20. (6 points) A financial institution has sold short three European call options on a stock
with the following characteristics:
Stock price: 100
Strike price: 110
Volatility: 20%
Expiry: 2 years
Contract size: 1,000 shares
Dividends: none
Risk-free rate: 5%
(a) (4 Points) Describe and evaluate strategies the financial institution can use to
manage the risk of this portfolio.
(b) (1 Point) Calculate delta, theta and gamma for the portfolio.
(c) (1 Point) Verify the calculations of delta, theta and gamma by showing that the
results for the portfolio satisfy the Black-Scholes-Merton differential equation.
COURSE 8: Fall 2003 - 17 - GO ON TO NEXT PAGE
Investment
Afternoon Session
21. (6 points) A model of the type GARCH(1,1) is being used to set volatility for integer
values of strike price for an option pricing model that is based on a lognormal
distribution. You have the following information on the model for a specific underlying
asset:
•A weight of 10% is given to a variance of 4% that is reached
asymptotically as the strike price increases toward infinity.
•The model assumes returns are a constant 20%.
•A weight of 80% is given to the variance determined for the prior strike
price.
•The variance is 9% for a strike price of 0.
(a) Calculate the change in volatility for each of strike prices of 1 and 2 assuming the
variance for a strike price of 0 decreases from 9% to 8%.
(b) Assess whether the model with the given parameters is more appropriate for an
equity option or a currency option. Support your answer by interpreting the
resulting implications for the volatility smile.
(c) Describe how to determine a maximum likelihood estimate for the above model,
assuming the distribution of returns conditioned on the variance is normal.
COURSE 8: Fall 2003 - 18 - GO ON TO NEXT PAGE
Investment
Afternoon Session
22. (7 points) You would like to calibrate a new Monte Carlo pricing model by comparing
the model price to the Black-Scholes price of a European call option on a non-dividend
paying stock. The European call option and the underlying stock have the following
characteristics:
Current stock value = 100
Expected growth rate of the stock price = 10%
Volatility = 25%
Exercise price = 95
Time to maturity = 6 months
Risk-free interest rate = 6%
Black-Scholes price of the option = 11.37
Assume that the percentage change in the stock price is normally distributed.
(a) Calculate the control variate technique adjustment using stratified sampling on 3
intervals.
(b) Describe additional techniques that can be used to improve the efficiency of the
Monte Carlo method.
(c) Describe how the Monte Carlo method can be adapted to calculate the price of
American options.
COURSE 8: Fall 2003 - 19 - GO ON TO NEXT PAGE
Investment
Afternoon Session
23. (6 points) You are given the following information about an interest rate collar with
quarterly resets. Assume quarterly compounding unless otherwise indicated.
Notional amount = 100,000,000
Strike interest rate for cap = 7.0%
Strike interest rate for floor = 4.5%
Time to expiration = 2 years
Time t Forward 3-month interest
rate starting at time t
Zero-coupon interest rate
to time t (continuous
compounding)
9 months 5.0% 4.49%
12 months 5.2% 4.64%
15 months 5.3% 4.77%
(a) Compare the collar to a portfolio of bond options with equivalent payoffs.
(b) Calculate, using Black’s model, the price of the floorlet that prevents the interest
rate from being lower than the strike price for 3 months starting in one year.
Assume that the volatility of the underlying three-month rate is 20%.
(c) Assess the suitability of Black’s model for valuing options embedded in a
mortgage backed security.
COURSE 8: Fall 2003 -20- STOP
Investment
Morning Session
24. (3 points) You have been asked to review the trading costs of the following stock:
Time Price Number of
shares traded
(Previous close is 44 1/8)
1:40 pm 44 1/8 1000
1:42 44 1/4 2000
1:45 44 500
1:46 43 7/8 5000
1:50 43 3/4 2000
1:51 43 7/8 400
1:54 44 1/4 6000
(a) Explain why the measurement of trading costs is important.
(b) Calculate the price impact of trading for the above stock using the “money flow”
system as adapted by Birinyi.
***END OF EXAMINATION***

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