Assignment 1.
1. The daily log return on a stock are independent and normally distributed
with mean 0.001 and standard deviation 0.015. Suppose you buy $1000
worth of this stock.
a) What is the probability that after
...
Assignment 1.
1. The daily log return on a stock are independent and normally distributed
with mean 0.001 and standard deviation 0.015. Suppose you buy $1000
worth of this stock.
a) What is the probability that after one trading day your investment is
worth less than $990?
Solution:
= 0.001, sd = 0.015, P0 = $1000
P (P1 < 990)
Log P1/P0 N (, δ 2)
P (log P1/P0 < log 990/1000) = ∮ (( log 990/1000 - 0.001)/0.015) =
0.23066
b) What is the probability that after five trading days your investment is
worth less than $990?
Solution:
P (P5 < 990)
Log P5/P0 = r5(5) N( 5, 5 δ 2) N (0.005,
√5 * 0.015)
P (log P5/P0 < log 990/1000) = ∮ ((log 990/1000 – 0.005)/ √5 *
0.015 = 0.3268
4. The prices and dividends of a stock are given in the table below.
a) What is R2?
1 + Rt = (Pt + Dt)/Pt-1
1 + R2 = (54 + 0.2)/52
R2 = 0.0423 = 4.23%
b) What is R4(3)?
1 + R4(3) = ((P4 + D4)/P3)*((P3+D3)/P2)*((P2 + D2)/P1)
1 + R4(3) =((59 + 0.25)/53)*((53+0.2)/54)*((54+0.2)/52)
R4(3) = 0.14796 = 14.796%
c) What is r3?
rt = log (Pt/Pt-1)
r3 = log (P3/P2) = - 0.01869 = - 1.869%
6. Suppose that X1, X2 … is lognormal geometric random walk with
parameters (, δ 2). More specifically, suppose that Xk = X0 exp (r1 +… + rk),
where X0 is a fixed constant.
a) Find P(X2 > 1.3 X0)
P(X2 > 1.3 X0) = P (X2/X0 > 1.3)
Log (X2/X0 ) N (2, 2 δ 2)
P ((log (X2/X0) - 2)/ √2 δ 2 > (log1.3 - 2)/ √2 δ 2) = P (Z >
(log1.3 - 2)/ √2 δ 2) = 1 – P(Z > (log1.3 - 2)/ √2 δ 2) = 1 -
∮ ( log1.3 - 2)/ √2 δ 2)
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