STAT/ACTSC 446/846
Assignment #2 (due February 9, 2012)
(13)
Problems from the book “Financial Economics ...”:
5.2 (do this by using the fundamental
theorem of asset pricing, not graphically as suggested), 5.5, and 5.8.
(4)
First we introduce definition of a predictable process.
Suppose that we are given a filtration
{F
n
}
n
≥
0
. A stochastic process
{
X
n
}
n
≥
1
is called
{F
n
}
n
≥
0
 predictable if
X
n
is
F
n

1
 measurable
for all
n
≥
1.
Prove that a predictable martingale is constant.
(5)
Suppose that
{
X
n
}
n
≥
0
is adapted to the filtration
{F
n
}
n
≥
0
and that
{
φ
n
}
n
≥
1
is
{F
n
}
n
≥
0

predictable. Define a new process
Z
n
=
Z
0
+
n

1
X
j
=0
φ
j
+1
(
X
j
+1

X
j
)
,
where
Z
0
is a constant. Show that if
{
X
n
}
n
≥
0
is a martingale with respect to the filtration
{F
n
}
n
≥
0
then so is
{
Z
n
}
n
≥
1
.
(6)
(This is Problem 6.5 from the “Financial Economics” book) A European derivative pays the
square of the asset price in 3 month’s time.
The current price is 20, and the asset price has a
volatility of 15% per year. The continuously compounded interest rate is 7% per year. Use the
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 Winter '09
 Adam
 Pricing, asset price

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