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Continuous Time Finance Notes, Spring 2004 – Section 3,
Feb. 4, 2004
Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec
tion with the NYU course Continuous Time Finance.
Announcement:
There will be no class on Wednesday, February 11.
These notes wrap up our treatment of the probabilistic framework for option pricing. We
conclude the discussion of problems with one source of randomness by discussing the “mar
ket price of risk” and the use of alternative numeraires (leading to consideration of equivalent
martingale measures other than the riskneutral one). Then we discuss the analogous theory
for complete markets with multiple sources of randomness. Finally we discuss two impor
tant examples where there are two sources of randomness: (a) exchange options, and (b)
quanto options. Except for exchange options, this material is covered in Sections 4.44.5 and
corresponding part of Hull (5th edition) is Chapter 21 (my discussion of exchange options
is from there). Hull is relatively terse, but well worth reading.
*****************
The market price of risk.
We continue a little longer the hypothesis that the market
has just one source of randomness (a scalar Brownian motion
w
). Recall what we achieved
in Section 1: let
r
t
be the riskfree rate, and
B
the associated moneymarket instrument
(characterized by
dB
=
r
t
B dt
with
B
(0) = 1); let
S
be tradeable, and assume
S
solves an
SDE of the form
dS
=
μ
t
S dt
+
σ
t
S dw
. We showed that if
S
t
/B
t
is a martingale relative to
Q
then any payoﬀ
X
at time
T
can be replicated by selfﬁnancing portfolio which invests in just
S
and
B
, and the time
t
value of this portfolio
V
t
is characterized by
V
t
=
B
t
E
Q
[
X/B
T
F
t
].
We also showed that
Q
exists (and is unique), by an application of Girsanov’s theorem.
Reviewing this: we need
S/B
to be a
Q
martingale. Under the original probability we have
d
(
S/B
) = (
μ
t

r
t
)(
S/B
)
dt
+
σ
t
(
S/B
)
dw.
According to Girsanov’s theorem changing the measure (without changing the sets of mea
sure zero) can change the drift but not the volatility. So
Q
is characterized by
d
(
S/B
) =
σ
t
(
S/B
)
d
˜
w
where ˜
w
is a
Q
Brownian motion. Comparing these equations we see that
dw
+
μ
t

r
t
σ
t
dt
=
d
˜
w.
The ratio
λ
t
=
μ
t

r
t
σ
t
1
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View Full Document is called the
market price of risk
, by analogy with the Capital Asset Pricing Model (notice
that
λ
is the ratio of excess return to volatility). It is important because it determines
Q
in terms of
P
, namely:
dQ/dP
= exp
±

Z
T
0
λ
t
dw

1
2
Z
T
0
λ
2
t
dt
!
.
This
Q
is called the “riskneutral measure.”
Finally, we observed that if
S
is
any
tradeable in this market, its value
S
t
must have the
property that
S
t
/B
t
is a
Q
martingale (for the
same
riskneutral measure
Q
). This shows
that
all tradeables must have the same market price of risk
.
It’s enlightening to give another proof that all tradeables have the same market price of
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.
 Fall '11
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 Finance

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