Pricing of XVA adjustments : from expected exposures to wrong-way risks

Abstract : The point of departure of this thesis is the valuation of the expected exposure which represents one of the major components of XVA adjustments. Under independence assumptions with credit and funding costs, we derive in Chapter 3 a new representation of the expected exposure as the solution of an ordinary differential equation w.r.t the default time variable. We rely on PDE arguments in the spirit of Dupire’s local volatility equation for the one dimensional problem. The multidimensional extension is addressed using the co-area formula. This forward representation gives an explicit expression of the exposure’s time value, involving the local volatility of the underlying diffusion process and the first order Greek delta, both evaluated only on finite set of points. From a numerical perspective, dimensionality is the main limitation of this approach. Though, we highlight high accuracy and time efficiency for standalone calculations in dimensions 1 and 2.The remaining chapters are dedicated to aspects of the correlation risk between the exposure and XVA costs. We start with the general correlation risk which is classically modeled in a joint diffusion process for market variables and the credit/funding spreads. We present a novel approach based on asymptotic expansions in a way that the price of an XVA adjustment with correlation risk is given by the classical correlation-free adjustment to which is added a sum of explicit correction terms depending on the exposure Greeks. Chapter 4 is consecrated to the technical derivation and error analysis of the expansion formulas in the context of pricing credit contingent derivatives. The accuracy of the valuation approach is independent of the smoothness of the payoff function, but it is related to the regularity of the credit intensity model. This finding is of special interest for pricing in a real financial context. Pricing formulas for CVA and FVA adjustments are derived in Chapter 5, along with numerical experiments. A generalization of the asymptotic expansions to a bilateral default risk setting is addressed in Chapter 6.Our thesis ends by tackling the problem of modeling the specific Right-Way Risk induced by rating trigger events within the collateral agreements. Our major contribution is the calibration of a rating transition model to market implied default probabilities
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Marouan Iben Taarit. Pricing of XVA adjustments : from expected exposures to wrong-way risks. General Mathematics [math.GM]. Université Paris-Est, 2018. English. ⟨NNT : 2018PESC1059⟩. ⟨tel-01939269⟩

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