Monte Carlo option model

In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features.

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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 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 general [1], 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 value (i.e. "payoff") of the option for each path. These payoffs are then averaged and discounted to today, and this result is the value of the option today.

This approach allows for increasing complexity:

• An option on equity may be modelled with one source of uncertainty: the price of the underlying stock in question.

• In other cases, the source of uncertainty may be at a remove. For example, for bond options [2] the underlying source of uncertainty is the annualized interest rate (i.e. the short rate). Here, for each possible evolution of the interest rate we observe a different resultant bond price. This bond price is then the input for the valuation of the bond option's payoff. The same approach is used in valuing swaptions [3], where the value of the underlying swap is also a function of the evolving interest rate.

• Monte Carlo Methods allow for a 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.

• Further complications - such as the impact of commodity prices or inflation on the underlying, or multiple underlying assets - can also be introduced. Correlation between prices and assets is once again incorporated.

• 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.

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 options and in real options analysis.

Conversely, however, if an 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.

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