TY - JOUR
T1 - Nonlinear Valuation with XVAs: Two Converging Approaches
AU - Brigo, Damiano
AU - Buescu, Cristin
AU - Francischello, Marco
AU - Pallavicini, Andrea
AU - Rutkowski, Marek
N1 - Funding Information:
Funding: The research of Damiano Brigo and Marek Rutkowski was supported by the EPSRC Mathematics Platform Grant EP/I019111/1 Mathematical Analysis of Funding Costs at Imperial College London. The research of M. Rutkowski was supported by the Australian Research Council Discovery Project DP200101550.
Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.
PY - 2022/3/2
Y1 - 2022/3/2
N2 - When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each factor or cost in isolation, while seemingly ignoring the combined effects. Instead, risk factors and costs can be jointly accounted for ab initio in the pricing mechanism at the level of cash flows, and this “adjusted cash flow" approach leads to a nonlinear valuation formula. While for practitioners this made more sense because it showed which discount factor is used for which cash flow (recall the multi-curve environment post-crisis), for academics, the focus was on checking that the resulting nonlinear valuation formula is consistent with the theoretical arbitrage-free “replication approach” that we also analyse in the paper. We formulate specific reasonable assumptions, which ensure that the valuation formulae obtained by the two approaches coincide, thus reinforcing both academics’ and practitioners’ confidence in adopting such nonlinear valuation formulae in a multi-curve setup.
AB - When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each factor or cost in isolation, while seemingly ignoring the combined effects. Instead, risk factors and costs can be jointly accounted for ab initio in the pricing mechanism at the level of cash flows, and this “adjusted cash flow" approach leads to a nonlinear valuation formula. While for practitioners this made more sense because it showed which discount factor is used for which cash flow (recall the multi-curve environment post-crisis), for academics, the focus was on checking that the resulting nonlinear valuation formula is consistent with the theoretical arbitrage-free “replication approach” that we also analyse in the paper. We formulate specific reasonable assumptions, which ensure that the valuation formulae obtained by the two approaches coincide, thus reinforcing both academics’ and practitioners’ confidence in adopting such nonlinear valuation formulae in a multi-curve setup.
KW - risk-neutral valuation
KW - replication
KW - funding costs
KW - default
KW - collateral
UR - http://www.scopus.com/inward/record.url?scp=85126275052&partnerID=8YFLogxK
U2 - 10.3390/math10050791
DO - 10.3390/math10050791
M3 - Article
SN - 2227-7390
VL - 10
JO - Mathematics
JF - Mathematics
IS - 5
M1 - 791
ER -