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Review of Quantitative Finance and Accounting

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01-08-2016 | Original Research

Alternative methods to derive option pricing models: review and comparison

Authors: Cheng-Few Lee, Yibing Chen, John Lee

Published in: Review of Quantitative Finance and Accounting | Issue 2/2016

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Abstract

The main purposes of this paper are: (1) to review three alternative methods for deriving option pricing models (OPMs), (2) to discuss the relationship between binomial OPM and Black–Scholes OPM, (3) to compare Cox et al. method and Rendleman and Bartter method for deriving Black–Scholes OPM, (4) to discuss lognormal distribution method to derive Black–Scholes OPM, and (5) to show how the Black–Scholes model can be derived by stochastic calculus. This paper shows that the main methodologies used to derive the Black–Scholes model are: binomial distribution, lognormal distribution, and differential and integral calculus. If we assume risk neutrality, then we don’t need stochastic calculus to derive the Black–Scholes model. However, the stochastic calculus approach for deriving the Black–Scholes model is still presented in Sect. 6. In sum, this paper can help statisticians and mathematicians understand how alternative methods can be used to derive the Black–Scholes option model.

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In this section, we follow the notations used by Cox et al. (1979).

To sell the call option means to write the call option. If a person writes a call option on stock A, then he or she is obliged to sell at exercise price X during the contract period.

Here, we distinguish udS and duS, and count them as two possible outcomes.

This uses the property that the variance of a variable Q can be calculated by: E(Q)² – [E(Q)]², where E(·) denotes the expected value.

Here, Taylor series expansion is used: \( e^{x} = 1 + x + \frac{{x^{2} }}{2!} + \frac{{x^{3} }}{3!} + \cdots \).

\( e^{{r\Delta t}} = 1 + r\Delta t \) and \( e^{{2r\Delta t}} = 1 + 2r\Delta t \) when higher powers of \( \Delta t^{2} \) are ignored. The solution to u and d implies that \( u = 1 + \sigma \sqrt{\Delta t} + \frac{1}{2}\sigma^{2}\Delta t \), \( d = 1 - \sigma \sqrt{\Delta t} + \frac{1}{2}\sigma^{2}\Delta t \). All of these expansions satisfy Eq. (19).

The details of program presentation for stock price, call option price, and also put option price using both binomial model and Black–Scholes model are shown in Chapter 18 in Lee et al. (2013a).The Excel VBA Code for binomialBS_OPM.xls can be found in Appendix 18A in Lee et al. (2013a). The readers can read them if interested. Due to space limit, we only present the illustrative decision trees for call options using binomial model and Black–Scholes model in the main text.

Please note that notation T used here is the number of periods rather than calendar time.

Please note that some of the variables used in this section are different from those used in Sects. 2.1 and 4.1.

We first solve equality \( S_{0} H^{{ +^{i} }} H^{{ -^{T - i} }} = X \). This yields \( i = \frac{{\ln (X/S_{0}) - T\ln (H^{ - } )}}{{\ln H^{ + } - \ln H^{ - } }} \). To get a, the minimum integer value of i for which \( S_{0} H^{{ +^{i} }} H^{{ -^{T - i} }} > X \) will be satisfied, we should note a as: \( a = 1 + INT\left[ {\frac{{\ln (X/S_{0}) - T\ln (H^{ - } )}}{{\ln H^{ + } - \ln H^{ - } }}} \right] \).

In “Appendix”, we will use de Moivre–Laplace theorem to show that the best fit between the binomial and normal distributions occurs when the binomial probability (or pseudo probability in this case) is \( \frac{1}{2} \).

Using Taylor expansion, we have \( e^{x} = 1 + x + \frac{{x^{2} }}{2!} + O(x^{2} ) \).

The presentation and derivation of this section follow Garven (1986), Lee et al. (2013a, b).

Now that \( x = e^{y} \), then \( dx = d(e^{y} ) = e^{y} dy = xdy \).

The second equality is obtained by substituting the PDF of normal distribution into \( \int_{{\ln ({\text{a}})}}^{\infty } {f(y)e^{y} dy} \) and does the appropriate transformation.

Black and Scholes have used two alternative methods to derive this equation. In addition, the careful derivation of this equation can be found in Chapter 27 of Lee et al. (2013a), which was written by Professor A.G. Malliaris, Loyola University of Chicago. Beck (1993) has proposed an alternative way to derive this equation, and raised questions about the methods used by Black and Scholes. In the summary of his paper, he mentioned that the traditional derivation of the Black–Scholes formula is mathematically unsatisfactory. The hedge portfolio is not a hedge portfolio since it is neither self-financing nor riskless. Due to compensating inconsistencies, the final result obtained is nevertheless correct. In his paper, these inconsistencies, which abound in the literature, were pointed out and an alternative, more rigorous derivation avoiding these problems is presented.

The following procedure has closely related to Kutner (1988). Therefore, we strongly suggest the readers read his paper.

The solution is obtained as an application of the general Fourier integral. See Churchill (1963, pp. 154–155) for more details.

The lower limit exists since if \( u < 0,\,f(u) = 0 \). Therefore, we require \( u + 2\eta \sqrt{v} \geq 0\), i.e. \( \eta \ge - {u \mathord{\left/ {\vphantom {u {2\sqrt{v} }}} \right. \kern-0pt} {2\sqrt{v} }} \).

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Title: Alternative methods to derive option pricing models: review and comparison
Authors: Cheng-Few Lee
Yibing Chen
John Lee
Publication date: 01-08-2016
Publisher: Springer US
Published in: Review of Quantitative Finance and Accounting / Issue 2/2016
Print ISSN: 0924-865X
Electronic ISSN: 1573-7179
DOI: https://doi.org/10.1007/s11156-015-0505-5

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