338 |doi=10.1016/0304-405x(77)90005-8}}
*{{cite journal|last1=Broadie |first1=M. |first2=P. |last2=Glasserman |url=http://www.columbia.edu/~mnb2/broadie/Assets/bg_ms_1996.pdf |access-date=June 28, 2012 |title=Estimating Security Price Derivatives Using Simulation |journal=Management Science |volume=42 |issue=2 |year=1996 |pages=269–285 |doi=10.1287/mnsc.42.2.269|citeseerx=10.1.1.196.1128 }}
*{{cite journal|last1=Longstaff |first1=F.A. |first2=E.S. |last2=Schwartz |url=http://repositories.cdlib.org/anderson/fin/1-01/ |access-date=June 28, 2012 |title=Valuing American options by simulation: a simple least squares approach |journal=Review of Financial Studies |volume=14 |year=2001 |pages=113–148 |doi=10.1093/rfs/14.1.113|citeseerx=10.1.1.155.3462 }}

'''Bibliography'''
* {{cite book | title = Monte Carlo:methodologies and applications for pricing and risk management | author = Bruno Dupire | year = 1998 | publisher = Risk | author-link = Bruno Dupire }}
* {{cite book | title = Monte Carlo methods in financial engineering | author = Paul Glasserman | year = 2003 | publisher = [[Springer-Verlag]] | isbn = 978-0-387-00451-8 }}
* {{cite book | title = Monte Carlo methods in finance | author = Peter Jaeckel | year = 2002 | publisher = John Wiley and Sons | isbn = 978-0-471-49741-7 | author-link = Peter Jaeckel }}
* {{cite book | title = Monte Carlo Simulation & Finance | author = Don L. McLeish| year = 2005 |isbn = 978-0-471-67778-9}}
* {{cite book | title = Monte Carlo Statistical Methods| url = https://archive.org/details/springer_10.1007-978-1-4757-4145-2| author = Christian P. Robert, George Casella| year = 2004 |isbn = 978-0-387-21239-5}}

==External links==
''' Online tools'''
*[http://25yearsofprogramming.com/blog/20070412c-montecarlostockprices.htm Monte Carlo simulated stock price time series and random number generator] (allows for choice of distribution), Steven Whitney

'''Discussion papers and documents'''
<!-- alphabetical by author -->
*[http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-03.pdf Monte Carlo Simulation], Prof. Don M. Chance, [[Louisiana State University]]
*[http://www.quantnotes.com/publications/papers/Fink-montecarlo.pdf Pricing complex options using a simple Monte Carlo Simulation], Peter Fink (reprint at quantnotes.com)
*[http://www.global-derivatives.com/maths/k-o.php MonteCarlo Simulation in Finance], global-derivatives.com
*[http://spears.okstate.edu/home/tlk/legacy/fin5883/notes6_s05.doc Monte Carlo Derivative valuation], [http://spears.okstate.edu/home/tlk/legacy/fin5883/notes7_s05.doc contd.], Timothy L. Krehbiel, [[Oklahoma State University–Stillwater]]
*[http://www.smartquant.com/references/MonteCarlo/mc6.pdf Applications of Monte Carlo Methods in Finance: Option Pricing], Y. Lai and J. Spanier, [[Claremont Graduate University]]
*[http://finance-old.bi.no/~bernt/gcc_prog/recipes/recipes/node12.html Option pricing by simulation], Bernt Arne Ødegaard, [[Norwegian School of Management]]
*[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.194.9001 Pricing and Hedging Exotic Options with Monte Carlo Simulations], Augusto Perilla, Diana Oancea, Prof. Michael Rockinger, [[HEC Lausanne]]
*[http://www.riskglossary.com/link/monte_carlo_method.htm Monte Carlo Method], riskglossary.com

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[[Category:Monte Carlo methods in finance]]
[[Category:Options (finance)]]