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MAT1856/APM466: Assignment #2

 MAT1856/APM466: Mathematical Finance Winter 2021

Assignment #2: Credit Risk
Professor: Luis Seco, TA: Jonathan Mostovoy
Please bring any questions about this assignment to your TA’s, Jonathan’s, weekly (virtual) office hour on
the Facebook group.
Expectations
1. Please have your final report typeset using LATEX and using this template: https://www.overleaf.
com/read/ttgrjvwchhct. Please also structure your answers in line with the mock answers provided.
2. You may, and are encouraged, to discuss how to do these questions with your peers. However, your
write-up must be done individually, and the sharing of your write-up before April 13th is prohibited.
Additional Notes: Marks will be awarded for each question as either full-, half-, or zero-marks according to if
the question was answered with a few small mistakes, substantial mistakes but fundamental idea still correct, or
fundamental idea wrong / no answer respectively. -10 marks if not typeset in LATEXusing the template provided as
intended.
2 Questions- 100 points
1. (40 points) Suppose that company X has four states of solvency: good, bad, crisis, and default.
Suppose also that the one year transition (between solvency states) probability matrix is given by:
P =
state good bad crisis default
good 7/10 2/10 1/10 0
bad 1/10 5/10 2/10 2/10
crisis 1/10 3/10 3/10 3/10
default 0 0 0 1
For the following questions, feel free to use a computer to aid your calculations. For part a)&b),
you must state your final answer with a small explanation (explicit calculations discouraged in your
report). For parts c)& d), a formal proof is not needed, just a 1 or 2 sentence explanation.
(a) (10 points) What is the three year transition probability matrix?
(b) (10 points) What is the probability that if company X is currently in a “crisis” solvency state,
they will default within the next two months?
(c) (10 points) What is limt→∞ Pt?
(d) (10 points) If t ∈ N, (t < ∞), given that the company X has not yet defaulted, is it guaranteed
(/with probability 1) that company X will default within t years?
(Hint: Either use induction or show that ∃t < ∞ for which Pt
ij = 0 ∀j = 4, Pt
ij = 1 if j = 4.)
2. (40 points) Assume that Germany’s bonds are risk-free and Italy’s bonds are risk-prone, and that
each country issues zero coupon bonds with a face value of 1. We denote a German bond with an
outstanding term of i years simply by its current price P Gi
, and an Italian bond with outstanding
term of i years also simply by PIi
. Finally, assume everything henceforth is priced using continuous
discounting, and a 25% recovery rate under default.
1
Assignment 2: Credit Risk, Luis Seco 2
(a) (10 points) Given {P G1
, . . . , P Gn } and {PI1
, . . . , PIn }, derive a closed form formula for the credit
spread, hi
, at time i ∈ {1, . . . , n} for Italy in terms of i, P Gi
, and PIi .
(b) (10 points) Under a two state markov chain model (solvency and default), write Italy’s ith-year
probability transition matrix, Pi
, in terms of just i and hi.
(c) (10 points) If the Italian government issues a one-off asset, A, that pays Ci
, i = 1, . . . , n, at time
i, find the price of this asset in terms of {1, . . . , n}, {h1, . . . , hn}, and {P G1
, . . . , P Gn }.
(d) (10 points) First find ∂hiA, then use this to say what would happen to the price of A given
Italy’s probability of default decreases.
3. (20 points) First, list 2 simplifications (I.e., assumptions that might not be true in real life) that are
made under Merton’s Credit Risk Model. Then, list 2 assumptions that are made under Merton’s
Credit Risk Model that seem reasonable from a practical perspective.
Max 1 sentence per simplification.
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