. 1
( 2 .)



>>

434 OTHER METHODS AND CONCLUSIONS

Notes on the Bibliography.
Here we point out references which complement and extend the material
in the text. There are several excellent books (many of which did not exist
when this book began) whose material intersects with that of several chap-
ters including the comprehensive book Glasserman (2004) and Jaeckel (2003).
Both of these texts deal speci¬cally with Monte Carlo methods in Finance.
There is a number of excellent more general Monte Carlo books as well, includ-
ing Gentle(2003), Fishman(1996) and the succinct but classic Hammersley and
Handscomb(1964) book, from which much of modern variance reduction has
grown and the pioneering paper of Hammersley(1960). More specialized books
on Monte Carlo are Robert and Cassella (1999) and Liu (2001). Boyle (1977)
is usually credited as the ¬rst use of simulation methodology in Finance. The
remainder of these notes are broken down by Chapter. There is discussion of
much of the ¬nance and econometrics material of this book in Campbell, Lo,
MacKinlay(1996), a book which takes a more statistical (hence somewhat more
critical) view of the standard models in Finance and econometrics.
Chapter 2. Background material on the types and uses of options can be
found in Boyle and Boyle (2001) and an excellent and popular discussion the
mathematical theory of Finance in Björk (1998). Bachelier(1900) deserves credit
as the founder of stochastic models in ¬nance. Fundamental papers on the valu-
ation of options include Black and Scholes (1973), Merton(1973), Cox and Ross
(1976a, 1976b), Cox, Ross and Rubinsten (1979), Hull and White(1987) and a
standard text on option pricing, Hull(1993). References for the material on En-
tropy in ¬nance are the theses of Samperi(1998), Gulko(1998) and Reesor(2002)
as well as papers by Gerber and Shiu(1994) and Avellaneda(1998).
Chapter 3
Uniform random number generators and their uses are comprehensively
treated in Fishman(1996), Fox(1986) with older but valuable treatments of non-
uniform generators in Devroye(1986) and Rubinstein(1981) and a very clear and
concise discussion of random number generation in Ripley(1983, 1987, 1988).
Gentle(2003) also has a nice exposition of di¬erent methods of random number
generation. For speci¬c generators, see Tadikamalla(1978), Cheng(1977, 1979)
for the Gamma and Beta,
For the adequacy of various models for ¬nancial data and alternatives see
Blattberg and Gonedes (1974), Press(1967), Fama(1965) and Fama and Roll,(1968,
1971), Fielitz and Rozelle (1983) (stable distributions), Chan et. al. (1992),
Clark(1973),
Chapter 4
The variance reduction in this chapter is heavily dependent on Hammersley
and Handscomb(1964) and Glasserman(2004) provides a number of applications
speci¬c to ¬nance.

Chapter 5
The material in the last part of this chapter is largely drawn from McLeish
(2002) but there are related results in a number of standard texts and references,
7.13. ALTERNATIVE MODELS 435

including Karatzas and Shreve(1999), Redekop(1995), Shepp(1979).
Chapter 6
An excellent reference for the use of Quasi-Random sequences is Niederre-
iter(1992) and applications to option pricing in the university of Waterloo thesis
of K.S. Tan and Tan and Boyle(2001).

Chapter 7
The problems of ¬nding a root or a maxima of a function when evaluations
are subject to noise is an old one, dating at least to the paper of Robbins
and Monro (1951). The work on Monte Carlo optimization is largely due to
Geyer(1995). Material on the EM algorithm and data augmentation is in the
books of Liu(2001) and Robert and Casella(1999). The estimation of parameters
for di¬usion processes is relatively well known, but in this case much is borrowed
from McLeish and Kolkiewicz(1997) and Kloeden and Platten(1992). The last
section on estimating volatility is largely in McLeish(2002).
Chapter 8
There is a considerable literature on sensitivity analysis in simulation, much
of it for discrete event simulations such as networks and queues. See for example,
Cao(1987) and Cao and Ho(1987), Arsham et. al. (1989), McLeish and Rollans
(1992), Glynn(1989). The treatment here of Monte Carlo optimization is similar
to Rollans(1991) and Reesor(2002).
436 OTHER METHODS AND CONCLUSIONS
Bibliography

[1] Acklam, P.J. (2003) An algorithm for computing the inverse
normal cumulative distribution function (Updated 2003-05-06)
http://home.online.no/˜pjacklam/notes/invnorm/

[2] Ahrens, J.H. and Dieter, U. (1974) Computer methods for sampling from the
gamma, beta, Poisson and binomial distributions. Computing 12, 223-246.

[3] Atkinson (1979) Recent Developments in the computer generation of Poisson
random variables Applied Statist. 28, 29-35

[4] Avellaneda, M. (1998) Minimum Entropy Calibration of Asset-Pricing Mod-
els. Int. J. Theor. App. Finance. 1. 447-472

[5] Bachelier, Louis Jean Baptiste Alphonse, (1900), Th©orie de la Speculation,
Annales de l™Ecole Normale Sup©rieure, Series 3, XVII, 26-81, Gautier Vil-
lars: Paris.

[6] P. Billingsley (1968). Convergence of Probability Measures. Wiley, New York

[7] Björk, T. (1998) Arbitrage Theory in Continuous Time Oxford U. Press,
Oxford

[8] Blattberg, R.C. and Gonedes, N.J. (1974), A comparison of stable and stu-
dent distributions as statistical models for stock prices, Journal of Business,
47(2), 244-280.

[9] Black, F. & M. Scholes (1973): The Pricing of Options and Corporate Lia-
bilities Journal of Political Economy, no 81, pp.637-659

[10] T. Bollerslev, T. (1986) Generalized Autoregressive Conditional Het-
eroscedasticity. Journal of Econometrics 31, 307”327.

[11] Boyle, P. (1977) Options: A Monte-Carlo Approach J. Financial Economics
4. 323-338

[12] Boyle, P. Broadie, M. and Glasserman, P. (1997) Monte Carlo Methods
for Security Pricing. J. of Economics Dynamics and Control 21, 1267-1321

437
438 BIBLIOGRAPHY

[13] Boyle, P. and Boyle, F. (2001 ) Derivatives: The Tools that Changed Fi-
nance Risk Books, London

[14] Box, G.E.P and Jenkins, G. (1976) Time Series Analysis: Forecasting and
Control. Holden-Day

[15] Box, G.E.P. and Tiao, G.C.(1973) Bayesian Inference in Statistical Analy-
sis. Addison-Wesley

[16] Bratley, P., B.L. Fox, L.E. Schrage, (1983) A Guide to Simulation, Springer-
Verlag, New York.

[17] Brennan, M.J. and Schwartz, E.S. (1979) A continuous time approach to
the pricing of bonds, Journal of Banking and Finance, 3, 133-155.

[18] Campbell, J.Y., Lo, A.W., MacKinlay, A.C. (1996) The Econometrics of
Financial Markets, Princeton U. Press, Princeton

[19] Cao, X., (1987) Sensitivity estimates based on one realization of a stochastic
system. Journal of Stat. Comp. and Simulation 27, 211-232.

[20] Cao, X. and Ho., Y.C. (1987). Estimating the sojourn time sensitivity in
queueing networks using perturbation analysis. JOTA 53, 353-375.

[21] Chambers, J.M. , Mallows, C.L. and Stuck, B.W. (1976) A method for
simulating stable random variables, Journal of the American Statistical As-
sociation, 71, 340-344 (corrections, 1987, ibid. 82. 704 and 1988, ibid. 83,
581)

[22] Chan, K. C., Karolyi, G.A., Longsta¬, F.A., Sanders, A.B. (1992) An Em-
pirical Comparison of Alternative Models of the Short-Term Interest Rates.
J. of Finance 47, 1209-1227

[23] Cheng, R.C.H. (1977) The generation of gamma variables with non-integral
shape parameter. Applied Stat. 26, 71-75

[24] Cheng, R.C.H. (1978) Generating beta variates with non-integral shape
parameters. Communications of the ACM 21, 317-322.

[25] Cheng, R.C.H. and Feast (1979) Some simple gamma variate generators,
Applied Statistics, 28, 290-295.

[26] Clark, P.K. (1973), A Subordinated Stochastic Process Model with Finite
Variance for Speculative Prices Econometrica, 41(1), pp135-155

[27] Cox, J.C. (1996), The constant elasticity of variance option pricing model
J.P.M. , Special Issue

[28] Cox, J.C. and A. Ross (1976) The Valuation of Options for Alternative
Stochastic Processes J. of Financial Economics, .3, 145-166
BIBLIOGRAPHY 439

[29] Cox, J.C., Ross, S.A. and Rubinstein, M. (1979) Option Pricing: A Sim-
pli¬ed Approach J. Financial Economics, 7, 229-263

[30] Cox, J.C., Ingersoll J.E., and Ross, S. A. (1985) A theory of the term
structure of interest rates, Econometrica, 53, 385-406.

[31] Cox, D.R. and Miller, H.D. (1965) The Theory of Stochastic Processes J.
Wiley, N.Y.

[32] Crane, M.A. and D.L. Iglehart, (1975). Simulating stable stochastic sys-
tems III. Regenerative processes and discrete-event simulations. Operations
Research 23, 33-45.

[33] Daniels, H.E. (1954) Saddlepoint Approximations in Statistics Ann. Math.
Statist. 25. 631-650

[34] Daniels, H.E. (1980) Exact Saddlepoint Approximations. Biometrika 67,
59-63

[35] Devroye, L. (1986) Non-Uniform Random Variate Generation. Springer,
New York

[36] Diebold, F.X. and Nerlove, M. (1989), The Dynamics of Exchange Rate
Volatility: A Multivariate Latent Factor ARCH Model, J of Applied Econo-
metrics.

[37] Duan, 1995, The GARCH Option Pricing Model, Mathematical Finance.

[38] Duan, J-C. (1995) The Garch Option Pricing Model. Mathematical Finance
5. 13-32

[39] Du¬e, D. (1996) Dynamic Assest Pricing Theory. (second edition) Prince-
ton University Press, Princeton, N.J.

[40] Engle, R.F. (1982): Autoregressive Conditional Heteroscedasticity with Es-
timates of the Variance of United Kingdom In¬‚ation. Econometrica 50, 987”
1008.

[41] Engle, R.F. and Rosenberg, J.V. (1995) Garch Gamma. J. of Derivatives
47-59.

[42] Fama, Eugene F., (1965) The Behaviour of Stock Market Prices”, Journal
of Business, 38, 34-105.

[43] Fama, E.F. and Roll, R. (1968) Properties of Symmetric Stable Distribu-
tions J. American Statistical Association, 817-836.

[44] Fama, E.F. and Roll, R., Parameter Estimates for Symmetric Stable Dis-
tributions,” (1971), J. American Statistical Association, 331-338.
440 BIBLIOGRAPHY

[45] Fielitz, B.D. and Rozelle, J.P. (1983), Stable Distributions and the Mix-
tures of Distributions Hypotheses for Common Stock Returns, Journal of
the American Statistical Association, 78, pp28-36.
[46] Fishman, G.S. (1996) Monte Carlo: Concepts, Algorithms and Applica-
tions. Springer, New York.
[47] Fox, B.L. (1986) Implementation and Relative E¬ciency of Quasirandom
sequence generators A.C.M. Trans. Math. Software 12, 362-376
[48] French, K.R., Schwert, G.W. and Stambaugh, R. (1987), Expected Stock
Returns and Volatility, J. Financial Economics. 19, 3-29
[49] Galanti, S. and Jung, A (1997) Low-discrepancy sequences: Monte Carlo
simulation of option prices. Journal of Derivatives 63-83
[50] Gentle, J.E. (2003) Random Number Generation and Monte Carlo Methods,
Second Ed. Springer, New York
[51] Garman, M. and Klass, M. (1980) On the estimation of security price
volatilities from historical data Journal of Business 53, 67-78.
[52] Gerber, H. and Shiu, E. (1994) Option Pricing by Esscher Transforms.
Trans. Soc. Actuaries, 46. 99-140
[53] Geyer, C. J. (1995). Estimation and Optimization of Functions. In Markov
Chain Monte Carlo in Practice, eds. W. R. Gilks, S. Richardson, and D. J.
Spiegelhalter, London: Chapman and Hall, 241-258.
[54] Glasserman, P. (2004) Monte Carlo Methods in Financial Engineering
Springer, New York.
[55] Gliek, J. (1987) Chaos: Making a new Science. Viking Press, New York.
[56] Glynn, P.W. (1989). Likelihood ratio derivative estimators for stochastic
systems. Proceedings of the 1989 Winter Simulation Conference, E.A. Mac-
Nair, K.J. Musselman, P. Heidelberger (eds.), 374-380.
[57] Gulko, B. L. (1998) The Entropy Pricing Theory. PhD Thesis, Yale Uni-
versity.
[58] Hammersley, J.M., (1960). Monte Carlo Methods for solving multivariable
problems, Annals of the New York Academy of Science, 86, 844-874.
[59] Hammersley, J.M. and Handscomb, D. C. (1964) Monte Carlo Methods .
Methuen, London
[60] Hesterberg, T. (1995) Weighted average importance sampling and defensive
mixture distributions. Technometrics 37, 185-194.
[61] Ho, T.S.Y. and Lee, S.B. (1986), Term Structure Movements and Pricing
Interest Rate Contingent Claims, J. Finance 41, 1011-1029
BIBLIOGRAPHY 441

[62] Hsieh, D. A. (1991) Chaos and Nonlinear Dynamics: Application to Finan-
cial Markets. J. of Finance 46. 1839-1877.

[63] Hull, J. C. (1993) Options, Futures, and Other Derivatives Securities Pren-
tice Hall, second ed. 1993, Englewood Cli¬s, NJ

[64] Hull, J. and White, A. (1987), The Pricing of Options on Assets with
Stochastic Volatilities, J. Finance 42, 281-300

[65] Jaeckel, P. (2003) Monte Carlo Methods in Finance, Wiley, New York

[66] Jöhnk, M.D. (1964) Erzeugung von Betverteilten and Gammaverteilten
Zufallszahlen Metrike 8. 5-15

. 1
( 2 .)



>>