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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?.