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### Cox, Ross, & Rubinstein Option-Pricing Model (CRR model) Definition

The two-item option-pricing model, also known as CRR, is a mathematical formula used to estimate the value of an American option’s value. It is exercisable at any given time up to the expiration date. CRR assumes that the underlying asset’s price follows the binomial distribution, also known as the Binomial tree. This pricing option was developed by three mathematicians; Ross, Cox, and Rubinstein in 1979.

**A Little More on What is The Cox, Ross, & Rubinstein Option-Pricing Model (CRR model)**

The pricing option model assumes that the stock price volatility follows the downward and upward directions only. The stock price’s magnitude and the probability of rising or falling fluctuation are constant throughout the period of inspection.

As per the stock price’s historical volatility, there is all possible development paths’ stimulation of all the stock during a life-time period. It calculates the right of warrants and benefits for each node and path. The warrant’s price is usually calculated by the law.

The exercises can be done in advance for American warrants. For this reason, the theoretical price on each node is supposed to be greater for the two warrant exercises income, including the discounted calculated price.

**Cox-Ross-Runistein Binomial Option Pricing Model**

There are two complementary methods when it comes to the CRR model; the Black-Hughes option pricing and binomial option pricing model. For the binomial option pricing model, its derivation is relatively simple. It is suitable when it comes to explaining the option pricing’s basic theory.

The CRR model has based securities on the theory that price movement follows two possible directions in a given time interval (upward or downward). The assumption seems to be simple, but the model of binomial option pricing is appropriate when dealing with more complex options. It is because the time period can be divided into smaller time units.

**References for Cox, Ross, & Rubinstein Option Pricing Model**

https://www.investopedia.com/university/options-pricing/cox-rubenstein-binomial.asp

https://en.wikipedia.org/wiki/Binomial_options_pricing_model

**Academic Research for Cox, Ross, & Rubinstein Option Pricing Model**

**Option pricing**: A simplified approach, **Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. ***Journal of financial Economics***, ***7***(3), 229-263. **This paper presents a simple discrete-time model for valuing options. The fundamental economic principles of option pricing by arbitrage methods are particularly clear in this setting. Its development requires only elementary mathematics, yet it contains as a special limiting case the celebrated Black-Scholes model, which has previously been derived only by much more difficult methods. The basic model readily lends itself to generalization in many ways. Moreover, by its very construction, it gives rise to a simple and efficient numerical procedure for valuing options for which premature exercise may be optimal.

On the relation between binomial and trinomial **option pricing models**, **Rubinstein, M. (2000). On the relation between binomial and trinomial option pricing models. **This paper shows that the binomial option pricing model, suitably parameterized, is a special case of the explicit finite difference method. To prepare for writing the sequel volume of my new book Derivatives: A PowerPlus Picture Book, I recently reviewed the work on trinomial option pricing since Boyle’s 1988 JFQA paper. I found myself attracted to the Kamrad and Ritchken (1991) trinomial model because it seemed to be the “natural” generalization of the binomial model described by Cox, Ross and Rubinstein (1979). In that model, as is quite well known, the underlying asset price moves by return x over each period of elapsed time h, where x equals either u or d, while cash earns return r for sure. The resulting corresponding binomial tree is designed to emulate continuoustime risk-neutral geometric Brownian motion with annualized logarithmic mean log(r/d) — 2 and variance 2 , where r is the annualized riskless return (discrete) and d is the annualized payout return (discre…

On the relation between binomial and trinomial **option pricing models**, **Rubinstein, M. (2000). On the relation between binomial and trinomial option pricing models. **This paper shows that the binomial option pricing model, suitably parameterized, is a special case of the explicit finite difference method. To prepare for writing the sequel volume of my new book Derivatives: A PowerPlus Picture Book, I recently reviewed the work on trinomial option pricing since Boyle’s 1988 JFQA paper. I found myself attracted to the Kamrad and Ritchken (1991) trinomial model because it seemed to be the “natural” generalization of the binomial model described by Cox, Ross and Rubinstein (1979). In that model, as is quite well known, the underlying asset price moves by return x over each period of elapsed time h, where x equals either u or d, while cash earns return r for sure. The resulting corresponding binomial tree is designed to emulate continuoustime risk-neutral geometric Brownian motion with annualized logarithmic mean log(r/d) — 2 and variance 2 , where r is the annualized riskless return (discrete) and d is the annualized payout return (discre…

Foreign currency option values, **Garman, M. B., & Kohlhagen, S. W. (1983). Foreign currency option values. ***Journal of international Money and Finance***, ***2***(3), 231-237. **Foreign exchange options are a recent market innovation. The standard Black-Scholes option-pricing model does not apply well to foreign exchange options, since multiple interest rates are involved in ways differing from the Black-Scholes assumptions. The present paper develops alternative assumptions leading to valuation formulas for foreign exchange options. These valuation formulas have strong connections with the commodity-pricing model of Black (1976) when forward prices are given, and with the proportional-dividend model of Samuelson and Merton (1969) when spot prices are given.

Term structure movements and pricing interest rate contingent claims, **Ho, T. S., & Lee, S. B. (1986). Term structure movements and pricing interest rate contingent claims. ***the Journal of Finance***, ***41***(5), 1011-1029. **This paper derives an arbitrage‐free interest rate movements model (AR model). This model takes the complete term structure as given and derives the subsequent stochastic movement of the term structure such that the movement is arbitrage free. We then show that the AR model can be used to price interest rate contingent claims relative to the observed complete term structure of interest rates. This paper also studies the behavior and the economics of the model. Our approach can be used to price a broad range of interest rate contingent claims, including bond options and callable bonds.

Computing the constant elasticity of variance **option pricing formula**, **Schroder, M. (1989). Computing the constant elasticity of variance option pricing formula. ***the Journal of Finance***, ***44***(1), 211-219. **This paper expresses the constant elasticity of variance option pricing formula in terms of the noncentral chi-square distribution. This allows the application of well-known approximation formulas and the derivation of a whole class of closed-form solutions. In addition, a simple and efficient algorithm for computing this distribution is presented. Copyright 1989 by American Finance Association.

The valuation of uncertain income streams and the pricing of options, **Rubinstein, M. (2005). The valuation of uncertain income streams and the pricing of options. In ***Theory of Valuation*** (pp. 25-51). **A simple formula is developed for the valuation of uncertain income streams consistent with rational investor behavior and equilibrium in financial markets. Applying this formula to the pricing of an option as a function of its associated stock, the Black–Scholes formula is derived even though investors can trade only at discrete points in time.

**Option pricing **when the variance is changing, **Johnson, H., & Shanno, D. (1987). Option pricing when the variance is changing. ***Journal of Financial and Quantitative Analysis***, ***22***(2), 143-151. **The Monte Carlo method is used to solve for the price of a call when the variance is changing stochastically.

A lattice framework for **option pricing **with two state variables, **Boyle, P. P. (1988). A lattice framework for option pricing with two state variables. ***Journal of Financial and Quantitative Analysis***, ***23***(1), 1-12. **A procedure is developed for the valuation of options when there are two underlying state variables. The approach involves an extension of the lattice binomial approach developed by Cox, Ross, and Rubinstein to value options on a single asset. Details are given on how the jump probabilities and jump amplitudes may be obtained when there are two state variables. This procedure can be used to price any contingent claim whose payoff is a piece-wise linear function of two underlying state variables, provided these two variables have a bivariate lognormal distribution. The accuracy of the method is illustrated by valuing options on the maximum and minimum of two assets and comparing the results for cases in which an exact solution has been obtained for European options. One advantage of the lattice approach is that it handles the early exercise feature of American options. In addition, it should be possible to use this approach to value a number of financial instruments that have been created in recent years.

The accelerated binomial **option pricing model**, **Breen, R. (1991). The accelerated binomial option pricing model. ***Journal of Financial and Quantitative Analysis***, ***26***(2), 153-164. **This paper describes the application of a convergence acceleration technique to the binomial option pricing model. The resulting model, termed the accelerated binomial option pricing model, also can be viewed as an approximation to the Geske-Johnson model for the value of the American put. The new model is accurate and faster than the conventional binomial model. It is applicable to a wide range of option pricing problems.

Option replication in discrete time with transaction costs, **Boyle, P. P., & Vorst, T. (1992). Option replication in discrete time with transaction costs. ***The Journal of Finance***, ***47***(1), 271-293. **Option replication is discussed in a discrete‐time framework with transaction costs. The model represents an extension of the Cox‐Ross‐Rubinstein binomial option pricing model to cover the case of proportional transaction costs. The method proceeds by constructing the appropriate replicating portfolio at each trading interval. Numerical values of these prices are presented for a range of parameter values. The paper derives a simple Black‐Scholes type approximation for the option prices with transaction costs and demonstrates numerically that it is quite accurate for plausible parameter values.