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Before the 2008 financial crisis, most research in financial mathematics focused on pricing options without considering the effects of counterparties' defaults, illiquidity problems, and the role of the sale and repurchase agreement (Repo) market. Recently, models were proposed to address this by computing a total valuation adjustment (XVA) of derivatives; however without considering a potential crisis in the market.…
Crises challenge client XVA management when continuous collateralization is not possible because a derivative locks in the client credit level and the provider's funding level, on the trade date, for the life of the trade. We price XVA reduction strategies from the client point of view comparing multiple trade strategies using Mandatory Breaks or Restructuring, to modifications of a single trade using a Reset. We ana…
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 sell…
We propose a structural default model for portfolio-wide valuation adjustments (xVAs) and represent it as a system of coupled backward stochastic differential equations. The framework is divided into four layers, each capturing a key component: (i) clean values, (ii) initial margin and Collateral Valuation Adjustment (ColVA), (iii) Credit/Debit Valuation Adjustments (CVA/DVA) together with Margin Valuation Adjustment…
Exposure simulations are fundamental to many xVA calculations and are a nested expectation problem where repeated portfolio valuations create a significant computational expense. Sensitivity calculations which require shocked and unshocked valuations in bump-and-revalue schemes exacerbate the computational load. A known reduction of the portfolio valuation cost is understood to be found in polynomial approximations, …
Valuation adjustments, collectively named XVA, play an important role in modern derivatives pricing to take into account additional price components such as counterparty and funding risk premia. They are an exotic price component carrying a significant model risk and computational effort even for vanilla trades. We adopt an industry-standard realistic and complete XVA modelling framework, typically used by XVA tradin…
Every "x"-adjustment in the so-called xVA financial risk management framework relies on the computation of exposures. Considering thousands of Monte Carlo paths and tens of simulation steps, a financial portfolio needs to be evaluated numerous times during the lifetime of the underlying assets. This is the bottleneck of every simulation of xVA. In this article, we explore numerical techniques for improving the simula…
XVAs denote various counterparty risk related valuation adjustments that are applied to financial derivatives since the 2007--09 crisis. We root a cost-of-capital XVA strategy in a balance sheet perspective which is key in identifying the economic meaning of the XVA terms. Our approach is first detailed in a static setup that is solved explicitly. It is then plugged in the dynamic and trade incremental context of a r…
In this paper, we present a novel computational framework for portfolio-wide risk management problems, where the presence of a potentially large number of risk factors makes traditional numerical techniques ineffective. The new method utilises a coupled system of BSDEs for the valuation adjustments (xVA) and solves these by a recursive application of a neural network based BSDE solver. This not only makes the computa…
The X-valuation adjustment (XVA) problem, which is a recent topic in mathematical finance, is considered and analyzed. First, the basic properties of backward stochastic differential equations (BSDEs) with a random horizon in a progressively enlarged filtration are reviewed. Next, the pricing/hedging problem for defaultable over-the-counter (OTC) derivative securities is described using such BSDEs. An explicit suffic…
This paper investigates calculations of robust XVA, in particular, credit valuation adjustment (CVA) and funding valuation adjustment (FVA) for over-the-counter derivatives under distributional uncertainty using Wasserstein distance as the ambiguity measure. Wrong way counterparty credit risk and funding risk can be characterized (and indeed quantified) via the robust XVA formulations. The simpler dual formulations a…
In this paper we extend the existing literature on xVA along three directions. First, we enhance current BSDE-based xVA frameworks to include initial margin in presence of defaults. Next, we solve the consistency problem that arises when the front-office desk of the bank uses trade-specific discount curves (CSA discounting) which differ from the discount rate adopted by the xVA desk. Finally, we clarify the impact of…
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 (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 t…
XVA is a material component of a trade valuation and hence it must impact the decision to exercise options within a given netting set. This is true for both unsecured trades and secured / cleared trades where KVA and MVA play a material role even if CVA and FVA do not. However, this effect has frequently been ignored in XVA models and indeed in exercise decisions made by option owners. This paper describes how XVA im…
Banks must manage their trading books, not just value them. Pricing includes valuation adjustments collectively known as XVA (at least credit, funding, capital and tax), so management must also include XVA. In trading book management we focus on pricing, hedging, and allocation of prices or hedging costs to desks on an individual trade basis. We show how to combine three technical elements to radically simplify XVA m…
This study contributes to understanding Valuation Adjustments (xVA) by focussing on the dynamic hedging of Credit Valuation Adjustment (CVA), corresponding Profit & Loss (P&L) and the P&L explain. This is done in a Monte Carlo simulation setting, based on a theoretical hedging framework discussed in existing literature. We look at hedging CVA market risk for a portfolio with European options on a stock, first in a Bl…
Financial decision systems require fast surrogate models for pricing, calibration, hedging, XVA, stress testing, and portfolio optimization. Standard neural surrogates reproduce prices or risk quantities, but downstream tasks depend as much on derivatives: deltas, vegas, curve and credit-spread sensitivities, exposure and objective gradients. We formulate a derivative-informed operator-learning framework in which the…
Credit Valuation Adjustment — Adjustment to derivative value for counterparty default risk.
Debt Valuation Adjustment (Credit).
Funding Valuation Adjustment — Funding cost adjustment in uncollateralized derivative books.