I am paid 20 million every time a bond drops to a new low over a 120 month period. I need to know how to find the present value of such an arrangement if there is a continuously compound interest of 5 percent.
The bond starts at a A rating and is only paid the first time it reaches any rating lower than A. So A -> AA-> A-> B->CCC would pay 40 million.
I am given the following probability transition matrix (the probability of the rating on the left becoming the rating on the right):
p[AAA][AAA] = 0.9725; p[AAA][AA] = 0.0275; p[AA][AAA] = 0.0020; p[AA][AA] = 0.9742; p[AA][A] = 0.0238; p[A][AA] = 0.0020; p[A][A] = 0.9825; p[A][BBB] = 0.0155; p[BBB][A] = 0.0073; p[BBB][BBB] = 0.9819; p[BBB][BB] = 0.0108; p[BB][BBB] = 0.0030; p[BB][BB] = 0.9783; p[BB][B] = 0.0187; p[B][BB] = 0.0010; p[B][B] = 0.9751; p[B][C] = 0.0239; p[C][B] = 0.0066; p[C][C] = 0.9852; p[C][D] = 0.0082; p[D][D] = 1.0;
When the bond reaches a rating of D, the bond defaults and cannot recover from that position thus the swap arrangement is closed and we receive the most money possible at $100 million. Otherwise the arrangement continues until the bond matures at 120 months.
I need to find a fair monthly premium, which I believe involves finding the present value of this arrangement.