# Monte Carlo methods for option pricing

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. 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.[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 used in valuing American options. The technique works in 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).
• 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.

## 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.

## 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

Primary references

• Boyle, Phelim P. (1977). "Options: A Monte Carlo Approach". Journal of Financial Economics. 4 (3): 323–338. doi:10.1016/0304-405x(77)90005-8. Retrieved June 28, 2012.
• Broadie, M.; Glasserman, P. (1996). "Estimating Security Price Derivatives Using Simulation" (PDF). Management Science. 42 (2): 269–285. CiteSeerX 10.1.1.196.1128. doi:10.1287/mnsc.42.2.269. Retrieved June 28, 2012.
• Longstaff, F.A.; Schwartz, E.S. (2001). "Valuing American options by simulation: a simple least squares approach". Review of Financial Studies. 14: 113–148. CiteSeerX 10.1.1.155.3462. doi:10.1093/rfs/14.1.113. Retrieved June 28, 2012.

Bibliography