Explanation of Black-Scholes Option Valuation Equation
The main equation for the Black-Scholes Option Valuation is as follows:
C = S.N(d1. - X.e-RT. N(d2. (12)
C is the value of call option
S is the current market price of shares in question
X is the future exercise price
R is the risk free interest rate per annum
T is the time to expiry (in years)
e is the mathematical constant 2.718, used to calculate value on a continuous compounding basis.
(X.eRT is the future value or amount, at time T, of a sum of money X invested @ R% on continuous compounding basis. If compounded annually, the amount at R% compound rate of interest at time T is X(1 + r/100)T, which is roughly equal to X.eRT)
N(d1) and N(d2) represent the cumulative area under the normal distribution curve for a 'z'* value of d1 and d2 where
d1 = log(S / X) RT
+ +. V.T
d2= d1– V.T
And log (SIX) is the natural log (to the base 'e') of SIX.
*'z' is the standard normal variable (observation value) in a normal distribution with a mean (μ) = 0, and a standard deviation 6 = 1. The value of z is given by:
z = X − μ (A)
X is a variable, which can take any value in a normal distribution.
The value of z reflects the deviation from the mean (vertical line drawn at the mean value 0 and hence determines the area under the curve between - z and + z.
Suppose a normal distribution has a mean (μ) of 10 and a standard deviation σ of 5. If we want to know the probability of x having a value between 0.2 and 19.8, we shall find z for x = 0.2 and x = 19.8 by using the formula (A). The value of z works out as - 1.96 and + 1.96, and for z = 1.96, the area under the normal curve is a 0.95 from the tables. There is thus 95% probability that x will have a value between 0.2 and 19.8.