Finance 534 Final Exam
Instructor: Dermot Hayes
You have 90 minutes to answer five of the following six questions. Each question is worth 20
points.
Question 1
(a) Describe the portfolio you would construct if you are selling calls with a delta of 1/3.
Now explain intuitively or mathematically why the derivative of this portfolio with
respect to the asset price (the delta of the portfolio) is zero.
(b) Now describe how you would construct a portfolio with a delta of zero and a gamma of
zero.
Question 2
(a) Use an arbitrage table to show the relationship between put and call premiums for options
on the same asset and with the same strike price. Your answer should indicate the call
premium is equal to the put premium plus the difference between the current asset price
and the discounted value of the strike price.
(b) The Chicago mercantile exchange now offers futures on individual stocks. Show how
you would replicate a long futures position on stocks with an appropriate options
position.
Question 3
(a) Use arbitrage to show the relationship that must exist between the exchange rate today
and a futures contract on this exchange rate.
(b) Use the result from 3 (a) to motivate an adjustment to the Black Scholes equation when it
is used to price options on the physical delivery of currency.
Question 4
(a) Describe how you would create a histogram of the revenue of an Iowa corn producer
assuming that you know the parameters of the yield distribution and the degree of
negative correlation.
(b) Explain why Iowa corn producers should not consider the sale of their expected corn
production on futures or forward markets a risk reducing proposition.
Question 5
(a) The Black Scholes and Black option pricing formulas assume that the variance of prices
is directly proportional to time. Show what this assumption implies using a chart with a
price range on the y axis and with time on the x axis.
(b) Assume that the supply of crude oil depends on the price of crude oil. Does the
assumption described in 5 (a) make sense for the crude oil market.
(c) Assume that traders on crude oil futures markets use the Black options pricing equation;
describe a speculative position that would allow you to take advantage of their mistake.
Question 6
(a) Draw a graph showing the relationship between a call premium and the expected
volatility with options that are (1) at the money (2) out of the money and (3) in the
money.
(b) Draw a graph showing the relationship between the delta of a call and the asset price.
