This new website section focuses real option models using Monte Carlo simulations, a very flexible way to model and to combine the uncertainties.

The Monte Carlo method solves a problem by simulating directly the physical process, and is not necessary to write down the differential equations that describe the behavior of the system. This is very general and is valid not only for our real options problem as in other areas of knowledge like physics, chemistry, etc. In our case, the Monte Carlo simulation permits simulate several sources of uncertainties that affect the value of our real option, given an optimal rule of exercise.

For several state variable (several sources of uncertainties), real
options models suffer the problem of the ** curse of dimensionality**,
which limit the model solution with others methods. For example, for more
than three or four state variables, both lattice and finite-difference
methods face several difficulties and are not practical.

However, for simpler real options models, Monte Carlo is not the better solution because is very time-consuming in terms of computation. So, the interest on Monte Carlo simulation approach is related to solve complex real options models.

Most real options models are American-type options (earlier exercise
feature) so that the __Monte Carlo simulation must be used together with
some optimization method__ in order to get the threshold curve (earlier
exercise free-boundary).

This section includes the topics (many topics are placed in other pages):

**Bibliography on Monte Carlo Simulation:**

1) Monte Carlo for American Options;

2) Quasi-Monte Carlo (Low Discrepancy Sequences) Methods; and

3) Books, Classics and Miscellaneous in Monte Carlo Methods

**Monte Carlo for European Real Options with
Animation**

**Research Report: Real Options with Monte Carlo +
Optimization with Genetic Algorithms**

**Quasi-Monte Carlo Simulation (practical use
of low-discrepancy sequences)**.

**Monte Carlo and Quasi-Monte Carlo Internet
Links**.

**Monte
Carlo Simulation of Stochastic Processes**.

The name "Monte Carlo" appeared in the World War II times, and
sometimes is attributed to the researcher Nicholas Metropolis, inspired in
the interest of Stanislaw Ulam, his colleague of Manhattan Project at Los
Alamos, in the poker game. Monte Carlo, the capital of Monaco, was a known
reference for gambling.

According Eckhardt, Ulam invented the Monte Carlo method in 1946 while
pondering the probabilities of winning a card game of *solitaire*
(see the
Ulam
description on this event). However, Metropolis "attributes the
germ of this statistical method to Enrico Fermi, who had used such ideas
some 15 years earlier" see the
webpage
on Metropolis.

According Liu (2001, p.vii-viii): "*The basic idea underlying the
method was first brought up by Ulam and deliberated between him and von
Neumann in a car when they drove together from Los Alamos to Lamy.
Allegedly, Nick Metropolis coined the name 'Monte Carlo', which played an
essential role in popularizing the method*". Liu comments that the
Los Alamos scientists aiming to take advante of the first "super"
computer MANIAC, invented a statistical sampling-based technique to solve
problems related to stochastic neutron diffusion in atomic bomb project
and for estimating eigenvalues of the Schrödinger equation.

Winston (1996, p.22) wrote that the term was coined by mathematicians S.
Ulam and J. von Neumann in the feasibility project of atomic bomb by
simulations of nuclear fission, and they given the *code name* Monte
Carlo for these simulations.

The first Monte Carlo paper, "*The Monte Carlo Method*"
by Metropolis & Ulam, was published in 1949 in the *Journal of the
American Statistical Association*.

Since then, several different areas has been using the Monte Carlo
simulations. With the advent of personal computers and the popularization
of faster computational machines, the Monte Carlo simulations has been
increasing popular as an important alternative for the solution of complex
problems.

The main interest in real options applications are for *American*
type options applications, so I set one special bibliographical topic for
this.

The use of more advanced techniques of sampling, improving simulation
speed and accuracy, is the second topic of our bibliography.

There is a last topic, which comprises more general references, like
text-books, on this fascinating topic.

1) Monte Carlo for American Options

2) Quasi-Monte Carlo (Low Discrepancy Sequences) Methods

3) Books, Classics and Miscellaneous in Monte Carlo Methods

Acworth, P. & M. Broadie & P. Glasserman (1996): "A
Comparison of Some Monte Carlo and Quasi-Monte Carlo Techniques for Option
Pricing"

in Niederreiter et al. (Eds.), *Monte Carlo and Quasi-Monte Carlo
Methods* 1996 - Springer-Verlag New York, Lectures Notes in Statistics,
1998, pp.1-18

Andersen, L. (2000): "A Simple Approach to the Pricing of Bermudan
Swaptions in the Multifactor Libor Market Model"

Journal of Computational Finance, Winter 1999/2000, vol.3, n__ ^{o}__
2, pp.5-32

Andersen, L. & M. Broadie (2004): "Prial-Dual Simulation Algorithm
for Pricing Multidimensional American Options"

Management Science, vol.50, n__ ^{o}__ 9, September 2004, pp.1222-1234

Averbukh, V.Z. (1997): "Pricing American Options Using Monte Carlo
Simulation"

Doctoral Dissertation, Cornell University, August 1997, 53 pp.

Barraquand, J. & D. Martineau (1995): "Numerical Valuation of
High Dimensional Multivariate American Securities"

Journal of Financial and Quantitative Analysis, vol.30, n__ ^{o}__
3, pp.383-405

Barraquand, J. & T. Pudet (1994): "Pricing of American Path
Dependent Contingent Claims"

Digital/PRL Research Report n__ ^{o}__ 37, January 1994, 44
pp.

Boyle, P. & M. Broadie & P. Glasserman (1997): "Monte Carlo
Methods for Security Pricing"

Journal of Economic Dynamics and Control, June 1997, vol.21, n__ ^{o}__
8-9, pp.1267-1321

Boyle, P.P. & A.W. Kolkiewicz & K.S. Tan (2002): "Pricing
American Derivatives Using Simulation: A Biased Low Approach"

in Fang, K.-T. & F.J. Hickernell & H. Niederreiter, Eds., *Monte
Carlo and Quasi-Monte Carlo Methods 2000* - Springer-Verlag Berlin
Heidelberg, 2002, pp.181-200

Broadie, M. & P. Glasserman (1997): "Pricing American-Style
Securities Using Simulation"

Journal of Economic Dynamics and Control, June 1997, vol.21, n__ ^{o}__
8-9, pp.1323-1352

See an online calculator with the method of this paper, using Java applet at Meier & Haug's American Monte Carlo calculator webpage.

Broadie, M. & P. Glasserman & G. Jain (1997): "Enhanced
Monte Carlo Estimates for American Option Prices"

Journal of Derivatives, vol.5, pp.25-44

Carr, P. & G. Yang (1998): "Simulating American Bond Options in
an HJM Framework"

Working Paper, Morgan Stanley & Open Link Financial, February 1998, 22
pp.

Carriere, J.F. (1996): "Valuation of the Early-Exercise Price for
Options Using Simulations and Nonparametric Regression"

Insurance: Mathematics and Economics, vol.19, 1996, pgs.19-30

Castillo-Ramírez, A. (2000): "An Application of Natural
Resource Evaluation Using a Simulation-Dynamic Programming Approach"

Journal of Computational Finance, Winter 1999/2000, vol.3, n__ ^{o}__
2, pp.91-107

Castro, A.L. (2000): "Avaliação de Investimento de
Capital em Projetos de Geração Termoelétrica no Setor
Elétrico Brasileiro Usando Teoria das Opções Reais"
(*Evaluation of Capital Investment in Thermoelectric Generation
Projects in the Brazilian Electricity Sector Using Real Options Theory*)

M.Sc. Dissertation, Dept. of Industrial Engineering, PUC-Rio, April 2000,
106 pp. (in __Portuguese__)

Charnes, J.M. (2000): "Using Simulation for Option Pricing"

Proceedings of the 2000 Winter Simulation Conference, 2000, 7 pp.

Cortazar, G. (2000): "Simulation and Numerical Methods in Real
Options Valuation"

Working Paper, PUC de Chile, 2000, 24 pp.

Cortazar, G. & E.S. Schwartz (1998): "Monte Carlo Evaluation
Model of an Undeveloped Oil Field"

Journal of Energy Finance & Development, vol.3, n__ ^{o}__
1, pp.73-84; and also Working Paper UCLA #1-98, PUC-Chile and UCLA,
January 1998, 20 pp.

Dias, M.A.G. (2002): "Investment in Information in Petroleum: Real
Options and Revelation"

Working Paper, Dept. of Industrial Engineering, PUC-Rio, April 2002, 38
pp.

Dias, M.A.G. (2001): "Selection of Alternatives of Investment in
Information for Oilfield Development Using Evolutionary Real Options
Approach"

Working Paper, Dept. of Electrical Engineering, PUC-Rio, January 2001, 29
pp., presented at the *5th Annual International Conference on Real
Options*, UCLA, Los Angeles, July 2001

Dias, M.A.G. (2000): "Real Options Evaluation: Optimization under
Uncertainty with Genetic Algorithms and Monte Carlo Simulation"

Working Paper, Dept. of Electrical Engineering, PUC-Rio, July 2000, 23
pp.

Douady, R. (2000): "Bermudan Option Pricing with Monte-Carlo
Methods"

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- Volume III*, World Scientific Publishing Co., 2001, pp.314-328

Fu, M.C. & S.B. Laprise & D.B. Madan & Y. Su & R. Wu
(2000/1): "Pricing American Options: A Comparison of Monte Carlo
Simulation Approaches"

Working Paper, University of Maryland at College Park, April 2000, 44
pp., and Journal of Computational Finance, Vol.4, n__ ^{o}__
3, Spring 2001, pp.39-88

Gamba, A. (2002): "Real Options Valuation: a Monte Carlo Simulation
Approach"

Working Paper 2002/03, Faculty of Management, University of Calgary, 2002,
40 pp.

Garcia, D. (2000): "A Monte Carlo Method for Pricing American
Options"

Working Paper, University of California at Berkeley, January 2000, 43 pp.

Grant, D. & G. Vora & D.E. Weeks (1996): "Path-Dependent
Options: Extending the Monte Carlo Simulation Approach"

Management Science, vol.43, n__ ^{o}__ 11, November 1997,
pp.1589-1602

Grant, D. & G. Vora & D.E. Weeks (1996): "Simulation and
Early-Exercise of Option Problem"

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September 1996, pp.211-227

Ibáñez, A. (2004): "Valuation by Simulation of Contingent Claims with Multiple Early Exercises Opportunities"

Mathematical Finance, vol.14, n.2, April 2004, pp.223-248

Ibáñez, A. & F. Zapatero (1999): "Monte Carlo
Valuation of American Options Through Computation of the Optimal Exercise
Frontier"

Working Paper, Instituto Tecnológico Autónomo de México
& University of South California, August 1999, 30 pp.

Longstaff, F.A. & E.S. Schwartz (1998/2001): "Valuing American
Options By Simulation: A Simple Least-Square Approach"

Review of Financial Studies, vol.14, n^{o} 1, Spring 2001,
pp.113-147, and Working Paper n__ ^{o}__ 25-98, UCLA, November
1998, 38 pp.

Moreno, M. & J.F. Navas (2001): "On the Robustness of
Least-Squares Monte Carlo (LSM) for Pricing American Derivatives"

Working Paper, Universitat Pompeu Fabra and Instituto de Empresa (Spain),
April 2001, 36 pp.

Picazo, J.A. (2002): "American Option Pricing: A
Classification-Monte Carlo (CMC) Approach"

in Fang, K.-T. & F.J. Hickernell & H. Niederreiter, Eds., *Monte
Carlo and Quasi-Monte Carlo Methods 2000* - Springer-Verlag Berlin
Heidelberg, 2002, pp.422-433

Pizzi, C. & P. Pellizzari (2002): "Monte Carlo Pricing of American Options - Using Nonparametric Regression"

Working Paper, University of Venice, Italy, August 2002, 15 pp.

Raymar, S.B. & M.J. Zwecher (1997): "Monte Carlo Estimation of
American Call Option on the Maximum of Several Assets"

Journal of Derivatives, Fall 1997, vol.5, pp.7-24

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Induction with Monte Carlo Simulations: A Fast Fourier Transform (FFT)
Approach"

Revision of July 1997 and IFA Working Paper n__ ^{o}__
236-1996, London Business School, August 1996, 26 pp.

Rogers, L.C.G. (2001/2): "Monte Carlo Valuation of American Options"

Mathematical Finance, vol.12, n^{o} 3, July 2002, pp.271-286, and
Working Paper, University of Bath (UK), 2001, 19 pp.

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There are some real options applications which can be modeled as European
type options. The main example is for single stage **R&D projects**
(which is not possible to exercise earlier, the research and tests need to
be complete before some the development phase) __and__ for markets
with strong **first-mover advantages** (in the sense of Lieberman &
Montgomery, 1988), so that typically the product development option will
be exercise immediately after the R&D ending (conditional to a
positive NPV for the development). In this case, the waiting value is zero
for the development phase and the R&D can be modeled as European
option with time to expiration equal to the estimate time to complete the
R&D project.

If the R&D expected cost is lower than its European option value, the
R&D project is valuable and shall be started.

For European real options case, sometimes is possible to use the
Black-Scholes solution (without dividends, because there is no opportunity
cost to retain the option). In this case only one underlying market
uncertainty is relevant.

In others case is better to consider other underlying uncertainties, like
costs, preferences, demand, price, etc. For more complex cases of several
sources of uncertainties (but remaining European), the Monte Carlo
simulation is a good alternative.

Let us see one simple example, with only one source of uncertainty (market uncertainty), using a risk-neutral simulation. The case is divided into two sequential animations and described below.

The first and second steps are to simulate the risk-neutral sample paths
for the underlying asset (project value or output price) and to take the
cross-section distribution at the expiration T. You get the * risk
neutral distribution* for the underlying asset.

The figure below illustrates these initial steps:

For details on the risk-neutral simulation, including the difference of a real simulation, see the FAQ 4.

With the risk-neutral distribution of the project value V
at the expiration (T), the next step is to apply the options thinking.
Rational managers will exercise the option only if this exercise results
in positive expected values:

**F (T)= max. (NPV, 0)**. In other
words, managers are not obligated to exercise negative NPV projects. This
equation creates an asymmetry in the distribution of the real option value
F(T). This asymmetric risk-neutral distribution for the European-type real
options at T displaces the expected value to the higher value and
represents the active management of real options.

The current value (t = 0) of the real options in this case is just to take
the present value using the risk-free discount rate. The figure below
shows this process.

In short, the European type (real) option valuation with Monte Carlo simulation is performed with the steps:

- Simulate
for the underlying asset (project value or output price);**risk-neutral sample paths** - Take the cross-section distribution at the expiration T, getting
for the underlying asset;**risk neutral distribution** - Apply the options thinking
**F (T)= max. (NPV, 0)**; - Calculate the expected value of the option value at the expiration;
- Use the
**risk-free discount rate**and calculate the present value of the option value.

For more realistic cases, other stochastic process can be easily added into the Monte Carlo framework, for example for the development cost, and also technical uncertainties that will be revealed along the R&D project.

Monte Carlo + Optimization with Genetic Algorithms

This paper started in my doctoral discipline "Applied Evolutionary
Computation" that I performed in the first semester of 2000, in the
Electrical Department of PUC-Rio.

Evolutionary approach like genetic algorithms, is a very flexible
technique that has been used lately in several applications of engineering
and few but growing applications in finance.

The main idea is: the genetic algorithm (GA) evolves the earlier
exercise free-boundary (threshold curve) of the American option and a
risk-neutral Monte Carlo simulation is used to evaluated the GA guess.

In the GA terminology, each guess is an organism (or chromosome) and the
most adapted organisms (that is, with higher option value evaluated by the
Monte Carlo simulation) has more probability to disseminate their genetic
material, throughout the new generations. The genes are characteristics of
the proposed solution, and here are or points in the threshold curve
and/or parameters of a multipiece function threshold curve.

This approach is more general in the sense that, in a near future, it
can get good solutions (near the optimal) for* any free-boundary
problem*, not only in financial-economics.

The main drawback of this procedure is the necessity to run a simulation for every new threshold generated by the GA algorithm, so that to get a near optimal threshold demand a lot of time.

However, for the user point of view, the evolution of a solution (like
the GA approach) is a very flexible method for Monte Carlo with American
options because is a * direct maximization method*. So that
with the passage of time, the faster computation environment this method
will become very promising.

The alternative is the famous Bellman dynamic-programming, which typically performed

Nowadays, there are a lot of research in this area (see bibliography), but not for using a modern direct maximization method like evolutionary computing. This research aims to fill this lacuna.

The potential of this method is its flexibility for the user modeling,
reducing the problem of "curse of modeling", permitting a large
popularization of real options modeling for people without a comprehensive
finance knowledge.

The only commercial tool available combining Monte Carlo simulation and
optimization with genetic algorithms is the Excel add-in RiskOptimizer,
from Palaside. This tool has advantages like its great flexibility, but
also important limitations (user cannot choose the fraction of the
population to be replaced in the steady-state reproduction among others).

These issues are discussed in the paper below.

Complex real options models for project economics evaluation suffer the
** curse of dimensionality**, with several sources of
uncertainties and with several options to invest in information. Details
from the changing practical reality highlight the

This paper presents a model of optimization under uncertainty with genetic algorithms and Monte Carlo simulation. This approach permits to get new insights for the real options theory.

Using the Excel-based software RiskOptimizer for a simple case (with a known value) and for a more complex real options model with investment in information. Some results from several experiments are presented with improvement suggestions. The strengths and weaknesses of RiskOptimizer are pointed out.

Download the compressed (zip) Word for Windows 97 file with the complete paper:

genetic_alg-real_options-marco_dias.zip, with 816 KB.