The correction in value of an over-the-counter derivative contract due to counterparty risk under funding constraints is represented as the value of a dividend-paying option on the value of the contract clean of counterparty risk and excess funding costs. This representation allows one to analyze the structure of this correction, the so-called Credit Valuation Adjustment (CVA for short), in terms of replacement cost/benefits, credit cost/benefits, and funding cost/benefits. We develop a reduced-form backward stochastic differential equations (BSDE) approach to the problem of pricing and hedging the CVA. In the Markov setup, explicit CVA pricing and hedging schemes are formulated in terms of semilinear partial differential equations.

This and the follow-up paper deal with the valuation and hedging of bilateral counterparty risk on over-the-counter derivatives. Our study is done in a multiple-curve setup reflecting the various funding constraints (or costs) involved, allowing one to investigate the question of interaction between bilateral counterparty risk and funding. The first task is to define a suitable notion of no arbitrage price in the presence of various funding costs. This is the object of this paper, where we develop an "additive, multiple curve" extension of the classical "multiplicative (discounted), one curve" risk-neutral pricing approach. We derive the dynamic hedging interpretation of such an "additive risk-neutral" price, starting by consistency with pricing by replication in the case of a complete market. This is illustrated by a completely solved example building over previous work by Burgard and Kjaer.KEY WORDS: counterparty risk, funding costs, nonlinear pricing and hedging, arbitrage, backward stochastic differential equation.

Following an approach introduced by Lagnado and Osher (1997), we study Tikhonov regularization applied to an inverse problem important in mathematical finance, that of calibrating, in a generalized Black-Scholes model, a local volatility function from observed vanilla option prices. We first establish W 1,2 p estimates for the Black-Scholes and Dupire equations with measurable ingredients. Applying general results available in the theory of Tikhonov regularization for ill-posed nonlinear inverse problems, we then prove the stability of this approach, its convergence towards a minimum norm solution of the calibration problem (which we assume to exist), and discuss convergence rates issues.

It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the so-called Mokobodski condition. As for uniqueness of solutions, this holds under mild integrability conditions. However, for practical purposes, existence and uniqueness are not enough. In order to further develop these results in Markovian set-ups, one also needs a (simply or doubly) reflected BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and one further needs a suitable comparison theorem. In this paper, we derive such estimates and comparison results. In the last section, applicability of the results is illustrated with a pricing problem in finance. . This reprint differs from the original in pagination and typographic detail. 1 2 S. CRÉPEY AND A. MATOUSSI bonds in finance (see Section 6 and [5,6,8]). In this case, the state process (first component) Y of a solution to a related R2BSDE may be interpreted in terms of an arbitrage price process for the bond. As demonstrated in [7], the mere existence of a solution to the related R2BSDE is a result with important theoretical consequences in terms of pricing and hedging the bond. Yet, in order to further develop these results in Markovian set-ups, we also need the R2BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and we also need a suitable comparison theorem. Now, as opposed to the situation prevailing for RBSDEs (see, e.g., El Karoui et al. [17]), universal a priori estimates cannot be obtained for R2BSDEs. In order to get estimates for R2BSDEs, one needs to specialize the problem somewhat. Likewise, universal comparison theorems do not hold in models with jumps (see [2] for a counterexample in the simple case of a BSDE without barriers).Section 2 presents an abstract set-up in which our results are derived, as well as the BSDEs under consideration (Section 2.1). In Sections 3 and 4, we establish the a priori bound and error estimates (Theorem 3.2) and our comparison theorem (Theorem 4.2). The a priori error estimates immediately imply uniqueness of a solution to our problems (Section 5.1). Assuming an additional martingale representation property and the quasi-left-continuity of the barriers, we then give existence results (Section 5.2). In Section 6, we show that all of the required assumptions are satisfied in the case of the convertible-bonds-related R2BSDEs, in a rather generic Markovian specification of our abstract set-up. These R2BSDEs thus admit (unique) solutions.These results can be used to develop a related variational inequality approach in the Markovian case (see [10,11,12]).2. Set-up. Throughout the paper we work with a finite time horizon T > 0, a probability space (Ω, F, P) and a filtration F = (F t ) t∈[0,T ] , with F T = F, satisfying the usual conditions of right-conti...

In Crépey (2015, Part II), a basic reduced-form counterparty risk modeling approach was introduced, under a rather standard immersion hypothesis between a reference filtration and the filtration progressively enlarged by the default times of the two parties, also involving the continuity of some of the data at default time. This basic approach is too restrictive for application to credit derivatives, which are characterized by strong wrong-way risk, i.e. adverse dependence between the exposure and the credit riskiness of the counterparties, and gap risk, i.e. slippage between the portfolio and its collateral during the so called cure period that separates default from liquidation. This paper shows how a suitable extension of the basic approach can be devised so that it can be applied in dynamic copula models of counterparty risk on credit derivatives. More generally, this method is applicable in any marked default times intensity setup satisfying a suitable integrability condition. The integrability condition expresses that no mass is lost in a related measure change. The changed probability measure is not needed algorithmically. All one needs in practice is an explicit expression for the intensities of the marked default times.