Page 116 - DMGT513_DERIVATIVES_AND_RISK_MANAGEMENT

Unit 8: Option Pricing

3. The value of (1+r) is greater than d , but smaller than u i.e., u<1+r<d. This condition Notes
or assumption ensures that there is no arbitrage opportunity.
4. The investors are prone to wealth maximization and lose no time in exploiting the
arbitrage opportunities.

8.3.2 The Black and Scholes Model

In the early 1970s, Fischer Black, Myron Scholes, and Robert Merton made a major breakthrough
in the pricing of stock options by developing what has become known as the Black-Scholes
Model. The Black-Scholes model, often simply called Black-Scholes, is a model of the varying
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:
C = S N (d ) – E e N (d )
–rt
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)
I (S/K) (r   2 /2)

d  n
1
 T

I (S/K) (r   2 /2)T
d  n  d   T
2 1
 T
C (S, T) = price of the European call option,
P (S, T) = price of the European put option,
 = the annualized 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
P( S, T) Ke N(  d 2 ) SN(d 1 ).



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