Derivatives Pricing Under Bilateral Counterparty Risk
28 Pages Posted: 3 Apr 2015 Last revised: 12 Apr 2015
Date Written: April 12, 2015
We consider risk-neutral valuation of a contingent claim under bilateral counterparty risk in a reduced-form setting similar to that of Duffie and Huang  and Duffie and Singleton . The probabilistic valuation formulas derived under this framework cannot be usually used for practical pricing due to their recursive path-dependencies. Instead, finite-difference methods are used to solve the quasi-linear partial differential equations that equivalently represent the claim value function. By imposing restrictions on the dynamics of the risk-free rate and the stochastic intensities of the counterparties' default times, we develop path-independent probabilistic valuation formulas that have closed-form solution or can lead to computationally efficient pricing schemes. Our framework incorporates the so-called wrong way risk (WWR) as the two counterparty default intensities can depend on the derivatives values. Inspired by the work of Ghamami and Goldberg  on the impact of WWR on credit value adjustment (CVA), we derive calibration-implied formulas that enable us to mathematically compare the derivatives values in the presence and absence of WWR. We illustrate that derivatives values under unilateral WWR need not be less than the derivatives values in the absence of WWR. A sufficient condition under which this inequality holds is that the price process follows a semimartingale with independent increments.
Keywords: Reduced-Form Modeling, Counterparty Risk, Wrong Way Risk, Credit Value Adjustment, Basel III
JEL Classification: G12, G2, C63
Suggested Citation: Suggested Citation