Problem Set 5
Question 1
The stock price St (in HKD) of the HK company ABC Inc and the exchange rate Ct (in
CDN/HKD) follow the following dynamics:
dSt
= 0.1 dt + 0.25 dWtf
St
dCt
= −0.02 dt + 0.02 dWtc
Ct
Where Wtf and Wtc are correlated Brownian motions under the real world probability measure
with E[dWtf dWtc ] = 0.12 dt. The Canadian and HK risk-free interest rates are r = 1% and
u = 0.5%, respectively. The current stock price is S0 = 800 HKD and one CDN is currently
worth 8 HKD.
(a) Find today’s price in CDN of the European option which 3 years from now gives the
holder the right, but not the obligation, to sell one share of ABC Inc for 800 HKD;
(b) Find today’s price in CDN of the European option which 3 years from now gives the
holder the right, but not the obligation, to sell one share of ABC Inc for 90 CDN.
Question 2
Assume the following Vasicek short rate model under the risk-neutral probability measure:
˜ t.
drt = 0.8[0.02 − rt ]dt + 0.06 dW
Today’s short rate is 3%.
(a) Determine the distributions, including their parameter values, of r0.5 and r1 . What is
the co-variance between the two short rates?
(b) Determine the price of zero coupon bonds that pays 1 dollar at the end of 6 months and
one year, respectively, and the price of a coupon bond with a face value of 1000, semiannual
coupons and a maturity of one year. The coupon rate is 4%. What is the yield rate of this
coupon bond?
(c) Derive the forward rate curve f (0, t), 0 ≤ t ≤ 1.
Question 3
Again, you are given the following Vasicek short rate model under the risk-neutral probability
1
measure:
˜ t.
drt = 0.8[0.02 − rt ]dt + 0.06 dW
Today’s short rate is 3%.
(a) Write a VBA program to generate 500 short rate paths over a one-year time period,
using daily time steps and 250 days a year;
(b) Calculate the sample mean and the (unbiased) sample variance of the data generated in
(a) for r1 , the short rate at the end of the year;
(c) Display a Q-Q plot of the data for r1 against a normal distribution with mean and
variance being the sample mean and sample variance obtained in (b).
When you complete the Excel file and the VBA codes, please send them separately to
[email protected]. And put your name and student number in the title, the format
for the title should be
Assignment 5 + N ame + Student ID
Question 4
Assume the following Vasicek short rate model under the risk-neutral probability measure:
˜ t.
drt = 0.8[0.03(1 − e−t ) − rt ]dt + 0.06 dW
Today’s short rate is 3%.
(a) Find the mean and variance of r1 ;
(b) Determine the probability that the short rate one year from now is greater than 3%.
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