Monte Carlo methods for option pricing
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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 1940s. 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.
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[edit] 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 [1]. Here the price of the underlying instrument St is usually modelled such that it follows a geometric Brownian motion with constant drift μ and volatility σ. So:
, where dWt 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 of a european option 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 [2] the underlying 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 [3], 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 [4]. 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 where management's decision at any point is a function of multiple underlying variables.
- Simulation can be used to value options where the payoff depends on the value of multiple underlying assets [5] 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.
[edit] 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 [6] and in real options analysis [4] .
Conversely, however, if an analytical technique for valuing the option exists - or even a numeric technique, such as a (modified) pricing tree [6] - Monte Carlo methods will usually be too slow to be competitive. They are, in a sense, a method of last resort [6] . See further under Monte Carlo methods in finance.
[edit] References
[edit] Notes
- ^ a b Don Chance: Teaching Note 96-03: Monte Carlo Simulation
- ^ Peter Carr and Guang Yang: Simulating American Bond Options in an HJM Framework
- ^ Carlos Blanco, Josh Gray and Marc Hazzard: Alternative Valuation Methods for Swaptions: The Devil is in the Details
- ^ a b Gonzalo Cortazar, Miguel Gravet and Jorge Urzua: The valuation of multidimensional American real options using the LSM simulation method
- ^ global-derivatives.com: Basket Options - Simulation
- ^ a b c Rich Tanenbaum: Battle of the Pricing Models: Trees vs Monte Carlo
[edit] 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
[edit] 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
[edit] Software
- Fairmat (freeware) modeling and pricing complex options
- MG Soft (freeware) valuation and Greeks of vanilla and exotic options
[edit] External links
- Monte Carlo Simulation, Prof. Don M. Chance, Louisiana State University
- Pricing complex options using a simple Monte Carlo Simulation, Peter Fink (reprint at quantnotes.com)
- MonteCarlo Simulation in Finance, global-derivatives.com
- Monte Carlo Derivative valuation, contd., Timothy L. Krehbiel, Oklahoma State University–Stillwater
- Applications of Monte Carlo Methods in Finance: Option Pricing, Y. Lai and J. Spanier, Claremont Graduate University
- Option pricing by simulation, Bernt Arne Ødegaard, Norwegian School of Management
- The Longstaff-Schwartz algorithm for American options, Astrid Prajogo, Princeton University
- Monte Carlo Method, riskglossary.com
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