The Black-Scholes 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 risk-free 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.
Price of a call
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 Black-Scholes formula:
Price of a put
The theoretical price of a put , pay off is given by:
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 Black-Scholes 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 self-fulfilling 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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ALEA AND FORECASTING
INFORMATION THEORY AND FRACTAL MATHEMATICS