Notes 7
Finance 5108
Learning Objectives
1. Modification of the BlackScholes Formula to accommodate continuously paid
dividends
2. Understand how to use the binomial with continuous dividend payments.
3. Valuation of index options using the modified BlackScholes formula
4. What is portfolio insurance and how options can be used to set up this type of
hedge
5. Valuation of currency options
6. Options on the futures contract and their value
7. The Black model for the evaluation of options on futures contracts
Bounds of options that pay a continuous dividend
1. The argument that was put forward in previous proofs would hold in the pricing
of options on a stock that pays a continuous dividend.
It could be shown that the
only modification would be to the stock price.
Since the dividend will be assumed
to have been paid over the life of the option the value of the stock would be
0
0
0
0
= S
Where q is the continous dividend this leads us to
c
max(0, S
)
max(0,
S
)
Which are the lower bounds of the call and put option
Put/Call parity is then
S

qT
adjusted
qT
rT
rT
qT
qT
S
e
e
Ke
p
Ke
e
e






≥

≥

c + p =
rT
Ke

We can use the same logic to modify the
qT
0
1
2
qT
2
0
1
2
0
1
2
1
e
(
)
(
)
(
)
e
(
)
ln
+
q +
2
=
rT
rT
c
S
N d
Ke
N d
p
Ke
N
d
S
N
d
Where
S
r
T
K
d
T
d
d
T
σ
σ
σ


=

=




=

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The same modification can be made to the stochastic differential equation and
the risk neutral process by just modifying the interest rate.
The logic is that the
net interest rate would be the risk free rate minus the continuous dividend yield
which is rq.
This follows from the fact that you would receive the dividend rate of
q which offset the riskfree rate.
In addition this same logic can be used to apply to the binomial and the interest
rate would be e
(rq)T
. This can be used to value index options using the Black
Scholes or the binomial.
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 Fall '09
 Rader
 Finance, Derivatives, Options, Valuation, Writer

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