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BLACK and SCHOLES OPTION PRICING MODEL (BSOPM):
BSOPM has been referred to as a milestone in the development of financial theory on the grounds that
it significantly raised the level of sophistication in the quantitative valuation of complex financial
instruments. While options have been around for hundreds of years, their market value was mostly
determined by the forces of supply and demand prior to BSOPM. Now, options are valued based on
empirical data, which tends to give them a truer value (perhaps closer to an inherent value). The result
of BSOPM is the price (or value) of a European call, or an American call on a non-dividend-paying stock.
The input variables are:
S = Current price ("spot" price) of the underlying stock. As this pps changes through time, so too will the
price of the option.
K = The "strike" price of the call. The strike is fixed (does not change) throughout the life of the option.
It is the price that the buyer of the option will pay for the stock if the call is exercised.
r = The "risk free rate". As with many derivative models, the current LIBOR (for a time period equivalent
to the remaining life of the option) is used as the risk free rate. However, note that there is always
some academic debate as to what quoted rate should be used as the “risk free rate”.
T = The amount of time remaining to the expiration date. T is in years, so an option with 3-months
remaining would have T = .25.
σ = The standard deviation of the underlying asset's annualized returns based on historical data. This
variable is necessarily based on historical data, and as such, is an imperfect estimator of future volatility
– a weak link in the model.
The intermediate variables are:
d1 = the z-score, or number of standard deviations away from the mean of the distribution of stock
prices. This is a measure of the spot price relative to the mean stock price.
d2 = the z-score of the strike price relative to the mean stock price.
N(d1) = the cumulative standard normal distribution for d1, or the area under the curve given d1, or the
probability related the spot price. To read more about converting z-scores (d1) to the cumulative
standard normal distributions (N(d1)), see stats.doc.
N(d2) = the cumulative standard normal distribution for d2.
The output variable is:
C = The price of the call.
The BSOPM Model is:
C = S x N(d1) – K x (e^-rT ) x N(d2)
A more detailed model with the calculations are listed on a separate pages at bsopm1.jpg , and
bsopm2.jpg.
© 2009 Edward Harding