Page 118 - DMGT513_DERIVATIVES_AND_RISK_MANAGEMENT

Unit 8: Option Pricing

Black-Scholes European model except that it checks to see if the value returned is below the Notes
intrinsic value of the option. If this is the case, then the Modified Black-Scholes model returns the
intrinsic value of the option.
Black-Scholes American = Max (Black-Scholes European, Intrinsic Value)
The Modified Black-Scholes American model makes the following additional assumptions:
1. The price the option may be exercised prior to its expiration date.

2. Changes of the underlying asset are lognormally distributed.
3. The risk-free interest rate is fixed over the life of the option.
4. Dividend payments are not discrete; rather, the underlying asset yields a continuous
constant amount.

Example: Consider the situation where the stock price six months from the expiration of
an option is $42, the exercise price of the option is $40, the risk free interest rate is 10 % per
annum and the volatility is 20 % per annum. This means that,

Current price of the share, S = ` 42
0
Exercise price of the option, E = ` 40

Time period to expiration = 6 months. Thus, t = 0.5 years.
Standard deviation of the distribution of continuously compounded rates of return, = 0.2
Continuously compounded risk-free interest rate, r = .10

Ln (42/40) (0.10 0.5 0.2 2 )(0.50)



d   0.7693
1
0.2 0.50
Ln (42/40) (0.10 0.5 0.2 2 )(0.50)



d   0.6278
2
0.2 0.50
-rt
And Ke = 40e –(0.1*0.5) = 38.049
Hence, if the option is a European call, its value C is given by
C = 42N(0.7693) – 38.049N(-0.7693)
If the option is European Put, its value P is given by
P = 38.049N(–0.6278) – 42N(–0.7693)
Using the Polynomial approximation
N (0.7693)=0.7791 N (–0.7693)=0.2209

N (0.6278)=0.7349 N (–0.6278)=0.2651
So that,
C= 4.76 P= 0.81

The value of European call is ` 4.76 and the value of European put option is ` 0.81.

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