A Non-Linear PDE for XVA by Forward Monte Carlo

Vladimir Piterbarg

We consider the following semi-linear partial differentiation equation (PDE)

$$\left(\frac{\partial }{\partial t}+L\right)V\left(t,x\right)=r\max \left(V\left(t,x\right),0\right)$$

(30.1)

where V(t, x) is a function of time t, t ∈ [0, T] and space x (one- or multi-dimensional), L is an infinitesimal generator of some diffusion and r ≥ 0. The equation is combined with the terminal condition V(T, x) = ψ(x) and is solved to obtain V(0, x).

Having efficient numerical methods for this PDE is important, as the equation (or its close cousins) appears in a number of related but distinct applications. These include pricing derivatives contracts with a one-way Credit Support Annex (CSA) (Piterbarg 2010), risky close-outs for credit value adjustments (CVAs) (Burgard and Kjaer 2011), derivatives pricing with differential borrowing and lending rates (Mercurio 2014) and accounting-consistent valuation (Albanese et al 2015; Burgard and Kjaer 2015). And, as far back as 2005, a similar non-linear PDE appeared in Andreasen (2005) in the context of fixed physical cash supply and negative rates. More details can be found in Piterbarg (2015).

In a low number of space dimensions (one to three, say) the PDE can be