Chapter 7: Valuation and Scenario Analysis: The Risk-Neutral and Real
Worlds
7.10
A stock price has an expected return of 9% and a volatility of 25%. It is currently $40. What is
the probability that it will be less than $30 in 18 months?
The probability is N(−d2) where
2274.1
5.125.0
5.1)2/25.009.0()30/40ln(
2
2
d
It is 0.11.
7.11
An investor owns 10,000 shares of a particular stock. The current market price is $80. What is
the “worst case” value of the portfolio in six months. For the purposes of this question, define
the worst case value of the portfolio as the value which is such that there is only a 1% chance of
the actual value being lower. Assume that the expected return and volatility of the stock price
are 8% and 20%, respectively.
From equation (7.5), the “worst case” stock price is
33.595.02.0)01.0(5.0)2/2.008.0(exp80
12
N
The worst case value of the portfolio is therefore $593,300.
7.12
A binary option pays off $500 if a stock price is greater than $60 in three months. The current
stock price is $61 and its volatility is 20%. The risk-free rate is 2% and the expected return on
the stock is 8%. What is the value of the option? What is the real-world expected payoff?
The value of the option is 500N(d2)e−0.02×0.25 where
1653.0
25.02.0
25.0)2/2.002.0()60/61ln(
2
2
d
The value is $281.4.
The real-world expected payoff is 500N(d2) where
3153.0
25.02.0
25.0)2/2.008.0()60/61ln(
2
2
d