SYLLABUS Previous: 4.3.2 Solution of the
Up: 4.3 Methods for European
Next: 4.4 Methods for European
4.3.3 BlackScholes formula
[ SLIDE
formula 
N(x)  same
VIDEO as previous section
modem 
LAN 
DSL]
In the case of plain vanilla call and put options, the price can be
evaluated in terms of the cumulative normal distribution
and
yields the well known BlackScholes formula

(4.3.3#eq.1) 

(4.3.3#eq.2) 

(4.3.3#eq.3) 
Remember that
denotes the (spot) price of an underlying share that
pays a dividend
and has a historical volatility
,
is
the strike price of the option evolving in time
from the
present to the expiry date and
the riskfree interest (spot) rate.
Note that the last relation (4.3.3#eq.3) is nothing more
than the putcall parity previously obtained in (2.1.3#eq.2),
where the guaranteed payoff has been discounted back in time to achieve
the risk free return of the spot rate.
The cumulative normal distribution is related with the socalled error
function
, which
is available in Matlab and can be approximated with 6 digits
accuracy using the polynomial expansion [1]
with the coefficients
g=0.2316419, a_{1}=0.319381530,
a_{2}=0.356563782, a_{3}=1.781477937,
a_{4}=1.821255978, a_{5}=1.330274429.
SYLLABUS Previous: 4.3.2 Solution of the
Up: 4.3 Methods for European
Next: 4.4 Methods for European