Continuous Time Finance Notes, Spring 2004 – Section 2,
Jan. 28, 2004
Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec
tion with the NYU course Continuous Time Finance.
In Section 1 we discussed how Girsanov’s theorem and the martingale representation the
orem tell us how to price and hedge options.
This short section makes that discussion
concrete by applying it to (a) options on a stock which pays dividends, and (b) options on
foreign currency. We close with a brief discussion of Siegel’s paradox. For topics (a) and (b)
see also Baxter and Rennie’s sections 4.1 and 4.2 – which are parallel to my discussion, but
different enough to be well worth reading and comparing to what’s here. For a discussion
of Siegel’s paradox and some related topics, see chapter 1 of the delightful book
Puzzles
of Finance: Six Practical Problems and their Remarkable Solutions
by Mark Kritzman (J.
Wiley & Sons, 2000, available as an inexpensive paperback).
********
Options on a stock with dividend yield.
You probably already know from Derivative
Securities how to price an option on a stock with continuous dividend yield
q
. If the stock
is lognormal with volatility
σ
and the riskfree rate is (constant)
r
then the “riskneutral
process” is
dS
= (
r

q
)
S dt
+
σS dw
, and the time0 value of an option with payoff
f
(
S
T
)
and maturity
T
is
e

rT
E
RN
[
f
(
S
T
)]. From the SDE we get
dE
RN
[
S
]
/dt
= (
r

q
)
E
RN
[
S
], so
E
RN
[
S
](
T
) =
e
(
r

q
)
T
S
0
.
This is the
forward price
, i.e. the unique choice of
k
such that a
forward with strike
k
and maturity
T
has initial value 0. (Proof: apply the pricing formula
to
f
(
S
T
) =
S
T

k
.) When the option is a call, we get an explicit valuation formula using the
fact that if
X
is lognormal with mean
E
[
X
] =
F
and volatility
s
(defined as the standard
deviation of log
X
), then
E
[(
X

K
)
+
] =
FN
(
d
1
)

KN
(
d
2
)
(1)
with
d
1
=
ln(
F/K
) +
s
2
/
2
s
,
d
2
=
ln(
F/K
)

s
2
/
2
s
.
There is of course a similar formula for a call (easily deduced by putcall parity).
What does it mean that the “riskneutral process is
dS
= (
r

q
)
S dt
+
σS dw
?”
What
happens when the dividends are paid at discrete times? We can clarify these points by using
the framework of Section 1. Remember the main points: (a) there is a unique “equivalent
martingale measure”
Q
, obtained by an application of Girsanov’s theorem; and (b) this
Q
is characterized by the property that the value
V
t
of any tradeable asset satisfies
V
t
/B
t
=
E
[
V
T
/B
T
 F
t
] where
B
t
is the value of a riskfree moneymarket account, i.e.
it solves
dB
=
rB dt
with
B
(0) = 1. Put differently:
V/B
is a
Q
martingale.
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 Fall '11
 Bayou
 Finance, United States dollar, Mathematical finance, Rational pricing, Forward measure

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