given S < EX: as t
®
0, then C
®
0
given S > EX: as t
®
0, then C
®
S - EX
as t
®
∞
, then C
®
S
4. Volatility:
given S < PV(EX): as
s ®
0, then C
®
0
given S > PV(EX): as
s ®
0, then C
®
S – PV(EX)
as
s ®
∞
, then C
®
S
5. Interest rate:
r
®
∞
, then C
®
S

Some important definitions:
”Moneyness” of an option
A Call is
“in-the-money” if
S
t
>
EX
“out-of-the-money” if
S
t
<
EX
“at-the-money” if
S
t
≈
EX
A Put is
“in-the-money” if
S
t
<
EX
“out-of-the-money” if
S
t
>
EX
“at-the-money” if
S
t
≈
EX

Review questions for lecture 3:
Q1:
Bullmart Inc.’s common stock is currently trading at 330p, and can go up by a factor of 1.2 or down by a factor of
1/1.2 each period.
The periodic riskfree rate is 5%.
What is the value of a two-period European put on Bullmart with a
strike price of 300p?
(Bullmart does not pay any dividends.)
Q2:
Insert the Black and Scholes formula to Excel [the cumulative standard normal distribution for a variable x is given
by NORMSDIST(x)]
and verify the calculations done in page 7. Examine what effects do changes in parameters have
on the option price. Verify that the call option price is increasing in S
0
, the interest rate, r
c
, time to maturity, T, and price
volatility,
s
, and that it is decreasing in the exercise price, EX.
Q3:
You have estimated from financial market data that the annual volatility of XYZ is 15%.
Today the price of XYZ
is $100. Assume that XYZ does not pay any dividends. You are interested in valuing a call option on XYZ maturing in
two year’s time. Assume that the annual interest rate is 10%.
a)
Construct a two period binomial tree (t= 0,1,2) for XYZ (that is, let
D
t =1)
b) Now consider a call option on XYZ with exercise price EX = $110 that expires in period 2. Find the replicating
portfolio for this option first at time t=1 and then at t=0.
c) What are the rebalancing trades?
d) Price the same call option using the “risk neutral pricing” method.
Q4:
You have estimated from financial market data that the annual volatility of QWE is also 15%. Similarly to the
company in the previous example, the price of QWA is also $100. One difference exists, however, in that QWE is
expected to pay a $6 dividend at time 1. Remember that the option holders are not entitled to receive the dividend.
a) How does the binomial tree for QWE’s stock look like in the presence of this known $6 dollar dividend?
b) Use risk neutral pricing method to value a call option on QWE’s stock assuming that the exercise price of the option
is still $110. Explain the difference between QWE’s option price and the price of the XYZ’s option.

Answers:
Q1.
The two-period stock price tree for Bullmart is:
330
To price the put option, we first determine the terminal payoffs. Then, because all we need to do is
to price the option, the quickest method is to use risk neutral pricing, working backwards from t=2.

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- Summer '14
- sam
- Derivatives, Valuation