Problem Set 5
Question 1
The exchange rate Ct (in CDN/HKD) follow the following dynamics:
dCt
= −0.02 dt + 0.02 dWt
Ct
Where Wt is a standard Brownian motions under the real world probability measure. The
Canadian and HK risk-free interest rates are r = 1% and u = 0.5%, respectively. Currently
one CDN is currently worth 8 HKD.
Find today’s price in CDN of a European option which 3 years from now gives the holder
the right to buy Canadian dollars with 100,000 HKD at rate 1/9 (9 HKD for 1 CDN).
Question 2
A stock with price St following a geometric Brownian motion with drift α = 0.2 and volatility
σ = 0.2 under the real world probability measure P . The current stock price is S0 = 80 and
the risk-free interest rate is r = 5%. The stock pays dividends continuously at an annual
rate of 3%. Consider a European option having the payoff Φ(S) six months from now, where

0
if S ≤ 75




 2S − 150 if 75 ≤ S ≤ 85
20
if 85 ≤ S ≤ 95
Φ(S) =


210 − 2S if 95 ≤ S ≤ 105



0
if S ≥ 105
Compute the price, and Delta and Rho of this option at time 0.
Question 3
Assume the following Vasicek short rate model under the risk-neutral probability measure:
drt = 0.8[0.02 − rt ]dt + 0.06 dW̃t .
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 two zero coupon bonds that pays 1,000 dollars at the end of 6
months and one year, respectively, and the price of a coupon bond with a face value of 1000,
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semiannual coupons and a maturity of one year. The coupon rate is 4%.
(c) Derive the forward rate curve f (0, t), 0 ≤ t ≤ 1.
Question 4
Assume the following CIR short rate model under the risk-neutral probability measure:
√
drt = 0.8[0.02 − rt ]dt + 0.1 rt dW̃t .
Today’s short rate is 3%.
(a) Find the mean and variance of rt , t = 3, using the gamma approximation;
(b) Determine the price of two zero coupon bonds that pays 1,000 dollars 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%.
Question for Fun! Will not be graded.
Again, you are given the same Vasicek short rate model under the risk-neutral probability
measure:
drt = 0.8[0.02 − rt ]dt + 0.06 dW̃t .
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).
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