Two Alternative Binomial Option Pricing Model Approaches to Derive Black-Scholes Option Pricing Model

Abstract

In this chapter, we review two famous models on binomial option pricing, Rendleman and Barter (RB 1979) and Cox et al. (CRR 1979). We show that the limiting results of the two models both lead to the celebrated Black-Scholes formula. From our detailed derivations, CRR is easy to follow if one has the advanced level knowledge in probability theory but the assumptions on the model parameters make its applications limited. On the other hand, RB model is intuitive and does not require higher level knowledge in probability theory. Nevertheless, the derivations of RB model are more complicated and tedious. For readers who are interested in the binomial option pricing model, they can compare the two different approaches and find the best one that fits their interests and is easier to follow.

Appendices

Appendix 26A The Binomial Theorem

$${(x+y)}^{n} ={\sum \nolimits}_{k=0}^{n}\left (\begin{array}{*{20}c} n\\ k\\ \end{array} \right ){x}^{k}{y}^{n-k}$$

Lindberg-Levy Central Limit Theorem

If x ₁, …, x _n are a random sample from a probability distribution with finite mean μ and finite variance σ² and $\bar{x} = \left ( \frac{1} {n}\right ){\sum \nolimits}_{i=1}^{n}{x}_{i}$, then

$$\sqrt{n}(\bar{{x}}_{n} - \mu )\mathop{\longrightarrow}\limits_{}^{d}N\left [0,{\sigma}^{2}\right ].$$

Proof of Theorem 1

Since

$$\begin{array}{@{}ll} p{\left \vert \log u -\hat{\mu}\right \vert}^{3} & = p{\left \vert \log u - p\log \dfrac{u} {d} -\log d\right \vert}^{3} \\ & = p{(1 - p)}^{3}{\left \vert \log \dfrac{u} {d}\right \vert}^{3} \end{array}$$

And

$$\begin{array}{@{}ll} (1 - p){\left \vert \log d -\hat{\mu}\right \vert}^{3} & = (1 - p){\left \vert \log d - p\log \dfrac{u} {d} -\log d\right \vert}^{3} \\ & = {p}^{3}(1 - p){\left \vert \log \dfrac{u} {d}\right \vert}^{3}, \end{array}$$

we have $p{\left \vert \log u -\hat{\mu}\right \vert}^{3} + (1 - p){\left \vert \log d -\hat{\mu}\right \vert}^{3} = p(1 - p)[{(1 - p)}^{2} - {p}^{2}]{\left \vert \log \frac{u} {d}\right \vert}^{3}$ .

Thus

$$\begin{array}{l} \dfrac{p{\left \vert \log u -\hat{\mu}\right \vert}^{3} + (1 - p){\left \vert \log d -\hat{\mu}\right \vert}^{3}} {\hat{{\sigma}}^{3}\sqrt{n}} \\ = \dfrac{p(1 - p)[{(1 - p)}^{2} - {p}^{2}]{\left \vert \log \frac{u} {d}\right \vert}^{3}} {{\left (\sqrt{p(1 - p)}\log \left (\dfrac{u} {d}\right )\right )}^{3}\sqrt{n}} \\ = \dfrac{{(1 - p)}^{2} + {p}^{2}} {\sqrt{\mathit{np} (1 - p)}} \rightarrow 0\quad \mathrm{as}\ n \rightarrow \infty \end{array}.$$

Hence the condition for the theorem to hold as stated in Equation (26.24) is satisfied.