In mathematical finance, a Monte Carlo option model uses Monte Carlo methods[Notes 1] to calculate the value of an option with multiple sources of uncertainty or with complicated features.[1] 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. An important development was the introduction in 1996 by Carriere of Monte Carlo methods for options with early exercise features.


In terms of theory, Monte Carlo valuation relies on risk neutral valuation.[1] 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.[2]

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

Least Square Monte Carlo

Least Square Monte Carlo is a technique for valuing early-exercise options (i.e. Bermudan or American options). It was first introduced by Jacques Carriere in 1996.[11]

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


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 lattice based computation. The technique is thus widely used in valuing path dependent structures like lookback- and Asian options[9] and in real options analysis.[1][7] Additionally, as above, the modeller is not limited as to the probability distribution assumed.[9]

Conversely, however, if an analytical technique for valuing the option exists—or even a numeric technique, such as a (modified) pricing tree[9]—Monte Carlo methods will usually be too slow to be competitive. They are, in a sense, a method of last resort;[9] see further under Monte Carlo methods in finance. With faster computing capability this computational constraint is less of a concern.[according to whom?]

See also



  1. ^ 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.


  1. ^ a b c d Marco Dias: Real Options with Monte Carlo Simulation
  2. ^ a b Don Chance: Teaching Note 96-03: Monte Carlo Simulation
  3. ^ Peter Carr and Guang Yang: Simulating American Bond Options in an HJM Framework
  4. ^ Carlos Blanco, Josh Gray and Marc Hazzard: Alternative Valuation Methods for Swaptions: The Devil is in the Details Archived 2007-12-02 at the Wayback Machine
  5. ^ Frank J. Fabozzi: Valuation of fixed income securities and derivatives, pg. 138
  6. ^ Donald R. van Deventer (Kamakura Corporation): Pitfalls in Asset and Liability Management: One Factor Term Structure Models Archived 2012-04-03 at the Wayback Machine
  7. ^ a b Gonzalo Cortazar, Miguel Gravet and Jorge Urzua: The valuation of multidimensional American real options using the LSM simulation method
  8. ^ Basket Options – Simulation
  9. ^ a b c d e Rich Tanenbaum: Battle of the Pricing Models: Trees vs Monte Carlo
  10. ^ Les Clewlow, Chris Strickland and Vince Kaminski: Extending mean-reversion jump diffusion
  11. ^ a b Carriere, Jacques (1996). "Valuation of the early-exercise price for options using simulations and nonparametric regression". Insurance: Mathematics and Economics. 19: 19–30. doi:10.1016/S0167-6687(96)00004-2.
  12. ^ Longstaff, Francis. "Valuing American Options by Simulation: A Simple Least-Squares Approach" (PDF). Retrieved 18 December 2019.

Primary references


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Discussion papers and documents