[Paper Review] Arbitrage-Free Pricing of XVA -- Part I: Framework and Explicit Examples

Maxim Bichuch, Agostino Capponi+1 more|arXiv (Cornell University)|Jan 23, 2015

Credit Risk and Financial Regulations4 references4 citations

TL;DR

This paper develops an arbitrage-free framework for pricing XVA—accounting for funding costs, counterparty credit risk, and collateralization—using nonlinear backward stochastic differential equations (BSDEs). It derives buyer’s and seller’s XVA as bounds, and provides an explicit closed-form XVA expression when borrowing and lending rates are equal, extending Piterbarg (2010) by incorporating premature default risk.

ABSTRACT

We develop a novel 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 the nonlinear backward stochastic differential equations (BSDEs) 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. When borrowing and lending rates coincide we provide a fully explicit expression for the uniquely determined price of XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. This extends the result of Piterbarg by incorporating the effect of premature contract termination due to default risk of the trader and of his counterparty.

Motivation & Objective

• To establish a no-arbitrage valuation framework for XVA that incorporates funding spreads, counterparty credit risk, and collateralization.
• To model the impact of premature contract termination due to default of either the trader or counterparty on derivative pricing.
• To derive nonlinear BSDEs for replicating portfolios of long and short positions in a European claim under rate asymmetry.
• To define buyer’s and seller’s XVA as arbitrage-free bounds, ensuring market consistency under default and funding risk.
• To provide a fully explicit XVA formula in the special case where borrowing and lending rates coincide.

Proposed method

• Formulates a dynamic trading framework with a default-free stock and two risky bonds (trader and counterparty), incorporating asymmetric funding and repo rates.
• Uses nonlinear BSDEs to model the wealth process of replicating portfolios for long and short positions, with drivers capturing funding, credit, and collateral costs.
• Introduces a change of measure to a forward measure under which the discounted wealth process becomes a local martingale, enabling arbitrage-free pricing.
• Applies comparison theorems for BSDEs to establish the existence of a no-arbitrage interval defined by buyer’s and seller’s XVA.
• Derives explicit hedging strategies via the Malliavin derivative and Markovian representation, expressing the Z-process as a function of the underlying stock price.
• Reduces the solution to a Markovian function of time and stock price on the survival set, enabling tractable computation of hedging sensitivities.

Experimental results

Research questions

• RQ1 How can XVA be priced in a way that ensures no-arbitrage when funding costs, counterparty credit risk, and collateralization are all present?
• RQ2 What is the impact of asymmetric borrowing and lending rates on the valuation and hedging of derivative claims?
• RQ3 How does the risk of premature default by the trader or counterparty affect the XVA and the resulting no-arbitrage bounds?
• RQ4 Under what conditions does a unique XVA price exist, and what is its explicit form?
• RQ5 How do the hedging strategies for the long and short positions differ when funding and default risks are asymmetric?

Key findings

• The paper establishes that the absence of arbitrage implies the existence of a no-arbitrage interval defined by the buyer’s and seller’s XVA, derived from nonlinear BSDEs.
• When borrowing and lending rates are equal, the XVA is uniquely determined and expressed as a percentage of the traded claim’s price, extending Piterbarg (2010) to include default risk.
• The explicit XVA formula is derived in closed form under the equal rates assumption, with the solution depending on the claim’s payoff, default intensities, and funding spreads.
• The hedging strategies for the long and short positions are not symmetric due to rate asymmetry, and are explicitly given via the Malliavin derivative of the solution process.
• The solution process is Markovian on the survival set, allowing the value function to be represented as a measurable function of time and the underlying stock price.
• The comparison theorem for BSDEs confirms that higher default risk or funding costs lead to wider no-arbitrage intervals, reflecting increased uncertainty in unilateral pricing.