Lecture_7_CVA_201820180402201111

CVA
Unilateral CVA
Unilateral CVA – Proof I
Startingfrom thedefinitionofCVA andexpression (5)of theriskypayoff, we derive formula
(6), through the following steps:
1 we express the sum of all discounted payoff terms between t and τ, i.e. Π(t,τ), as a
function of Π(t,T) as follows:
2 Plugging this result into eq. (5) we get:
Π(t,τ) = Π(t,T)−D(t,τ)Π(τ,T)
Π D (t,T) = 1 {τ>T} Π(t,T)+1 {τ≤T} Π(t,T)
+ 1 {τ≤T} D(t,τ) { −Π(τ,T)+Rec (E τ[Π(τ,T)]) + +(E τ[Π(τ,T)]) −}
and, using the fact that 1 {τ>T} Π(t,T)+1 {τ≤T} Π(t,T) = Π(t,T),
Π D (t,T) = Π(t,T)
+ 1 {τ≤T} D(t,τ) { −Π(τ,T)+Rec (E τ[Π(τ,T)]) + +(E τ[Π(τ,T)]) −}
Paola Mosconi 20541 – Lecture 10-11 32 / 86