Monte Carlo option model
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In [[mathematical finance]], a '''Monte Carlo option model''' uses [[Monte Carlo method]]s to calculate the value of an [[Option (finance)|option]] with multiple sources of uncertainty or with complicated features.
The term 'Monte Carlo method' was coined by [[Stanislaw Ulam]] in the 1940's. 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. In 2001 F. A. Longstaff and E. S. Schwartz developed a practical Monte Carlo method for pricing [[American option|American-style options]].
==Methodology==
In general [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-03.pdf], the technique is to generate several thousand possible (but random) price paths for the [[underlying]] (or underlyings) via [[simulation]], and to then calculate 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.
This approach allows for increasing complexity:
*An [[option (finance)|option on equity]] may be modelled with one source of uncertainty: the price of the underlying [[stock (finance)|stock]] in question. Here the price of the [[underlying instrument]] ''S''<sub>''t''</sub> 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]]. (Of course, for enough price paths, the value here will be the same as under Black Scholes).
*In other cases, the source of uncertainty may be at a remove. For example, for [[bond option]]s [http://www.math.nyu.edu/research/carrp/papers/pdf/hjm.pdf] the underlying source of uncertainty is the annualized [[interest rate]] (i.e. the [[Short-rate model#The short rate|short rate]]). Here, for each possible evolution of the [[interest rate]] we observe a different 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 [http://www.fea.com/resources/pdf/swaptions.pdf], where the value of the underlying [[swap (finance)|swap]] is also a function of the evolving interest rate. For the models used to simulate the interest-rate see [[Short-rate_model#Particular_short-rate_models|further]] under [[Short-rate model]].
*Monte Carlo Methods allow for a [[Joint probability |compounding in the uncertainty]]. 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 it is common to model [[correlation]] between the underlying sources of risk; see [[Cholesky_decomposition#Monte_Carlo_simulation|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.
*Simulation can be used to value options where payoff depends on the value of multiple underlying assets such as a [[Basket option]] or [[Rainbow option]]. Here, correlation between assets is similarly incorporated.
*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]] of the underlying changes with time.
==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 computation. The technique is thus widely used in valuing [[Asian option]]s and in [[real options analysis]].
Conversely, however, if an [[Closed-form expression|analytical technique]] for valuing an option exists, Monte Carlo methods will usually be too slow to be competitive. They are, in a sense, a method of last resort. See [[Monte_Carlo_methods_in_finance#Level_of_complexity|further]] under [[Monte Carlo methods in finance]].
== References ==
*Don L. McLeish, ''Monte Carlo Simulation & Finance'' ([[2005]]) ISBN 0471677787
*Broadie, M. and P. Glasserman, ''Estimating Security Price Derivatives Using Simulation'', Management Science, 42, (1996) 269-285.
*Longstaff F.A. and E.S. Schwartz, ''Valuing American options by simulation: a simple least squares approach'', Review of Financial Studies 14 (2001), 113-148
*Boyle, Phelim P., ''Options: A Monte Carlo Approach''. Journal of Financial Economics 4, (1977) 323-338
*Christian P. Robert, George Casella, ''Monte Carlo Statistical Methods'' ([[2005]]) ISBN 0-387-21239-6
==External links==
*[http://www.global-derivatives.com/maths/k-o.php MonteCarlo Simulation in Finance], global-derivatives.com
*[http://www.riskglossary.com/link/monte_carlo_method.htm Monte Carlo Method], riskglossary.com
*[http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-03.pdf Monte Carlo Simulation], Prof. Don M. Chance, [[Louisiana State 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://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://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.quantnotes.com/publications/papers/Fink-montecarlo.pdf Pricing complex options using a simple Monte Carlo Simulation], Peter Fink - reprint at quantnotes.com
*[http://repositories.cdlib.org/anderson/fin/1-01/ The Longstaff-Schwartz algorithm for American options], repositories.cdlib.org
*[http://www.crystalball.com/articles/download/charnes-options.pdf Using simulation for option pricing], John Charnes
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[[Category:Monte Carlo methods]]
[[Category:Mathematical finance]]