# Monte Carlo methods for option pricing

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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 early exercise features.

## Methodology

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:

• In other cases, the source of uncertainty may be at a remove. For example, for bond options[3] 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,[4] 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.[5]) For the models used to simulate the interest-rate see further under Short-rate model; "to create realistic interest rate simulations" Multi-factor short-rate models are sometimes employed.[6] To apply simulation to IRDs, the analyst must first "calibrate" the model parameters, such that bond prices produced by the model best fit observed market prices.
• Monte Carlo Methods allow for a compounding in the uncertainty.[7] 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[1] 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[8] such as a Basket option or Rainbow option. Here, correlation between asset returns is likewise incorporated.[according to whom?]

## 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:

• 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 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).[12]
• 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.[11]

## 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 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?]

## References

Notes

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.

Sources

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. ^
6. ^ Donald R. van Deventer (Kamakura Corporation): Pitfalls in Asset and Liability Management: One Factor Term Structure Models
7. ^ a b Gonzalo Cortazar, Miguel Gravet and Jorge Urzua: The valuation of multidimensional American real options using the LSM simulation method
8. ^ global-derivatives.com: Basket Options – Simulation
9. 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

Bibliography