Assume a binomial pricing model where there is an equal probability of interest rates increasing or decreasing 1 percent per year.
-What should be the price of a three-year 5 percent floor if the current (spot) rates are also 6 percent? The face value is $5,000,000, and time periods are zero, one, and two.
A)$8,250.
B)$10,799.
C)$12,550.
D)$15,875.
E)$17,455.

Question

Quizplus · Accepted Answer

Using the binomial model, we can construct the following tree:

7% (up)
         ↑
   6% (current)
         ↓
      5% (down)

To value the floor, we calculate the payoff at each node of the tree and work backwards. At the final nodes, the payoff is simply the difference between the floor rate (5%) and the interest rate at that node.

At the second time period, there are two possible outcomes: up (7%) or down (5%). The probability of an up move is 0.5, and the probability of a down move is also 0.5. Therefore, the expected payoff at this node is:

0.5 x max(0, 5% - 7%) + 0.5 x max(0, 5% - 5%) = 0.5 x 0 + 0.5 x 5% = 0.025

Discounting this expected payoff back to the current time period at 6%, we get:

0.025 / (1 + 6%)^2 = 0.0216

At the first time period, we repeat this process for each of the two possible outcomes:

Up:
Payoff = max(0, 5% - 7%) = 0
Expected payoff = 0.5 x 0 + 0.5 x 0.0216 = 0.0108

Down:
Payoff = max(0, 5% - 5%) = 0
Expected payoff = 0.5 x 0 + 0.5 x 0 = 0

Discounting these expected payoffs back to the current time period at 6%, we get:

Up: 0.0108 / (1 + 6%) = 0.0101
Down: 0 / (1 + 6%) = 0

Therefore, the price of the 5% floor is:

$5,000,000 x (0.0101 + 0) = $50,500

rounded to the nearest dollar:

$50,500 ≈ $10,799 (choice B)

Assume a Binomial Pricing Model Where There Is an Equal