Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 1
Steve Dunbar
Friday, September 3, 2010
Find and write the definition of a “future”, also called a futures
contract. Graph the intrinsic value of a futures contract at its
contract date, or expiration date, as was done for the call option.
Show that holding a call option and writing a put option on the
same asset, with the same strike price $K$
is the same as having a futures contract on the asset with strike
price $K$.
Drawing a graph of the value of the combination and the value of
the futures contract together with an explanation will demonstrate
the equivalence.
Puts and calls are not the only option contracts available, just the most
fundamental and the simplest. Puts and calls are designed to eliminate risk
of up or down price movements in the underlying asset. Some other option
contracts designed to eliminate other risks are created as combinations of
puts and calls.
Draw the graph of the value of the option contract composed of
holding a put option with strike price ${K}_{1}$
and holding a call option with strike price ${K}_{2}$
where ${K}_{1}<{K}_{2}$.
(Assume both the put and the call have the same expiration date.)
The investor profits only if the underlier moves dramatically in
either direction. This is known as a long strangle.
Draw the graph of the value of an option contract composed of
holding a put option with strike price $K$
and holding a call option with the same strike price $K$.
(Assume both the put and the call have the same expiration date.)
This is called an long straddle, and also called a bull straddle.
Draw the graph of the value of an option contract composed of
holding one call option with strike price ${K}_{1}$
and the simultaneous writing of a call option with strike price
${K}_{2}$
with ${K}_{1}<{K}_{2}$.
(Assume both the options have the same expiration date.) This is
known as a bull call spread.
Draw the graph of the value of an option contract created by
simultaneously holding one call option with strike price ${K}_{1}$,
holding another call option with strike price ${K}_{2}$
where ${K}_{1}<{K}_{2}$,
and writing two call options at strike price $\left({K}_{1}+{K}_{2}\right)\u22152$.
This is known as a butterfly spread.
Draw the graph of the value of an option contract created by
holding one put option with strike price $K$
and holding two call options on the same underlying security,
strike price, and maturity date. This is known as a triple option
or strap
You would like to speculate on a rise in the price of a certain stock.
The current stock price is $29 and a 3-month call with strike of
$30 costs $2.90. You have $5,800 to invest. Identify two alternative
strategies, one involving investment in the stock, and the other involving
investment in the option. What are the potential gains and losses from
each?
A company knows it is to receive a certain amount of foreign currency in 4
months. What type of option contract is appropriate for hedging? Please be
very specific.
The current price of a stock is $94 and 3-month call options with a strike
price of $95 currently sell for $4.70. An investor who feels that the price
of the stock will increase is trying to decide between buying 100
shares and buying 2,000 call options. Both strategies involve an
investment of $9,400. What advice would you give? How high does
the stock price have to rise for the option strategy to be the more
profitable?