In this paper, we investigate conditionsto convert nonlinear equations of XVA to linear equations. As we consider different borrowing/lendingrates, pricing derivatives encounters a non-linear equation and this makes finding an analytic solution for the price with the XVA difficult. Thus, in general, we have to resort to expensive numerical computation. In most previous works, the attempts to find an analytic solution were conducted on a restrictive assumption that the borrowing / lending rates are same. We find conditions to relax the assumption. Moreover, the conditions are mild in the sense that it is often satisfied in practice .
We discuss a binary nature of funding impacts. Under some conditions, funding is either cost or benefit, i.e., one of the lending / borrowing rates does not play any role in pricing derivatives. When we price derivatives, considering different lending/borrowing rates leads to semi-linear BSDEs and PDEs, so we need to solve the equations numerically. However, once we can guarantee that only one of the rates affects pricing, we can recover linear equations and derive analytic formulae. Moreover, as a byproduct, our results explain how debt value adjustment (DVA) and funding benefits are different. It is often believed that DVA and funding benefits are overlapped but it will be shown that the two components are affected by different mathematical structures of derivative transactions. We will see later that FBA occurs where the payoff is non-increasing, but this relationship becomes weaken as the funding choices of underlying assets are transferred to repo markets .
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In this paper, we investigate conditionsto convert nonlinear equations of XVA to linear equations. As we consider different borrowing/lendingrates, pricing derivatives encounters a non-linear equation and this makes finding an analytic solution for the price with the XVA difficult. Thus, in general, we have to resort to expensive numerical computation. In most previous works, the attempts to find an analytic solution were conducted on a restrictive assumption that the borrowing
We discuss a binary nature of funding impacts. Under some conditions, funding is either cost or benefit, i.e., one of the lending
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lending rates are same. We find conditions to relax the assumption. Moreover, the conditions are mild in the sense that it is often satisfied in practice
borrowing rates does not play any role in pricing derivatives. When we price derivatives, considering different lending/borrowing rates leads to semi-linear BSDEs and PDEs, so we need to solve the equations numerically. However, once we can guarantee that only one of the rates affects pricing, we can recover linear equations and derive analytic formulae. Moreover, as a byproduct, our results explain how debt value adjustment (DVA) and funding benefits are different. It is often believed that DVA and funding benefits are overlapped but it will be shown that the two components are affected by different mathematical structures of derivative transactions. We will see later that FBA occurs where the payoff is non-increasing, but this relationship becomes weaken as the funding choices of underlying assets are transferred to repo markets