Fixed Income Risk Management
Second Assignment

QUESTION 1

Maturity 0.5 1 1.5

Interest Rate 0.01 0.016 0.02 0.022

Discount Factor 0.9940 0.9841 0.9704 0.9570

Pz(0.T) 99.4018 98.4127 97.0446 95.6954

2 Table 1- Zero option prices

A. Given the current term structure of interest rates, we can easily derive the prices of zero-coupon bonds. As the interest rates are continuously compounded, the formulas used to calculate the discount factors and the zero prices are: ������(������, ������) = ������ −������(������,������)(������− ������) ������������ (������, ������) = 100 × ������(������, ������)

B. Assuming Ɵi=1% and σ=1.50%, we can set up the Ho-Lee model, which implies the following 3-step interest rate tree (where each step is  = 0.5):
1 1.20% 2.76% 0.64% 2 4.32% 2.20% 0.08%
Table 2- Ho-Lee interest rate tree

3 5.88% 3.76% 1.64% -0.48%

At each node, the interest rate was calculated using ������������+1,������ = ������������,������ + ������������ × ∆ + ������ × √∆ For upward movements, and ������������+1,������+1 = ������������,������ + ������������ × ∆ − ������ × √∆ For downward movements. The risk neutral probability of an upward or downward movement is set at p*=1/2. The model implied zero-coupon prices are then computed using the different step bond trees (Table 14 in the Appendix). The following table shows the comparison between the implied and the term structure zero-coupon prices for each maturity: the Check row contains the difference between the two.

1 Term structure Implied Check 99.4018 99.4018 0.0000

2 98.4127 98.5618 -0.1491

3 97.0446 97.4891 -0.4445

4 95.6954 96.1940 -0.4986

Table 3 - Comparison on bond prices

We can see that the implied Ho-Lee zero-coupon bond prices do not agree with the observed term structure, due to the arbitrary Ɵ values. C. In order to find the drift parameters that exactly fit