The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

APPENDIX D

Multiple Bond Survival Probabilities Given Correlated Default Probability Rates

Let the survival probability of bond 1 be s₁ and the default probability be p₁Δt over a time period Δt, then

s₁ = 1 − p₁Δt.

If we introduce the idea of an expectation probability for default, then

p₁⁽¹⁾ = p₁Δt,

s₁⁽¹⁾ = s₁ = 1 − p₁Δt

where the superscript denotes the fact that the probability is in a system with only one variable (bond).

We introduce correlation for a pair of bonds 1 and 2, by making the double default pobability different to the product of the uncorrelated probabilities by introducing a “mixing parameter” α₁₂,

p₁p₂⁽²⁾ = (p₁Δt)(p₂Δt) + α₁₂(p₁Δts₂ + s₁p₂Δt),

p₁⁽²⁾ = p₁Δt + α₁₂s₁p₂Δt.

This ...