Page 98 - DCOM510_FINANCIAL_DERIVATIVES

Unit 6: Valuation and Pricing of Options

Model. The Black-Scholes model, often simply called Black-Scholes, is a model of the varying Notes
price over time of financial instruments, and in particular stock options. The Black-Scholes
formula is a mathematical formula for the theoretical value of so-called European put and call
stock options that may be derived from the assumptions of the model. The equation was derived
by Fischer Black and Myron Scholes in their paper “The Pricing of Options and Corporate
Liabilities” published in 1973.
The Black and Scholes Option Pricing Model didn’t appear overnight, in fact, Fisher Black started
out working to create a valuation model for stock warrants. This work involved calculating a
derivative to measure how the discount rate of a warrant varies with time and stock price. They
built on earlier research by Paul Samuelson and Robert C. Merton. The fundamental insight of
Black and Scholes is that the call option is implicitly priced if the stock is traded. The use of the
Black-Scholes model and formula is pervasive in financial markets.
The value of call option is calculated as follows:
–rt
C = S N (d ) – E e N (d )
0 1 2
C = Theoretical Call Premium
S = Current Stock Price
T = Time until option expiration
K = Option Striking Price

r = Risk-Free Interest Rate
N = Cumulative standard normal Distribution
e = exponential term (2.7183)

2
I (S/K)  (r   /2)
d  n
1
 T
2
I (S/K)  (r   /2)T
d 2  n  d 1   T
 T
C (S, T) = price of the European call option,
P (S, T) = price of the European put option,
s = the annualised standard deviation of underlying asset price.

The price of a put option may be computed from this by put-call parity and simplifies to
–rT
d
P( S, T)  Ke N( d )  SN(  ).

2 1
6.4.1 Assumptions Underlying Black-Scholes Model
The key assumptions of the Black-Scholes model are:
1. The risk-free interest rate exists and is constant (over the life of the option), and the same
for all maturity dates.

2. The short selling of securities with full use of proceeds is permitted.
3. There is no transactions cost and there are no taxes. All securities are perfectly divisible
(e.g. it is possible to buy 1/100th of a share).

4. There is no risk less arbitrage opportunities; security trading is continuous.

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