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Math 623, W 2007: Homework 3.
For full credit, your solutions must be clearly presented and all code included.
All the problems in this homework set deal with the following (Heston) stochastic volatility model
for the price
S
t
of a stock
±
dY
t
= 2(0
.
04

Y
t
)
dt
+ 0
.
2
√
Y
t
dW
t
dS
t
S
t
= 0
.
01
dt
+
√
Y
t
(

0
.
7
dW
t
+
√
0
.
51
dZ
t
)
(1)
for 0
≤
t
≤
0
.
25. Here
W
t
and
Z
t
are independent (uncorrelated) Brownian motions under the risk
neutral measure
Q
. Time is counted in years unless otherwise stated. By convention, one year is 360
days. Today is
t
= 0 and
Y
0
= 0
.
04. The stock pays no dividends.
(1) First some general questions.
(a) What is the riskfree interest rate
r
?
(b) The current stock price
S
0
is such that the
E
Q
[
S
0
.
25
] = 100. What is
S
0
?
(c) What is (formally) the expected value and variance of the relative stock return
dS
t
/S
t
(conditioned on
S
t
and
Y
t
)?
(d) What is (formally) the expected value and variance of
dY
t
(conditioned on
S
t
and
Y
t
)?
(e) Why is it reasonable to view
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 Fall '08
 CONLON
 Math

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