Page 241 - DCOM504_SECURITY_ANALYSIS_AND_PORTFOLIO_MANAGEMENT

Security Analysis and Portfolio Management

Notes Then, the equation for the call looks like this:
c = 35N(0.6) – 31.6693N(0.4)
Here d is a standardised normal random variable N(d ) is a cumulative standardised normal
1 1
probability distribution. It represents the area under the standardised normal curve from Z.
By referring to mathematical table given at the end of book on the standardised normal
distribution we can arrive at the values of –N(d ) and N(d ) as follows:
1 2
The value of N(d ) when d = 0.6 is 0.7257
1 1
The value of N(d ) when d = 0.4 is 0.6554
2 2
When the above values are substituted in the equation, then
c = 35 (0.7257) – 31.6693 (0.6554) – 4.6434

Valuing Put Options with the Black-scholes Model

An alternative form of valuation is to use the Black-Scholes formula for a put, which is:
P = Xe –r(T – t) [(1 – N(d ) ] – S[1 – N(d )]
1 1 1
Where d and d are as given in the section deriving a call option.
1 2
Note that [1 – N(d )] is the same as N(–d ) and [1 – N(d )] is the same as N(–d ).
2 2 1 1
Using the same data that we used in valuing the call, the put option value is calculated as
follows:

P = 31.6693 (0.3446) – 35(0.2743) = 1.3127

Example: Calculate the value of option from the following information
S = 20, K = 20, t = 3 months or 0.25 years
r = 1296 = 0.12, 2 = 0.16
Solution:

Since d and d are required inputs for Black-Scholes Option Pricing Model.
1 2
ln(20/20) + (0.12+(0.16/2)(0.25)
d =
1 0.40(0.50)

0 0.05
= = 0.25
0.20
d = d – 0.20 = 0.05
2 1
N(d ) = N(0.25)
1
N(d ) = N(0.05)
2
The above two represent area under a standard normal distribution function.
From table given at the end of the book, we see that value d = 0.25 implies a probability of
1
0.0987 + 0.5000 = 0.5987, so N(d ) = 0.5987. Similarly, N(d ) = 0.5199. We can use those values to
1 2
solve the equation in Black-Scholes Option Pricing Model

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