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Preliminaries on Financial Markets Read chapter Read first chapter

2016 | OriginalPaper | Chapter

5. Binomial Trees and Security Pricing Modeling

Authors : Arlie O. Petters, Xiaoying Dong

Published in: An Introduction to Mathematical Finance with Applications

Publisher: Springer New York

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Abstract

We introduce a discrete-time model of a risky security’s future price using a binomial tree. By increasing the number of time-steps in the tree, the assumption is that one obtains a more and more accurate model of the random future price of a security. This chapter covers: the general binomial tree model of future security prices, the Cox-Ross-Rubinstein (CRR) tree in the real world and risk-neutral world, the Lindeberg Central Limit Theorem with applications to the continuous-time limit of the CRR tree, and statistical and probability formulas for continuous-time security prices.

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previous chapter Capital Market Theory and Portfolio Risk Measures
next chapter Stochastic Calculus and Geometric Brownian Motion Model
Footnotes
1
For readers familiar with measure theory, it suffices for the functions f and g to be measurable, which includes the continuous functions. In fact, all the functions you will encounter in our financial applications are measurable. Some measure theory will be introduced in Chapter 6
 
2
Proof. \(\mathbb{P}(f(X) \in A,\ g(Y ) \in B) = \mathbb{P}(X \in f^{-1}(A),\ Y \in g^{-1}(B)) = \mathbb{P}(X \in f^{-1}(A))\ \mathbb{P}(Y \in g^{-1}(B)) = \mathbb{P}(f(X) \in A)\ \mathbb{P}(g(Y ) \in B)\), where the independence of X and Y was used in the second to the last equality.
 
3
Strictly speaking, we are considering the conditional expectation \(\mathbb{E}\left (S(t_{1})\vert S(t_{0}) = S_{0}\right ) \approx S_{0}\,\mathrm{e}^{(m-q)\,h_{n}}\) for n sufficiently large.
 
4
We remind readers to be mindful of the usage of the terminology real world; see the introductory paragraph to this section on page 219.
 
5
See page 19.
 
6
Recall that it is essentially impossible to determine a reliable value for m in the marketplace.
 
7
Recall that \(\mathsf{r}\) is the risk-free rate per annum (by default).
 
8
Theorem. (Classical CLT) Assume that X 1, X 2, … are i.i.d. random variables with each having a finite mean \(\mathbb{E}(X_{i}) =\mu _{0}\) and finite variance \(\mathrm{\,\,Var}(X_{i}) = \sigma _{0}^{2}> 0\). Let Z n be the standardization of the sample mean \(\bar{X}_{n} = \frac{1} {n}\left (X_{1} + \cdots + X_{n}\right )\), namely,
$$\displaystyle{Z_{n} = \frac{\bar{X}_{n} -\mu _{0}} {(\sigma _{0}/\sqrt{n})} = \frac{1} {\sqrt{n}}\,\sum _{j=1}^{n}\left (\frac{X_{j} -\mu _{0}} {\sigma _{0}} \right ).}$$
Then Z n converges in distribution to a standard normal Z as \(n \rightarrow \infty\).
 
9
The classical CLT follows from the Lindeberg CLT through the Dominated Convergence Theorem (expectation form) from measure theory; see Durrett [6, pp. 29,129].
 
10
Recall that two intervals arenonoverlapping if their interiors are disjoint.
 
11
The median of an absolutely continuous random variable X is a number, denoted \(\mathop{\mathrm{Med}}\nolimits (X)\), such that \(\mathbb{P}\left (X>\mathop{ \mathrm{Med}}\nolimits (X)\right ) = \frac{1} {2} = \mathbb{P}\left (X <\mathop{ \mathrm{Med}}\nolimits (X)\right ).\) Additionally, if \(X \sim \mathcal{N}(a,b^{2})\), where a and b > 0 are constants, then the mean and median of X satisfy \(\mathbb{E}\left (\mathrm{e}^{X}\right ) =\mathrm{ e}^{a+b^{2}/2 }\) and \(\mathop{\mathrm{Med}}\nolimits \left (A\,\mathrm{e}^{a\,X}\right ) = A\,\mathrm{e}^{a\,\mathop{\mathrm{Med}}\nolimits (X)}\).
 
12
Recall that the conditional expectation of an absolutely continuous random variable V given an event A with probability ℙ(A) > 0 is defined by \(\mathbb{E}\left (V \vert A\right ) = \frac{\mathbb{E}\left (V \,\boldsymbol{1}_{A}\right )} {\mathbb{P}(A)} = \frac{1} {\mathbb{P}(A)}\,\int _{A}v\,f_{V }(v)dv,\) where f V is the p.d.f. of V.
 
13
Replace m by \(\mathsf{r}\) and (hence) μ RW by μ ∗ in the formulas.
 
Literature
[1]
Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)MATH
[2]
Chance, D.: Proofs and derivations of binomial models. Technical Finance Notes, Louisiana State University (2007)
[3]
Chance, D.: Risk neutral pricing in discrete time. Teaching Note 96-02, Louisiana State University (2008)
[4]
Cox, J., Ross, S., Rubinstein, M.: Option pricing: a simplified approach. J. Financ. Econ. 7, 229 (1979)CrossRefMATH
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Cox, J., Rubinstein, M.: Options Markets. Prentice Hall, Upper Saddle River (1985)
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Jarrow, R., Rudd, A.: Option Pricing. Richard Irwin, Homewood (1983)
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Jarrow, R., Turnbull, S.: Derivative Securities. South-Western College Publishing, Cincinnati (2000)
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Leunberger, D.: Investment Science. Oxford University Press, Oxford (1998)
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McDonald, R.: Derivative Markets. Addison-Wesley, Boston (2006)
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Neftci, S.: An Introduction to the Mathematics of Financial Derivatives. Academic, San Diego (2000)MATH
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Pitman, J.: Setup for the Central Limit Theorem. STAT 205 Lecture Note 10, scribe D. Rosenberg. University of California, Berkeley (2003)
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Wilmott, P., Dewynne, N., Howison, S.: Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press, Cambridge (1995)CrossRefMATH
Metadata
Title
Binomial Trees and Security Pricing Modeling
Authors
Arlie O. Petters
Xiaoying Dong
Copyright Year
2016
Publisher
Springer New York
Book
An Introduction to Mathematical Finance with Applications

Print ISBN: 978-1-4939-3781-3

Electronic ISBN: 978-1-4939-3783-7

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-1-4939-3783-7_5

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