# Classical models for estimating the fair price of derivative financial assets - Investments

## Classical models for estimating the fair price of derivative financial assets

The theory of evaluation of derivative financial assets was based on the fundamental work of F. Black and M. Scholes. With the help of R. Merton, the work was succeeded through the stochastic process attributed to the share price, and a number of assumptions about the options characteristics to obtain an analytical expression for the evaluation of the European option as a solution to the stochastic equation. Later, for this work, the authors were awarded the Nobel Prize in Economics.

In the future, the approach based on stochastic processes developed (significant progress was made on the basis of the application of the mathematical theory of martingales), but many options (including American ones) can not be described analytically. The modern high level of development of computing power and numerical methods has allowed to obtain solutions for the evaluation of options and in the absence of a compact analytical formula.

More popular and available for understanding and using in various areas the methods of "binomial" (and also "trilogical") trees appeared later (in the 1980s). It is interesting that these methods are a discrete approximation of stochastic differential equations.

Three American economists: J. Cox, R. Ross and M. Rubinshtein1 managed to prove that the binomial option evaluation under certain conditions in the limit is equal to the evaluation of the Black-Scholes model.

The main idea of ​​options evaluation is the possibility of a complete replacement (duplication) of the analyzed derivative asset by a certain portfolio (called portfolio-copy, replicative, duplicating portfolio). A duplicate portfolio is made up of a basic asset and a risk-free security. The main characteristic of the duplicating portfolio is that the cash flows on it completely copy the flows of the derivative in question. The second important characteristic is that the duplicate portfolio is risk-free.

The evaluation of financial options for today is based on three fundamental works:

1) the Black-Scholes model of 1973 (obtained with the help of R. Merton, an analytical solution of the stochastic equation for a simple European option per share) ( Black - Scholes model, BSM );

2) the Koch and Ross model of valuation in 1976 with the assumption of a (replicating portfolio model) ,

3) the binomial Cox-Ross-Rubinstein estimation algorithm of 1979 ( Cox - Ross - Rubinstein model).

The key premise of all models is that the benefits of the option can be completely duplicated through the possession of other assets existing on the market. The option is in a sense a degenerate financial asset. Fair (equilibrium) option evaluation should be equal to the present value of the constructed replacement portfolio (replicating portfolio or portfolio-copy).

Consideration of evaluation logic is better to start with a binomial approach. Although it appeared later than the classical Black-Scholes model, but it captivates with simplicity and the possibility of visual interpretation.

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