Page 240 - DCOM504_SECURITY_ANALYSIS_AND_PORTFOLIO_MANAGEMENT

Unit 8: Derivatives

The variables are: Notes
S = Stock price
X = Strike price
t = Time remaining until expiration, expressed as a percent of a year
r = Current continuously compounded risk-free interest rate
v = Annual volatility of stock price (the standard deviation of the short-term
returns over one year). See below for how to estimate volatility.
ln = Natural logarithm
N(x) = Standard normal cumulative distribution function
e = Exponential function
or
The Black-Scholes model for valuing a European call is:
C = SN(d ) – Xe –r(T–t) N(d )
1 2
Where,
ln(S/X) (r 2 /2(T t)
D =
1 T t

D = d T t
2 1
C = Call option premium
S = Current asset price
X = Exercise price
T-t = Time to expiry in decimals of a year
= The annualized standard deviation of the natural log of the asset price
relative in decimals
ln = Natural logarithm
N(d ) = Cumulative standard normal probability distribution
1
d and d = Standardised normal variables
1 2
r = Risk-free rate on interest in decimals (continuously compounded)

Example: The current asset price is 35.0, the exercise price is 35.0, the risk-free rate of
interest is 10%, the volatility is 20% and the time to expiry is one year. Thus S = 35, X = 35, (T – t)
= 1.0, r = 0.1 and = 0.2.
Solution:

First, we calculate d , then d and, finally, the present value of the exercise price Xe –r(T – t)
1 2
2
ln(35/35) + (0.1 + 0.2 /2) × 1.0
d = = 0.60
1 0.2 1.0
d – d – 0.2 1.0 = 0.4
2 1
Xe –r(T – t) = 35e –(0.1 × 1.0) = 31.66934

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