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# 2-3 - Two Alternative Binomial Option Pricing Model...

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Two Alternative Binomial Option Pricing Model Approaches to Derive Black-Scholes Option Pricing Model 1 CHENG-FEW LEE Department of Finance and Economics, Rutgers Business School Rutgers University, New Brunswick New Jersey, U.S. CARL S. LIN Department of Economics Rutger University, New Brunswick New Jersey, U.S. Abstract In this chapter, we review two famous models on binomial option pricing, Rendleman and Barter (RB, 1979) and Cox, Ross, and Rubinstein (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 which fits their interests and is easier to follow. 1. Introduction The main purpose of this chapter is to review two famous binomial option pricing model: Rendleman and Barter (RB, 1979) and Cox, Ross, and Rubinstein (CRR, 1979). First, we will give an alternative detailed derivation of the two models and show that the limiting results of the two models both lead to the celebrated Black- 1 Section 3 of this chapter is essentially drawing from the paper by Lee et al.(2004). 1

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Scholes formula. Then we will make comparisons of the two different approaches and analyze the advantages of each approach. Hence, this chapter can help to understand the statistical aspects of option pricing models for Economics and Finance professions. Also, it gives important financial and economic intuitions for readers in statistics professions. Therefore, by showing two alternative binomial option pricing models approaches to derive the Black-Scholes model, this chapter is useful for understanding the relationship between the two important optional pricing models and the Black-Scholes formula. 2. The Two-State Option Pricing Model of Rendleman and Bartter In Rendleman and Bartter (1979), a stock price can either advance or decline during the next period. Let T H + and T H - represent the returns per dollar invested in the stock if the price rises (the + state) or falls (the - state), respectively, from time T-1 to time T. And T V + and T V - the corresponding end-of-period values of the option. Let R be the riskless interest rate, Rendleman and Bartter (1979) show that the price of the option can be represented as a recursive form 2
1 (1 ) ( 1 ) ( )(1 ) T T T T T T V R H V H R P H H R + - - + - - + - + - + - - = - + that can be applied at any time T-1 to determine the price of the option as a function of its value at time T.

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2-3 - Two Alternative Binomial Option Pricing Model...

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