Abstract
This book is about the nature of stock prices and its attendant consequences in terms of risk and money management. What we know today is the culmination of many years of observation and profound insight by many influential thinkers. It is rightfully so that there should be such devotion to the subject for understanding the mysteries of the ups and downs of stock prices has far reaching consequences. We will encounter many such examples in this text.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The amount of a dividend is subtracted from the close price. See the discussion on page 25.
- 2.
On the basis of his derivation, Einstein was able to predict the size of water molecules. At the time the existence of atoms and molecules was still in doubt.
- 3.
It is customary to use upper case letters to denote random variables and lower case letters to denote an instance of the random variable.
- 4.
Note that \(\sqrt{\mathbb{E}({X}^{2 } )}\) is not necessarily equal to \(\mathbb{E}(\vert X\vert )\) and usually they are not equal. More generally, for a given function f, \(f(\mathbb{E}(X))\) is not necessarily equal to \(\mathbb{E}(f(X))\). A simple demonstration is provided by the function f(x) = x 2 in the present case since \({(\mathbb{E}(X))}^{2} = 0\) but \(\mathbb{E}({X}^{2}) = n{(\Delta x)}^{2}\).
- 5.
By definition the variance is \(\mbox{ var}(X) = \mathbb{E}({(X - \mathbb{E}(X))}^{2})\); see Section A.4.
- 6.
Stirling’s formula is \(n! \approx \sqrt{2\pi }{n}^{n+{ 1 \over 2} }{e}^{-n}\).
- 7.
The notation Z ∼ N(0, 1) means that the random variable Z is a sample from the density indicated, here, the normal density with mean 0 and variance 1.
- 8.
If only uniform samples U ∼ U(0, 1) are available, normal samples can be generated from them, see Section A.9.
- 9.
Theoretically this is so. But estimating the drift encounters a fundamental problem known as statistical blur. Error in the drift is given by the standard deviation. As Δ t is reduced, the per period value of μ decreases by the same factor, but the per period value of standard deviation decreases by the square root of Δ t. Hence for small periods, the error in drift exceeds the value of the drift itself.
- 10.
Approximately, it depends on the year.
- 11.
The symbol \(\prod _{i}^{n}a_{i}\) means the product of the a i from i = 1 to i = n.
- 12.
See Section A.1.
- 13.
The mean of a constant is the constant itself, the variance of a constant is zero. Of course the mean and variance of Z is 0 and 1 respectively.
- 14.
See Section A.3.
- 15.
The density may be plotted either as y(x) = f(x) with α as in (1.27) or y(x) = (1 ∕ S 0)f(x ∕ S 0) with \(\alpha = (\mu -{{1 \over 2}\sigma }^{2})T\).
- 16.
See Appendix C.
- 17.
A binomial tree graph is a directed graph with two outward edges at every node or none for leaf nodes. If the tree re-connects as here, it is called a binomial lattice.
- 18.
See Section A.1 for a refresher on the sum of geometric series.
- 19.
See Section A.8 for the rationale for batching statistical data.
- 20.
See Section A.6.
References
Achelis, S.B.: Technical Analysis from A to Z. McGraw-Hill, New York (2000)
Bailey, R.W.: Polar generation of random variaes with the t-distribution. Math. Comput. 62(206), 779–781 (1994)
Bernstein, P.L.: Against the Gods. Wiley, New York (1996)
Björk, T.: Arbitrage Theory in Continuous Time. Oxford University Press, New York (2004)
Borokin, A., Salminsen, P.: Handbook of Brownian Motion – Facts and Formulae. Birkhäuser Verlag, Basel (2002)
Bratley, P., Fox, B.L., Schrage, L.E.: A Guide to Simulation. Springer, New York (1983)
Capiński, M., Zastawniak, T.: Mathematics for Finance, an Introduction to Financial Engineering, Springer, New York (2003)
Carmona, R., Durrleman, V.: Pricing and hedging spread options. SIAM Rev. 45(4), 627–685 (2003)
Cont, R., Tankov, P.: Financial Modeling with Jump Processes. Chapman & Hall, New York (2004)
Derman, E.: Outperformance options. In: Nelken, I. (ed.) The Handbook of Exotic Options. Irwin, Chicago (1996). Chapter Nine
Fries, C.: Mathematical Finance. Wiley–Interscience, New York (2007)
Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, New York (2003)
Graybill, F.: Theory and Application of the Linear Model. Wadsworth Publishing, Belmont (1976)
Haugen, R.A.: The New Finance, the Case Against Efficient Markets. Prentice Hall, Upper Saddle River (1999)
Hull, J.C.: Options, Futures, and Other Derivatives. Prentice Hall, Upper Saddle River (2011)
Joshi, M.: The Concepts and Practice of Mathematical Finance. Cambridge University Press, New York (2003)
Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes. Academic, New York (1975)
Kelly, J.L. Jr.: A new interpretation of information rate. Bell Syst. Tech. J. 35, 917–926 (1956)
Kemna, A.G.Z., Vorst, A.C.F.: A Pricing Method for Options Based on Average Asset Values. Econometrisch Instituut, Erasmus University, Rotterdam (1990). http://ideas.repec.org/a/eee/jbfina/v14y1990i1p113-129.html
Longstaff, F.A.: Pricing options with extendible maturities: analysis and applications. J. Financ. XLV(3), 935–957 (1990)
Longstaff, F.A., Schwartz, E.S.: Valuing American options by simulation: a simple least squares approach. Rev. Financ. Stud. 14(4), 557–583 (2001)
Luenberger, D.G.: Investment Science. Oxford University Press, New York (1998)
Malkiel, B.G.: A Random Walk Down Wall Street. W.W. Norton, New York (2003)
Margrabe, W.: The value of an option to exchange one asset for another. J. Financ. 33, 177–186 (1978)
Merton, R.: Option pricing when the underlying stock returns are discontinous. J. Financ. Econ. 3, 124–144 (1976)
Meyer, G.H.: Pricing options with transaction costs with the method of lines. In: Otani, M. (ed.) Nonlinear Evolution Equations and Applications. Kokyuroku, vol. 1061. RIMS Kyoto University (1998)
Nelken, I. (ed.), Handbook of Exotic Options. McGraw-Hill, New York (1996)
Papapantoleon, A.: An introduction to Lévy processes with applications in finance. Web document http://page.math.tu-berlin.de/~papapan/papers/introduction.pdf (2005)
Park, C.-H., Irwin, S.H.: What do we know about the profitability of technical analysis. J. Econ. Surv. 21(4), 786–826 (2007)
Poundstone, W.: Fortunes Formula. Hill and Wang, New York (2005)
Ripley, B.D.: Stochastic Simulation. Wiley, New York (1987)
Roman, S.: Introduction to the Mathematics of Finance. Springer, New York (2012)
Schoutens, W.: Lévy Processes in Finance, Pricing Financial Derivatives. Wiley, New York (2003)
Shonkwiler, R., Mendivil, F.: Explorations in Monte Carlo Methods. Springer, New York (2009)
Yudaken, L.: Numerical pricing of shout options. MSc thesis, Trinity College, University of Oxford, Oxford, UK (2010)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Shonkwiler, R.W. (2013). Geometric Brownian Motion and the Efficient Market Hypothesis. In: Finance with Monte Carlo. Springer Undergraduate Texts in Mathematics and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8511-7_1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-8511-7_1
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-8510-0
Online ISBN: 978-1-4614-8511-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)