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ECON5065 Applied Computational Finance Practice Problems Set 1
发布时间:2022-02-16
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ECON5065
Applied Computational Finance
Practice Problems Set 1
1. Let W(t), t > 0, be a Brownian motion, and let F(t), t > 0, be a ltration for this Brownian motion. Show that W2(t)
t is a martingale.
(Hint: For 0 < s < t, write W2(t) as (W(t) W(s))2 + 2W(t)W(s)
W2(s).)
2. Let the interest rate r and the volatility > 0 be constant. Let S(t) = S(0)e (r
2)t+
W(t)
be a geometric Brownian motion with mean rate of return r, where the initial stock price S(0) is positive. Let K be a, positive constant. Show that, for T > 0,
E[e rT(S(T)
K)+] = S(0)N(d+(T,S(0))
Ke
rTN(d
(T,S(0)),
where
d (T,S(0)) =
[log
+ (r
)] ,
and N is the cumulative standard normal distribution function
N(y) =
e
z2 dz =
e
z2 dz .
3. Let W be a Brownian motion. Fix m > 0 and ∈ R. For 0 < t < ∞, de
ne X(t) =
t + W(t),
m = min{t
0;X(t) = m}.
As usual, we set m = ∞ if X(t) never reaches the level m. Let
be a positive number and set
Z(t) = exp (X(t)
(
+
2) t) .
(a) Show that Z(t), t 0, is a martingale
(b) Use (a) to conclude that
E [exp (X(t ∧
m)
(
+
2)(t ∧
m))] = 1, t
0.
(c) Now suppose
0. Show that for
> 0,
E [exp (m
(
+
2)
m) 1{
m<∞}] = 1.
Use this fact to show P(m < ∞) = 1 and to obtain the Laplace transform Ee
m = em
m
for all
> 0.