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In mathematical finance, a Monte Carlo option model uses Monte Carlo methods 〔Although the term 'Monte Carlo method' was coined by Stanislaw Ulam in the 1940s, some trace such methods to the 18th century French naturalist Buffon, and a question he asked about the results of dropping a needle randomly on a striped floor or table. See Buffon's needle.〕 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 (for European options). 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. ==Methodology== In terms of theory, Monte Carlo valuation relies on risk neutral valuation.〔Marco Dias: (Real Options with Monte Carlo Simulation )〕 Here the price of the option is its discounted expected value; see risk neutrality and 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 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.〔Don Chance: (Teaching Note 96-03: Monte Carlo Simulation )〕 This approach, although relatively straightforward, allows for increasing complexity: *An option on equity may be modelled with one source of uncertainty: the price of the underlying stock in question.〔 Here the price of the underlying instrument is usually modelled such that it follows a geometric Brownian motion with constant drift and volatility . So: , where is found via a random sampling from a normal distribution; see 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 the same as under Black Scholes. More generally though, simulation is employed for path dependent exotic derivatives, such as Asian options. *In other cases, the source of uncertainty may be at a remove. For example, for bond options 〔Peter Carr and Guang Yang: (Simulating American Bond Options in an HJM Framework )〕 the underlying is a bond, but the source of uncertainty is the annualized interest rate (i.e. the short rate). Here, for each randomly generated yield curve 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 swaptions,〔Carlos Blanco, Josh Gray and Marc Hazzard: (Alternative Valuation Methods for Swaptions: The Devil is in the Details )〕 where the value of the underlying swap is also a function of the evolving interest rate. (Whereas these options are more commonly valued using lattice based models, as above, for path dependent interest rate derivatives – such as CMOs – simulation is the ''primary'' technique employed.〔Frank J. Fabozzi: (''Valuation of fixed income securities and derivatives'', pg. 138 )〕) For the models used to simulate the interest-rate see further under Short-rate model; note also that "to create realistic interest rate simulations" Multi-factor short-rate models are sometimes employed.〔Donald R. van Deventer (Kamakura Corporation): (Pitfalls in Asset and Liability Management: One Factor Term Structure Models )〕 *Monte Carlo Methods allow for a compounding in the uncertainty.〔Gonzalo Cortazar, Miguel Gravet and Jorge Urzua: (The valuation of multidimensional American real options using the LSM simulation method )〕 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 Cholesky decomposition #Monte Carlo simulation. Further complications, such as the impact of 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 〔 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 〔global-derivatives.com: (Basket Options – Simulation )〕 such as a Basket option or Rainbow option. Here, correlation between asset returns is likewise incorporated. *As required, Monte Carlo simulation can be used with any type of probability distribution, including changing distributions: the modeller is not limited to normal or lognormal returns;〔 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 jumps or mean reversion or both; this feature makes simulation the primary valuation method applicable to energy derivatives.〔Les Clewlow, Chris Strickland and Vince Kaminski: (Extending mean-reversion jump diffusion )〕 Further, some models even allow for (randomly) varying statistical (and other) parameters of the sources of uncertainty. For example, in models incorporating stochastic volatility, the volatility of the underlying changes with time; see Heston model. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Monte Carlo methods for option pricing」の詳細全文を読む スポンサード リンク
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