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.

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 [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-03.pdf] , 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. Here the price of the underlying instrument "S"_"t" is usually modelled such that it follows a geometric Brownian motion with constant drift $mu$ and volatility $sigma$. So: $dS\_t\; =\; mu\; S\_t,dt\; +\; sigma\; S\_t,dW\_t\; ,$, where $dW\_t$ 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 here should be the same as under Black Scholes).

*In other cases, the source of uncertainty may be at a remove. For example, for bond options [http://www.math.nyu.edu/research/carrp/papers/pdf/hjm.pdf] 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 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 [http://www.fea.com/resources/pdf/swaptions.pdf] , where the value of the underlying swap is also a function of the evolving interest rate. For the models used to simulate the interest-rate see further under Short-rate model.
*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 . Further complications, such as the impact of commodity prices or inflation on the underlying, can also be introduced.

*Simulation can be used to value options where payoff depends on the value of multiple underlying assets such as a Basket option or Rainbow option. Here, correlation between assets is similarly 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 further under Monte Carlo methods in finance.

References

Articles

*Boyle, Phelim P., "Options: A Monte Carlo Approach". Journal of Financial Economics 4, (1977) 323-338
*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

Books

*Don L. McLeish, "Monte Carlo Simulation & Finance" (2005) ISBN 0471677787
*Christian P. Robert, George Casella, "Monte Carlo Statistical Methods" (2005) ISBN 0-387-21239-6

External links

* [http://www.global-derivatives.com/maths/k-o.php MonteCarlo Simulation in Finance] , global-derivatives.com
* [http://www.riskglossary.com/link/monte_carlo_method.htm Monte Carlo Method] , riskglossary.com
* [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-03.pdf Monte Carlo Simulation] , Prof. Don M. Chance, Louisiana State University
* [http://finance-old.bi.no/~bernt/gcc_prog/recipes/recipes/node12.html Option pricing by simulation] , Bernt Arne Ødegaard, Norwegian School of Management
* [http://www.smartquant.com/references/MonteCarlo/mc6.pdf Applications of Monte Carlo Methods in Finance: Option Pricing] , Y. Lai and J. Spanier, Claremont Graduate University
* [http://spears.okstate.edu/home/tlk/legacy/fin5883/notes6_s05.doc Monte Carlo Derivative valuation] , [http://spears.okstate.edu/home/tlk/legacy/fin5883/notes7_s05.doc contd.] , Timothy L. Krehbiel, Oklahoma State University–Stillwater
* [http://www.quantnotes.com/publications/papers/Fink-montecarlo.pdf Pricing complex options using a simple Monte Carlo Simulation] , Peter Fink - reprint at quantnotes.com
* [http://repositories.cdlib.org/anderson/fin/1-01/ The Longstaff-Schwartz algorithm for American options] , repositories.cdlib.org
* [http://www.crystalball.com/articles/download/charnes-options.pdf Using simulation for option pricing] , John Charnes