7.1 A fixed income analyst needs to estimate the price of an interest rate caplet that pays $1,000,000 next year if the one-year Treasury rate exceeds 3% and pays nothing otherwise. Using a macroeconomic model developed in another area of the firm, the analyst estimates that the one-year Treasury rate will exceed 3% with a probability of 25%. Since the current 1-year rate is 1%, the analyst prices the caplet as follows: 25% $1,000,000 $247,525 1.01 = Comment on this pricing procedure. 7.2 Assume that the true 6-month rate process starts at 5% and then increases or decreases by 100 basis points every 6 months. The probability of each increase or decrease is 50%. The prices of 6-month, 1-year, and 1.5-year zeros are 97.5610, 95.0908, and 92.5069. Find the risk-neutral probabilities for the six-month rate process over the next year (i.e., two steps for a total of three dates, including today). Assume, as in the text, that the risk-neutral probability of an up move from date 1 to date 2 is the same from both date 1 states. As a check to your work, write down the price trees for the 6-month, 1-year, and 1.5-year zeros. 7.3 Using the risk-neutral tree derive for Question 7.2, price $100 face amount of the following 1.5-year collared floater. Payments are made every six months according to this rule. If the short rate on date i is ri then the interest payment of the collared floater on date i +1 is 1 3.50% 2 if ri < 3.50% ; 1 $ 2 ir if 6.50% 3.50% i r ; 1 6.50% 2 if ri > 6.50%. In addition, at maturity, the collared floater returns the $100 principal amount.