Math 441 Spring 2015 First Midterm (Practice) Problem 1. (10 points

Math 441 Spring 2015
First Midterm
(Practice)
Problem 1. (10 points) Answer two of the three questions below
(a) What is a European call option?.
(b) What does it mean for a (long position) on an American put option to be “in the money”?.
(c) What is the “payo↵” of a portfolio? How is it di↵erent from the “profit”?.
Problem 2. (15 points) Suppose the interest rate for a savings account at a 1% compounded bi-monthly. Then,
say a) what is the e↵ective annual rate? b) what rate rq compounded quarterly would lead to the
same e↵ective annual rate?.
Problem 3. (20 points) Today you see an European call option with a premium of 5$, strike price of 100$ and
maturity of 1 years. You know the associated stock is selling today at 100$ and you know with
absolute certainty that in 1 year the stock is guaranteed to be worth either 75$ or 125$. Assume
a zero risk rate of 10% per annum. Show this situation violates the “No Arbitrage Principle” by
constructing an arbitrage opportunity.
Hint: In 1 year the stock price will definitely move away from the current price, meaning there
is volatility. Pick a portfolio that takes advantage of this and estimate its payo↵.
Problem 4. (20 points) Consider a portfolio consisting of 1 (short) stock St and 2 (long) Straddle with expiration in 6 months and common strike price K = 50$.
Suppose a zero risk rate of 5% per annum (continuously compounded). Suppose also that S0
is trading today at 75$, and that the premium you pay today for the call and put options on the
straddle are respectively 10$ for the call and 5$ for the put.
(a) In 6 months, the stock is trading at 102$. Computethe the payo↵ and the profit.
(b) Repeat the caclulation assuming instead that in 6 months the stock is trading at 7$.
Problem 5. (15 points) A stock St behaves in the following way (t here representing months)
St = X t St
1
where Xt is a random variable that takes the values 1/2, 2 with probability 1/4 each and the value
1 with probability 1/2. Then,
(a) Find a recurrence relation between E[St ] and E[St 1 ].
(b) Find r such that if r is the (continuously compounded, annual) zero risk rate then the expected rate of return for the stock St a year from now is the same as the zero risk rate.
Problem 6. (20 points) For a given stock St consider a derivative comprised of the following:
(
2 (long) straddle with
T = 6 months, K = 15$
h=
1 (short) European put option with T = 6 months, K = 10$.
Assume a zero risk rate of 7% (compounded quarterly) and that the stock St is such that
S0 = 10$ and S1/2 = 25$ or 6$.
What is the payo↵ of the derivative at time t = 0?.