# Binomial And Black And Scholes Pricing Models

Strengths and weakness of the Binomial and Black and Scholes Pricing models and their applications:

Introduction:

The binomial and the Black and Schole models are option valuing models, the Binomial model involves determining the value of options using a tree like format whereby the value of the option is determined by the expiration time period of the option and volatility, for the Black and Schole model the value of options is determined by simply getting a derivative that helps get the discount rates of options.

Binomial pricing model:

The binomial pricing model was introduced by Ross, Cox and Rubinstein in 1979; it provides a numerical method, in which valuation of options can be undertaken.

Application:

This model breaks down the option into many potential outcomes during the time period of the option, this steps form a tree like format where by the model assumes that the value of the option will rise or go down, this value is calculated and it is determined by the expiration time and volatility. Finally at the end of the tree of the option the final possible value is determined because the value is equal to the intrinsic value.

Assumptions:

- The model also assumes that the market is efficient in that people cannot predict the direction of change in the stock prices.
- The interest rates are constant and known and therefore they do not change in the time we consider an option.
- The model assumes that there are no dividends paid during the period in which one considers the option.
- The model assumes that the returns on the stocks are normally distributed.
- It also assumes that no commission is paid when buying or selling stock.

Strengths:

- This valuation method is simple mathematics and therefore it is used incorporated in computer software for the valuation of options making it easy to use the model.
- This model is considered to be more accurate than the Black Schole model, it is considered more accurate especially when valuing options for a long period of time.
- The binomial model solves an option using a computational procedure while the black and Schole solves the same option using an analytical procedure. Therefore the binomial method is much easier to use.

Weaknesses:

- The binomial model is too slow compared to the Black and Schole model and takes a much longer time to calculate options using this model.

The black and scholes model:

The Black and Scholes model was introduced by Myron Scholes and Fisher Black in the year 1973, this model involves calculation of a derivative that aids in showing how the discount rates of various options vary with time and at the same time stock price.

Application:

The black and Schole model entails the calculation of a theoretical price ignoring the presence of dividends and where the value of the options is determined by the price of stock, volatility, time period to expiration and interest rates.

The model:

C = SN(d1) - Ke(-rt)N(d2)

Where C is the call premium

S is the present stock price

T is the time period being considered

K is the option striking price

R is the risk minus interest rate

N is the cumulative normal distribution

E is the exponential value

The model can be divided into two parts and where the first step which is SN(d1) which is the expected benefits and the second part which is Ke(-rt)N(d2) is the present value of the value paid on the expiry time of the option.

Assumptions:

- The model assumes that there are no dividends paid during the period in which one considers the option.
- The model also assumes that the market is efficient in that people cannot predict the direction of change in the stock prices.
- It also assumes that no commission is paid when buying or selling stock.
- The interest rates are constant and known and therefore they do not change in the time we consider an option.
- Finally the model assumes that the returns on the stocks are normally distributed.

Strength:

- This model is much faster in the calculation of option value as compared to the binomial model which is slow and time consuming.

Weaknesses:

- This model uses analytical procedures in the calculation of option value and therefore it is not as accurate as the binomial model.
- This method of option valuation is much more difficult than the binomial model which is much simpler because it entails simple mathematical calculations.

Conclusion:

The binomial and the Black and Schole model have similar assumptions, however the their calculation differs where the Black and Schole model involves the use of a derivative while the binomial model involves the use of a tree which analysis all possible outcomes, the Binomial model however is more accurate than the Scholes and Black model.

Therefore the best model to use in valuation of models is the Binomial model which is more accurate than the Black and Schole model, the model is also easier to use than the Black and Schole model because it uses a computational procedure where all outcomes are analyzed in steps for the time period.

Reference:

Simon Benninga (1997) Financial Models, MIT Publishers, New York

Steinmetz R. and Stroughair J. (1993) "Implementing the numerical option pricing models" the mathematical Journal, volume 3 issue 4, Page 66 to 73

Ross, Cox and Rubinstein (1979) "Option pricing model: A simplified approach" Financial Economics Journal, issue 7, page 229 to 263

Hull J. (1997) Options, Future and Derivatives, Prentice hall publishers, New York

Omberg E (1987) "the convergence of binomial pricing and compound option model" Journal of Finance, volume 42 issue 2, page 463 to 469