Arbitrage‐free XVA

Abstract

We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no‐arbitrage arguments, we derive backward stochastic differential equations associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buyer's and seller's XVA, which in turn identify a no‐arbitrage interval. In the case that borrowing and lending rates coincide, we provide a fully explicit expression for the unique XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. In the general case of asymmetric funding, repo, and collateral rates, we study the semilinear partial differential equations characterizing buyer's and seller's XVA and show the existence of a unique classical solution to it. To illustrate our results, we conduct a numerical study demonstrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.

Arbitrage-free pricing of derivatives in nonlinear market models

“…Let us now comment on the existing approaches to the nonlinear valuation of derivatives, as first developed by El and El and later applied by several authors to particular financial models or classes on contracts (see, for instance, Bichuch et al (2018), Brigo and Pallavicini (2014), Crépey (2015a, b), Dumitrescu et al (2017), Mercurio (2013) or Pallavicini et al (2012a, b)). The most common approach to the valuation problem in a nonlinear framework seems to hinge, at least implicitly, on the following steps in which it is usually assumed that the hedger's initial endowment is immaterial and thus it may be set to zero.…”

“…In contrast, only a few papers devoted to nonlinear models of credit risk are available. More recently, Crépey (2015a, b), Dumitrescu et al (2017) and Bichuch et al (2018) used BSDEs with jumps to solve the valuation and hedging problems for derivative contracts exposed to the counterparty credit risk. In Bichuch et al (2018) and Dumitrescu et al (2017), the authors focus on valuation of the full contract, whereas Crépey (2015a, b) examines the problem of the approximate additivity for the credit valuation adjustments.…”

“…unilateral price and the Black-Scholes price (see Definition 4.8 in Bichuch et al (2018), which is hence supposed to play the role of the clean price. Using our notation, the definition of the total valuation adjustment adopted in Bichuch et al (2018) Bichuch et al (2018) do not advocate the practical approach where the clean price and valuations adjustment are supposed to be first independently computed by separate trading and CVA desks and subsequently aggregated into the full price.…”

“…Using our notation, the definition of the total valuation adjustment adopted in Bichuch et al (2018) Bichuch et al (2018) do not advocate the practical approach where the clean price and valuations adjustment are supposed to be first independently computed by separate trading and CVA desks and subsequently aggregated into the full price. It is also observed in Bichuch et al (2018) that the total valuation adjustments are equal and thus unilateral prices collapse to a single full price when the pricing BSDE is linear. Otherwise, the full prices computed independently by the two counterparties, who are supposed to use identical trading model, are likely to differ.…”

“…Particular instances of such unilateral and bilateral valuation problems were previously studied in Nie and Rutkowski (2015, 2016a (published online on 18 April 2017)) where it was shown that a non-empty interval of either fair bilateral prices or bilaterally profitable prices can be obtained in some nonlinear models for contracts with either an exogenous or an endogenous collateralization. It should be acknowledged that there exists a vast body of literature devoted to valuation and hedging of financial derivatives under differential funding costs, collateralization, the counterparty credit risk and other trading adjustments (see, for instance, Bichuch et al (2018), Brigo and Pallavicini (2014), Brigo et al (2018Brigo et al ( , 2017, Kjaer (2011, 2013), Crépey (2015a, b), Mercurio (2013), Pallavicini et al (2012b, a), and Piterbarg (2010). In view of limited space, we cannot present here these works in detail.…”