FE-680 – Qizz 10
Problem 1
Consider the Gaussian Latent Variable Model. Recall that credit i defaults before time T if 𝐴𝐴𝑖𝑖 =𝛽𝛽𝑖𝑖𝑍𝑍+�1−𝛽𝛽𝑖𝑖2𝜀𝜀𝑖𝑖 <𝐶𝐶𝑖𝑖(𝑇𝑇)
Assuming for simplicity a flat and deterministic conditional hazard rate, then the conditional hazard rate is given by
𝜆𝜆 𝑖𝑖 ( 𝑇𝑇 | 𝑍𝑍 ) = − 𝑇𝑇1 l n Φ ⎛ 𝛽𝛽 𝑖𝑖 𝑍𝑍 − 𝐶𝐶 𝑖𝑖 ( 𝑇𝑇 ) ⎞ ⎝ �1−𝛽𝛽𝑖𝑖2 ⎠
Plot the conditional hazard rate distribution for 𝛽𝛽 = 0, 0.2, 0.4, 0.6, 0.8, 1. Assume that the unconditional hazard rate is 3% and one-year horizon.
Construct a trinomial tree for the Ho and Lee model where 𝜎𝜎 = 0.025. Suppose that the initial zero-coupon interest rate for a maturities of 0.5, 1.0, and 1.5 years are 6.5%, 7%, and 7.5%. Use two time steps, each six months long. Calculate the value of a zero-coupon bond with a face value of $100 and a remaining life of six months at the ends of the final nodes of the tree. Use the tree to value a one-year European put option with a strike price of 95 on the bond.
Problem 2