Let and be the values of American calls and puts when
the stock price is and there are periods to go.
(A) What is the value of and ?
(B) What is
the value of and ?
(C) What is
the lowest price your firm could charge and still break even?
Problem 2
See the payoff diagram below.
Use the
following assets to replicate the payoff diagram.
(i) shares of stock
(ii) $50strike calls
(iii) $100strike puts
(iv) $100strike calls
In this question, denote purchased quantities with a positive number and
written quantities with a negative number. You may not use bonds.
Problem 3
Below is information (pershare) for a $102.00strike purchased put with one year to expiration, where interest rate r=7.13%, volatility σ=0.21, and dividend yield δ=0.00%
A dealer sells this put option, on 100 shares. You are to describe the dealer's hedge and evaluate profit or loss after 3 days.
(A) On day 0, how many shares does the dealer buy or sell to deltahedge the written put? (Fractional shares are permissible.) In this question, denote quantities to buy with a positive number and quantities to sell with a negative number.
(B) On day O, taking into account the option premium received and the share transaction from part (A), what position in the riskfree asset will give the dealer zero net investment? In this question, denote amount to lend with a positive number and amount to borrow with a negative number.
(C) On day 3, what is the dealer's profit on this hedged position? (Use a negative number to denote losses.)
Problem 4
You have the information listed below on three options on a normalized stock price index, including the current index value S, the riskfree rate r, the dividend yield δ, the time to expiration in years T, and the call price C.
(A) Based on this information, and using an Excel spreadsheet with the BlackScholesMerton model, estimate the implied volatility from each option. Write your answers in decimal numbers.
σ1
σ2
σ3
(B) If you want to take a longshort position that will profit if the implied volatilities of these options tend to converge over time, which of the following positions might you choose? There may be more than one correct answer; indicate all that apply.
 long call 1 and short call 2
 long call 1 and short call 3

short call 1 and long call 2
 short call 1 and long call 3
 long call 3 and short call 2
Problem 5
The twoyear riskfree interest rates in Germany and in the United States are, respectively, 0.11% and 0.93% per annum with continuous compounding. The spot price of the Euro is $1.15. What is the twoyear forward price of $1.00 in Euros?
Problem 6
Merit Inc. has zero coupon debt outstanding that will have to be repaid in three years, with a face value of $80.00 million. The riskfree interest rate is 5.13% per annum (continuously compounded). You estimate that the firm's asset volatility is 16.29% per annum and that its assets are currently worth $102.00 million. It doesn't pay dividends.
(A) Use Merton's model to estimate the value of Merit's equity. Write your answer in millions of dollars. million
(B) What is the implied value of the debt? Write your answer in millions of dollars. What is the perannum yield (on a continuously compounded basis) on the debt? Write your answer in units of percentage points.
Problem 7
The graph below represents the estimated price/yield relationship for two mortgagebacked securities, taking into expected account prepayment and default behavior. Both securities are derivatives of a pool of underlying fixedrate mortgages, each with a final maturity of 15 years and that can be prepaid without penalty at any time.
(A) Which is the interest only (IO)?
MBS 1
MBS 2
(B) Which of these securities has a negative effective duration when interest rate is at 6% ?
MBS 1
MBS 2
(C) Which of these securities has a negative convexity when interest rate is at 6% ?
MBS 1
MBS 2
Problem 8
Imagine that you manage an investment fund that is long in $2.00 million pool of whole mortgages. You estimate that the effective duration of those holdings is 6.03 years. You are concerned that the central bank is going to tighten and you want to hedge your exposure using an interest rate swap.
The swap has the follow terms:
 Type: fixed for floating
 Frequency: annual
 Maturity: 10 years
 Fixed rate: 3.13% (annual percentage rate)

Floating rate: LIBOR
 Notional principal: one million
Currently the yield curve is flat at 3.13%, and the first floating rate payment will be based on 3.13%.
(A) To hedge the interest rate exposure of the mortgage portfolio, would you be the fixed or floating rate payor in the swap?
Fixed rate payor
Floating rate payor
(B) What is the dollar duration of the swap?
Write your answer in unit of (million dollars × year). (Hint: Be specific about the sign of the duration.)
(million dollars × year)
(C) How many swaps you need to deltahedge the interest rate exposure of the fund's mortgage holdings?
(Fractional swaps are permissible.)