BLACK SCHOLES MODEL
OPTION PRICE EVALUATION BlackScholes Model
Principles of calculation
The model of Black and Scholès defines the value of an option at time t as being the average of the possible intrinsic values of the latter weighted by their respective probability of occurrence.
The model calculates the possible prices of the underlying asset at maturity, as well as their respective probability of occurrence, on the basis of the fundamental assumption that it is a random variable whose distribution law follows a Gaussian curve. It determines the present value at the money market rate of the option on the date t of the calculations.
Calculation assumptions
The BlackScholes model is based on a number of cumulative assumptions

the price of the underlying asset S _{t} follows a geometric Brownian motion with a constant volatility σ and a constant μ derivative

,

there are no arbitration opportunities,

time is a continuous function

it is possible to make short sales

there are no transaction costs,

there is a riskfree rate, known in advance and constant,

all the underlying are perfectly divisible (for example, you can buy 1/100 ^{th} of a share),

the share does not pay dividends between the time the option is valued and the option expires.
The BlackScholes formula makes it possible to calculate the theoretical value of an option from the following five data:

the current value of the underlying stock,

the time remaining in the option before it expires (expressed in years),

the exercise price set by the option,

the riskfree interest rate,

the volatility of the share price.
If the first four data is obvious, the volatility of the asset is difficult to assess. Two analysts may have a different opinion on the value of to choose.
The theoretical price of a call giving the right but not the obligation to buy the asset S at the value K on the date T, is characterized by its pay off :
It is given by the expectation under neutral risk probability of the discounted terminal pay off
,
either the BlackScholes formula:
The theoretical price of a put , pay off is given by:
with

the distribution function of the reduced e centered normal law , i.e.


The formula could be reversed, so as to calculate on the basis of the price of the option which is quoted in the markets the value of so that the BlackScholes formula gives exactly this price. This makes it possible to calculate the implied volatility.
Reliability of the Black and Scholès model
The model retains simplifying assumptions which are likely to cause significant differences between the reality of the market and the values given by the model. In fact, in a period of stability, the majority of operators using the Black and Scholtès formula to set their price, the model has a selffulfilling value. As traders in the market use the same valuation benchmark the market value tends to be the value set by the model.
BLACK SCHOLES MODEL
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